892’ - fakultet strojarstva i brodogradnje · 0(+$1,.$)/8,’$ 892’ 892’...

65
0(+$1,.$)/8,’$892’ 892’ 0HKDQLNDIOXLGDMHGLRIL]LNHNRMLVHEDYLJLEDQMHPIOXLGDLVLODPDNRMHGMHOXMXQDIOXLG 0HKDQLND IOXLGD VH GLMHOL QD VWDWLNX IOXLGD NRMD SURXþDYD UDYQRWHåX IOXLGD X PLURYDQMX NLQHPDWLNXIOXLGDNRMDVHEDYL]DNRQLPDJLEDQMDIOXLGDLGLQDPLNXIOXLGDNRMDVHEDYLVLODPD NRMHGMHOXMXQDIOXLGDLJLEDQMLPDNRMHQDVWDMXGMHORYDQMHPWLKVLODWHLQWHUDNFLMDL]PH XþYUVWLK WLMHODLIOXLGD )OXLGLOLWHNXüLQD ’HILQLFLMDIOXLGD)OXLGMHWYDUNRMDVWUXMLWMQHSUHNLGQRVHGHIRUPLUDSRGGMHORYDQMHP PDNDNRPDORJVPLþQRJQDSUH]DQMD)OXLGPRåHELWLXNDSOMHYLWRPLOLSOLQRYLWRPVWDQMX ,]GHILQLFLMHIOXLGDVOLMHGL8IOXLGXXPLURYDQMXQHPDVPLþQLKQDSUH]DQMD )OXLG GLMHOLPR )OXLGH GLMHOLPR V RE]LURP QD YHOLþLQX GHIRUPDFLMH NDR SRVOMHGLFX WODþQRJQDSUH]DQMDQDNDSOMHYLQHLSOLQRYH 2VQRYQHGLPHQ]LMHXPHKDQLFLIOXLGD ’LPHQ]LMXIL]LNDOQHYHOLþLQHRGUH XMXQMHQXNDWHJRULMXVRE]LURPQDPMHUHQMHNYDOLWDWLYQR .YDQWLWDWLYQRPMHUHQMHIL]LNDOQHYHOLþLQH]DKWLMHYDMXXYR HQMHPMHUQHMHGLQFHPMHULODHWDORQD 9HOLþLQD 2]QDNDGLPHQ]LMH -HGLQLFDX6,VXVWDYX PDVD 0 NJ GXOMLQD / P YULMHPH 7 V WHPSHUDWXUD . +LSRWH]DNRQWLQXXPD .RQWLQXXPMHPDWHPDWLþNLPRGHOPDWHULMHSUHPDNRMHPRQD]DGUåDYDVYRMDIL]LNDOQDVYRMVWYD SULVPDQMLYDQMXYROXPHQDXWRþNXýHVWLFDNRQWLQXXPDPDWHULMDOQDWRþNDLPDLQILQLWH]LPDOQL YROXPHQ G9 D VYDND þHVWLFD ]DX]LPD VDPR MHGQX WRþNX SURVWRUD D X MHGQRM WRþNL SURVWRUD VH PRåH QDOD]LWL VDPR MHGQD þHVWLFD NRQWLQXXPD +LSRWH]D NRQWLQXXPD RPRJXüXMH SULPMHQX LQWHJUDOQRJLGLIHUHQFLMDOQRJUDþXQDXPHKDQLFLIOXLGD 3ULPMHU*XVWRüDþHVWLFHIOXLGDVHL]UDåDYDGHULYDFLMRP d d m V S [] [] 3 3 SI kg ML ; m ρ ρ - = =

Upload: ngominh

Post on 15-May-2018

228 views

Category:

Documents


4 download

TRANSCRIPT

Page 1: 892’ - Fakultet strojarstva i brodogradnje · 0(+$1,.$)/8,’$ 892’ 892’ 0hkdqlndioxlgdmhglril]lnhnrmlvhedyljledqmhpioxlgdlvlodpdnrmhgmhoxmxqdioxlg 0hkdqlndioxlgdvhglmholqdvwdwlnxioxlgdnrmdsurxþdydudyqrwhåxioxlgdxplurydqmx

0(+$1,.$�)/8,'$���892'� �

����892'��0HKDQLND�IOXLGD�MH�GLR�IL]LNH�NRML�VH�EDYL�JLEDQMHP�IOXLGD�L�VLODPD�NRMH�GMHOXMX�QD�IOXLG��0HKDQLND� IOXLGD� VH� GLMHOL� QD� VWDWLNX� IOXLGD� NRMD� SURXþDYD� UDYQRWHåX� IOXLGD� X� PLURYDQMX��NLQHPDWLNX�IOXLGD�NRMD�VH�EDYL�]DNRQLPD�JLEDQMD�IOXLGD��L�GLQDPLNX�IOXLGD�NRMD�VH�EDYL�VLODPD�NRMH�GMHOXMX�QD�IOXLGD��L�JLEDQMLPD�NRMH�QDVWDMX�GMHORYDQMHP�WLK�VLOD�WH�LQWHUDNFLMD�L]PH X�þYUVWLK�WLMHOD�L�IOXLGD��������)OXLG�LOL�WHNXüLQD�� ��� 'HILQLFLMD�IOXLGD��)OXLG�MH�WYDU�NRMD�VWUXML��WM��QHSUHNLGQR�VH�GHIRUPLUD��SRG�GMHORYDQMHP�PD�NDNR�PDORJ�VPLþQRJ�QDSUH]DQMD��)OXLG�PRåH�ELWL�X�NDSOMHYLWRP�LOL�SOLQRYLWRP�VWDQMX����� ,]�GHILQLFLMH�IOXLGD�VOLMHGL��8�IOXLGX�X�PLURYDQMX�QHPD�VPLþQLK�QDSUH]DQMD����� )OXLG� GLMHOLPR�� )OXLGH� GLMHOLPR� V� RE]LURP� QD� YHOLþLQX� GHIRUPDFLMH� NDR� SRVOMHGLFX�WODþQRJ�QDSUH]DQMD�QD�NDSOMHYLQH�L�SOLQRYH�������2VQRYQH�GLPHQ]LMH�X�PHKDQLFL�IOXLGD��'LPHQ]LMX�IL]LNDOQH�YHOLþLQH�RGUH XMX�QMHQX�NDWHJRULMX�V�RE]LURP�QD�PMHUHQMH�–�NYDOLWDWLYQR��.YDQWLWDWLYQR�PMHUHQMH�IL]LNDOQH�YHOLþLQH�]DKWLMHYDMX�XYR HQMH�PMHUQH�MHGLQFH��PMHULOD��HWDORQD�� 9HOLþLQD� 2]QDND�GLPHQ]LMH� -HGLQLFD�X�6,�VXVWDYX�PDVD� 0� NJ�GXOMLQD� /� P�YULMHPH� 7� V�

WHPSHUDWXUD� � .������+LSRWH]D�NRQWLQXXPD��.RQWLQXXP�MH�PDWHPDWLþNL�PRGHO�PDWHULMH�SUHPD�NRMHP�RQD�]DGUåDYD�VYRMD�IL]LNDOQD�VYRMVWYD�SUL�VPDQMLYDQMX�YROXPHQD�X�WRþNX��ýHVWLFD�NRQWLQXXPD��PDWHULMDOQD�WRþND��LPD�LQILQLWH]LPDOQL�YROXPHQ�G9��D�VYDND�þHVWLFD�]DX]LPD�VDPR�MHGQX� WRþNX�SURVWRUD��D�X� MHGQRM� WRþNL�SURVWRUD�VH�PRåH� QDOD]LWL� VDPR� MHGQD� þHVWLFD� NRQWLQXXPD�� +LSRWH]D� NRQWLQXXPD� RPRJXüXMH� SULPMHQX�LQWHJUDOQRJ�L�GLIHUHQFLMDOQRJ�UDþXQD�X�PHKDQLFL�IOXLGD��3ULPMHU��*XVWRüD�þHVWLFH�IOXLGD�VH�L]UDåDYD�GHULYDFLMRP� d

d

m

VS� ���[ ] [ ]3

3SI

kgML ;

mρ ρ−= = ��

������������

Page 2: 892’ - Fakultet strojarstva i brodogradnje · 0(+$1,.$)/8,’$ 892’ 892’ 0hkdqlndioxlgdmhglril]lnhnrmlvhedyljledqmhpioxlgdlvlodpdnrmhgmhoxmxqdioxlg 0hkdqlndioxlgdvhglmholqdvwdwlnxioxlgdnrmdsurxþdydudyqrwhåxioxlgdxplurydqmx

0(+$1,.$�)/8,'$���892'� �

����9LVNR]QRVW�IOXLGD��9LVNR]QRVW�IOXLGD�MH�PMHUD�RWSRUD�WHþHQMX�IOXLGD��1HZWRQRY�]DNRQ�YLVNR]QRVWL���vA

Fh

µ= ���*HQHUDOL]LUDQL�1HZWRQRY�]DNRQ�YLVNR]QRVWL��

F v

A yτ µ ∂= =

∂�

�8�QHZWRQVNLP� IOXLGLPD�YLVNR]QD� QDSUH]DQMD� VX� OLQHDUQR� UD]PMHUQD�EU]LQL�GHIRUPDFLMH� IOXLGD��.RHILFLMHQW� UD]PMHUQRVWL� VH� QD]LYD� �GLQDPLþND�� YLVNR]QRVW� IOXLGD� µ ��[ ] 1 -1ML Tµ −= �[ ]SI

Pa sµ = ⋅ �� 9LVNR]QRVW� MH� IL]LNDOQR� VYRMVWYR� IOXLGD�� L� ]DYLVL� RG�WHUPRGLQDPLþNRJ�VWDQMD� IOXLGD��.RG�SOLQRYD�V�SRUDVWRP� WHPSHUDWXUH� UDVWH� L�YLVNR]QRVW��D�NRG�NDSOMHYLQD�RSDGD��9LVNR]QRVW�SRND]XMH�RWSRU�IOXLGD�ND�WHþHQMX���.LQHPDWLþND�YLVNR]QRVW� [ ] [ ]

22 -1

SI

m L T ;

s,

µυ υ υρ

= = = ���Fluidi koji poštuju zakonitost

v

yτ µ ∂=

∂QD]LYDMX�VH�1HZWRQRYVNL�IOXLGL�

������6LOH�X�IOXLGX��$��0DVHQH� VLOH� VX� SRVOMHGLFD� SRORåDMD� PDVH� X� SROMX� f

G � PDVHQH� VLOH�� � fG � MH� VSHFLILþQD�

PDVHQD� VLOD� � VLOD� SR� MHGLQLþQRM� PDVL��2

2SI

mLT ;

sf f− = = G G ��

�0DVHQD�VLOD�dF

G �QD�þHVWLFX�IOXLGD��d d dF f m f V

G GGS� � �

�6LOD�FG �QD�XNXSQL�YROXPHQ�9�

dV

F f VGG

S� ¨ �2

SIMLT ; NF F− = =

G G ��3ULPMHUL��VLOD�JUDYLWDFLMH�� f gk

G G�� �

�����������������LQHUFLMVNH�VLOH�� f aG G�� �

��������6OLND�X]�GHILQLFLMX�PDVHQLK�VLOD�

2�

]�

\�[�

df mG¸

9�fG

Y� )�$� K�

µ�

Page 3: 892’ - Fakultet strojarstva i brodogradnje · 0(+$1,.$)/8,’$ 892’ 892’ 0hkdqlndioxlgdmhglril]lnhnrmlvhedyljledqmhpioxlgdlvlodpdnrmhgmhoxmxqdioxlg 0hkdqlndioxlgdvhglmholqdvwdwlnxioxlgdnrmdsurxþdydudyqrwhåxioxlgdxplurydqmx

0(+$1,.$�)/8,'$���892'� �

� %�� Površinske sile� VX� VLOH� GRGLUD� L]PH X�þHVWLFD� IOXLGD� LOL� L]PH X� þHVWLFD� IOXLGD� L�VWLMHQNH�� 'HILQLUDQH� VX� VSHFLILþQRP�YHNWRURP�QDSUH]DQMD� GT ���[ ] [ ]-1 2

SIML T ; Paσ σ−= =G G ��

6LOD�dFG

na elementarnu površinu dS �� d = dF S

G GT ��6LOD�FG na ukupnu površinu S �� = d

S

F SG GT¨ �

�Za površinske sile vrijedi III Newtonow zakon �SULQFLS�DNFLMH�L�UHDNFLMH���WM��� n n

G GG GT T�� � ��þLWDM�YHNWRU�QDSUH]DQMD�QD�SRYUãLQL�RULMHQWLUDQRM�MHGLQLþQLP�YHNWRURP�QRUPDOH� n

G �MHGQDN�MH�SR�YHOLþLQL�L�VXSURWDQ�SR�VPMHUX�YHNWRUX�QDSUH]DQMD�QD�SRYUãLQL�RULMHQWLUDQRM�QRUPDORP� n

G� �����6WDQMH� QDSUH]DQMD� X� WRþNL� SURVWRUD� MHGQR]QDþQR� MH� GHILQLUDQR� WHQ]RURP� QDSUH]DQMD��.RPSRQHQWH� WHQ]RUD� QDSUH]DQMD� GHILQLUDQH� VX� NRPSRQHQWDPD� WULMX� YHNWRUD� QDSUH]DQMD� NRML�

djeluju na površinama orijentiranim normalama u VPMHUX� RVL� NRRUGLQDWQRJ� VXVWDYD�� NDR� QD� VOLFL�� 6YDNL�YHNWRU� QDSUH]DQMD� LPD� MHGQX� QRUPDOQX� NRPSRQHQWX��RNRPLWX� QD� SRYUãLQX�� L� GYLMH� WDQJHQFLMDOQH� �VPLþQH��NRPSRQHQWH�� 7DEOLþQL� ]DSLV� NRPSRQHQWL� WHQ]RUD�QDSUH]DQMD�

�xx xy xz

ji yx yy yz

zx zy zz

T T TT T T T

T T T� �

3UYL� LQGHNV� R]QDþXMH� UHGDN�� WM�� VPMHU� QRUPDOH� QD�površinu, a drugi stupac RGQRVQR� SUDYDF� GMHORYDQMD�NRPSRQHQWH�WHQ]RUD�QDSUH]DQMD��7HQ]RU� QDSUH]DQMD� MH� VLPHWULþDQ� ij jiT T� � �RVLP� DNR�postoje maseni i površinski momenti).��9H]D�L]PH X�YHNWRUD�L�WHQ]RUD�QDSUH]DQMD��� �YHNWRU�QDSUH]DQMD�MH�SURMHNFLMD�WHQ]RUD�QDSUH]DQMD�QD�VPMHU�QRUPDOH��

( ) ( )

( ) ( )

ji x xx y yx z zx

x xy y yy z zy x xz y yz z zz

n n n n n i

n n n j n n n k

GG GG

GGT T T T T

T T T T T T

� � � � � �

� � � � � ��

���'RJRYRU�R�SUHG]QDFLPD�QDSUH]DQMD��

[�

\�

]�

xxTxyT

xzT yyTyzT

yxT

zzT

zxTzyT

6OLND�X]�GHILQLFLMX�NRPSRQHQWL�WHQ]RUD�QDSUH]DQMD�

6�

Slika uz definiciju površinskih sila�

2�

]�

\�[�

9�nG

dSGT

Page 4: 892’ - Fakultet strojarstva i brodogradnje · 0(+$1,.$)/8,’$ 892’ 892’ 0hkdqlndioxlgdmhglril]lnhnrmlvhedyljledqmhpioxlgdlvlodpdnrmhgmhoxmxqdioxlg 0hkdqlndioxlgdvhglmholqdvwdwlnxioxlgdnrmdsurxþdydudyqrwhåxioxlgdxplurydqmx

0(+$1,.$�)/8,'$���892'� �

Pozitivna naprezanja na površinama orijentiranim normalama u pozitivnom smjeru NRRUGLQDWQLK� RVL� WDNR HU� JOHGDMX� X� SR]LWLYQH� VPMHURYH� WLK� RVL� L� REUQXWR�� SR]LWLYQD�QDSUH]anja na površinama orijentiranim normalama koje gledaju u negativnom smjeru NRRUGLQDWQLK�RVL��WDNR HU�JOHGDMX�X�QHJDWLYQH�VPMHURYH�WLK�RVL����3RYUãLQVND�VLOD��YHNWRU�QDSUH]DQMD��]D�VOXþDM�VWDQMD�WODþQRJ�QDSUH]DQMD�� tpn

GG GT T�� � ��

6OXþDM� 9HNWRU�QDSUH]DQMD�5HDOQL�IOXLG�X�JLEDQMX��SRVWRMH�YLVNR]QH�VLOH��

tji jin pn n pn

G G G GG GT T 4 T� � �� � � �� � �ji4 �WHQ]RU�YLVNR]QD�QDSUH]DQMD�

S� �WODN� �QRUPDOQR�QDSUH]DQMH�� tGT �YLVNR]QR�QDSUH]DQMH� �WDQJHQFLMDOQR�Q��

��5HDOQL�IOXLG�X�PLURYDQMX�LOL�UHODWLYQRP�PLURYDQMX���,GHDOQL�IOXLG��QHYLVNR]DQ�� jin pnG GGT T� � �� �

Page 5: 892’ - Fakultet strojarstva i brodogradnje · 0(+$1,.$)/8,’$ 892’ 892’ 0hkdqlndioxlgdmhglril]lnhnrmlvhedyljledqmhpioxlgdlvlodpdnrmhgmhoxmxqdioxlg 0hkdqlndioxlgdvhglmholqdvwdwlnxioxlgdnrmdsurxþdydudyqrwhåxioxlgdxplurydqmx

0(+$1,.$�)/8,'$��–��+,'5267$7,.$� �

���+,'5267$7,.$�� �2VQRYQD�MHGQDGåED�JLEDQMD��,,��1HZWRQRY�]DNRQ��þHVWLFH�LGHDOQRJ�IOXLGD�L�UHDOQRJ�IOXLGD�X�UHODWLYQRP�PLURYDQMX��d d d

V V S

a V f V Sρ ρ σ= +∫ ∫ ∫GG G �

( )d d dSt

V V S

a V f V pnρ ρ σ= + − +∫ ∫ ∫GG G G �

8]�]DQHPDUHQMH�YLVNR]QLK�VLOD�d d dS= d dV

V V S V V

a V f V pn f V pρ ρ ρ= − − ∇∫ ∫ ∫ ∫ ∫G GG G � ili grada f p a f pρ ρ ρ ρ= − ∇ = −

G GG G ���������2VQRYQD�MHGQDGåED�VWDWLNH�IOXLGD���2VQRYQD�MHGQDGåED�JLEDQMD�]D�VOXþDM� 0a =G L]UDåDYD�UDYQRWHåX�PDVHQLK�VLOD�L�VLOD�WODND��� ili gradf p f pρ ρ= ∇ =

G G ���������,]�RVQRYQH�MHGQDGåEH�VWDWLNH�LPDMXüL�QD�XPX�VYRMVWYD�JUDGLMHQWD�]DNOMXþXMH�VH������$NR�QHPD�PDVHQLK�VLOD�� 0f =

G ��VOLMHGL�GD�MH�WODN�S�NRQVWDQWDQ�����7ODN�QDMEUåH� UDVWH�X� VPMHUX�JUDGS� WM��X�VPMHUX�PDVHQH�VLOH��D�QDMEUåH�RSDGD�X�VPMHUX��

–�JUDGS�WM��X�VPMHUX�VXSURWQRP�RG�PDVHQH�VLOH�����%XGXüL�GD�MH�JUDGp okomit na površinu p=konst. promjena tlaka u okomitom smjeru na YHNWRU�PDVHQH� VLOH� MH� MHGQDND� QXOL��'UXJLP� ULMHþLPD�� YHNWRU�PDVHQH� VLOH� MH�RNRPLW�QD�površine konstantnog tlaka (izobare).�7DNR HU�YULMHGL�����*UDQLFD� GYDMX� IOXLGD� X� PLURYDQMX� SRNODSD� VH� V� L]REDURP�� WH� MH� YHNWRU�PDVHQH� VLOH� X�VYDNRM�WRþNL�RNRPLW�QD�UD]GMHOQX�SRYUãLQX�����9HNWRU�PDVHQH�VLOH�MH�XVPMHUHQ�RG�UD]GMHOQH�SRYUãLQH�SUHPD�IOXLGX�YHüH�JXVWRüH����� 1D� JUDQLFL� GYDMX� IOXLGD� WODN� MH� QHSUHNLGDQ�� DNR� VH� ]DQHPDUH� XþLQFL� SRYUãLQVNH�QDSHWRVWL�������������

6�

������6OLND�X]�GHILQLFLMX�VLOH�WODND�2�

]�

\�[�

9�nG

dSGT

G9�fdVρG

Page 6: 892’ - Fakultet strojarstva i brodogradnje · 0(+$1,.$)/8,’$ 892’ 892’ 0hkdqlndioxlgdmhglril]lnhnrmlvhedyljledqmhpioxlgdlvlodpdnrmhgmhoxmxqdioxlg 0hkdqlndioxlgdvhglmholqdvwdwlnxioxlgdnrmdsurxþdydudyqrwhåxioxlgdxplurydqmx

0(+$1,.$�)/8,'$��–��+,'5267$7,.$� �����3URPMHQD�WODND�X�PLUXMXüHP�IOXLGX�X�SROMX�VLOH�WHåH��3URPMHQD�WODND�L]PH X�GYLMH�WRþNH���X]� konst.ρ = �L� konst.f =

G ��,]�RVQRYQH�MHGQDGåEH�VWDWLNH�VOMHGL��

f p

f dr p dr dp

ρρ

= ∇

⋅ = ∇ ⋅ =

G

G G G ��� 2 1p p f rρ= + ⋅

G G �LOL�

( )2 1 1 cosp p f r p f rρ ρ α= + ⋅ = +G GG G �

,]� VYRMVWYD� VNDOUQRJ�SURGXNWD� MH� MDVQR�GD� VH�SUL�RGUH LYDQMX�SURPMHQH� WODND�PRåH� LOL�SXW�SURMLFLUDWL�QD�VLOX�LOL�VLOX�QD�SXW��2þLWR�MH�GD�DNR�VH�SRYHüD�WODN�S��X�WRþNL����GD�üH�VH�RQ�SRYHüDWL�L�X�WRþNL����RGQRVQR�X�VYLP�GUXJLP�WRþNDPD��ãWR�MH�ELW�3DVFDORYD�]DNRQD�NRML�NDåH�GD�VH�WODN�QDULQXW�L]YDQD�QD�IOXLG�X�mirovanju širi jednoliko u svim smjerovima.��3URPMHQD�WODND�X�PLUXMXüHP�IOXLGX�X�SROMX�VLOH�WHåH�� 29,80665 m/sg = ��� 0 0p p gz p ghρ ρ= − = + ���LOL���� 0 konst.

p pz

g gρ ρ+ = = �

JGMH�]�R]QDþXMH�YLVLQX��K�GXELQX��D� 0p tlak u ishodištu koordinatnog sustava.�

SI

visina tlaka, L, m stupca fluida,p p p

g g gρ ρ ρ

= = =

�piezometriþND�YLVLQD�p

zgρ

+ = ��3ULQFLS�VSRMHQLK�SRVXGD��

S

p� p�p�

g

� z=konst.O

$NR�KRPRJHQD�NDSOMHYLQD�PLruje u više PH XVREQR� VSRMHQLK� SRVXGD�� WDGD� üH�slobodne površine otvorene prema LVWRP� DWPRVIHUVNRP� WODNX� 0p � OHåDWL� X�LVWRM� L]REDUL� �]D� PLUXMXüL� IOXLG� WR� MH��KRUL]RQWDOQD�UDYQLQD���

����������

��

��rG

U�fG

B�

Page 7: 892’ - Fakultet strojarstva i brodogradnje · 0(+$1,.$)/8,’$ 892’ 892’ 0hkdqlndioxlgdmhglril]lnhnrmlvhedyljledqmhpioxlgdlvlodpdnrmhgmhoxmxqdioxlg 0hkdqlndioxlgdvhglmholqdvwdwlnxioxlgdnrmdsurxþdydudyqrwhåxioxlgdxplurydqmx

0(+$1,.$�)/8,'$��–��+,'5267$7,.$� �����+LGURVWDWVNL�PDQRPHWUL��3RVWXSDN� ]D� SRVWDYOMDQMH� MHGQDGåEH�PDQRPHWUD� �MHGQDGåEH� SURPMHQH� WODND� L]PH X� GYLMX�WRþDND�NRMH�VH�PRJX�PH XVREQR�VSRMLWL�NUR]�IOXLG���3ROD]L� VH� V� WODNRP�X� MHGQRM� WRþNL� L� WRP� VH� WODNX�GRGDMX� VYH�SURPMHQH� WODND�REOLND� ghρ ���LGXüL�RG�PHQLVNXVD�GR�PHQLVNXVD��L�WR�V�SR]LWLYQLP�SUHG]QDNRP�DNR�VH�LGH�SUHPD�GROMH��D�V�QHJDWLYQLP�DNR�VH�LGH�SUHPD�JRUH��.DGD�VH�GR H�GR�GUXJH�WRþNH�WDNR�GRELYHQL�L]UD]�VH�L]MHGQDþXMH�V�WODNRP�X�WRM�WRþNL���

h�

h�

h�

A

B

S�S�

S� ��

3ULPMHU�GLIHUHQFLMDOQRJ�PDQRPHWUD�����MHGQDGåED�RG�WRþNH�%�GR�WRþNH�$�B 2 2 0 0 1 1 Ap gh gh gh pρ ρ ρ+ − − = �

���MHGQDGåED�RG�WRþNH�$�GR�WRþNH�%�A 1 1 0 0 2 2 Bp gh gh gh pρ ρ ρ+ + − = �

�$SVROXWQL�WODN�VH�PMHUL�RG�DSVROXWQH�QXOH�������YDNXXP���0DQRPHWDUVNL�WODN�S0�MH�UD]OLND�DSVROXWQRJ�S�L�DWPRVIHUVNRJ�WODND�SD��PMHUL�VH�X�RGQRVX�QD�DWPRVIHUVNL� WODN��S0� �S� ��SD��3R]LWLYQL�PDQRPHWDUVNL� WODN� VH�QD]LYD�SUHWODN�� D�QHJDWLYQL�SRGWODN���2.4 Sila tlaka na ravne površine��

F p A0= 0

F p Ah= C

h

p0O

x

Y

CH

%y

%x

A

hC

S=konst.

p p gh= +0 S

n

C

H

y

g

yC =

sin +

hC

+

�N

0 0 0 0( ) sin( )

c

c

S S S S

y S

F pdS p gh dS p S g hdS p S g ydS p S gh Sρ ρ ρ α ρ= = + = + = + ⋅ = +∫ ∫ ∫ ∫ �

Page 8: 892’ - Fakultet strojarstva i brodogradnje · 0(+$1,.$)/8,’$ 892’ 892’ 0hkdqlndioxlgdmhglril]lnhnrmlvhedyljledqmhpioxlgdlvlodpdnrmhgmhoxmxqdioxlg 0hkdqlndioxlgdvhglmholqdvwdwlnxioxlgdnrmdsurxþdydudyqrwhåxioxlgdxplurydqmx

0(+$1,.$�)/8,'$��–��+,'5267$7,.$� ��

( ) ( ) 2sin

xx

h c

S S

I

F y y g h ydS g y dSρ ρ α+ ∆ = ⋅ ⋅ ⋅ = ⋅ ⋅∫ ∫�

( ) ( ) ( )( )2sin sin

xx

c c c

I

g y S y y g I y Sξξρ α ρ α⋅ ⋅ ⋅ ⋅ ⋅ + ∆ = ⋅ ⋅ +���

c

Iy

y Sξξ∆ = �

• 6LOD� 0F �XVOLMHG�NRQVWDQWQRJ� WODND� 0p okomita je na ravnu površinu A � L�GMHOXMH�X�QMHQRP�WHåLãWX��D�SR�YHOLþLQL�MH��� 0 0F p A= �

• 6LOD� hF � XVOLMHG� SURPMHQMLYRJ� KLGURVWDWVNRJ� WODND� hp ghρ= � RNRPLWD� MH� QD� UDYQX�površinu A � L� GMHOXMH� X� WRþNL� +�� D� SR� YHOLþLQL� MH�� C ChF p A gh Aρ= = � JGMH� MH� Ch �GXELQD�QD�NRMRM�VH�QDOD]L�WHåLãWH�&�SRYUãLQH� A ��

• 3RORåDM�WRþNH�+�MH�X�RGQRVX�QD�WHåLãWH�&�SRYUãLQH�$�GHILQLUDQ�SRPDFLPD�∆[�L�∆\�]D�NRMH�YULMHGL��

C

Iy

y Aξξ= � L�

C

Ix

y Aξη= � JGMH� MH� C C siny h ϑ= � XGDOMHQRVW�

WHåLãWD� &� RG� VORERGQH� SRYUãLQH�� PMHUHQR� X� UDYQLQL� X� NRMRM� VH� QDOD]L� SRYUãLQD��XGDOMHQRVW� OC �SUHPD�VOLFL���D� Iξξ � L� Iξη �VX�JODYQL�L�FHQWULIXJDOQL�PRPHQW�LQHUFLMH�površine $�X�RGQRVX�QD�RVL�ξ�L�η�NUR]�WHåLãWH��SUHPD�VOLFL��3RPDN� x je za površine V�EDUHP�MHGQRP�RVL�VLPHWULMH�MHGQDN�QXOL��YLGMHWL�NDR�SULPMHU�WDEOLFX�NRMD�SULND]XMH�SRGDWNH�R�FHQWULIXJDOQRP�PRPHQWX�LQHUFLMH� Iξη ���

• Za vertikalno uronjenu površinu prema slici vrijedi \& K&�� =D� KRUL]RQWDOQR�uronjenu površinu ( 0ϑ = �� Cy → ∞ �SD�VX�SUHPD�JRUQMLP�L]UD]LPD�∆[ ∆\ ���WH�üH�VLOD� hF GMHORYDWL�X�WHåLãWX�SRYUãLQH��NDR�L�]D�VOXþDM�NRQVWDQWQRJ�WODND�S����

A

+

F gh Ah S C

S

p0O

C

H%h

y hC C=

g

AS

F gh Ah S C

p0

C

hC

g

�0RPHQWL�0[� L�0\� VLOH� KLGURVWDWVNRJ� WODND� X� RGQRVX� QD� WHåLãWH� &� SRYUãLQH� QH� ]DYLVH� RG�GXELQH�QD�NRMRM�VH�WHåLãWH�QDOD]L�� h C

C

sinx

IM F y gh A gI

y Aξξ

ξξ∆ ρ ρ ϑ= ⋅ = ⋅ =⋅

�� h C

C

siny

IM F x gh A gI

y Aξη

ξη∆ ρ ρ ϑ= ⋅ = ⋅ =⋅

��

Page 9: 892’ - Fakultet strojarstva i brodogradnje · 0(+$1,.$)/8,’$ 892’ 892’ 0hkdqlndioxlgdmhglril]lnhnrmlvhedyljledqmhpioxlgdlvlodpdnrmhgmhoxmxqdioxlg 0hkdqlndioxlgdvhglmholqdvwdwlnxioxlgdnrmdsurxþdydudyqrwhåxioxlgdxplurydqmx

0(+$1,.$�)/8,'$��–��+,'5267$7,.$� �Geometrijska svojstva nekih površina�

*HRPHWULMVNL�OLN� Površina� Iξξ � Iηη � Iξη ��

b�� b��

a YI

C

��

A ab= � 3

12

ba � 3

12

ab � ��

Y

I

CR

��

2A R= Q � 4

4

RQ � 4

4

RQ � ��

�Y

I

C

R R

4R3Q

��

21

2A R= Q � 40 1098, R � 40 3927, R � ��

Y

I

C

b+d3

a3

d

b

a

��

2

abA = � 3

36

ba � � ( )2

272

bab d− �

Y

IR

4R3Q

4R3Q C

R

��

21

4A R= Q � 40 05488, R � 40 05488, R � 40 01647, R− �

������

Page 10: 892’ - Fakultet strojarstva i brodogradnje · 0(+$1,.$)/8,’$ 892’ 892’ 0hkdqlndioxlgdmhglril]lnhnrmlvhedyljledqmhpioxlgdlvlodpdnrmhgmhoxmxqdioxlg 0hkdqlndioxlgdvhglmholqdvwdwlnxioxlgdnrmdsurxþdydudyqrwhåxioxlgdxplurydqmx

0(+$1,.$�)/8,'$��–��+,'5267$7,.$� ��3RORåDM�UH]XOWDQWQH�VLOH�)5 )K�)��]D�VOXþDM�LVWRVPMHUQLK�L�PLPRVPMHUQLK�VLOD�)��L�)K����

F0

Fh

+F0FhFR=

%y

%yR

C

H

% %y yR= +F0Fh

Fh

�D��LVWRVPMHUQH�VLOH��

F0

Fh -F0FhFR=

%y

%yR

C

H% %y yR= -F0Fh

Fh

�� E��PLPRVPMHUQH�VLOH��

Fiktivna slobodna površina�$NR�MH�WODN�V�REMH�VWUDQH�SRYUãLQH�LVWL��VOXþDM�RWYRUHQRJ�VSUHPQLND���VLOH�NRQVWDQWQRJ�WODND�VH� SRQLãWDYDMX�� =D� VOXþDM� ]DWYRUHQRJ� VSUHPQLND� UH]XOWDWQWQD� VLOD� NRQVWDQWQRJ� WODND� VH�UDþXQD� V�PDQRPHWDUVNLP� WODNRP� M0p � X� VSUHPQLNX��5DþXQDQMH� VLOH� NRQVWDQWQRJ� WODND� �X�VOXþDMX� GD� MH� SRYUãLQD� SRWSXQR� XURQMHQD� X� IOXLG�� PRåH� VH� L]EMHüL� XYR HQMHP� ILNWLYQH�slobodne površine. Fiktivna slobodna površina je udaljena od stvarne slobodne površine za YLVLQX�PDQRPHWDUVNRJ�WODND� f M0h p gρ= ��]D�VOXþDM�SUHWODND�MH�L]QDG��D�]D�VOXþDM�SRGWODND�ispod stvarne slobodne površine). $NR� ILNWLYQD� VORERGQD� SRYUãLQD� SDGQH� LVSRG� WHåLãWD�&�površine, dubina K�SRVWDMH�QHJDWLYQD��D�VYL�L]UD]L�L�GDOMH�YULMHGH��)LNWLYQD� VORERGQD� SRYUãLQD� VH� PRåH� XYHVWL� L� ]D� VOXþDM� PLURYDQMD� GYDMX� IOXLGD� UD]OLþLWLK�JXVWRüD�SUHPD�VOLFL���

pa

pa

C

A

g

h hf = 1

h1

SS1

fiktivna slobodnapovr inaš

p p gh g h h= + + ( - )a 0 1 1S Sp p gh= +a S

O

p p gh= +a 0S

p ghM0 0=S

S S0<h

h

���2.5 Sila tlaka na zakrivljene površine��6LOD�WODND�QD�]DNULYOMHQX�SRYUãLQX�VH�UD]ODåH�QD�NRPSRQHQWH�X�VPMHURYLPD�RVL��=DNULYOMHQD�površina 6 se projicira na koordinatne ravnine. Projekcija površine je pozitivna ako je kut L]PH X�YHNWRUD�QRUPDOH� L�SR]LWLYQRJ�VPMHUD�RVL�PDQML�RG����� �IOXLG� MH� LVSUHG�SRYUãLQH�6�JOHGDQR�L]�SR]LWLYQRJ�VPMHUD�RVL���

Page 11: 892’ - Fakultet strojarstva i brodogradnje · 0(+$1,.$)/8,’$ 892’ 892’ 0hkdqlndioxlgdmhglril]lnhnrmlvhedyljledqmhpioxlgdlvlodpdnrmhgmhoxmxqdioxlg 0hkdqlndioxlgdvhglmholqdvwdwlnxioxlgdnrmdsurxþdydudyqrwhåxioxlgdxplurydqmx

0(+$1,.$�)/8,'$��–��+,'5267$7,.$� ��

y

z

Sz

O

h z= -

x

Cy

x

z

Hy

%h yx%hyh

hy

slobodna povr inaš

Vd = dV h Sz

dSz

dS

n

S z

hHx %hxh

%hxy

g

Sx

Cx

Sy

hx

�• ,]UD]L�]D�NRPSRQHQWH 0

xF �� 0yF �� 0

zF �VLOH� 0FG �XVOLMHG�NRQVWDQWQRJ�WODND��

0x 0 xF p S= − �������������� 0

y 0 yF p S= − ������������������� 0z 0 zF p S= − �

• ,]UD]L�]D�KRUL]RQWDOQH�NRPSRQHQWH�)[� L�)\� VLOH�XVOLMHG�SURPMHQMLYRJ�KLGURVWDWVNRJ�WODND� hp ghρ= �L�]D�SRPDNH�KYDWLãWD�WLK�NRPSRQHQWL�X�RGQRVX�QD�WHåLãWD�SURMHNFLMD�VX��

� Cx x x x xF p S gh Sρ= − ⋅ = − � Cy y y y yF p S gh Sρ= − ⋅ = − �� xh

x x

Ih

h Sηη∆ =⋅

� yh

y y

Ih

h Sξξ∆ =⋅

� xyx x

Ih

h Sηξ∆ =⋅

� yx

y y

Ih

h Sξς∆ =⋅

��

• 9HUWLNDOQD� NRPSRQHQWD� )] sile hidrostatskog tlaka na površinu 6� MH� SR� YHOLþLQL�MHGQDND� WHåLQL� IOXLGD� NRML� VH� QDOD]L� X� YROXPHQX�9� L]PH X� SRYUãLQH� 6� L� VORERGQH�površine. Sila )]�SUROD]L�WHåLãWHP�YROXPHQD�9��3UHG]QDN�NRPSRQHQWH�VLOH�)]�RYLVL�R�SUHG]QDNX�SURMHNFLMH�6]��WH�VH�PRåH�SLVDWL�GD�MH� �� � � zF gVρ= B �1HJDWLYQL� SUHG]QDN� VH� RGQRVL� QD� VOXþDM� SR]LWLYQH� SURMHNFLMH� SRYUãLQH�6]� �IOXLG� MH�iznad površine 6��� D�SR]LWLYQL�SUHG]QDN�]D� VOXþDM�QHJDWLYQH�SURMHNFLMH�6]� �IOXLG� MH�ispod površine 6���3ULPMHU�� 9HUWLNDOQa i horizontalna komponenta sile na zakrivljenu površinu ABDEF

(prema slici) širine %��RNRPLWR�QD�UDYQLQX�VOLNH���)OXLG�MH�R]QDþHQ�VLYRP�ERMRP��D�WRþNH�*��H i I su na slobodnoj površini.�

Page 12: 892’ - Fakultet strojarstva i brodogradnje · 0(+$1,.$)/8,’$ 892’ 892’ 0hkdqlndioxlgdmhglril]lnhnrmlvhedyljledqmhpioxlgdlvlodpdnrmhgmhoxmxqdioxlg 0hkdqlndioxlgdvhglmholqdvwdwlnxioxlgdnrmdsurxþdydudyqrwhåxioxlgdxplurydqmx

0(+$1,.$�)/8,'$��–��+,'5267$7,.$� ��

$

% % %

' '( () )

V

+ +, , **

+

��9HUWLNDOQD� NRPSRQHQWD� MHGQDND� MH� SR� YHOLþLQL� WHåLQL� IOXLGD� X� RVMHQþDQRP� YROXPHQX� 9��GMHOXMH� SUHPD� GROMH� L� SUROD]L� WHåLãWHP� WRJ� YROXPHQD�� 1D� GLMHOX� SRYUãLQH� %'()� IOXLG� MH�L]QDG�SRYUãLQH��WH�VLOD�GMHOXMH�SUHPD�GROMH��D�SR�YHOLþLQL�MH�MHGQDND�WHåLQL�IOXLGD�X�YROXPHQX�BDEFIGB. Na dijelu površine AB fluid je ispod površine pD�VLOD�GMHOXMH�SUHPD�JRUH��D�SR�YHOLþLQL�MH�MHGQDND�WHåLQL�IOXLGD�X�YROXPHQX�$+*%$��+RUL]RQWDOQH� NRPSRQHQWH� VLOH� WODND� QD� GLMHORYLPD� SRYUãLQH� ()� L� ('� VH� PH XVREQR�SRQLãWDYDMX��3URMHNFLMD�SRYUãLQH�V�NRMRP�VH�UDþXQD�KRUL]RQWDOQD�VLOD�WODND�MH�GDNOH�MHGQDND�Xmnošku visine AD sa širinom % površine.������6LOD�X]JRQD��Sila uzgona je rezultat djelovanja sila tlaka po površini tijela uronjenog u fluid. Sila uzgona MH�MHGQDND�WHåLQL�IOXLGD�LVWLVQXWRJ�WLMHORP��WHåLQL�LVWLVQLQH���GMHOXMH�YHUWLNDOQR�X�YLV�L�SUROD]L�WHåLãWHP�LVWLVQLQH���6LOD�X]JRQD�QD�JUDQLFL�GYDMX�IOXLGD�6OLND� SULND]XMH� VOXþDM� SOLYDQMD� WLMHOD� PDVH� m �� JXVWRüH� 0ρ na razdjelnoj površini dvaju

IOXLGD�JXVWRüD�ρ�� L�ρ���7RþNH�&�� L�&�� VX� WHåLãWD�YROXPHQD� LVWLVQLQH�9�� L�9���D�7� MH� WHåLãWH�WLMHOD���

Fb1

Fb2

V1

V2C2

S0

S1

mg

g

T

��6LOD�)E�X]JRQD�MH�]EURM� b b1 b2 1 1 2 2F F F gV gVρ ρ= + = + �8YMHW�SOLYDQMD��UDYQRWHåH�� MH�GD�VX�UH]XOWDQWQD�VLOD��WM�� bF mg= �� L� UH]XOWDQWQL�PRPHQW�QD�WLMHOR� MHGQDNL�QXOL� �WM�� VXPD�PRPHQDWD� VLOD� b1F � L� b2F �X�RGQRVX�QD� WHåLãWH� WLMHOD�PRUD�ELWL�MHGQDND�QXOL���-DVQR�MH�GD�YULMHGL� 2 0 1ρ ρ ρ> > ��=D�VOXþDM� 2 1ρ ρ>> �VLOD� b2F �VH�]DQHPDUXMH��

Page 13: 892’ - Fakultet strojarstva i brodogradnje · 0(+$1,.$)/8,’$ 892’ 892’ 0hkdqlndioxlgdmhglril]lnhnrmlvhedyljledqmhpioxlgdlvlodpdnrmhgmhoxmxqdioxlg 0hkdqlndioxlgdvhglmholqdvwdwlnxioxlgdnrmdsurxþdydudyqrwhåxioxlgdxplurydqmx

0(+$1,.$�)/8,'$��–��.,1(0$7,.$�)/8,'$� ��

���.,1(0$7,.$�)/8,'$�.LQHPDWLND�IOXLGD�MH�GLR�IL]LNH�NRML�VH�EDYL�JLEDQMHP�IOXLGD��

• 3UHPD� KLSRWH]L� NRQWLQXXPD� YULMHGL� SUDYLOR� GD� VYDND� þHVWLFD� IOXLGD� �PDWHULMDOQD�WRþND�� ]DX]LPD� VDPR� MHGQX� WRþNX� SURVWRUD�� D� X� MHGQRM� WRþNL� SURVWRUD� VH� PRåH�QDOD]LWL�VDPR�MHGQD�þHVWLFD�NRQWLQXXPD��������2SLV�JLEDQMD�IOXLGD� /DJUDQJHRY�RSLV�JLEDQMD�IOXLGD��• 3RORåDML�WRþDND�SURVWRUD�L�SRORåDML�þHVWLFD�IOXLGD�RSLVXMX�VH�UDGLMXV�YHNWRURP� rG ��þLMH�VX� NRPSRQHQWH� SURVWRUQH� LOL� (XOHURYH� NRRUGLQDWH� [�� \�� ]��� 8� DSVROXWQRP�NRRUGLQDWQRP� VXVWDYX� MH� SRORåDM� WRþNH� SURVWRUD� MH� VWDODQ� X� YUHPHQX� �SURVWRUQH�NRRUGLQDWH� [�� \�� ]� QLVX� IXQNFLMH� YUHPHQD��� D� SRORåDM� JLEDMXüH� þHVWLFH� IOXLGD� VH�PLMHQMD� V� YUHPHQRP�� ãWR� ]QDþL� GD� VX� NRPSRQHQWH� � UDGLMXV� YHNWRUD� r

G � �YHNWRUD�SRORåDMD��NRMH�RSLVXMX�SRORåDM�þHVWLFH�IOXLGD�MHVX�IXQNFLMD�YUHPHQD��*LEDQMH�þHVWLFH�GHILQLUDQR� MH� YUHPHQVNRP� SURPMHQRP� QMHQD� YHNWRUD� SRORåDMD� X� REOLNX� ( )r r t

G G� �

�MHGQDGåED�JLEDQMD�þHVWLFH�IOXLGD���• %U]LQD� þHVWLFH� IOXLGD� MHVW� YUHPHQVND� GHULYDFLMD� YHNWRUD� SRORåDMD� ( )v r t

G G�� � �WRþNLFD�R]QDþXMH�YUHPHQVNX�GHULYDFLMX���D�XEU]DQMH�þHVWLFH�IOXLGD�MHVW�YUHPHQVND�GHULYDFLMD�EU]LQH� ( ) ( )a v t r t

G G G� ��� � ��

�• 0DWHULMDOQL� YROXPHQ� VH� VDVWRML� RG� EHVNRQDþQRJ� EURMD� þHVWLFD� IOXLGD�� D� NRMH� VX� WR�

þHVWLFH� GHILQLUDQR� MH� XRþHQRP� NRQILJXUDFLMRP� ( )0M tV � X� SRþHWQRP� YUHPHQVNRP�WUHQXWNX� 0t ��=D�SRWUHEH�RSLVD�QMLKRYD�JLEDQMD�QXåQR�LK�MH�UD]OLNRYDWL��6�RE]LURP�GD�VH� X� MHGQRM� WRþNL� SURVWRUD�PRåH� QDOD]LWL� VDPR� MHGQD� þHVWLFD� IOXLGD�� ýHVWLFH� üH� VH�UD]OLNRYDWL� SR� SRORåDMX� NRMHJ� ]DX]LPDMX� X� SRþHWQRM� NRQILJXUDFLML�� =D� NRRUGLQDWH�SRþHWQRJ� SRORåDMD� þHVWLFD� IOXLGD� VH� XYRGL� SRVHEQD� R]QDND� 0 0( )r r t=G G � L� WH� VH�NRRUGLQDWH� QD]LYDMX� PDWHULMDOQLP� LOL� /DJUDQJHRYLP� NRRUGLQDWDPD�� -DVQR� MH� GD� VX�PDWHULMDOQH�NRRUGLQDWH�YUHPHQVNL�QH]DYLVQH��

0t

0 0( )r r tG G�

0( , )r r tG G

2�

90�W��$�

$�

t90�W���

x

y

z0( , )r r r t=G G G �

6OLND�X]�RSLV�JLEDQMD�þHVWLFD�IOXLGD�

Page 14: 892’ - Fakultet strojarstva i brodogradnje · 0(+$1,.$)/8,’$ 892’ 892’ 0hkdqlndioxlgdmhglril]lnhnrmlvhedyljledqmhpioxlgdlvlodpdnrmhgmhoxmxqdioxlg 0hkdqlndioxlgdvhglmholqdvwdwlnxioxlgdnrmdsurxþdydudyqrwhåxioxlgdxplurydqmx

0(+$1,.$�)/8,'$��–��.,1(0$7,.$�)/8,'$� ��

• *LEDMXüL�PDWHULMDOQL�YROXPHQ�üH�X�WUHQXWNX� t �]DX]HWL�QRYL�SRORåDM��D�EXGXüL�GD�VH�UDGL� R�PDWHULMDOQRP�YROXPHQX� X� WRP� WUHQXWNX� üH� VH� X� QMHPX� QDOD]LWL� LVWH� þHVWLFH�NRMH� VX� X� QMHPX� ELOH� L� X� WUHQXWNX� 0t �� 1D� SULPMHU� WRþND� $� NRMD� MH� X� SRþHWQRM�NRQILJXUDFLML�ELOD�QD�SRORåDMX�GHILQLUDQRP�NRRUGLQDWDPD� 0r

G ��üH�X�WUHQXWNX� t �ELWL�X�WRþNL�V�NRRUGLQDWDPD� r

G �� -DVQR� MH�GD�üH�YULMHGQRVWL�NRRUGLQDWD�[��\��]�]DYLVLWL� L�RG�YUHPHQD�L�RG�WRþNH�X�SRþHWQRM�NRQILJXUDFLML��WDNR�GD�YULMHGL��

�� 0 ,r r r tG G G� ��RGQRVQR�

( )( )( )

1 0 0 0

2 0 0 0

3 0 0 0

, , ,

, , ,

, , ,

x x y z t

y x y z t

z x y z t

ξξξ

=

=

=

� �

*RUQMH�MHGQDGåEH�RSLVXMX�YUHPHQVNL�SURPMHQOMLYL�SRORåDM�RQH�þHVWLFH�IOXLGD�NRMD�MH�X�WUHQXWNX� 0t �ELOD�QD�SR]LFLML�RSLVDQRM�YHNWRURP�SRORåDMD� 0r

G ��0LMHQMDMXüL�YHNWRU� 0rG �

GRELYDMX�MH�MHGQDGåEH�JLEDQMD�UD]OLþLWLK�þHVWLFD�PDWHULMDOQRJ�YROXPHQD��• %U]LQD�þHVWLFH�IOXLGD�MHVW�YUHPHQVND�GHULYDFLMD�YHNWRUD�SRORåDMD� �

�0

00 konst.

( , ) D( , )

Dr

r r t rv r t

t tG

G G GG G

�s

� �s

� �8�PHKDQLFL� VH� RQD� QD]LYD�PDWHULMDOQRP�GHULYDFLMRP�� D� ]ERJ� SRVHEQH� YDåQRVWL� VH�R]QDþXMH� V� D

Dt�� 0DWHULMDOQRP� GHULYDFLMRP� VH� L]UDåDYD� YUHPHQVND� SURPMHQD�

IL]LNDOQRJ�VYRMVWYD�þHVWLFH�IOXLGD��RQDNR�NDNR�EL�WR�RVMHüDR�SURPDWUDþ�NRML�VH�JLED�]DMHGQR� V� þHVWLFRP�� *RUQML� L]UD]� RSLVXMH� SURPMHQX� EU]LQH� þHVWLFD� IOXLGD� L]UDåHQX�/DJUDQJHRYLP�NRRUGLQDWDPD��3URPMHQRP�NRRUGLQDWD� 0r

G �GRELMX�VH�EU]LQH�UD]OLþLWLK�þHVWLFD�PDWHULMDOQRJ�YROXPHQD��

• 8EU]DQMH�þHVWLFH�IOXLGD�MHVW�PDWHULMDOQD�GHULYDFLMD�EU]LQH� ��

0

00 konst.

( , ) D( , )

Dr

v r t va r t

t tG

G G GG G

�s

� �s

� �3RQRYR� VH� SURPMHQRP� /DJUDQJHRYLK� NRRUGLQDWD� GROD]L� GR� XEU]DQMD� UD]OLþLWLK�þHVWLFD�NRQWLQXXPD��X�ELOR�NRMHP�WUHQXWNX���

• 8�/DJUDQJHRYRP� RSLVX� VWUXMDQMD� IOXLGD� VH� IXQNFLMDPD� /DJUDQJHRYLK� NRRUGLQDWD� L�YUHPHQD� PRJX� RSLVDWL� L� GUXJD� IL]LNDOQD� VYRMVWYD� þHVWLFD� IOXLGD�� $NR� VH� VD� ' �R]QDþL� QHNR� IL]LNDOQR� VYRMVWYR� NRQWLQXXPD� �JGMH� ]D� ' � PRåH� VWDMDWL� VNDODUQR�IL]LNDOQR�VYRMVWYR�SRSXW�JXVWRüH� L� WHPSHUDWXUH��YHNWRUVNR�SRSXW�SRORåDMD��EU]LQH�L�XEU]DQMD�LOL�WHQ]RUVNR�VYRMVWYR���RSüHQLWR�VH�PRåH�SLVDWL�� �� L

0 ,r tG

' '� � �5LMHþLPD�EL� VH� UHNOR�GD�JRUQMD� MHGQDGåED�RSLVXMH�YUHPHQVNX�SURPMHQX� IL]LNDOQRJ�VYRMVWYD� ' � þHVWLFH� 0r

G �� 1DGLQGHNV� /� X� R]QDFL� IXQNFLMH� XND]XMH� GD� MH� IL]LNDOQR�VYRMVWYR�L]UDåHQR�/DJUDQJHRYLP�NRRUGLQDWDPD��� (XOHURY�RSLV�JLEDQMD�IOXLGD��

• 8�PHKDQLFL�IOXLGD�VH�XJODYQRP�NRULVWL�(XOHURY�RSLV�VWUXMDQMD�IOXLGD��NRML�VH�WHPHOML�QD�SROMLPD�IL]LNDOQLK�YHOLþLQD��$NR�VH�VYDNRM�WRþNL�SURVWRUD�X�VYDNRP�YUHPHQVNRP�WUHQXWNX� SULGUXåL� IL]LNDOQR� VYRMVWYR� RQH� þHVWLFH� IOXLGD� NRMD� VH� X� SURPDWUDQRP�WUHQXWNX�QDOD]L�X�SURPDWUDQLP� WRþNDPD�SURVWRUD�GRELMH� VH�SROMH� IL]LNDOQH�YHOLþLQH�L]UDåHQR�SURVWRUQLP��(XOHURYLP��NRRUGLQDWDPD� �� E ,r t

G' '� � �

Page 15: 892’ - Fakultet strojarstva i brodogradnje · 0(+$1,.$)/8,’$ 892’ 892’ 0hkdqlndioxlgdmhglril]lnhnrmlvhedyljledqmhpioxlgdlvlodpdnrmhgmhoxmxqdioxlg 0hkdqlndioxlgdvhglmholqdvwdwlnxioxlgdnrmdsurxþdydudyqrwhåxioxlgdxplurydqmx

0(+$1,.$�)/8,'$��–��.,1(0$7,.$�)/8,'$� ��

• =D� SROMH� NRMH� QLMH� IXQNFLMD� YUHPHQD� NDåH� VH� GD� MH� VWDFLRQDUQR�� LQDþH� MH�QHVWDFLRQDUQR��• 9H]X� PH X� /DJUDQJHRYLP� L� (XOHURYLP� RSLVRP� QHNRJ� IL]LNDOQRJ� VYRMVWYD� X�VWUXMDQMX�IOXLGD�GHILQLUDMX�LQYHU]QH�MHGQDGåEH�JLEDQMD��� �

0 0

0 0

0 0

, , ,

, , ,

, , ,

x x x y z t

y y x y z t

z z x y z t

������LOL�NUDüH����� 0 0 ,r r r tG G G� � �

*RUQMH�MHGQDGåEH�GDMX�SRþHWQL�SRORåDM��X�WUHQXWNX 0t ��RQH�þHVWLFH�IOXLGD�NRMD�VH�X�WUHQXWNX�W�QDOD]L�QD�SR]LFLML�GHILQLUDQRM�SURVWRUQLP�NRRUGLQDWDPD� rG . Uvrštavanjem JRUQMHJ� L]UD]D� X� /DJUDQJHRY� ]DSLV� IL]LNDOQRJ� VYRMVWYD� ' � VOLMHGL� (XOHURY� ]DSLV�SROMD� ' � �� L E E

0 0, ( , ), ,r t r r t t r tG G G G

' ' ' '� � � � �• %H]�RE]LUD�ãWR�VX�IL]LNDOQD�VYRMVWYD�L]UDåHQD�SURVWRUQLP�NRRUGLQDWDPD�MDVQR�MH�GD�VX� QRVLWHOML� IL]LNDOQLK� VYRMVWDYD� þHVWLFH� IOXLGD�� D� QH� WRþNH� SURVWRUD�� 8� WRþNDPD�SURVWRUD�X�NRMLPD�QHPD�þHVWLFD�IOXLGD�SROMH�IL]LNDOQH�YHOLþLQH�QLMH�GHILQLUDQR��� 0DWHULMDOQD�GHULYDFLMD�• 0DWHULMDOQD�GHULYDFLMD� L]UDåDYD�EU]LQX�SURPMHQH�IL]LNDOQRJ�VYRMVWYD�þHVWLFH� IOXLGD��WM�� SURPMHQX� NRMX� EL� RVMHWLR� SURPDWUDþ� NRML� EL� VH� JLEDR� ]DMHGQR� V� þHVWLFRP�� =D�IL]LNDOQR�VYRMVWYR�]DSLVDQR�/DJUDQJHRYLP�NRRUGLQDWDPD�RQD�MH�GHILQLUDQD�NDR� �

0

L0

konst.

,

r

r tD

Dt t G

G''

s�

s� �

0DWHULMDOQD� GHULYDFLMD� LVWRJ� WRJ� IL]LNDOQRJ� VYRMVWYD� ]DSLVDQRJ� X� (XOHURYLP�NRRUGLQDWDPD�JODVL� �� E

E Ekons . kons .

D ( , )( , ) ( , )

D r t i t t

r tv r t r t

t t = =Φ ∂Φ= + ⋅∇Φ

∂G

GG G � �

3UYL� þODQ� GHVQH� VWUDQH� JRUQMHJ� L]UD]D� R]QDþXMH� ORNDOQX� SURPMHQX� IL]LNDOQRJ�VYRMVWYD�� NRMX� EL� RVMHWLR� SURPDWUDþ� X� ILNVQRM� WRþNL�SURVWRUD�� GRN�GUXJL� þODQ�GHVQH�VWUDQH� R]QDþXMH� NRQYHNWLYQX� LOL� SULMHQRVQX� EU]LQX� SURPMHQH� IL]LNDOQRJ� VYRMVWYD��XVOLMHG� SRPLFDQMD� þHVWLFH� IOXLGD� X� SROMX�Φ�� ,VSXãWDMXüL� R]QDNX� (� ]D� (XOHURYVNR�SROMH� L� L]EMHJDYDMXüL� HNVSOLFLWQR� QDYR HQMH� ]DYLVQRVWL� SROMD� Φ� RG� SURVWRUQLK� L�YUHPHQVNH�NRRUGLQDWH��JRUQML�L]UD]�X�UD]YLMHQRP�REOLNX�SRSULPD�REOLN�� ��

Nokalna konvektivna promjenapromjena

D

D x y z

l

v v vt t x y z

Φ ∂Φ ∂Φ ∂Φ ∂Φ= + + +∂ ∂ ∂ ∂���������

� �

0RJXüH�MH�GHILQLUDWL�L�RSHUDWRU�PDWHULMDOQH�GHULYDFLMH��NRML�JODVL�� �� D

Dv

t t

• ∂ •= + ⋅∇ •∂

G � �*GMH�XPMHVWR�R]QDNH�•�PRåH�VWDMDWL�VNDODUQR��YHNWRUVNR�LOL�WHQ]RUVNR�SROMH�L]UDåHQR�X�IXQNFLML�SURVWRUQLK�NRRUGLQDWD�L�YUHPHQD��

���������������������������������������������������1XåDQ�L�GRYROMDQ�XYMHW�]D�SRVWRMDQMH�LQYHU]QH�IXQNFLMH�MH�GD�MH�GHWHUPLQDQWD� 0/r r∂ ∂G G �UD]OLþLWD�RG�QXOH�L�NRQDþQD��

Page 16: 892’ - Fakultet strojarstva i brodogradnje · 0(+$1,.$)/8,’$ 892’ 892’ 0hkdqlndioxlgdmhglril]lnhnrmlvhedyljledqmhpioxlgdlvlodpdnrmhgmhoxmxqdioxlg 0hkdqlndioxlgdvhglmholqdvwdwlnxioxlgdnrmdsurxþdydudyqrwhåxioxlgdxplurydqmx

0(+$1,.$�)/8,'$��–��.,1(0$7,.$�)/8,'$� ��

• 'RN�VH�X�/DJUDQJHRYRP�RSLVX�VWUXMDQMD�IOXLGD�SROD]L�RG�MHGQDGåEL�JLEDQMD��þLMLP�VH�GHULYLUDQMHP� GROD]L� GR� EU]LQH� L� XEU]DQMD��� X� (XOHURYRP� VH� RSLVX� SROD]L� RG� SROMD�EU]LQH��MHU�VH�SROMH�EU]LQH�SRMDYOMXMH�X�RSHUDWRUX�PDWHULMDOQH�GHULYDFLMH��������6WUXMQLFH��• 6WUXMQLFH�VX�]DPLãOMHQH�NULYXOMH�NRMLPD�VH�X�VYDNRM�WRþNL�VPMHU�WDQJHQWH�SRNODSD�VD�

smjerom vektora brzine. Na strujnicama se ucrtava smjer strujanja kao što SULND]XMH�VOLND��=D�QHVWDFLRQDUQR�SROMH�EU]LQH��VOLND�VWUXMQLFD�VH�PLMHQMD�RG�WUHQXWND�GR�WUHQXWND��SD�VH�VOLND�VWUXMQLFD�RGQRVL�QD�MHGDQ�L]DEUDQL�YUHPHQVNL�WUHQXWDN��QSU��W W��� $NR� VH� SUDYDF� YHNWRUD� EU]LQH� SRNODSD� V� WDQJHQWRP� QD� VWUXMQLFX�� WDGD� MH�XVPMHUHQL� HOHPHQW� OXND� VWUXMQLFH� drG � SDUDOHODQ� YHNWRUX� EU]LQH� v

G �� WH� MH� QMLKRY�YHNWRUVNL� SURGXNW� MHGQDN� QXOL�� RGQRVQR� SULSDGDMXüH� NRPSRQHQWH� LP� VH� UD]OLNXMX�LVWLP�IDNWRURP��WDNR�GD�YULMHGL���d d d

( , , , ) ( , , , ) ( , , , )x y z

x y z

v x y z t v x y z t v x y z t= = � �

�• 2VQRYQR�VYRMVWYR�VWUXMQLFD�MH�GD�VH�RQH�QH�PRJX�SUHVLMHFDWL��MHU�EL�WR�]QDþLOR�GD�X�WRþNL�SUHVMHND�YHNWRU�EU]LQH�LPD�GYD�UD]OLþLWD�VPMHUD��ãWR�MH�QHIL]LNDOQR��,]X]HWDN�þLQH�WRþNH�]DVWRMD�X�NRMLPD�MH�EU]LQD�MHGQDND�QXOL�������7UDMHNWRULMH��• 7UDMHNWRULMD� MH� SURVWRUQD� NULYXOMD� NRMX� VYRMLP� JLEDQMHP� RSLVXMH� þHVWLFD� IOXLGD��-HGQDGåEH�JLEDQMD�þHVWLFH�IOXLGD�]DSLVDQH�X�/DJUDQJHRYLP�NRRUGLQDWDPD�R]QDþXMX�SDUDPHWDUVNL� ]DSLV� MHGQDGåEH� WUDMHNWRULMH�� 8� (XOHURYRP� RSLVX� VWUXMDQMD�� JGMH� VH�SROD]L� RG� SROMD� EU]LQH�� GR� MHGQDGåEH� WUDMHNWRULMD� VH� GROD]L�� SROD]HüL� RG� GHILQLFLMH�

EU]LQH�þHVWLFH�NRQWLQXXPD��$NR�MH� drG �XVPMHUHQL�LQILQLWH]LPDOQL�HOHPHQW�SXWD�NRMHJ�SUHYDOL� þHVWLFD� NRQWLQXXPD� JLEDMXüL� VH� SR� VYRMRM� WUDMHNWRULML� ]D� LQILQLWH]LPDOQR�YULMHPH�GW��WDGD�]D�WDM�XVPMHUHQL�HOHPHQW�OXND�WUDMHNWRULMH��L]�VDPH�GHILQLFLMH�EU]LQH�

VOLMHGL�d ( , ) dr v r t t= ⋅G G G, šWR� VH� PRåH� SULND]DWL� L� X� REOLNX� VXVWDYD� GLIHUHQFLMDOQLK�

MHGQDGåEL���� � d d d

d( , , , ) ( , , , ) ( , , , )x y z

x y zt

v x y z t v x y z t v x y z t= = = � �

þLMLP� VH� UMHãDYDQMHP� X]� SRþHWQH� XYMHWH� ]D� W W��� ( )0 0r t r=G G �� GROD]L� GR� MHGQDGåEL�WUDMHNWRULMD���

• .ULYXOMD�RELOMHåHQLK�þHVWLFD�X�GDQRP�YUHPHQVNRP�WUHQXWNX�VSDMD�VYH�þHVWLFH�IOXLGD�NRMH�VX�SURãOH�]DGDQRP�WRþNRP�SURVWRUD��• 8� VWDFLRQDUQRP� VWUXMDQMX� WUDMHNWRULMH�� VWUXMQLFH� L� NULYXOMH� RELOMHåHQLK� þHVWLFD� VH�SRNODSDMX���

Page 17: 892’ - Fakultet strojarstva i brodogradnje · 0(+$1,.$)/8,’$ 892’ 892’ 0hkdqlndioxlgdmhglril]lnhnrmlvhedyljledqmhpioxlgdlvlodpdnrmhgmhoxmxqdioxlg 0hkdqlndioxlgdvhglmholqdvwdwlnxioxlgdnrmdsurxþdydudyqrwhåxioxlgdxplurydqmx

0(+$1,.$�)/8,'$��–��.,1(0$7,.$�)/8,'$� ��

� �YHNWRUL�EU]LQH�]D�VWUXMDQMH�X�EOL]LQL�WRþNH�]DVWRMD� VWUXMQLFH�]D�VWUXMDQMH�X�EOL]LQL�WRþNH�]DVWRMD��

� �6WUXMQLFH�SUL�RSWMHFDQMX�FLOLQGUD� VOLND�VWUXMQLFD�]D�VOXþDM�QDJORJ�SURãLUHQMD��3.4 Strujna površina i strujna cijev��

Strujna površina je sastavljena od strujnica koje SUROD]H�WRþNDPD�QHNH�NULYXOMH�&��9HNWRU� EU]LQH� MH� WDQJHQFLMDODQ na površinu 0v n⋅ =G G

, pa kroz strujnu površinu nema protoka

d 0S

Q v n S= ⋅ =∫G G ��

Ako je krivulja C zatvorena, strujna površina prelazi u plašt strujne cijevi, kroz kojeg nema SURWRND� IOXLGD�� NDR� L� NUR]� SODãW� QHNH� IL]LþNH�FLMHYL��$NR� MH� SRYUãLQD� SRSUHþQRJ� SUHVMHND� FLMHYL� dS �LQILQLWH]LPDOQD�� JRYRUL� VH� R� HOHPHQWDUQRM�VWUXMQRM� FLMHYL�� 8� JUDQLþQRP� SULMHOD]X� d 0S → �HOHPHQWDUQD�VWUXMQD�FLMHY�SUHOD]L�X�VWUXMQLFX���

Page 18: 892’ - Fakultet strojarstva i brodogradnje · 0(+$1,.$)/8,’$ 892’ 892’ 0hkdqlndioxlgdmhglril]lnhnrmlvhedyljledqmhpioxlgdlvlodpdnrmhgmhoxmxqdioxlg 0hkdqlndioxlgdvhglmholqdvwdwlnxioxlgdnrmdsurxþdydudyqrwhåxioxlgdxplurydqmx

0(+$1,.$�)/8,'$��–��.,1(0$7,.$�)/8,'$� ��������3URWRN��9ROXPHQVNL�SURWRN�LOL�MHGQRVWDYQR�SURWRN�Q �MHVW�YROXPHQ�þHVWLFD�IOXLGD�NRMH�X�MHGLQLþQRP�YUHPHQX�SUR X�NUR]�SURPDWUDQX�SRYUãLQX� S �RULMHQWLUDQX�MHGLQLþQLP�YHNWRURP�QRUPDOH� nG ��$NR� VH� þHVWLFH� IOXLGD� JLEDMX� EU]LQRP� v

G �� D� WRþNH� SRYUãLQH� EU]LQRP� uG �� WDGD� MH� UHODWLYQD�

EU]LQD� JLEDQMD� þHVWLFD� IOXLGD� X� RGQRVX� QD� SRYUãLQX� w v u= −G G G �� D� SURWRN� Q � MH� GHILQLUDQ�L]UD]RP� �� ( )d d

S S

Q w n S v u n S= ⋅ = − ⋅∫ ∫G G G G G ��

�3ULPMHU� ��� 3URWRN� NUR]� PLUXMXüX� SRYUãLQX� � 0u =G �� MH�SUHPD�RSüRM�IRUPXOL� d

S

Q v n S= ⋅∫G G ��

ýHVWLFD�IOXLGD�7�VH�X�WUHQXWNX�W nalazi na površini dS ��D�X� WUHQXWNX� W�GW� üH� ]DX]HWL�QRYL�SRORåDM�X�SURVWRUX��SUL�þHPX� üH� SUHYDOLWL� SXW� dv t

G �� RGQRVQR� VYRMLP� JLEDQMHP�RSLVDWL� NRVX� SUL]PX�� NRMRM� MH�YLVLQD� MHGQDND�SURMHNFLML�YHNWRUD� SXWD� QD� VPMHU� QRUPDOH� d dh n v t

G G� ¸ �� 9ROXPHQ�

þHVWLFD� IOXLGD� NRMH� X� YUHPHQX� dt � SUR X�NUR]�SRYUãLQX�G6� MHGQDN� MH� YROXPHQX� SUL]PH� d d d d dV S h v n S t

G G� ¸ � ¸ ¸ ��

Elementarni protok kroz površinu dS � MHGQDN� MH� SR�GHILQLFLML� RPMHUX� YROXPHQD� dV � L� YUHPHQD� dt �� WM��

dd = d

d

VQ v n S

t

G G� ¸ , a ukupni protok kroz površinu 6�

jednak je zbroju svih elementarnih protoka, što se opisuje integralom dS

Q v n S= ⋅∫G G ��

� 3RVHEDQ�VOXþDM��EU]LQD�RNRPLWD�QD�UDYQX�površinu)�

d dA A

Q v n A v A= ⋅ =∫ ∫G G �

vdA

Brzina je okomita na ravnu površinu i NRQVWDQWQD�d

A

Q v A vA= =∫ �vA

��Primjer 2: Protok kroz površinu koja se giba brzinom uG �X�PLUXMXüHP�IOXLGX�� 0v =G ��MH�SUHPD�RSüRM�IRUPXOL�

dS

Q u n S= − ⋅∫G G ��

Gibanjem površine 6�� HOHPHQW� G6� RSLVXMH� NRVX�SUL]PX� NRMRM� MH� GXOMLQD� EULGD� du t

G �� D� YROXPHQ�d d dV u n S t

G G� ¸ ¸ . Dakle gibanjem površine 6�PLUXMXüH�þHVWLFH� IOXLGD� SUHOD]H� V� GHVQH� QD� OLMHYX� VWUDQX�

površine, pa gledano relativno u odnosu na površinu WR� MH� LVWR� NDR� GD� MH� SRYUãLQD� PLURYDOD�� D� þHVWLFH�EU]LQRP� u�

G prolazile kroz površinu. Zato je protok GHILQLUDQ�L]UD]RP� d

S

Q u n S= − ⋅∫G G ��

0LUXMXüD�SRYUãLQD�6�

nG

G6�7�W��

dv tG7�W�GW��

*LEDMXüD�SRYUšina 6�

G6� 6�W�GW��

nG6�W��

du tG

Page 19: 892’ - Fakultet strojarstva i brodogradnje · 0(+$1,.$)/8,’$ 892’ 892’ 0hkdqlndioxlgdmhglril]lnhnrmlvhedyljledqmhpioxlgdlvlodpdnrmhgmhoxmxqdioxlg 0hkdqlndioxlgdvhglmholqdvwdwlnxioxlgdnrmdsurxþdydudyqrwhåxioxlgdxplurydqmx

0(+$1,.$�)/8,'$��–��.,1(0$7,.$�)/8,'$� ���3ULPMHU����3URWRN�NUR]�PDWHrijalnu površinu ( u v=G G ��

( ) d 0S

Q v u n S= − ⋅ =∫G G G ��-DVQR� MH�GD�NUR]�PDWHULMDOQX�SRYUãLQX�QHPD�SURWRND�þHVWLFD�IOXLGD�MHU�

VH�RQD�VDVWRML�VWDOQR�RG�MHGQLK�WH�LVWLK�þHVWLFD���3URWRN�IL]LNDOQH�YHOLþLQH��ýHVWLFH� IOXLGD� RVLP� YROXPHQD� LPDMX� PDVX�� HQHUJLMX�� NROLþLQX� JLEDQMD�� LWG�� 3URODVNRP�þHVWLFH�IOXLGD�NUR]�QHNX�SRYUãLQX��RQD�SURQRVL�IL]LNDOQH�YHOLþLQH��SD�VH�JRYRUL�R�SURWRFLPD��YROXPHQD��ãWR�MH�JRUH�GHILQLUDQR�MHGQRVWDYQR�NDR�SURWRN���PDVH��HQHUJLMH��NROLþLQH�JLEDQMD�L� VO�� $NR� VH� VD� ) � R]QDþL� IL]LNDOQD� YHOLþLQD�� D� VD� Φ � YROXPHQVNX� JXVWRüX� WH� IL]LNDOQH�YHOLþLQH��NRMD�MH�GHILQLUDQD�L]UD]RP� ��

GGV

limV V

Φ→

= =) ) ��RGQRVQR�VDGUåDM�IL]LNDOQH�YHOLþLQH�XQXWDU�þHVWLFH�IOXLGD��XQXWDU�LQILQLWH]LPDOQRJ�YROXPHQD�dV �� MHVW� d dVΦ) �� D� VDGUåDM� WH� IL]LNDOQH� YHOLþLQH� XQXWDU� RGUH HQRJ� YROXPHQD� V � MH�GHILQLUDQ�LQWHJUDORP� �� d

V

VΦ∫) ��3ULPMHUL�� 1V Φ⇒ =) ������ m Φ ρ⇒ =) ������ mv vΦ ρ⇒ =G G) ������ 2 21

2

mv vΦ ρ⇒ =�) � ��'DNOH�]D�VOXþDM�JLEDMXüH�SRYUãLQH�X�JLEDMXüHP�IOXLGX��YROXPHQVNL�SURWRN�NUR]�HOHPHQWDUQX�površinu dS � üH� ELWL� ( )d dQ v u n S= − ⋅G G G �� D� SURWRN� IL]LNDOQH� YHOLþLQH� SURQHVHQH� NUR]� WX�površinu je ( )d dQ v u n SΦ= − ⋅G G G

) ��RGQRVQR�SURWRN�IL]LNDOQH�YHOLþLQH�NUR]�XNXSQX�SRYUãLQX�MH�� ( ) d

S

Q v u n SΦ= − ⋅∫G G G

) ��3ULPMHUL��

D�� 0DVHQL� SURWRN�� ( ) dm

S

Q m v u n Sρ= = − ⋅∫G G G� � �� [ ] [ ]1

SIMT , kg/sm m−= =� � �� =D� VOXþDM�PLUXMXüH�

površine: dS

m v n Sρ= ⋅∫G G� ��=D� konst.S� �YULMHGL�m QS�� ��

E�� 7HåLQVNL� SURWRN� ( ) dG

S

Q G g v u n Sρ= = − ⋅∫G G G� � �� 3

SIMLT , N/sG G− = =

� � �� =D� VOXþDM�PLUXMXüH�SRYUãLQH�� d

S

G gv n Sρ= ⋅∫G G� ��=D� konst.S� �L konst.g � �YULMHGL�G mg gQS� �� � ��

F�� 3URWRN� NROLþLQH� JLEDQMD�� ( ) ( )KG dkkS

Q v v u n Sρ= − ⋅∫G G G � �� ( ) ( )2

KG KG SIMLT , N

k kQ Q− = = ��

=D� VOXþDM� PLUXMXüH� SRYUãLQH�� ( )KG dkkS

Q v v n Sρ= ⋅∫G G �� �3URWRN� NROLþLQH� JLEDQMD� MH�

YHNWRUVND�YHOLþLQD���G�� 3URWRN�NLQHWLþNH�HQHUJLMH�� ( )2

EK

1d

2S

Q v v u n Sρ= − ⋅∫G G G ���[ ] [ ]2 3

EK EK SIML T , WQ Q−= = ��

� ��

Page 20: 892’ - Fakultet strojarstva i brodogradnje · 0(+$1,.$)/8,’$ 892’ 892’ 0hkdqlndioxlgdmhglril]lnhnrmlvhedyljledqmhpioxlgdlvlodpdnrmhgmhoxmxqdioxlg 0hkdqlndioxlgdvhglmholqdvwdwlnxioxlgdnrmdsurxþdydudyqrwhåxioxlgdxplurydqmx

0(+$1,.$�)/8,'$��–��.,1(0$7,.$�)/8,'$� ������/HLEQLW]RY�WHRUHP� %U]LQD�SURPMHQH�YHOLþLQH�YROXPHQD���

D�� 2SüL� VOXþDM� YROXPHQD� V þLMD� VH� JUDQLFD�S �JLED�EU]LQRP�uG �%U]LQD�SURPMHQH�YROXPHQD�MH�SR�GHILQLFLML�

( ) ( )dd

d d

V t t V tV

t t

+ −= , a element površine

dS � RSLVXMH� HOHPHQW� YROXPHQD�( ) d dd dV u n t S

G G� ¸ , što iQWHJULUDQR� SR�

površini S � GDMH� UD]OLNX� YROXPHQD�( ) ( )dV t t V t+ − ��WH�MH�NRQDþQR��

�������������� dd d

d S V

Vu n S u V

t= ⋅ = ∇ ⋅∫ ∫

G G G ���

� %U]LQD�SURPMHQH�VDGUåDMD�IL]LNDOQH�YHOLþLQH�XQXWDU�YROXPHQD��0

( ) ( )

0( ) ( ) ( ) ( )

( ) ( ) ( ) (

1lim ( , ) ( , )

1lim ( , ) ( , ) ( , ) ( , )

( , )

tV V t t V t

tV t V t V t t V t

V t V t t V t V

df dV f r t t dV f r t dV

dt t

f r t t dV f r t dV f r t t dV f r t t dVt

f fdV f r t t dV dV

t t

φ∆ →

+∆

∆ →+∆

+∆ −

= + ∆ − =

+ ∆ − + + ∆ − + ∆ = ∆

∂ ∂+ + ∆ =∂ ∂

∫ ∫ ∫

∫ ∫ ∫ ∫

∫ ∫

G G

G G G G

G

) ( ) ( ) ( )

( )t S t V t S t

ffu ndS dV fu dV

t

∂+ ⋅ = + ∇ ⋅∂∫ ∫ ∫ ∫

G G G

��D��2SüL�VOXþDM�JLEDMXüHJ�YROXPHQD�( )

lokalna promjena promjena uslijedgibanja volumena

dd d d d

d V V S V

f ff V V fu n S fu V

t t t

∂ ∂ = + ⋅ = + ∇ ⋅ ∂ ∂ ∫ ∫ ∫ ∫G G G

�� ����

E��0DWHULMDOQL�YROXPHQ��u vG G� � d D

d Dt t→ ��

( )M M M M

Dd d d d

D V V S V

f ff V V fv n S fv V

t t t

∂ ∂ = + ⋅ = + ∇ ⋅ ∂ ∂ ∫ ∫ ∫ ∫G G G �

F��0LUXMXüL�YROXPHQ�� 0uG� ��

dd d

d V V

ff V V

t t

∂=∂∫ ∫ �

�����

*LEDMXüD�površina�6�

G6� 6�W�GW��

nG

6�W��

du tG

9�W�GW��9�W��

Page 21: 892’ - Fakultet strojarstva i brodogradnje · 0(+$1,.$)/8,’$ 892’ 892’ 0hkdqlndioxlgdmhglril]lnhnrmlvhedyljledqmhpioxlgdlvlodpdnrmhgmhoxmxqdioxlg 0hkdqlndioxlgdvhglmholqdvwdwlnxioxlgdnrmdsurxþdydudyqrwhåxioxlgdxplurydqmx

0(+$1,.$�)/8,'$��–��.,1(0$7,.$�)/8,'$� �������0DWHULMDOQL�YROXPHQ��

• 0DWHULMDOQL�YROXPHQ� MV ��IOXLGQR�WLMHOR��MH�XRþHQL�GLR�SURVWRUD�LVSXQMHQ�IOXLGRP�NRML�VH�WLMHNRP�JLEDQMD�VDVWRML�VWDOQR�RG�MHGQLK�WH�LVWLK�þHVWLFD��0DWHULMDOQL�YROXPHQ�MH�RG�okoline odijeljen materijalnom površLQRP� MS � NRMD� VH� WDNR HU� VDVWRML� VWDOQR� RG�MHGQLK� WH� LVWLK� þHVWLFD�� -DVQR� MH� GD� MH� EU]LQD� JLEDQMD�PDWHULMDOQH� SRYUãLQH� MHGQDND�EU]LQL�JLEDQMD�þHVWLFD�IOXLGD��NRMH�þLQH�PDWHULMDOQX�SRYUãLQX��

• 8�RSüHP�VOXþDMX�PDWHULMDOQL�YROXPHQ�WLMHNRP�JLEDQMD�PLMHQMD�VYRM�SRORåDM��REOLN�L�YHOLþLQX�� SD� MH� ]D� RSLV� QMHJRYD� JLEDQMD�� SRWUHEQR� RSLVDWL� JLEDQMH� VYDNH� QMHJRYH�þHVWLFH��• Nema protoka kroz materijalnu površinu ( v u=G G ��� %U]LQD� SURPMHQH� VDGUåDMD�IL]LNDOQH�YHOLþLQH�]D�PDWHULMDOQL�YROXPHQ�MHGQDND�MH��

( ) ( ) ( )M M MV t V t S t

D ffdV dV fv ndV

Dt t

∂= + ⋅∂∫ ∫ ∫

G G

Page 22: 892’ - Fakultet strojarstva i brodogradnje · 0(+$1,.$)/8,’$ 892’ 892’ 0hkdqlndioxlgdmhglril]lnhnrmlvhedyljledqmhpioxlgdlvlodpdnrmhgmhoxmxqdioxlg 0hkdqlndioxlgdvhglmholqdvwdwlnxioxlgdnrmdsurxþdydudyqrwhåxioxlgdxplurydqmx

0(+$1,.$�)/8,'$��–�',1$0,.$�)/8,'$� ��

���',1$0,.$�)/8,'$��'LQDPLND� IOXLGD� MH� GLR� PHKDQLNH� IOXLGD� NRML� VH� EDYL� VLODPD� NRMH� GMHOXMX� QD� IOXLGH��JLEDQMLPD�NRMH�QDVWDMX�GMHORYDQMHP�WLK�VLOD�L�LQWHUDNFLMDPD�L]PH X�þYUVWLK�WLMHOD�L�IOXLGD�X�JLEDQMX��• 0DWHULMDOQL� YROXPHQ� �IOXLGQR� WLMHOR�� MH�HNYLYDOHQWQR�VXVWDYX�PDWHULMDOQLK� WRþDND�X�PHKDQLFL�� WH� ]DWYRUHQRP� WHUPRGLQDPLþNRP� VXVWDYX� X� WHUPRGLQDPLFL�� SD� üH� VYL�]DNRQL� PHKDQLNH� L� WHUPRGLQDPLNH� ELWL� GLUHNWQR� SULPMHQMLYL� L� QD� PDWHULMDOQL�YROXPHQ��• 8�PHKDQLFL� VX�GHILQLUDQL�1HZWRQRYL� ]DNRQL�JLEDQMD�� RG�NRMLK� VH�GUXJL�1HZWRQRY�]DNRQ��PRåH�]DSLVDWL� X�REOLNX�]DNRQD�NROLþLQH�JLEDQMD�� ]DNRQD�PRPHQWD�NROLþLQH�JLEDQMD�LOL�]DNRQD�NLQHWLþNH��PHKDQLþNH��HQHUJLMH��D�X�WHUPRGLQDPLFL�VX�GHILQLUDQL�SUYL�]DNRQ�WHUPRGLQDPLNH��]DNRQ�RþXYDQMD�HQHUJLMH��L�GUXJL�]DNRQ�WHUPRGLQDPLNH��6YL�VX�WL�]DNRQL��NDR�L�]DNRQ�RþXYDQMD�PDVH��RVQRYQL�]D�NODVLþQX�IL]LNX�SD�WDNR�L�]D�PHKDQLNX�IOXLGD��• 8� WHUPRGLQDPLFL� VH� XYRGL� NRQFHSW� WRSOLQH�� XQXWDUQMH� HQHUJLMH� L� HQWURSLMH�� D� UDGQL�PHGLM�MH�XJODYQRP�SOLQ��NRMHPX�VH�GMHORYDQMHP�VLOH�WODND�PRåH�PLMHQMDWL�YROXPHQ��=D�VPDQMLYDQMH�YROXPHQD�SOLQD�XQXWDU�WHUPRGLQDPLþNRJ�VXVWDYD��NDGD�VH�JRYRUL�R�NRPSUHVLML���SRWUHEQR�MH�XODJDWL�PHKDQLþNL�UDG��D�SUL�ãLUHQMX�SOLQD��HNVSDQ]LML��SOLQ�

vrši rad u odnosu na okolinu. U procesima pri konstantnom volumenu korisni PHKDQLþNL�UDG�MHGQDN�MH�QXOL��• 2VLP�WODþQLK�VLOD�X�VXVWDYX�GMHOXMX�L�VLOH�WUHQMD��X�IOXLGX�VX�WR�YLVNR]QH�VLOH���%XGXüL�VX�VLOH� WUHQMD�XYLMHN�VXSURWQH�SRPDNX��QMLKRYLP�VH�GMHORYDQMHP�XYLMHN�PHKDQLþND�HQHUJLMD� SUHWYDUD� X� XQXWDUQMX�� D� QLNDG� REUQXWR�� ,]� UHþHQRJ� VH� ]DNOMXþXMH� GD� VH� X�VXVWDYLPD� V� NRQVWDQWQLP� YROXPHQRP� QH� PRåH� SRYHüDWL� PHKDQLþND� HQHUJLMD� QD�UDþXQ� XQXWDUQMH�� =DWR� VH� X� PHKDQLFL� NUXWRJ� WLMHOD� �VXVWDYD� PDWHULMDOQLK� WRþDND��NRMLPD� MH� YROXPHQ� NRQVWDQWDQ�� QH� UD]PDWUDMX� WHUPRGLQDPLþNL� ]DNRQL�� RGQRVQR�XQXWDUQMD�HQHUJLMD��MHU�VH�L]�XQXWDUQMH�HQHUJLMH�QH�PRåH�GRELWL�PHKDQLþND�HQHUJLMD��RGQRVQR�QH�PRåH�VH�GMHORYDWL�QD�JLEDQMH�WLMHOD��8�PHKDQLFL�VH�UDG�VLOD�WUHQMD��NRMLP�VH�PHKDQLþND�HQHUJLMD��]EURM�NLQHWLþNH�L�SRWHQFLMDOQH�HQHUJLMH��SUHWYDUD�X�XQXWDUQMX�R]QDþXMH� NDR� JXELWDN� PHKDQLþNH� HQHUJLMH� �MHU� MH� MDVQR� GD� MH� WD� SUHWYRUED�MHGQRVPMHUQD��������2VQRYQL�]DNRQL��'LQDPLND� IOXLGD� ED]LUDQD� MH� QD� SHW� RVQRYQLK� ]DNRQD� NRML� VX� GHILQLUDQL� ]D� PDWHULMDOQL�YROXPHQ����� =DNRQ�RþXYDQMD�PDVH���� =DNRQ�RþXYDQMD�NROLþLQH�JLEDQMD���� =DNRQ�RþXYDQMD�PRPHQWD�NROLþLQH�JLEDQMD���� =DNRQ�RþXYDQMD�HQHUJLMH���� =DNRQ�SURGXNFLMH�HQWURSLMH��������

Page 23: 892’ - Fakultet strojarstva i brodogradnje · 0(+$1,.$)/8,’$ 892’ 892’ 0hkdqlndioxlgdmhglril]lnhnrmlvhedyljledqmhpioxlgdlvlodpdnrmhgmhoxmxqdioxlg 0hkdqlndioxlgdvhglmholqdvwdwlnxioxlgdnrmdsurxþdydudyqrwhåxioxlgdxplurydqmx

0(+$1,.$�)/8,'$��–�',1$0,.$�)/8,'$� ������=DNRQ�RþXYDQMD�PDVH���0DWHULMDOQL�YROXPHQ�VH� WLMHNRP�JLEDQMD�VDVWRML�VWDOQR�RG� MHGQLK�WH� LVWLK�þHVWLFD�IOXLGD��ãWR�]QDþL� GD� PX� MH� PDVD� NRQVWDQWQD�� ãWR� VH� PRåH� L]UD]LWL� ULMHþLPD�� ©%U]LQD� SURPMHQH� PDVH�PDWHULMDOQRJ�YROXPHQD�MHGQDND�MH�QXOLª�WM��PDWHPDWLþNL��

M

Dd 0

D V

Vt

ρ =∫ ����������=DNRQ�RþXYDQMD�NROLþLQH�JLEDQMD��'HILQLFLMD�]DNRQD�RþXYDQMD�NROLþLQH�JLEDQMD�]D�PDWHULMDOQL�YROXPHQ��%U]LQD�SURPMHQH�NROLþLQH�JLEDQMD�PDWHULMDOQRJ�YROXPHQD�MHGQDND�MH�]EURMX�YDQMVNLK�VLOD��masenih i površinskih��NRMH�GMHOXMX�QD�PDWHULMDOQL�YROXPHQ���0DWHPDWLþNL�]DSLV�]DNRQD�RþXYDQMD�NROLþLQH�JLEDQMD�]D�PDWHULMDOQL�YROXPHQ��

�8� VWUXMDQMX� IOXLGD�X�SROMX�PDVHQH�VLOH� f

G � XRþHQ� MH� PDWHULMDOQL�YROXPHQ� MV � NRML� MH� RG� RNROQRJ�IOXLGD� RGLMHOMHQ� PDWHULMDOQRP�površinom MS �� 1D� VYDNX� þHVWLFX�IOXLGD�GMHOXMH�HOHPHQWDUQD�PDVHQD�VLOD� df V

GS �� D� QD� VYDNL� GMHOLü�

površine MS � HOHPHQWDUQD�površinska sila dS

GT �� SUL� þHPX� MH�YHNWRU� QDSUH]DQMD� GT � GHILQLUDQ� V�SRPRüX� WHQ]RUD� QDSUH]DQMD�UHODFLMRP� jin

GGT T� � �� .ROLþLQD�

JLEDQMD�þHVWLFH�IOXLGD�MH� dv VG

S ����

M M M M M

M M M

Brzina promjene ukupna masena ukupna površinskakoliþLQH�JLEDQMD� VLOD�QD� � VLOD�QD�

Dd d d d d

D ji

V V S V S

V V V

v V f V S f V n St

ρ ρ σ ρ σ= + = + ⊗∫ ∫ ∫ ∫ ∫G GG G G

��� ��� �

��������

60�

6OLND�X]�GHILQLFLMX�]DNRQD�NROLþLQH�JLEDQMD�

2�

]�

\�[�

90�

nG

dSGT

fG

G6�

df VG

S

GP G9�vG

Page 24: 892’ - Fakultet strojarstva i brodogradnje · 0(+$1,.$)/8,’$ 892’ 892’ 0hkdqlndioxlgdmhglril]lnhnrmlvhedyljledqmhpioxlgdlvlodpdnrmhgmhoxmxqdioxlg 0hkdqlndioxlgdvhglmholqdvwdwlnxioxlgdnrmdsurxþdydudyqrwhåxioxlgdxplurydqmx

0(+$1,.$�)/8,'$��–�',1$0,.$�)/8,'$� ������=DNRQ�RþXYDQMD�PRPHQWD�NROLþLQH�JLEDQMD��'HILQLFLMD�]DNRQD�RþXYDQMD�PRPHQWD�NROLþLQH�JLEDQMD�]D�PDWHULMDOQL�YROXPHQ��%U]LQD� SURPMHQH� PRPHQWD� NROLþLQH� JLEDQMD� PDWHULMDOQRJ� YROXPHQD� MHGQDND� MH� ]EURMX�PRPHQDWD�YDQMVNLK�VLOD��masenih i površinskih��NRMH�GMHOXMX�QD�PDWHULMDOQL�YROXPHQ������0DWHPDWLþNL�]DSLV�]DNRQD�RþXYDQMD�PRPHQWD�NROLþLQH�JLEDQMD�]D�PDWHULMDOQL�YROXPHQ��

�8�VWUXMDQMX�IOXLGD�X�SROMX�PDVHQH�VLOH� f

G � XRþHQ� MH� PDWHULMDOQL�YROXPHQ� MV � NRML� MH� RG� RNROQRJ�IOXLGD� RGLMHOMHQ� PDWHULMDOQRP�površinom MS ��1D�VYDNX�þHVWLFX�IOXLGD� GMHOXMH� HOHPHQWDUQD�PDVHQD� VLOD� df V

GS �� 8GDOMHQRVW�

þHVWLFH� IOXLGD� RG� LVKRGLãWD� MH�GHILQLUDQD� UDGLMXV� YHNWRURP� rG ��þLMH� VX� NRPSRQHQWH� [�� \�� ]�� D�PRPHQW�PDVHQH�VLOH�X�RGQRVX�QD�

ishodište koordinatnog sustava je

dr f VGG

Sq �� 1D� VYDNL� GMHOLü�površine MS �GMHOXMH�HOHPHQWDUQD�površinska sila�

dSGT �� SUL� þHPX� MH�YHNWRU�QDSUH]DQMD� GT �GHILQLUDQ�V�SRPRüX� WHQ]RUD�QDSUH]DQMD� UHODFLMRP�

ijnGG

T T� � �� 0oment elementarne površinske sile u odnosu na ishodište je dr STqG G ��

0RPHQW�NROLþLQD�JLEDQMD�þHVWLFH�IOXLGD�MH�� dr v VSqG G ���

M M M M M

M M M

Brzina promjene momenta ukupni moment masenih ukupni moment površinskihkoliþLQH�JLEDQMD� VLOD�QD� � VLOD�QD�

Dd d d d d

D ij

V V S V S

V V V

r v V r f V r S r f V r n St

σ

ρ ρ σ ρ σ× = × + × = × + × ⊗∫ ∫ ∫ ∫ ∫G�����G GG G G G G G G

����� ��� ���

��������������������������������������������������������������� 3UHWSRVWDYOMD� VH� GD� X� IOXLGX� QHPD�PRPHQDWD� UDVSRGLMHOMHQLK� SR� PDWHULMDOQRP� YROXPHQX� LOL� PDWHULMDOQRM�površini.�

6OLND�X]�GHILQLFLMX�]DNRQD�PRPHQWD�NROLþLQH�JLEDQMD�

60�

2�

]�

\�[�

90�

nG

dSGT

fG

G6�

df VG

S

GP G9�vG

rG

rG

Page 25: 892’ - Fakultet strojarstva i brodogradnje · 0(+$1,.$)/8,’$ 892’ 892’ 0hkdqlndioxlgdmhglril]lnhnrmlvhedyljledqmhpioxlgdlvlodpdnrmhgmhoxmxqdioxlg 0hkdqlndioxlgdvhglmholqdvwdwlnxioxlgdnrmdsurxþdydudyqrwhåxioxlgdxplurydqmx

0(+$1,.$�)/8,'$��–�',1$0,.$�)/8,'$� ������=DNRQ�RþXYDQMD�HQHUJLMH��'HILQLFLMD�]DNRQD�RþXYDQMD�HQHUJLMH�]D�PDWHULMDOQL�YROXPHQ��%U]LQD� SURPMHQH� HQHUJLMH� PDWHULMDOQRJ� YROXPHQD� MHGQDND� MH� ]EURMX� VQDJD� YDQMVNLK� VLOD��masenih i površinskih��NRMH�GMHOXMX�QD�PDWHULMDOQL�YROXPHQ�L�EU]LQL�GRYR HQMD�WRSOLQH���0DWHPDWLþNL�]DSLV�]DNRQD�RþXYDQMD�HQHUJLMH�]D�PDWHULMDOQL�YROXPHQ��

�8� VWUXMDQMX� IOXLGD�X�SROMX�PDVHQH�VLOH� f

G � XRþHQ� MH� PDWHULMDOQL�YROXPHQ� MV � NRML� MH� RG� RNROQRJ�IOXLGD� RGLMHOMHQ� PDWHULMDOQRP�površinom MS �� 1D� VYDNX� þHVWLFX�IOXLGD�� NRMRM� MH� XNXSQD� HQHUJLMD�

de VS �� GMHOXMH� HOHPHQWDUQD�PDVHQD�VLOD� df V

GS ��D�VQDJD�WH�VLOH�

MH� df v VG G

S ¸ �� 1D� VYDNL� GMHOLü�površine MS � HOHPHQWDUQD�površinska sila dS

GT �� D� QMHQD�VQDJD� MH� dv S

GGT ¸ �� SUL� þHPX� MH�YHNWRU� QDSUH]DQMD� GT � GHILQLUDQ�]EURMHP� WODþQLK� L� YLVNR]QLK� VLOD�fpnσ σ= − +G G G ��

Površinske sile koje djeluju po materijalnoj površini su za materijalni volumen vanjske sile �VLOH�GRGLUD�L]PH X�þHVWLFD�PDWHULMDOQRJ�YROXPHQD�L�RNROLQH��8NXSQD�HQHUJLMD� de V �þHVWLFH�fluida definirana je kao zbroj unutrašnje du Vρ L�NLQHWLþNH�HQHUJLMH� 2

d2

vV

ρ � 2

2

ve u

= +

��

.UR]�VYDNL�GMHOLü�SRYUãLQH� MS �SUROD]L�WRSOLQVNL�WRN�GHILQLUDQ�YHNWRURP�JXVWRüH�WRSOLQVNRJ�WRND� qG ��0DWHPDWLþNL�]DSLV�]DNRQD�MH����

M M M M M

MM M

M

2

Brzina promjene energije snaga masenih snaga vanjskih brzina dovo HQMD�sila na površinskih topline na

sila na

D Dd d d d d

D D 2V V V S S

VV V

V

ve V u V f v V v S q n S

t tρ ρ ρ σ

= + = ⋅ + ⋅ − ⋅

∫ ∫ ∫ ∫ ∫G G G G G G

��� ��� ��� ���

�����=DNRQ�SURGXNFLMH�HQWURSLMH��'HILQLFLMD�]DNRQD�SURGXNFLMH�HQWURSLMH��,,�]DNRQ�WHUPRGLQDPLNH��]D�PDWHULMDOQL�YROXPHQ��3URGXNFLMD�HQWURSLMH�PDWHULMDOQRJ�YROXPHQD�YHüD�MH�LOL�MHGQDND�QXOL�D

0D

M MV S

q nsdV dS

t Tσ ρ ⋅= + ≥∫ ∫

G G�

JGMH�MH�σ SURGXNFLMD�HQWURSLMH��V�HQWURSLMD��D�7�DSVROXWQD�WHPSHUDWXUD���

60�

6OLND�X]�GHILQLFLMX�]DNRQD�NROLþLQH�JLEDQMD�

2�

]�

\�[�

90�

nG

dSGT

fG

G6�

df VG

S

GP G9�vGG6�

nG

qG

Page 26: 892’ - Fakultet strojarstva i brodogradnje · 0(+$1,.$)/8,’$ 892’ 892’ 0hkdqlndioxlgdmhglril]lnhnrmlvhedyljledqmhpioxlgdlvlodpdnrmhgmhoxmxqdioxlg 0hkdqlndioxlgdvhglmholqdvwdwlnxioxlgdnrmdsurxþdydudyqrwhåxioxlgdxplurydqmx

0(+$1,.$�)/8,'$�,�–�,17(*5$/1(�0(72'(� ���

5. INTEGRALNE METODE RJEŠAVANJA JEDNODIMENZIJSKIH 352%/(0$�0(+$1,.(�)/8,'$���)RUPXODFLMD�]D�MHGQRGLPHQ]LMVNR�VWUXMDQMH�

3UHWSRVWDYNH����� 6WUXMDQMH� MH� VWDFLRQDUQR�0

t

∂ =∂

����� )OXLG�MH�QHVWODþLY� .constρ = ���� 0DVHQD� VLOD� MH� VLOD�

JUDYLWDFLMH� f gk= −G G �

��� 6WUXMDQMH� MH� L]RWHUPLþNR�.T const= �

��� fpnσ σ= − +G G G �

������.RQFHSW�NRQWUROQRJ�YROXPHQD��6YL� ]DNRQL� PHKDQLNH� L� WHUPRGLQDPLNH� ELW� üH� SULPMHQMLYL� QD� PDWHULMDOQL� YROXPHQ� �X�PHKDQLFL� MH� WR� PDWHULMDOQR� WLMHOR� LOL� VXVWDY� PDWHULMDOQLK� WRþDND�� D� X� WHUPRGLQDPLFL� MH� WR�]DWYRUHQL� WHUPRGLQDPLþNL� VXVWDY���8�PHKDQLFL� IOXLGD� QLMH� LQWHUHV� SUDWLWL� ãWR� VH�GRJD D� VD�VDPLP�IOXLGRP��GDNOH�QHüH�VH�SUDWLWL�JLEDQMH�PDWHULMDOQRJ�YROXPHQD, kao što se u mehanici SUDWL� JLEDQMH� WLMHOD��� QHJR� MH� SRWUHEQR� RGUHGLWL� SRVOMHGLFH� VWUXMDQMH� IOXLGD� X� EOL]LQL� QHNH�NRQVWUXNFLMH��8� WRP�VPLVOX�üH�VH�GHILQLUDWL�NRQWUROQL�YROXPHQ�þLMH�VH�JUDQLFH�SRNODSDMX�V�SRYUãLQRP�NRQVWUXNFLMH� ]D� NRMX� VH� åHOL� LVWUDåLWL� XWMHFDM� VWUXMDQMD� IOXLGD��%XGXüL�GD� üH� VYL�]DNRQL� PHKDQLNH� IOXLGD� ELWL� IRUPXOLUDQL� ]D� PDWHULMDOQL� YROXPHQ� SRWUHEQR� LK� MH�SUHIRUPXOLUDWL�]D�NRQWUROQL�YROXPHQ��.RQWUROQL�MH�YROXPHQ�X�YHüLQL�VOXþDMHYD�V�PLUXMXüLP�JUDQLFDPD�� 0u

G� ���D�X�DQDOL]L�NRQVWUXNFLMD�V�SRPLþQLP�GLMHORYLPD�NRULVWL�VH�L�IRUPXODFLMD�NRQWUROQRJ�YROXPHQD�V�SRPLþQLP�JUDQLFDPD��������5H\QROGVRY�WUDQVSRUWQL�WHRUHP�%U]LQD�SURPMHQH�VDGUåDMD�IL]LNDOQH�YHOLþLQH�XQXWDU�PDWHULMDOQRJ�YROXPHQD�L]UDåHQD�SURPMHQRP�X�NRQWUROQRP�YROXPHQX��8�WUHQXWNX�SRNODSDQMD�PDWHULMDOQRJ�L�NRQWUROQRJ�YROXPHQD�EU]LQD�ORNDOQH�SURPMHQH�LP�MH�

ista, kao što su isti i površinski integrali, u gornjim izrazima, iz kojih slijedi:��D��VOXþDM�NRQWUROQRJ�YROXPHQD�.9�NRML�MH�RJUD HQ�PLUXMXüRP�NRQWUROQRP�SRYUãLQRP�.3�

M M M

Dd d d d d

D V V S KV KP

V V v n S V v n St t t

Φ ΦΦ Φ Φ∂ ∂= + ⋅ = + ⋅∂ ∂∫ ∫ ∫ ∫ ∫

G G G G ����

6Z�$��

$��

sds se=G G

$�

G9 $GV�v vn= −G G

v vn=G G

]�

[� \�

Page 27: 892’ - Fakultet strojarstva i brodogradnje · 0(+$1,.$)/8,’$ 892’ 892’ 0hkdqlndioxlgdmhglril]lnhnrmlvhedyljledqmhpioxlgdlvlodpdnrmhgmhoxmxqdioxlg 0hkdqlndioxlgdvhglmholqdvwdwlnxioxlgdnrmdsurxþdydudyqrwhåxioxlgdxplurydqmx

0(+$1,.$�)/8,'$�,�–�,17(*5$/1(�0(72'(� ��

X]�QDSRPHQX�GD�YULMHGL� dd d

dKV V

V Vt t

Φ Φ∂ =∂∫ ∫ �

E��VOXþDM�SURPMHQMLYRJ�NRQWUROQRJ�YROXPHQD�V �þLMD�VH�JUDQLFD� S �JLED�EU]LQRP�uG �( )

M

D dd d d

D dV V S

V V v u n St t

Φ Φ Φ= + − ⋅∫ ∫ ∫G G G �

������-HGQDGåED�NRQWLQXLWHWD��=DNRQ�RGUåDQMD�PDVH��

M

Dd 0

D V

Vt

ρ =∫ �3ULPMHQRP�5H\QROGVRYRJ�WUDQVSRUWQRJ�WHRUHPD�X]�Φ ρ= �]DNRQ�VH�IRUPXOLUD�]D�NRQWUROQL�YROXPHQ�d

d d d V KP

m

V v n St

ρ ρ= − ⋅∫ ∫�

G G

����

/LMHYD�VWUDQD�R]QDþXMH�EU]LQX�SURPMHQH�PDVH�IOXLGD�XQXWDU�NRQWUROQRJ�YROXPHQD��D�GHVQD�ukupni maseni protok kroz kontrolnu površinu. Na dijelu kontrolne površine kroz koju IOXLG� XOD]L� X� NRQWUROQL� YROXPHQ�YHNWRU� YDQMVNH� QRUPDOH� L� YHNWRU� EU]LQH� þLQH� NXW� YHüL�RG�90°, te je 0v n⋅ <G G � L� PDVHQL� SURWRN� MH� QHJDWLYDQ�� D� QHJDWLYQL� SUHG]QDN� LVSUHG� LQWHJUDOD�XND]XMH�GD�üH�WDM�SURWRN�SRYHüDYDWL�VDGUåDM�PDVH�XQXWDU�NRQWUROQRJ�YROXPHQD��1D�L]OD]QRM�JUDQLFL� MH� 0v n⋅ >G G �� SD�QHJDWLYQL�SUHG]QDN� LVSUHG� LQWHJUDOD�XND]XMH�QD� LVWMHFDQMH� IOXLGD� L]�NRQWUROQRJ� YROXPHQD� WM�� R]QDþXMH� VPDQMHQMH� VDGUåDMD�PDVH� XQXWDU� NRQWUROQRJ� YROXPHQD��.UR]� QHSURSXVQX� VWLMHQNX� QHPD� SURWRND�� ãWR� ]QDþL� GD� MH� EU]LQD� LOL� MHGQDND� QXOL� LOL� MH�WDQJHQFLMDOQD�QD�VWLMHQNX��$NR�VH�VD� Um� �R]QDþL�XNXSQL�PDVHQL�SURWRN�NRMLP�IOXLG�XOD]L�X�NRQWUROQL�YROXPHQ��D�VD� Im� �PDVHQL�SURWRN�NRMLP�IOXLG�L]�QMHJD�L]OD]L��WDGD�YULMHGL��

U I

dd

d V

V m mt

� �ρ = −∫ ��� D�� 6OXþDM�VWDFLRQDUQRJ�VWUXMDQMD��8�VWDFLRQDUQRP�VWUXMDQMX�IOXLGD�VH�VOLND�VWUXMDQMD�QH�PLMHQMD� V� YUHPHQRP�� ãWR� ]QDþL� GD� VH� QHüH� PLMHQMDWL� QLWL� VDGUåDM� PDVH� XQXWDU�NRQWUROQRJ�YROXPHQD�SD�YULMHGL�MHGQDNRVW�XOD]QRJ�L�L]OD]QRJ�PDVHQRJ�SURWRND�

U Im m� �= �E�� 6OXþDM� QHVWODþLYRJ� �VWDFLRQDUQRJ� LOL� QHVWDFLRQDUQRJ�� VWUXMDQMD� KRPRJHQRJ� IOXLGD�

� konst.S� ���6�RE]LURP�GD�MH�JXVWRüD�NRQVWDQWQD�X�NRQWUROQRP�YROXPHQX�üH�VH�X�VYDNRP� WUHQXWNX� QDOD]LWL� MHGQDND� PDVD� IOXLGD�� D� PDVHQL� SURWRN� MH� m QS�� �� WH�YULMHGL�

U IQ Q= ��3ULPMHU��6WUXMDQMH�NUR]�UDþYDVWX�FLMHY�

Page 28: 892’ - Fakultet strojarstva i brodogradnje · 0(+$1,.$)/8,’$ 892’ 892’ 0hkdqlndioxlgdmhglril]lnhnrmlvhedyljledqmhpioxlgdlvlodpdnrmhgmhoxmxqdioxlg 0hkdqlndioxlgdvhglmholqdvwdwlnxioxlgdnrmdsurxþdydudyqrwhåxioxlgdxplurydqmx

0(+$1,.$�)/8,'$�,�–�,17(*5$/1(�0(72'(� ��

Q2Q4

Q3Q1

1D� VOLFL� MH� XRþHQ� NRQWUROQL� YROXPHQ� NRML�REXKYDüD�XQXWDUQMRVW�UDþYDVWH�FLMHYL��.UR]�GYD�SUHVMHND� QHVWODþLYL� IOXLG� XOD]L� X� NRQWUROQL�YROXPHQ�SURWRFLPD� 1Q � L� 2Q �� D� NUR]�GYD� L]OD]L�SURWRFLPD� 3Q � L� 4Q �� .UR]� SODãW� UDþYH� QHPD�SURWRND�IOXLGD��3UHPD�MHGQDGåEL�NRQWLQXLWHWD�YULMHGL���������������������� 1 2 3 4Q Q Q Q� � � ��

� �����-HGQDGåED�NROLþLQH�JLEDQMD���3ULPMHQRP�5H\QROGVRYD� WUDQVSRUWQRJ� WHRUHPD�QD� OLMHYX� VWUDQX� ]DNRQD� RþXYDQMD� NROLþLQH�JLEDQMD�]D�PDWHULMDOQL�YROXPHQ��NRML�JODVL��

M M M

Dd d d

D V V S

v V f V St

ρ ρ σ= +∫ ∫ ∫GG G �

VOLMHGL�MHGQDGåED�NROLþLQH�JLEDQMD�]D�NRQWUROQL�YROXPHQ�V�PLUXMXüLP�JUDQLFDPD�� ( )

protok koliþLQH�JLEDQMD XNXSQD�SRYUãLQVNDbrzina promjene ukupna masenakroz kontrolnu površinu sila na koliþLQH�JLEDQMD�� �D VLOD�QD�

dd d d d

d KV KP KV KP

KVKV KV

v V v v n S f V St

ρ ρ ρ σ+ ⋅ = +∫ ∫ ∫ ∫GG G G G G

����� ���� ����

JGMH� VH� NRQWUROQD� SRYUãLQD�PRåH� RSüHQLWR� SULND]DWL� ]EURMHP� XOD]QRJ� GLMHOD� uS NRQWUROQH�SRYUãLQH� �NUR]� NRML� IOXLG� XWMHþH� X� NRQWUROQL� YROXPHQ��� L]OD]QRJ� GLMHOD� iS �NUR]� NRML� IOXLG�napušta kontrolni volumen) i površine� VWLMHQNH� wS � �NRMD� MH� GLR� QHNRJ� XUH DMD�� VWURMD� LOL�NRQVWUXNFLMH��NUR]�NRMX�QHPD�VWUXMDQMD�IOXLGD� 0v n⋅ =G G ��� u i wKP S S S= + + ��8]� SUHWSRVWDYNX� QHVWODþLYRJ� VWUXMDQMD�� X]LPDMXüL� GD� MH� PDVHQD� VLOD� MHGQDND� VLOL� WHåLQH�( )f gk= −G G �MHGQDGåED�NROLþLQH�JLEDQMD�VH�PRåH�QDSLVDWL�L�X�REOLNX�

( )( )brzina promjene WHåLQD�IOXLGD�X� - =sila stijenke koliþLQH�JLEDQMD�� �D

na fluid

dd d d d

d u i w

w

KV KV S S S

G KV FKV

v V gk V v v n S St

ρ ρ ρ σ σ+

= − − ⋅ − +∫ ∫ ∫ ∫G G

GG G G G G G

��� ��� ��

3RVOMHGQML�LQWHJUDO�X�JRUQMRM�MHGQDGåEL�GDMH�XNXSQX�SRYUãLQVNX�VLOX�L]PH X�VWMHQNH�L�IOXLGD�L�WR�VLOX�NRMRP�RNROLQD��VWMHQND��GMHOXMH�QD�IOXLG��7D�MH�VLOD�SR�WUHüHP�1HZWRQRYRP�]DNRQX�MHGQDND�QHJDWLYQRM�YULMHGQRVWL�VLOH� wF

G kojom fluid djeluje na stjenku. Vektor površinske

VLOH�VH�PRåH�SULND]DWL�]EURMHP�VLOH�WODND�L�YLVNR]QLK�VLOD�� fpnσ σ= − +G G G �SUL�þHPX�VH�YLVNR]QH�VLOH�QD�XOD]QRM�L�L]OD]QRM�SRYUãLQL�RELþQR�]DQHPDUXMX��WDQJHQFLMDOQH�YLVNR]QH�VLOH�VH�RELþQR�PH XVREQR�SRQLãWDYDMX��D�QRUPDOQH�NRPSRQHQWH�YLVNR]QLK�VLOD�VX�PDOH�X�RGQRVX�QD�WODþQH�VLOH���WDNR�GD�]DNRQ�NROLþLQH�JLEDQMD�]D�NRQWUROQL�YROXPHQ�SUHOD]L�X�REOLN�� ( )( )

brzina promjene koliþLQH�JLEDQMD�� �D

dd d

d u i

w

KV S S

KV

v V G v v n pn S Ft

ρ ρ+

= − ⋅ + −∫ ∫G GG G G G G

����

Page 29: 892’ - Fakultet strojarstva i brodogradnje · 0(+$1,.$)/8,’$ 892’ 892’ 0hkdqlndioxlgdmhglril]lnhnrmlvhedyljledqmhpioxlgdlvlodpdnrmhgmhoxmxqdioxlg 0hkdqlndioxlgdvhglmholqdvwdwlnxioxlgdnrmdsurxþdydudyqrwhåxioxlgdxplurydqmx

0(+$1,.$�)/8,'$�,�–�,17(*5$/1(�0(72'(� ��8�XYMHWLPD�VWDFLRQDUQRJ�VWUXMDQMD��NDGD�VH�VOLND�VWUXMDQMD�QH�PLMHQMD�V�YUHPHQRP��EU]LQD�SURPMHQH�NROLþLQH�JLEDQMD�NRQWUROQRJ�YROXPHQD��OLMHYD�VWUDQD�MHGQDGåEH��MH� MHGQDND�QXOL��WH�üH�]DNRQ�NROLþLQH�JLEDQMD�L]UDåHQ�]D�NRQWUROQL�YROXPHQ�VOXåLWL�]D�RGUH LYDQMH�VLOH�NRMRP�IOXLG�GMHOXMH�QD�VWMHQNX�� N

n

du i

w

vS S

F G v v n pn Sρ+

= − ⋅ +

∫GG G G G G �

2þLWR�MH�GD�üH�]D�RGUH LYDQMH�VLOH�NRMRP�IOXLG�GMHOXMH�QD�VWMHQNX�ELWL�SRWUHEQR�SR]QDYDQMH�SURILOD�EU]LQH�L�WODND�QD�XOD]QRP�L�L]OD]QRP�GLMHOX kontrolne površine.�� 3ULPMHQD�MHGQDGåEH�NROLþLQH�JLEDQMD�]D�RGUH LYDQMH�VLOH�IOXLGD�QD�SODãW�FLMHYL��

6OLND� SULND]XMH� MHGDQ�NRQWUROQL� YROXPHQ� NRML�REXKYDüD� XQXWUDãQMRVW� UDþ�YDVWH� FLMHYL�� D� QD� NRQWUROQRM�SRYUãLQL� VH� PRJX� XRþLWL� GYD�XOD]QD� SUHVMHND� �SUHVMHFL� �� L���� L�GYD� L]OD]QD�SUHVMHND���� L����� 8� WLP� VX� SUHVMHFLPD�VWUXMQLFH� PH XVREQR� SDUD�OHOQH�� D� YHNWRUL� EU]LQH� VX�RNRPLWL� QD�SUHVMHN�� SUL� þHPX�YULMHGL��=D�XOD]QL�SUHVMHN� =D�L]OD]QL�SUHVMHN�

� �v vn= −G G �

v n v⋅ = −G G �N ( )

u un

2d dvS S

v v n pn S n v p Sρ ρ

− ⋅ + = − + ∫ ∫

G G G G G �

v vn=G G �v n v⋅ =G G �

N ( )i in

2d dvS S

v v n pn S n v p Sρ ρ

− ⋅ + = − + ∫ ∫

G G G G G �3UL�VWUXMDQMX�YLVNR]QRJ�IOXLGD�EU]LQD�SR�SRSUHþQRP�SUHVMHNX�FLMHYL�QLMH�NRQVWDQWQD��DOL�VH�LQWHJUDO� NYDGUDWD� EU]LQH� SR� SUHVMHNX� PRåH� SULND]DWL� SRPRüX� NYDGUDWD� VUHGQMH� EU]LQH� L�IDNWRUD� LVSUDYND�NROLþLQH�JLEDQMD�X�REOLNX� 2 2

srdS

v S v Sβ=∫ �JGMH� MH� IDNWRU� LVSUDYND�NROLþLQH�JLEDQMD�GHILQLUDQ�L]UD]RP� 2

2sr

1d

S

v Sv S

β = ∫ ��9ULMHGQRVWL�IDNWRUD�C �VX��6WUXMDQMH�LGHDOQRJ�IOXLGD�–�MHGQROLNL�SURILO�EU]LQH�SR�SUHVMHNX��� 1C � �/DPLQDUQR� VWUXMDQMH� X� RNUXJOLP� FLMHYLPD� SROXPMHUD� 5� –� SRVWRML�DQDOLWLþNR�UMHãHQMH�]D�SURILO�EU]LQH� 2

max 21

rv v

R

� ¬­� ­� �� ­� ­�� ®��� 1,33C � �

7XUEXOHQWQR� VWUXMDQMH� X� RNUXJOLP� FLMHYLPD� –� SURILO� EU]LQH� ]DYLVL� RG�vDS

1,01 1,03C � � �

S��� 4vG �

[�

]�

\�2�

S�� 1vG �

S��� 2vG �

S��� 3vG �

��

��

����

f gk= −G G

nG

vG

6X�

nG

vG

6L�

Page 30: 892’ - Fakultet strojarstva i brodogradnje · 0(+$1,.$)/8,’$ 892’ 892’ 0hkdqlndioxlgdmhglril]lnhnrmlvhedyljledqmhpioxlgdlvlodpdnrmhgmhoxmxqdioxlg 0hkdqlndioxlgdvhglmholqdvwdwlnxioxlgdnrmdsurxþdydudyqrwhåxioxlgdxplurydqmx

0(+$1,.$�)/8,'$�,�–�,17(*5$/1(�0(72'(� ��

5H\QROGVRYD� EURMD� vDRe

SN

� �� D� NRHILFLMHQW� C � VH� NUHüH� X� UDVSRQX�1,01C � (pri višim vrijednostima 5H!����� GR� 1,03C � � �SUL� QLåLP�

YULMHGQRVWLPD�5H��8�SUDNVL�MH�VWUXMDQMH�QDMþHãüH�WXUEXOHQWQR�SD�VH�X]LPD�GD�MH� 1C � (bez da se bitno naruši WRþQRVW�UH]XOWDWD�����8�VWUXMDQMX�IOXLGD�NUR]�FLMHYL�VWUXMQLFH�VX�SDUDOHOQH��SD�üH�SURPMHQD�WODND�SR�SUHVMHNX�ELWL�LVWD� NDR� X� IOXLGX� X�PLURYDQMX�� WM�� ELW� üH� OLQHDUQD��$NR� VH� SURPDWUD� VWUXMQLFD� NRMD� SUROD]L�WHåLãWHP�SRSUHþQRJ�SUHVMHND� FLMHYL�� WDGD� MH� LQWHJUDO� WODND�SR�SRYUãLQL� SRSUHþQRJ�SUHVMHND�MHGQDN�XPQRãNX�WODND�QD�VWUXMQLFL�L�SRYUãLQL�SRSUHþQRJ�SUHVMHND� d

S

p S pS=∫ ���.RQDþDQ�L]UD]�]D�L]UDþXQDYDQMH�VLOH�NRMRP�IOXLG�GMHOXMH�QD�SODãW�FLMHYL�MHVW�� ( )

( )

( ) ( )2

= imulsna funkcijak

i

k kw

k k

I

F G n v p S G Iβρ = + − + = + ∑ ∑G GG GG

��������LOL� ( )kw

k

F G IGG G

= + ∑ �

JGMH�MH�N broj ulaznih i izlaznih dijelova kontrolne površine.��,PSXOVQD� IXQNFLMD� MH� YHNWRU�� NRML� MH� SR� YHOLþLQL� MHGQDN� ( )2I v p Sβρ= + �� RNRPLW� MH� QD�površinu 6� L� JOHGD� VXSURWQR� RG� YDQMVNH� QRUPDOH� �XYLMHN� JOHGD� X� NRQWUROQL� YROXPHQ� EH]�RE]LUD�UDGL�OL�VH�R�XOD]QRP�LOL�L]OD]QRP�GLMHOX�NRQWUROQH�SRYUãLQH���NDR�QD�VOMHGHüRM�VOLFL��

$NR� VH� LPSXOVQH� IXQNFLMH�VKYDWH� NDR� VLOH�� WDGD� VH�SUREOHP� RGUH LYDQMD� VLOH�kojom fluid djeluje na plašt FLMHYL�VYRGL�QD�SUREOHP�VWDWLNH�WM�� RGUH LYDQMH� VXPD� VLOD��=DNRQRP� NROLþLQH� JLEDQMD�GHILQLUDQD� MH� YHOLþLQD� L� VPMHU�sile fluida na plašt, a hvatište MH� GHILQLUDQR� ]DNRQRP�PRPHQWD�NROLþLQH�JLEDQMD��

3RVWXSDN�L]UDþXQD�VLOH��• 3ULPMHQRP� MHGQDGåEH� NRQWLQXLWHWD� L� %HUQRXOOLMHYH� MHGQDGåEH� RGUHGH� VH� EU]LQH� L�

tlakovi na ulaznim i izlaznim dijelovima kontrolne površine.�• ,]� L]UDþXQDWLK� EU]LQD� L� WODNRYD� UDþXQDMX� VH� YULMHGQRVWL� LPSXOVQLK� IXQNFLMD� QD�

ulaznim i izlaznim dijelovima kontrolne površine.�• 9HNWRUVNLP� ]EUDMDQMHP� �X� DQDOLWLþNRP� SRVWXSNX� VXPLUDQMHP� NRPSRQHQWL� VLOD� X�VPMHURYLPD�RVL�� LPSXOVQLK�IXQNFLMD� L� VLOH� WHåLQH�VH�GRELMH�VLOD�NRMRP�IOXLG�GMHOXMH�

na plašt cijevi.��7UHED�QDJODVLWL�GD�JRUQMD�IRUPXOD�YULMHGL�]D�ELOR�NDNDY�REOLN�NRQWUROQRJ�YROXPHQD��MHGLQR�MH�YDåQR�GD�QD�XOD]QLP�L�L]OD]QLP�SUHVMHFLPD�VWUXMQLFH�EXGX�PH XVREQR�SDUDOHOQH�L�GD�VX�YHNWRUL�EU]LQH�RNRPLWL�QD�SULSDGDMXüH�SUHVMHNH��

[�

]�

\�2�

1IG

f gk= −G G

4IG

2IG

3IG

1GG

Page 31: 892’ - Fakultet strojarstva i brodogradnje · 0(+$1,.$)/8,’$ 892’ 892’ 0hkdqlndioxlgdmhglril]lnhnrmlvhedyljledqmhpioxlgdlvlodpdnrmhgmhoxmxqdioxlg 0hkdqlndioxlgdvhglmholqdvwdwlnxioxlgdnrmdsurxþdydudyqrwhåxioxlgdxplurydqmx

0(+$1,.$�)/8,'$�,�–�,17(*5$/1(�0(72'(� ���,PSXOVQH� IXQNFLMH� UDþXQDWH� V� DSVROXWQLP� WODNRP� GHILQLUDMX� VLOX� IOXLGD� QD� VWLMHQNX� �GDNOH�silu na plašt samo s unutrašnje strane). Ako s vanjske strane plašta djeluje DWPRVIHUVNL�WODN��onda bi rezultantna sila na plašt bila jednaka zbroju unutarnje sile i vanjske sile DWPRVIHUVNRJ� WODND��'R�UH]XOWDQWQH�VLOH�VH�GROD]L� WDNR�GD�VH�X�LPSXOVQX�IXQNFLMX�XPMHVWR�apsolutnog tlaka uvrštava manometarski tlak, dakle vrijedi�� ( )2

MF G n v p Sβρ= + − +∑GG G �

JGMH�MH�FG rezultantna sila na plašt cijevi.�� 3ULPMHQD�MHGQDGåEH�NROLþLQH�JLEDQMD�]D�RGUH LYDQMH�VLOH�POD]D�IOXLGD�QD�ORSDWLFH��

6OLND� SULND]XMH� POD]� IOXLGD� SRYUãLQH� SRSUHþQRJ�SUHVMHND 1A �� NRML� EU]LQRP� 1v � L� SURWRNRP� 1 1 1Q v A� ��QDLOD]L� QD� UDYQX� ORSDWLFX� �SORþX� MHGLQLþQH� ãLULQH��NRMD�QD� VHEL� LPD� UD]GMHOQLN�VWUXMDQMD� �QRVLü��NRMLP�VH�POD]�GLMHOL�QD�GYLMH�JUDQH�R]QDþHQH�LQGHNVLPD���i 3. Ako je površina mlaza mala u odnosu na SRYUãLQX� ORSDWLFH� POD]� üH� WDQJHQFLMDOQR� QDSXãWDWL�ORSDWLFX��0OD]� VWUXML� X� DWPRVIHUL�� D� V� GUXJH� VWUDQH�ORSDWLFH� YODGD� DWPRVIHUVNL� WODN�� 1D� VOLFL� MH� XFUWDQ�RGDEUDQL� NRQWUROQL� YROXPHQ� �FUWD�WRþND� OLQLMD�� QD�þLMRM�VH�NRQWUROQRM�SRYUãLQL�PRåH�XRþLWL�XOD]QL��

presjek mlaza, dva izlazna presjeka, rub mlaza i površina lopatice.� $NR� VH� SUHWSRVWDYH�MHGQROLNL� SURILOL� EU]LQH� SR� SUHVMHFLPD� L� OLQHDUQX� SURPMHQX� WODND�� WDGD� üH� VH� LPSXOVQH�IXQNFLMH�UDþXQDWL�SR�LVWLP�IRUPXODPD�NDR�L�SUL�RGUH LYDQMX�VLOH�IOXLGD�QD�SODãW�FLMHYL��$NR�VH�WUDåL�UH]XOWDQWQD�VLOD�QD�ORSDWLFX��X]LPDMXüL�X�RE]LU�L�VLOX�DWPRVIHUVNRJ�WODND�V�YDQMVNH�VWUDQH��LPSXOVQH�IXQNFLMH�VH�UDþXQDMX�V�PDQRPHWDUVNLP�WODNRP��NRML�MH�X�VYLP�SUHVMHFLPD�MHGQDN�QXOL��WH�]D�YHOLþLQX�LPSXOVQH�IXQNFLMH�YULMHGL�� 2I v A QvS S� � ��

1D� XOD]QLP� L� QD� L]OD]QLP� GLMHORYLPD�NRQWUROQH� Sovršine impulsne funkcije JOHGDMX� X� NRQWUROQL� YROXPHQ�� D� RNRPLWH� VX�QD� SRYUãLQH�� 3R� UXEX� POD]D� WDNR HU� WUHED�L]UDþXQDWL�LPSXOVQX�IXQNFLMX��MHU�WD�SRYUãLQD�QLMH� GLR� SRYUãLQH� ORSDWLFH� QD� NRMRM� VH� åHOL�RGUHGLWL� VLOX�� 0H XWLP� EXGXüL� GD� NUR]� WX�površinu nema stUXMDQMD��D�QD�QMRM�MH�SUHWODN�MHGQDN� QXOL�� ]DNOMXþXMH� VH� GD� MH� L� LPSXOVQD�IXQNFLMD� MHGQDND� QXOL�� WH� SUHRVWDMX� VDPR�LPSXOVQH� IXQNFLMH�NDR�SUHPD� VOLFL��7UDåHQD�VLOD�MHGQDND�MH�YHNWRUVNRP�]EURMX�LPSXOVQLK�IXQNFLMD�L�VLOH�WHåLQH���$NR� EL� VWUXMDQMH� ELOR� QHYLVNR]QR� �QHPD� VPLþQLK� QDSUH]DQMD��� D� SORþD� ELOD� UDYQD� �QHPD�UD]GMHOQLND� VWUXMDQMD�� VLOD� IOXLGD� EL� ELOD� RNRPLWD�QD�SORþX� �MHU�SRVWRMH� VDPR� VLOH� WODND��� D�

SURWRFL� 2Q �L� 3Q �EL�ELOL�XSUDYR�WDNYL�GD�QHPD�WDQJHQFLMDOQH�NRPSRQHQWH�VLOH�QD�SORþX���

Page 32: 892’ - Fakultet strojarstva i brodogradnje · 0(+$1,.$)/8,’$ 892’ 892’ 0hkdqlndioxlgdmhglril]lnhnrmlvhedyljledqmhpioxlgdlvlodpdnrmhgmhoxmxqdioxlg 0hkdqlndioxlgdvhglmholqdvwdwlnxioxlgdnrmdsurxþdydudyqrwhåxioxlgdxplurydqmx

0(+$1,.$�)/8,'$�,�–�,17(*5$/1(�0(72'(� ��-HGQDGåED�PRPHQWD�NROLþLQH�JLEDQMD���3ULPMHQRP� 5H\QROGVRYD� WUDQVSRUWQRJ� WHRUHPD� QD� OLMHYX� VWUDQX� MHGQDGåEH� PRPHQWD�NROLþLQH� JLEDQMD� ]D� PDWHULMDOQL� YROXPHQ�� VOLMHGL� MHGQDGåED�PRPHQWD� NROLþLQH� JLEDQMD� ]D�NRQWUROQL�YROXPHQ�V�PLUXMXüLP�JUDQLFDPD�

� ( )Protok momenta koliþLQH XNXSQL��PRPHBrzina promjene momenta ukupni moment masenihgibanja kroz koliþLQH�JLEDQMD� VLOD�QD� �

dd d d d

d KV KP KV KP

KPKV KV

r v V r v v n S r f V r St

ρ ρ ρ σ× + × ⋅ = × + ×∫ ∫ ∫ ∫GG G G G G G G G G

������������ ���nt površinskih

sila na KP

����

�8]�VOMHGHüH�SUHWSRVWDYNH����� VWUXMDQMH�MH�QHVWODþLYR�L�VWDFLRQDUQR���� PDVHQD�VLOD�MH�VLOD�WHåLQH���� kontrolna površina se sastoji od ulaznog, izlaznog dijela i površine plašta, u i wKP S S S= + + �

��� YHNWRU�QDSUH]DQMD� fpnσ σ= − +G G G ��MHGQDGåED� PRPHQWD� NROLþLQH� JLEDQMD� SULPMHQMHQD� QD� NRQWUROQL� YROXPHQ�� VOXåL� ]D�RGUH LYDQMH�PRPHQWD�VLOH�NRMRP�IOXLG�GMHOXMH�QD�SODãW��� ( ) ( ) N

nmoment sile moment sile fluida na plašt WHåLQH

du i

w f

vS S

M F M G r v v n pn Sρ σ+

= − × ⋅ + −

∫GG G G G G G G G G

��� ��

��3ULPMHQD�MHGQDGåEH�PRPHQWD�NROLþLQH�JLEDQMD�]D�RGUH LYDQMH�momenta sile fluida na plašt FLMHYL��

6OLND� SULND]XMH� MHGDQ�NRQWUROQL� YROXPHQ� NRML�REXKYDüD� XQXWUDãQMRVW� UDþ�YDVWH� FLMHYL�� D� QD� NRQWUROQRM�SRYUãLQL� VH� PRJX� XRþLWL� GYD�XOD]QD� SUHVMHND� �SUHVMHFL� �� L���� L�GYD� L]OD]QD�SUHVMHND���� L����� 8� WLP� VX� SUHVMHFLPD�VWUXMQLFH� PH XVREQR� SDUD�OHOQH�� D� YHNWRUL� EU]LQH� VX�RNRPLWL�QD�SUHVMHN��

1D�JRUQMRM�VOLFL�MH�WDNR HU�XFUWDQ�UDGLMXV�YHNWRU�GR�WHåLãWD�SUYRJ�SUHVMHND��$NR�VH�]DQHPDUH�PRPHQWL� YLVNR]QLK� VLOD� QD� XOD]QLP� L� L]OD]QLP� SUHVMHFLPD�� WDGD� EL� MHGQDGåED� PRPHQWD�NROLþLQH� JLEDQMD� ]D� SULND]DQL� NRQWUROQL� YROXPHQ� �X]LPDMXüL� X� RE]LU� GD� MH� QD� XOD]QRP�SUHVMHNX� YHNWRU� EU]LQH� RULMHQWLUDQ� VXSURWQR� RG� YDQMVNH� QRUPDOH�� D� QD� L]OD]QRP�X� VPMHUX�QRUPDOH��JODVLOD��

S�� 4vG �

[�

]�

\�2�

S�� 1vG �

S�� 2vG �

S�� 3vG �

��

��

����

f gk= −G G

1rG

Page 33: 892’ - Fakultet strojarstva i brodogradnje · 0(+$1,.$)/8,’$ 892’ 892’ 0hkdqlndioxlgdmhglril]lnhnrmlvhedyljledqmhpioxlgdlvlodpdnrmhgmhoxmxqdioxlg 0hkdqlndioxlgdvhglmholqdvwdwlnxioxlgdnrmdsurxþdydudyqrwhåxioxlgdxplurydqmx

0(+$1,.$�)/8,'$�,�–�,17(*5$/1(�0(72'(� ��

� ( ) ( ) ( )( )

2

moment sile moment sile fluida na plašt WHåLQH

dk

w

kA

M F M G r n v p Sρ= − × +∑ ∫GG G G G G

��� ��

�Ako su površine poprHþQLK� SUHVMHND�PDOH� X� RGQRVX� QD� YHOLþLQX� UDGLMXV� YHNWRUD�� WDGD� VH�PRJX�]DQHPDULWL�SURPMHQH�UDGLMXV�YHNWRUD�SR�SRYUãLQL�SRSUHþQRJ�SUHVMHND�L�]DPLMHQLWL�JD�X�JRUQMHP� LQWHJUDOX� V� NRQVWDQWQLP� UDGLMXV� YHNWRURP� GR� WHåLãWD� SUHVMHND��8� WRP� VH� VOXþDMX�XPQRåDN� r n×G G � PRåH� L]OXþLWL� LVSUHG� LQWHJUDOD�� SD� LQWHJUDO� R]QDþXMH� LPSXOVQX� IXQNFLMX�GHILQLUDQX�X�]DNRQX�NROLþLQH�JLEDQMD��WH�YULMHGL��� ( ) ( ) ( )kk

k

w ( )M F M G r IGG G G GG= + ×∑ �

�'DNOH� ]D� VOXþDM� VWUXMDQMD� NUR]� FLMHYL�� QD� VYDNRP� XOD]QR�L]OD]QRP� SUHVMHNX� VH� SRVWDYOMD�LPSXOVQD�IXQNFLMD��NRMD�VH�]D�SRWUHEH�SURUDþXQD�VLOH�IOXLGD�QD�SODãW�FLMHYL�L�PRPHQWD�WH�VLOH�X� RGQRVX� QD� RGDEUDQX� WRþNX� �RELþQR� MH� WR� LVKRGLãWH� NRRUGLQDWQRJ� VXVWDYD��� WUHWLUD� NDR�YDQMVND�VLOD��3UHPD�MHGQDGåEL�NROLþLQH�JLEDQMD�VLOD�IOXLGD�QD�SODãW�MHGQDND�MH�VXPL�YDQMVNLK�VLOD� NRMH� GMHOXMX� QD� NRQWUROQL� YROXPHQ� �LPSXOVQH� IXQNFLMH� L� VLOD� WHåLQH��� D� PRPHQW� VLOH�kojom fluid djeluje na plašt cijevi je jednak sumi momenata vanjskih sila na kontrolni YROXPHQ� �VXPL� PRPHQDWD� LPSXOVQLK� IXQNFLMD� L� PRPHQWX� VLOH� WHåLQH��� 3UREOHP� VH� GDNOH�VYRGL� QD� SULPMHQX� XYMHWD� UDYQRWHåH� VLOD� L�PRPHQDWD�� NDR� X� NODVLþQRM�PHKDQLFL�� RGQRVQR�VWDWLFL�IOXLGD�������%HUQRXOOLMHYD�MHGQDGåED���0DWHPDWLþNL�]DSLV�]DNRQD�RþXYDQMX�HQHUJLMH��

M M M M M

MM M

M

2

Brzina promjene energije snaga masenih snaga vanjskih brzina dovo HQMD�sila na površinskih topline na

sila na

D Dd d d d d

D D 2V V V S S

VV V

V

ve V u V f v V v S q n S

t tρ ρ ρ σ

= + = ⋅ + ⋅ − ⋅

∫ ∫ ∫ ∫ ∫G G G G G G

��� ��� ��� ���

�8�QHVWODþLYRP�VWUXMDQMX�MH�VQDJD�VLOD�WODND�MHGQDND�QXOL��MHU�QHPD�SURPMHQH�REXMPD�þHVWLFD�IOXLGD��� WH� VQDJX� XQXWDUQMLK� VLOD� GHILQLUDMX� VDPR� YLVNR]QH� VLOH�� 9LVNR]QH� VLOH� XYLMHN�SUHWYDUDMX� PHKDQLþNX� HQHUJLMX� X� XQXWUDãQMX�� WH� üH� XYLMHN� YRGLWL� VPDQMLYDQMX� PHKDQLþNH�HQHUJLMH��$NR�VH�VQDJD�XQXWDUQMLK�VLOD�R]QDþL�V� FP �L�GHILQLUD�NDR�SR]LWLYQD�YHOLþLQD��WDGD�üH�VH�X� MHGQDGåEL�NLQHWLþNH�HQHUJLMH�RQD�SRMDYOMLYDWL�V�QHJDWLYQLP�SUHG]QDNRP��MHU�VPDQMXMH�NLQHWLþNX� HQHUJLMX�PDWHULMDOQRJ� YROXPHQD�� 8� VQDJX� XQXWDUQMLK� VLOD� FP REUDþXQDWL� üH� VH� L�snaga vanjskih površinskih sila trenja. Zakon prelazi u oblik:��� N

M M M

M M MM

2

F

snagaunutrašnjih

Brzina promjene snaga masenih snaga vanjskih sila unutarkinetiþNH�HQHUJLMH� VLOD�QD� � QRUPDOQLK

sila na

Dd d d

D 2V V S

V V VV

vV f v V pn v S P

tρ ρ= ⋅ + − ⋅ −∫ ∫ ∫

G G G G

����� ��� ���

��)RUPXODFLMD�]DNRQD�]D�NRQWUROQL�YROXPHQ�

Page 34: 892’ - Fakultet strojarstva i brodogradnje · 0(+$1,.$)/8,’$ 892’ 892’ 0hkdqlndioxlgdmhglril]lnhnrmlvhedyljledqmhpioxlgdlvlodpdnrmhgmhoxmxqdioxlg 0hkdqlndioxlgdvhglmholqdvwdwlnxioxlgdnrmdsurxþdydudyqrwhåxioxlgdxplurydqmx

0(+$1,.$�)/8,'$�,�–�,17(*5$/1(�0(72'(� ��

( )2 2F

d 1 1d d d d

d 2 2KV KP KV KP

v V v v n S f v V pn v S Pt

ρ ρ ρ+ ⋅ = ⋅ + − ⋅ −∫ ∫ ∫ ∫GG G G G G �

,QWHJUDFLMRP�MHGQDGåEH�NLQHWLþNH�HQHUJLMH��SR�NRQWUROQRP�YROXPHQX�SUHPD�VOLFL��GRELMH�VH�( )

( ) ( ) ( )2 2 2 2 12 12 1

1

2 2F

1d 2

d 1 1d d d d

d 2 2KV KP KV KP

Q p pQg z zv Q v vQ ss

v V v v n S f v V pn v S Pt

ρρρ

ρ ρ ρ

− −− −∂ −∂

+ ⋅ = ⋅ + − ⋅ −

∫ ∫ ∫ ∫GG G G G G

������� �������� ����

JGMH� VX� 1v � L� 2v � SURVMHþQH� EU]LQH� QD� SUHVMHFLPD� 1A � L� 2A �� D� Q � SURWRN� NUR]� FLMHY�� =DNRQ�NLQHWLþNH� HQHUJLMH� ]D� MHGQRGLPHQ]LMVNR� VWUXMDQMH� R]QDþXMH� PRGLILFLUDQX� %HUQRXOOLMHYX�MHGQDGåEX��NRMD�JODVL�

22 2

F

12 1

brzina promjene snaga na izlazu iz cijevi snaga na ulazu u cijevkinetiþNH�HQHUJLMH�

d2 2

KV

v v vp gz Q p gz Q P Q s

sαρ ρ αρ ρ ρ

∂+ + = + + − − ∂ ∫

������������ ���������

$NR� X� FMHYRYRGX� L]PH X� SUHVMHND� SRVWRML� VWURM� �SXPSD� NRMD� SUHGDMH� VQDJX� PP � IOXLGX� LOL�WXUELQD�NRMD�RGX]LPD�VQDJX� TP �RG�IOXLGD���RQGD�VH�PRGLILFLUDQD�MHGQDGåED�PRåH�SRRSüLWL�X�VOMHGHüL�REOLN�

22 2

F P T

12 1

brzina promjene snaga na izlazu iz cijevi snaga na ulazu u cijevkinetiþNH�HQHUJLMH�

d2 2

KV

v v vp gz Q p gz Q P Q s P P

sαρ ρ αρ ρ ρ

∂+ + = + + − − + − ∂ ∫

������������ ���������

3XPSD� MH� SRJRQMHQD� PRWRURP�� SUL� þHPX� PRWRU� SUHGDMH� SXPSL� VQDJX� MP �� SD� MH� IDNWRU�NRULVQRVWL�SXPSH� P

PM

P

Pη = ��7XUELQD�RELþQR�SRJRQL�JHQHUDWRU��SUL�þHPX�JHQHUDWRUX�SUHGDMH�

VQDJX� GP ��SD�MH�IDNWRU�NRULVQRVWL�WXUELQH�GHILQLUDQ�RGQRVRP� GG

T

P

Pη = ��

8�JRUH�SULND]DQRP�REOLNX�PRGLILFLUDQH�%HUQRXOOLMHYH�MHGQDGåEH��VYDNL�þODQ�LPD�GLPHQ]LMX�VQDJH��D�NRULVWH�VH�L�VOMHGHüL�REOLFL�WH�MHGQDGåEH�2EOLN� 'LPHQ]LMD�

22 2F P T

12 1

d2 2

v v P v P Pp gz p gz s

Q s Q Qαρ ρ αρ ρ ρ ∂+ + = + + − − + − ∂

∫ � snaga

volumenki protok�

22 2F P T

12 1

d2 2

v p v p P v P Pgz gz s

Q s Q Qα α

ρ ρ ρ ρ ρ ∂+ + = + + − − + − ∂

∫ � snaga

maseni protok�

22 2F P T

12 1

1d

2 2

v p v p P v P Pz z s

g g g g gQ g s gQ gQα α

ρ ρ ρ ρ ρ ∂+ + = + + − − + − ∂

∫ � snaga

WHåLQVNL��SURWRN �8�]DGQMHP�REOLNX�PRGLILFLUDQH�%HUQRXOOLMHYH�MHGQDGåEH�RELþQR�VH�XYRGH�R]QDNH�

PP

Ph

gQρ= YLVLQD�GREDYH�SXPSH��

TT

Ph

gQρ= SDG�YLVLQH�HQHUJLMH�X�WXUELQL�

FF

Ph

gQρ= YLVLQD�JXELWDND�PHKDQLþNH�HQHUJLMH��HQHUJLMH�SUHWYRUHQH�X�XQXWDUQMX�HQHUJLMX��

Page 35: 892’ - Fakultet strojarstva i brodogradnje · 0(+$1,.$)/8,’$ 892’ 892’ 0hkdqlndioxlgdmhglril]lnhnrmlvhedyljledqmhpioxlgdlvlodpdnrmhgmhoxmxqdioxlg 0hkdqlndioxlgdvhglmholqdvwdwlnxioxlgdnrmdsurxþdydudyqrwhåxioxlgdxplurydqmx

0(+$1,.$�)/8,'$�,�–�,17(*5$/1(�0(72'(� ��=D�VOXþDM�UDþYDQMD�FMHYRYRGD�REOLFL�PRGLILFLUDQH�%HUQRXOOLMHYH�MHGQDGåEH�L]�JRUQMH�WDEOLFH�SRVWDYOMDMX�VH�GXå�VWUXMQLFH���3ULPMHU��

6OLND� SULND]XMH� UDþYDVWX� FLMHY� V� GYD� XOD]QD�SUHVMHND� ��� L� ��� WH� GYD� L]OD]QD� SUHVMHND� ��� L� ����,]PH X�WRþDND���L���VH�QDOD]L�SXPSD�NRMD�SUHGDMH�IOXLGX� VQDJX� 33�� 3UHPD� MHGQDGåEL� NRQWLQXLWHWD�XNXSQL�SURWRN�NUR]�SXPSX�MH� 1 2 3 4Q Q Q Q Q= + = + ��8� WRþNDPD� �� L� �� YLVLQD� HQHUJLMH� MH� MHGQR]QDþQR�GHILQLUDQD�� EH]�RE]LUD� V�NRMH� VWUDQH� VH�X� WH� WRþNH�GROD]L��

,QWHJUDOQL�REOLN�]DNRQD�NLQHWLþNH�HQHUJLMH�]D�VWDFLRQDUQR�VWUXMDQMH�IOXLGD�NDåH�GD�MH�VQDJD�QD�L]OD]X�L]�.9� �SUHVMHFL���L���� MHGQDND�VQD]L�QD�XOD]X��SUHVMHFL���L����XYHüDQRM�]D�VQDJX�SXPSH�L�XPDQMHQRM�]D�VQDJX�YLVNR]QLK�VLOD��WM��2 2 2 23 4 1 2

3 3 3 3 4 4 4 4 1 1 1 1 2 2 2 2 P F2 2 2 2

v v v vp gz Q p gz Q p gz Q p gz Q P PB S S B S S B S S B S S

� ¬ � ¬ � ¬ � ¬­ ­ ­ ­� � � �­ ­ ­ ­� � � �� � � � � � � � � � � � �­ ­ ­ ­� � � �­ ­ ­ ­­ ­ ­­ � � ��� � � �� ® � ® � ®� ®

0RGLILFLUDQD�%HUQRXOOLMHYD�MHGQDGåED�SRVWDYOMHQD�L]PH X�WRþDND���GR��MH��2 25 5 1 1

5 5 1 1 F152 2

v p v pz z h

g g g gB B

S S

� ¬ � ¬­ ­� �­ ­� �� � � � � �­ ­� �­ ­­­ ��� �� ®� ®���JGMH�MH� F15

F151

Ph

gQS� �

0RGLILFLUDQD�%HUQRXOOLMHYD�MHGQDGåED�L]PH X�WRþDND���L���JODVL�2 26 6 5 5

6 6 5 5 P F562 2

v p v pz z h h

g g g gB B

S S

� ¬ � ¬­ ­� �­ ­� �� � � � � � �­ ­� �­ ­­ ­� �� �� ® � ®��JGMH�VX� F56

F56P

hgQS

� �L� PP

Ph

gQS� ��

D�L]PH X�WRþDND���L���2 23 3 61

3 3 6 6 F632 2

v p pvz z h

g g g gB B

S S

� ¬ � ¬­ ­� �­ ­� �� � � � � �­ ­� �­ ­­­ ��� �� ®� ®���JGMH�MH� F63

F633

Ph

gQS� �

,]�NRPELQDFLMH�SUHWKRGQLK�MHGQDGåEL�GRELMH�VH�PRGLILFLUDQD�%HUQRXOOLMHYD�MHGQDGåED�L]PH X�SUHVMHND���L����2 23 3 1 1

3 3 1 1 P F15 F56 F632 2

v p v pz z h h h h

g g g gB B

S S

� ¬ � ¬­ ­� �­ ­� �� � � � � � � � �­ ­� �­ ­­­ ��� �� ®� ®�

'DNOH�PRGLILFLUDQD�%HUQRXOOLMHYD�MHGQDGåED�YULMHGL�GXå�VWUXMQLFH��$QDORJQR�VH�GRELMH�L]UD]�]D�PRGLILFLUDQX�%HUQRXOOLMHYX�MHGQDGåEX�L]PH X�SUHVMHND���L���LOL�L]PH X�SUHVMHND���L���LOL�L]PH X�SUHVMHND� �� L� ���9DåQR� MH� ]DSDPWLWL� GD� VH� VQDJD�YLVNR]QLK� VLOD�GRELMH�PQRåHQMHP�YLVLQH� JXELWDND� Fh � V� SULSDGDMXüLP� WHåLQVNLP� SURWRNRP�� NDR� L� VQDJD� SXPSH� �X� RYRP�SULPMHUX� P 1 2 PP g Q Q hS� � ���������������

Page 36: 892’ - Fakultet strojarstva i brodogradnje · 0(+$1,.$)/8,’$ 892’ 892’ 0hkdqlndioxlgdmhglril]lnhnrmlvhedyljledqmhpioxlgdlvlodpdnrmhgmhoxmxqdioxlg 0hkdqlndioxlgdvhglmholqdvwdwlnxioxlgdnrmdsurxþdydudyqrwhåxioxlgdxplurydqmx

0(+$1,.$�)/8,'$�,�–�,17(*5$/1(�0(72'(� ���� 3URPMHQD�WODND�RNRPLWR�QD�VWUXMQLFH��LQWHJUDO�MHGQDGåEH�JLEDQMD�IOXLGD�SR�SXWX�RNRPLWRP�QD�VWUXMQLFH���

,]UD]� ]D� SURPMHQX� WODND� RNRPLWR�QD�VWUXMQLFH�MH���

2 2

2 1 2 1

1

dv

p p g z z nR

S S� � � �¨ �XGDOMHQRVW� Q se mjeri od središta ]DNULYOMHQRVWL�VWUXMQLFH���

���� 8� VWUXMDQMX� IOXLGD� V� UDYQLP� VWUXMQLFDPD� �R �d �� SURPMHQD� WODND� RNRPLWR� QD�VWUXMQLFH�LVWD�MH�NDR�X�IOXLGX�X�PLURYDQMX����� 8�VWUXMDQMX�IOXLGD�X�KRUL]RQWDOQRM�UDYQLQL�VD�]DNULYOMHQLP�VWUXMQLFDPD�WODN�UDVWH�RG�središta zakrivljenosti strujnica.�

6OLND�X]�GHILQLFLMX�SURPMHQH�WODND�RNRPLWR�QD�VWUXMQLFH��

5 UDGLMXV�]DNULYOMHQR2�

[��

[��[��

giv

��

��n�

Page 37: 892’ - Fakultet strojarstva i brodogradnje · 0(+$1,.$)/8,’$ 892’ 892’ 0hkdqlndioxlgdmhglril]lnhnrmlvhedyljledqmhpioxlgdlvlodpdnrmhgmhoxmxqdioxlg 0hkdqlndioxlgdvhglmholqdvwdwlnxioxlgdnrmdsurxþdydudyqrwhåxioxlgdxplurydqmx

0(+$1,.$�)/8,'$��–�35,0-(1(�21291,+�-('1$'ä%,�0(+$1,.(�)/8,'$� ��

���35,0-(1(�261291,+�-('1$'ä%(�0(+$1,.(�)/8,'$������3RMDYH�L�SULQFLSL�UDGD�QHNLK�XUH DMD�NRML�VH�PRJX�REMDVQLWL�%HUQRXOOLMHYRP�MHGQDGåERP�� 0MHUHQMH�EU]LQH��pa pa

pa

v1

21

z

A B

h

g

S

%h

6OXþDM�RWYRUHQRJ�VWUXMDQMD�V�UDYQLP�VWUXMQLFDPD��&MHYþLFD� $� �SLH]RPHWULþND� FLMHY�� PMHUL� YLVLQX� WODND� X�WRþNL����3URPMHQD�WODND�RNRPLWR�QD�UDYQH�VWUXMQLFH�LVWD�MH�NDR� X� IOXLGX� X�PLURYDQMX�� SD� üH� UD]LQD� IOXLGD� X� FMHYþLFL�biti u slobodnoj površini.�&MHYþLFD�%��3LWRWRYD�FLMHY��PMHUL�YLVLQX�WODND�X�WRþNL����X�NRMRM� MH� EU]LQD� MHGQDND� QXOL� �]DXVWDYQD� WRþND��� 3UHPD�%HUQRXOOLMHYRM�MHGQDGåEL�YLVLQD�]DXVWDYQRJ�WODND� 2p / gρ �MH�YHüD�RG�YLVLQH�WODND� 1p / gρ �X�WRþNL���]D�YLVLQX�EU]LQH�

21 �h v / g= ��

�ýODQRYL�%HUQRXOOLMHYH�MHGQDGåEH�VH�PRJX�WXPDþLWL�L�QD�VOMHGHüL�QDþLQ�N N

2

dinamiþNL WODN hidrostatski tlakstatiþNL WODN

zaustavni tlaktotalni tlak

1kost.

2p v gzS S� � �

���������������������������������������������������������������

%HUQRXOOLMHYD�MHGQDGåED�NDåH�GD�WRWDOQL�WODN�RVWDMH�NRQVWDQDWDQ�GXå�VWUXMQLFH���%h

v1

21

0MHUHQMH�EU]LQH�VWUXMDQMD�IOXLGD�X�FLMHYLPD��/LMHYD� FMHYþLFD� PMHUL� VWDWLþNL� WODN� X� WRþNL� ��� D�3LWRWRYD�FLMHY�]DXVWDYQL�WODN�X�WRþNL����5D]OLND�WD�GYD�WODND� MH�YLVLQD�EU]LQH�� SD�YULMHGL� 1 2v g h= ��2þLWR�MH� GD� VH� EU]LQD� UDþXQD� L]� PMHUHQH� UD]OLNH� WODNRYD��NRMD�VH�RELþQR�PMHUL�GLIHUHQFLMDOQLP�PDQRPHWURP����

S

%h�

v1

2

R

x

1

SS��

��6OXþDM� NDGD� MH� GLIHUHQFLMDOQL�PDQRPHWDU�LVSXQMHQ�IOXLGRP�PDQMH�

S

%h

v1

SS�!

1 2

R

x

��6OXþDM� NDGD� MH� GLIHUHQFLMDOQL� PDQRPHWDU� LVSXQMHQ�

Page 38: 892’ - Fakultet strojarstva i brodogradnje · 0(+$1,.$)/8,’$ 892’ 892’ 0hkdqlndioxlgdmhglril]lnhnrmlvhedyljledqmhpioxlgdlvlodpdnrmhgmhoxmxqdioxlg 0hkdqlndioxlgdvhglmholqdvwdwlnxioxlgdnrmdsurxþdydudyqrwhåxioxlgdxplurydqmx

0(+$1,.$�)/8,'$��–�35,0-(1(�21291,+�-('1$'ä%,�0(+$1,.(�)/8,'$� ��PDQRPHWDU�LVSXQMHQ�IOXLGRP�PDQMH�JXVWRüH�RG�IOXLGD�NRML�VWUXML�X�FLMHYL�

01 2 �v g h

ρρ

= −

��

IOXLGRP�YHüH�JXVWRüH�RG�IOXLGD�NRML�VWUXML�X�FLMHYL��0

1 2 �v g hρρ

= −

��3UDQGWO�3LWRWRYD�FLMHY��6DVWRML� VH� RG� GYLMH� NRDNVLMDOQH� FLMHYL�� SUL� þHPX� MH� XQXWDUQMD� FMHYþLFD� VYRMLP� RWYRURP�VXSURWVWDYOMHQD� VWUXMDQMX� L� PMHUL� ]DXVWDYQL� WODN� �WRþND� �� QD� VOLFL��� 9DQMVND� FLMHY� LPD� SR�RERGX� UXSLFH� V�RWYRULPD�SUHNR�NRMLK�þHVWLFH� IOXLGD�SUROD]H� WDQJHQFLMDOQR�NRMLPD�VH�PMHUL�VWDWLþNL� WODN� �WRþND� �� QD� VOLFL��� 'RQMD� VOLND� NYDOLWDWLYQR� SULND]XMH� SURPMHQX� WODND� GXå�VWUXMQLFH� ������� 8� WRþNL� ]DVWRMD� MH� EU]LQD� MHGQDND� QXOL�� D� WODN� MH� PDNVLPDODQ�� 2G� WRþNH�]DVWRMD� IOXLG�VH�SRQRYR�XEU]DYD��D� WODN�RSDGD��8�SRGUXþMX� L]PH X� WRþDND��� L���EU]LQD�QD�

QHNLP�PMHVWLPD� SUHPašuje brzinu 1v �� WH� WODN� RSDGD� LVSRG� WODND� 1p �� DOL� VH� QD� RGUH HQRM�XGDOMHQRVWL�RG�WRþNH���WODN�SRQRYR�YUDüD�QD�YULMHGQRVW� WODND� 1p ��$NR�VH�]DQHPDUL�XþLQDN�YLVNR]QLK�VLOD�X�QHRJUDQLþHQRP�VWUXMDQMX�IOXLGD�WODN� 3p �üH�ELWL�MHGQDN�WODNX� 1p ��SD�üH�VH�L]�PMHUHQH�YLVLQH� h �PRüL�L]UDþXQDWL�EU]LQD� 1v ��SUL�þHPX�YULMHGL�L]UD]� 0

1 2 �v g hρρ

= −

���

S

%h

v1

SS�!

p p3 1=

3

21

tlak

p1

�����������������

Page 39: 892’ - Fakultet strojarstva i brodogradnje · 0(+$1,.$)/8,’$ 892’ 892’ 0hkdqlndioxlgdmhglril]lnhnrmlvhedyljledqmhpioxlgdlvlodpdnrmhgmhoxmxqdioxlg 0hkdqlndioxlgdvhglmholqdvwdwlnxioxlgdnrmdsurxþdydudyqrwhåxioxlgdxplurydqmx

0(+$1,.$�)/8,'$��–�35,0-(1(�21291,+�-('1$'ä%,�0(+$1,.(�)/8,'$� ����0MHUHQMH�SURWRND�X�VWUXMDQMX�NUR]�FLMHYL��

2

21

1 ��

6OLND� VKHPDWVNL� SULND]XMH� WUL� UD]OLþLWD�PMHUQD� XUH DMD� ]D� PMHUHQMH� SURWRND� X�VWUXMDQMX� NUR]� FLMHYL�� UHGRP�PMHUQD�EOHQGD��PMHUQD� VDSQLFD� L�9HQWXULMHYD� FLMHY��8� VYLP�XUH DMLPD� MH� SULQFLS� PMHUHQMD� LVWL�� X�VXåHQRP� SUHVMHNX� WODN� MH� ]ERJ� SRYHüDQMD�EU]LQH�QLåL��5D]OLND�WODND�X�SUHVMHFLPD���L���UDVWH� V� SRUDVWRP� SURWRND�� WH� VH� L]� PMHUHQH�UD]OLNH�WODND�PRåH�]DNOMXþLWL�R�SURWRNX�NUR]�FLMHY�� 3ULPMHQRP� %HUQRXOOLMHYH� MHGQDGåEH�VH� GROD]L� GR� SURWRND� LGHDOQRJ� IOXLGD�� D�XYR HQMHP� IDNWRUD� NRUHNFLMH� EU]LQH� L�NRQWUDNFLMH� WODND� VH� GROD]L� GR� SURWRND�UHDOQRJ�IOXLGD��

��9HQWXULMHYD�FLMHY��

h0

h

xz=0

Q

D1

D2

1

2

S0

S �N

6OLND� VKHPDWVNL� SULND]XMH� 9HQWXULMHYX� FLMHY�SRVWDYOMHQX� X� NRVRP� FMHYRYRGX�� X� NRMRM� VH�GLIHUHQFLMDOQLP� PDQRPHWURP� PMHUL� UD]OLND�WODND� X� GYD� SUHVMHND�� ,]� MHGQDGåEH�NRQWLQXLWHWD�� %HUQRXOLMHYH� MHGQDGåEH� L�MHGQDGåEH�PDQRPHWUD� VOLMHGL� L]UD]� ]D� SURWRN�LGHDOQRJ�IOXLGD�

���������0

022

id 4

2

1

2 1

41

ghD

QD

D

SSQ

� ¬­� � ­� ­� ­�� ®�

� ¬­� ­�� ­� ­�� ®

3URWRN�UHDOQRJ�IOXLGD�YLVNR]QRVWL�µ �MH������������ c idvQ C C Q� ��

9HQWXULMHYD�FLMHY�VH� L]YRGL� WDNR�GD�MH�IDNWRU�NRQWUDNFLMH�POD]D� c 1C � ��D�IDNWRU�NRUHNFLMH�EU]LQH� vC � MH� IXQNFLMD� 5H\QROGVRYD� EURMD� 1 1v D

Reρ

µ= �� 3ULPMHU� ]DYLVQRVWL� IDNWRUD� vC � R�

5H\QROGVRYX�EURMX�5H�MH�GDQ�QD�VOMHGHüRM�VOLFL���

Page 40: 892’ - Fakultet strojarstva i brodogradnje · 0(+$1,.$)/8,’$ 892’ 892’ 0hkdqlndioxlgdmhglril]lnhnrmlvhedyljledqmhpioxlgdlvlodpdnrmhgmhoxmxqdioxlg 0hkdqlndioxlgdvhglmholqdvwdwlnxioxlgdnrmdsurxþdydudyqrwhåxioxlgdxplurydqmx

0(+$1,.$�)/8,'$��–�35,0-(1(�21291,+�-('1$'ä%,�0(+$1,.(�)/8,'$� ��

�� ��.DYLWDFLMD��

p2Sg

p1

Sg

2g2g

1 2

v1 v� G.L.

E.L.

p1

p2

A1A2

Q

Q Q1>

3RYHüDQMHP�SURWRND�X]�LVWX�XNXSQX�HQHUJLMX� VWUXMDQMD� GROD]L� GR�VPDQMHQMD� WODND�X�QDMXåHP�SUHVMHNX��QD� VOLFL� MH� SULND]DQ� SRPDN� +*/�NDGD�VH�SURWRN�SRYHüD�RG�4�QD�4���.DGD� VH� WODN� X� QDMXåHP� SUHVMHNX�VQL]L�QD�YULMHGQRVW�WODND�LVSDUDYDQMD�SRMDYOMXMX� VH� PMHKXULüL� SDUH��NDYLWDFLMD��� þLPH� VH� VPDQMXMH�SRSUHþQL� SUHVMHN� WH� GROD]L� GR�zagušivanja strujanja. Protok pri NRMHP� VH� SRMDYOMXMH� NDYLWDFLMD� MH�PDNVLPDOQR� PRJXüL� SURWRN� ]D�]DGDQX�YLVLQX�HQHUJLMH��0MHKXULüL� SDUH� ELYDMX� QRãHQL� X� SRGUXþMH� YLãHJ� WODND�� JGMH� LPSORGLUDMX� �SRQRYR� VH�SUHWYDUDMX� X� NDSOMHYLWX� ID]X��� 3RMDYD� NDYLWDFLMH� MH� SRSUDüHQD� YLEUDFLMDPD� L� EXNRP�� D� SUL�LPSOR]LML�PMHKXULüD�SDUH�X�EOL]LQL�VWLMHQNH�GROD]L� L�GR�QMHQD�RãWHüHQMD��8�QHVWDFLRQDUQRP�VWUXMDQMX�VH�NDYLWDFLMD�PRåH�SRMDYLWL�XVOLMHG�QDJORJ�XEU]DYDQMD�IOXLGD��� �������������

Page 41: 892’ - Fakultet strojarstva i brodogradnje · 0(+$1,.$)/8,’$ 892’ 892’ 0hkdqlndioxlgdmhglril]lnhnrmlvhedyljledqmhpioxlgdlvlodpdnrmhgmhoxmxqdioxlg 0hkdqlndioxlgdvhglmholqdvwdwlnxioxlgdnrmdsurxþdydudyqrwhåxioxlgdxplurydqmx

0(+$1,.$�)/8,'$��–�35,0-(1(�21291,+�-('1$'ä%,�0(+$1,.(�)/8,'$� ��(MHNWRU��

6WUXMDQMH� SULPDUQRJ� IOXLGD� SURWRNRP� 4�� X�VXåHQRP� SUHVMHNX� L]D]LYD� VPDQMHQMH� WODND�� NRMH�LPD� ]D� SRVOMHGLFX� XVLVDYDQMH� VHNXQGDUQRJ� IOXLGD��SURWRNRP� 4��� WDNR� GD� MH� QD� L]OD]X� L]� HMHNWRUD�SURWRN�4��4���2YDM� VH� SULQFLS� NRULVWL� QSU�� X� XUH DMLPD� ]D�ERMDQMH��X�NRMLPD�VH�X�VWUXMX�]UDND�XYODþL�ERMD��

�,VWMHFDQMH�L]�YHOLNRJ�VSUHPQLND��

Slika prikazuje zamišljenu strujnicu unutar VSUHPQLND�� $NR� VH� SUHWSRVWDYL� YHOLNL� VSUHPQLN��EU]LQD� IOXLGa na slobodnoj površini unutar VSUHPQLND� üH� ELWL� YUOR� PDOD�� %U]LQD� VH� SRYHüDYD�SULEOLåDYDQMHP� XOD]X� X� FLMHY�� =D� SRWUHEH� FUWDQMD�KLGUDXOLþNH�JUDGLMHQWQH� OLQLMH�üH�VH�SUHWSRVWDYLWL�GD�MH�X�VYDNRM�WRþNL�VSUHPQLND�EU]LQD�MHGQDND�QXOL��SD�üH�YLVLQD�XNXSQH�HQHUJLMD�X�VSUHPQLNX�ELWL�MHGQDND�SLH]RPHWULþNRM� YLVLQL� �NRMD� MH� ]D� VOXþDM� PLURYDQMD�MHGQDND�X�VYLP�WRþNDPD�VSUHPQLND���3UHPD�WRPH�%HUQRXOOLMHYX�MHGQDGåEX�PRåH�VH�SRVWDYOMDWL�RG�ELOR�NRMH�WRþNH�X�VSUHPQLNX��D�RELþQR�VH�ELUD�WRþND�QD�VORERGQRM�SRYUãLQL��%HUQRXOOLMHYD�MHGQDGåED�SRVWDYOMHQD�RG�WRþNH���QD�VORERGQRM�SRYUãLQL�GR�WRþNH���QD�L]OD]X�L]�FLMHYL�JODVL�� 2

a a

2

p v pH

g g gS S� � � �LOL� 2v gH� �

L]� NRMH� MH� MDVQR� GD� VH� SRWHQFLMDOQD� HQHUJLMD� IOXLGD� X� VSUHPQLNX� SUHWYRULOD� X� NLQHWLþNX�HQHUJLMX�mlaza na izlazu iz cjevovoda, što prikazuje i slika. (Iz mehanike je poznato da bi kuglica u slobodnom padu puštena iz stanja mirovanja na putu +� SRVWLJOD� EU]LQX�

2v gH� ����*XELWDN�XWMHFDQMD�X�YHOLNL�VSUHPQLN���

pa

pa

g

h2

3

Q

H=h -h1 2

h1

4

1

2

v

8� SUHWKRGQRP� SULPMHUX� MH� POD]�IOXLGD� LVWMHFDR� X� DWPRVIHUX�� SD� MH�X� QMHPX� YODGDR� DWPRVIHUVNL� WODN��D� RYGMH� POD]� LVWMHþH� X� PLUXMXüL�IOXLG� X� YHOLNRP� VSUHPQLNX�� D�HNVSHULPHQWL� SRND]XMX� GD� üH� X�POD]X� YODGDWL� WODN� GHILQLUDQ�MHGQDGåERP�KLGURVWDWLNH�4 a 2p p ghS� � �

%HUQRXOOLMHYD�MHGQDGåED�SRVWDYOMHQD�GXå�VWUXMQLFH�L]PH X�WRþDND���L����JGMH�MH�]� ���JODVL�

Page 42: 892’ - Fakultet strojarstva i brodogradnje · 0(+$1,.$)/8,’$ 892’ 892’ 0hkdqlndioxlgdmhglril]lnhnrmlvhedyljledqmhpioxlgdlvlodpdnrmhgmhoxmxqdioxlg 0hkdqlndioxlgdvhglmholqdvwdwlnxioxlgdnrmdsurxþdydudyqrwhåxioxlgdxplurydqmx

0(+$1,.$�)/8,'$��–�35,0-(1(�21291,+�-('1$'ä%,�0(+$1,.(�)/8,'$� ��

�4

2a a

1 2

/

2p g

p v ph h

g g gS

S S� � � �

����LOL�X]� 1 2h h H� � ���� 2

2

vH

g� �

3RQRYR� MH� MDVQR�GD�üH�EU]LQD�ELWL� IXQNFLMD� UD]OLNH�YLVLQD�X�VSUHPQLFLPD��$NR�VH�]D�GHVQL�VSUHPQLN� XVYRML�PRGHO�PLUXMXüHJ� IOXLGD� RQGD� üH� HQHUJLMD�GHVQRJ� VSUHPQLND�ELWL� MHGQDND�SLH]RPHWULþNRM�YLVLQL�L�ELW�üH�PDQMD�RG�HQHUJLMH�OLMHYRJ�VSUHPQLND��'DNOH�X�FLMHYL�üH�SUHPD�%HUQRXOOLMHYRM� MHGQDGåEL�YLVLQD�XNXSQH�HQHUJLMH�ELWL�MHGQDND�HQHUJLML�OLMHYRJ�VSUHPQLND��D�XODVNRP� X� GHVQL� VSUHPQLN� HQHUJHWVND� OLQLMD� VNRNRYLWR� RSDGD� ]D� YLVLQX� +�� RGQRVQR� ]D�YLVLQX�EU]LQH��WH�VH�JRYRUL�R�JXELWNX�XWMHFDQMD��LOL�LVWMHFDQMD��X�YHOLNL�VSUHPQLN���%HUQRXOOLMHYD�MHGQDGåED�VH�IRUPDOQR�SRVWDYOMD�RG�VORERGQH�SRYUãLQH�OLMHYRJ�VSUHPQLND�GR�VORERGQH�SRYUãLQH�GHVQRJ�VSUHPQLND��V�WLP�GD�VH�SUL�XODVNX�X�VSUHPQLN�REUDþXQD�JXELWDN�YLVLQH� XNXSQH� HQHUJLMH� NRML� MH� MHGQDN� YLVLQL� EU]LQH�� 7DNR� EL� %HUQRXOOLMHYD� MHGQDGåED�L]PH X�WRþDND���L����SUHPD�SUHWKRGQR�VOLFL���X]�]� ����JODVLOD��

N N

2a a

energija gubitakenergija u toþNL��u toþNL��

2

p p vH

g g g���S S� � � �

3 4

2gv2

=H E.L.

H.G.L.

pa

Sg

pa

Sg

/LMHYD� VOLND� SULND]XMH� HQHUJHWVNX�OLQLMX� �(/�� ]D� VWUXMDQMH� L]PH X� GYD�YHOLND� VSUHPQLND�� 2GX]LPDQMHP�YLVLQH�EU]LQH�RG�(/�GRELMH�VH�+*/��3UHPD� SULMH� UHþHQRP� SUHWSRVWDYOMD�VH� GD� VX� EU]LQH� X� VSUHPQLFLPD�MHGQDNH� QXOL�� WH� VH� +*/� VNRNRYLWR�PLMHQMD�SUL�XOD]X�X�FLMHY��X�NRMRM� MH�EU]LQD� ]D� VOXþDM� NRQVWDQWQRJ�SURPMHUD�FLMHYL�NRQVWDQWQD������6LIRQ��

0

1

pa

pa

U

H

hd=konst

2�

6WUXMDQMH�NUR]�VLIRQ�üH�VH�RVWYDULWL�DNR�MH�FLMHY�X�SRþHWQRP� WUHQXWNX�ELOD� LVSXQMHQD� IOXLGRP� LOL� MH�SRWUHEQR� VWYRULWL� SRGWODN� QD� L]OD]QRP� NUDMX�FLMHYL� �WRþND� ��� WDNR� GD� VH� IOXLG� SRGLJQH� SUHNR�WRþNH����,]�%HUQRXOOLMHYH�MHGQDGåEH�RG���GR���MH�2

2

vH

g� �LOL� 2v gH� �

6SXãWDQMHP� L]OD]QRJ� NUDMD� SRYHüDYD� VH� EU]LQD�LVWMHFDQMD��,]�%HUQRXOOLMHYH�MHGQDGåEH�RG���L���� 1 ap p g H hS� � � �

6SXãWDQMHP� L]OD]QRJ�NUDMD� LOL�SRGL]DQMHP�WRþNH��� VPDQMXMH� VH� WODN� S��� NRML� PRUD� ELWL� YHüL� RG�WODND�SDUD�SY�GD�QH�EL�QDVWXSLOD�NDYLWDFLMD��þLMRP�EL�VH�SRMDYRP�VWUXMDQMH�SUHNLQXOR����

Page 43: 892’ - Fakultet strojarstva i brodogradnje · 0(+$1,.$)/8,’$ 892’ 892’ 0hkdqlndioxlgdmhglril]lnhnrmlvhedyljledqmhpioxlgdlvlodpdnrmhgmhoxmxqdioxlg 0hkdqlndioxlgdvhglmholqdvwdwlnxioxlgdnrmdsurxþdydudyqrwhåxioxlgdxplurydqmx

0(+$1,.$�)/8,'$��–�35,0-(1(�21291,+�-('1$'ä%,�0(+$1,.(�)/8,'$� ��0DNVLPDOQD�YLVLQD�XVLVDYDQMD�SXPSH���

pah

1

0

pumpa

'D�EL�VH�XNOMXþLYDQMHP�SXPSH�XVSRVWDYLOR�VWUXMDQMH��XVLVQD�FLMHY�PRUD�ELWL�LVSXQMHQD�IOXLGRP��'D�EL�VH�L]EMHJOD�SRMDYD�NDYLWDFLMH�WODN�X�WRþNL���PRUD�ELWL�YLãL�RG�WODND�SDUD��,]�%HUQRXOOLMHYH�MHGQDGåEH�RG���GR���MH�2

1 1

2ap p v

hg g gS S

� � � �8]� SUHWSRVWDYNX� GD� VX� YLVLQH� 2

1 1

2

p v

g gS� � ]DQHPDULYH��

WHRULMVNL� PDNVLPDOQD� YLVLQD� XVLVDYDQMD� MH� MHGQDND� YLVLQL�DWPRVIHUNRJ�WODND��D�VWYDUQR�MH�WR�L�PDQMH���.RUHNFLMH�EU]LQH�L�SURWRND�SUL�LVWMHFDQMX�NUR]�RWYRUH��

pa

1A0A �

6WUXMQLFD�QH�PRåH�ELWL�VORPOMHQD�FUWD��MHU�EL�X�WRþNL�ORPD�UDGLMXV�]DNULYOMHQRVWL� VWUXMQLFH� ELR� MHGQDN� QXOL�� WH� EL� GHULYDFLMD� WODND�RNRPLWR�QD�VWUXMQLFX�ELOD�EHVNRQDþQD��ãWR�QH�EL�ELOR�IL]LNDOQR��Zbog toga pri istjecanju fluida kroz otvor površine $� s oštrim UXERP� GROD]L� GR� VXåHQMD� POD]D�� 6OLND� SULND]XMH� SUHVMHN� �� X�NRMHPX�VX�VWUXMQLFH�SDUDOHOQH��D�WODN�NRQVWDQWDQ��8�WRP�SUHVMHNX�se mjeri površina $�SRSUHþQRJ�SUHVMHND�POD]D��)DNWRU�NRQWUDNFLMH�POD]D�MH� c 0/C A A� ��

5HDOQL� IOXLGL� VX�YLVNR]QL� WH� üH� VH�GLR�PHKDQLþNH� HQHUJLMH�QD�SXWX�RG� WRþNH���GR� WRþNH���XVOLMHG�GMHORYDQMD�YLVNR]QLK�VLOD�SUHWYRULWL�X�XQXWDUQMX�HQHUJLMX��ãWR�]QDþL�GD�üH�PHKDQLþND�HQHUJLMD� �RGQRVQR� EU]LQD�� ]D� VOXþDM� UHDOQRJ� IOXLGD� ELWL� PDQMD�� 7R� VH� X]LPD� X� RE]LU�LVNXVWYHQLP� IDNWRURP� NRUHNFLMH� EU]LQH� &Y� �NRML� VH� RGUH XMH� HNVSHULPHQWDOQR�� SUHPD�IRUPXOL�� id 2v vv C v C gH� � ��-DVQR�MH�GD�MH�IDNWRU�EU]LQH�XYLMHN�PDQML�RG�MHGDQ���3URWRN�4�IOXLGD�NUR]�RWYRU�üH�ELWL�MHGQDN�XPQRãNX�VWYDUQH�EU]LQH�L�VWYDUQH�SRYUãLQH�POD]D��

NNd id

c id 0 d idv

C Q

Q vA C C v A C Q� � � �� JGMH� MH� d cvC C C� � IDNWRU� NRUHNFLMH� SURWRND� �þHVWR� VH�R]QDþXMH�L�V� QC ���3ULPMHUL�IDNWRUD�EU]LQH�L�IDNWRUD�NRQWUDNFLMH�]D�QHNH�WLSLþQH�VOXþDMHYH��

�7DQND�VWLMHQND�oštri rub: &F �������&Y ����� �/LMHSR�]DREOMHQL�UXE��&F ����&Y �����

Page 44: 892’ - Fakultet strojarstva i brodogradnje · 0(+$1,.$)/8,’$ 892’ 892’ 0hkdqlndioxlgdmhglril]lnhnrmlvhedyljledqmhpioxlgdlvlodpdnrmhgmhoxmxqdioxlg 0hkdqlndioxlgdvhglmholqdvwdwlnxioxlgdnrmdsurxþdydudyqrwhåxioxlgdxplurydqmx

0(+$1,.$�)/8,'$��–�35,0-(1(�21291,+�-('1$'ä%,�0(+$1,.(�)/8,'$� ��

�,VSXVW��&F ����&Y ����� �,VSXVW��&F ����&Y ��������)RUPXOD�]D�L]UDþXQDYDQMH�YUHPHQD�SUDåQMHQMD�SRVXGH��pa

pa

g

Cd

H0

A0

1

v

0A z( )

v0=- dzdt

zH1

t t= 0

t=t1

3UHWSRVWDYNH����� 3RVXGD� MH� RWYRUHQD� SUHPD�DWPRVIHUL����� 9LVLQD� ]� VH� PMHUL� RG� SUHVMHND�POD]D� X� NRMHP� VX� VWUXMQLFH�SDUDOHOQH��YHQD�FRQWUDFWD������ 3RYUãLQD� SRSUHþQRJ� SUHVMHND�SRVXGH� $�]��� MH� SXQR� YHüD� RG�površine $�� RWYRUD� QD� GQX��NYD]LVWDFLRQDUQR� VWUXMDQMH�

id 2v gz� ����9ULMHPH� t �SRWUHEQR�GD�VH�UD]LQD�IOXLGD�VSXVWL�V�YLVLQH� 0z H= �QD� 1z H= �

1

0

1 0

d 0

1 G2

H

H

A zt t t z

C A g z� � �� ¨ �

���� ,OXVWUDFLMD�VDGUåDMD�%HUQRXOOLMHYH�MHGQDGåEH��

N NN

2

geometrijskavisina =geodetska

visina brzine visina tlaka linija

piezometriþND�YLVLQD� hidrauliþND�JUDGLMHQWQD�OLQLMD

visina ukupne energije = energetska linija

konst.2

v pz

g gS� � �

�������

�����������

9LVLQD�XNXSQH�HQHUJLMH�RVWDMH�NRQVWDQWQD�GXå�VWUXMQLFH��=D� VWUXMDQMH� X� FLMHYLPD� VDGUåDM� %HUQRXOLMHYH� MHGQDGåEH� VH� SULND]XMH� ]D� VWUXMQLFX� NRMD�SUROD]L� VLPHWUDORP� FLMHYL�� WDNR� GD� VLPHWUDOD� R]QDþXMH� JHRGHWVNX� OLQLMX� �*/���+LGUDXOLþNX�JUDGLMHQWQX�OLQLMX��+*/��VH�GRELMH�RGX]LPDQMHP�YLVLQH�EU]LQH�RG�HQHUJHWVNH�OLQLMH��(/����3ULPMHUL�LOXVWUDFLMH�VDGUåDMD�%HUQRXOOLMHYH�MHGQDGåEH��

Page 45: 892’ - Fakultet strojarstva i brodogradnje · 0(+$1,.$)/8,’$ 892’ 892’ 0hkdqlndioxlgdmhglril]lnhnrmlvhedyljledqmhpioxlgdlvlodpdnrmhgmhoxmxqdioxlg 0hkdqlndioxlgdvhglmholqdvwdwlnxioxlgdnrmdsurxþdydudyqrwhåxioxlgdxplurydqmx

0(+$1,.$�)/8,'$��–�35,0-(1(�21291,+�-('1$'ä%,�0(+$1,.(�)/8,'$� ��

z1

z2

p2Sg

p1

Sg

2gv�

2gv�

z=0

G.L.

H.G.L.

E.L.

2

1

p2Sg

p3

Sgp1

Sg

H.G.L.

z z z

2g 2g 2gv3

2

z=0

1 2 3

��� 3URPMHU� FLMHYL� MH� NRQVWDQWDQ�� SD� MH�SUHPD� MHGQDGåEL� NRQWLQXLWHWD�NRQVWDQWQD�L�EU]LQD���� 'ROD]L� GR�SUHUDVSRGMHOH�YLVLQH� WODND�L� JHRGHWVNH�YLVLQH��D�SURPMHQD� WODND�MH�LVWD�NDR�X�IOXLGX�X�PLURYDQMX���� 6PMHU� VWUXMDQMD� QHRGUH HQ� �VOLND� MH�LVWD�]D�RED�VPMHUD�VWUXDMQMD����� 3RORåDM�] ��VH�RGDELUH�SURL]YROMQR���� (QHUJHWVND� OLQLMD� VH� PRåH� GHILQLUDWL�LOL� V� DSVROXWQLP� WODNRP� LOL� V�SUHWODNRP� �DNR� MH� GHILQLUDQD� V�DSVROXWQLP�WODNRP�� WDGD�YLVLQD�WODND�QH� PRåH� ELWL� QHJDWLYQD�� WM�� +*/� QH�PRåH�ELWL�LVSRG�*/��NDR�QL�(/���

�� 9LVLQD�]� MH�NRQVWDQWQD��SD�GROD]L�GR�SUHUDVSRGMHOH� L]PH X� YLVLQH� EU]LQH� L�YLVLQH�WODND���� ,]� MHGQDGåEH� NRQWLQXLWHWD�4 Y$ NRQVW��� VOLMHGL� GD� üH� X� SUHVMHNX�PDQMH�SRYUãLQH�$�ELWL�YHüD�EU]LQD��D�L]�%HUQRXXOLMHYH�MHGQDGåEH�MH�MDVQR�GD�üH�SUL�YHüRM�EU]LQL�ELWL�QLåL�WODN���� 0LQLPDOQD� YULMHGQRVW� WODND� MH� GDNOH�X� QDMXåHP� SUHVMHNX�� D� QH� PRåH� ELWL�PDQMD� RG� WODND� SDUD� �WODND� NRG� NRMHJ�IOXLG� SUL� ]DGDQRM� WHPSHUDWXUL� SRþLQMH�LVSDUDYDWL����� 0LQLPDOQLP� WODNRP� MH� GHILQLUDQD� L�PDNVLPDOQD� EU]LQD� VWUXMDQMD�� RGQRVQR�PDNVLPDOQL�SURWRN�4���

pa

pa

A

��pa pa

A

B

��� *HRGHWVND� YLVLQD� L]OD]QRJ� NUDMD�FLMHYL� MH� SUHYLVRND�� SD� QHPD�VWUXMDQMD�IOXLGD���� %HUQRXOLMHYD� MHGQDGåED� VH� VYRGL� QD�RVQRYQX� MHGQDGåEX� KLGURVWDWLNH��SULQFLS�VSRMHQLK�SRVXGD���

�� 6NUDüLYDQMHP� SULNOMXþQH� FLMHYL��GROD]L� GR� VWUXMDQMD� IOXLGD�� D� YLVLQD�POD]D� MHGQDND� MH� YLVLQL� IOXLGD� X�YHOLNRP� VSUHPQLNX�� �]D� VOXþDM�YLVNR]QRJ� VWUXMDQMD�� WD� EL� YLVLQD� ELOD�QHãWR�PDQMD�]ERJ�SUHWYRUEH�PHKDQLþNH�HQHUJLMH�X�XQXWDUQMX���

Page 46: 892’ - Fakultet strojarstva i brodogradnje · 0(+$1,.$)/8,’$ 892’ 892’ 0hkdqlndioxlgdmhglril]lnhnrmlvhedyljledqmhpioxlgdlvlodpdnrmhgmhoxmxqdioxlg 0hkdqlndioxlgdvhglmholqdvwdwlnxioxlgdnrmdsurxþdydudyqrwhåxioxlgdxplurydqmx

0(+$1,.$�)/8,'$��–�',0(1=,-6.$�$1$/,=$� ��

���',0(1=,-6.$�$1$/,=$��'LPHQ]LMVND� DQDOL]D� L� WHRULMD� VOLþQRVWL� SUHGVWDYOMDMX� ]QDQVWYHQL� WHPHOM� HNVSHULPHQWDOQRP�LVWUDåLYDQMX�VORåHQLK�IL]LNDOQLK�SRMDYD�NDNR�X�PHKDQLFL�IOXLGD��WDNR�L�X�RVWDOLP�SRGUXþMLPD�IL]LNH��3ULPMHQRP�GLPHQ]LMVNH�DQDOL]H�PLQLPL]LUD�VH�SRWUHEQL�EURM�PMHUHQMD�]D�LVWUDåLYDQMH�QHNH�SRMDYH�� D�RODNãDYDMX� VH�SULND]� L� WXPDþHQMH� UH]XOWDWD�PMHUHQMD��7HRULMD� VOLþQRVWL�GDMH�SRGORJX�]D�SULPMHQX�PRGHOVNLK�LVWUDåLYDQMD�L�SULPMHQX�DQDORJLMD�X�IL]LFL������2VQRYQD�MHGQDGåED�PHWURORJLMH�6DGUåDM�IL]LNDOQH�YHOLþLQH�4�L]UDåDYD�VH�SURGXNWRP�PMHUQRJ�EURMD�Q� �L�PMHUQH�MHGLQLFH< >Q ��

[ ]Q Q Q= � �1SU��XEU]DQMH� 24 /a m s= ������6NXS�RVQRYQLK�L�L]YHGHQLK�IL]LNDOQLK�MHGLQLFD��'LPHQ]LMD� < >Q � RGQRVQR� MHGLQLFD� < >

SIQ � VYDNH� IL]LNDOQH� YHOLþLQH� Q � X� PHKDQLFL� IOXLGD� VH�

PRåH�SULND]DWL�SURGXNWRP�SRWHQFLMD�RVQRYQLK�GLPHQ]LMD�RGQRVQR�MHGLQLFD�X�REOLNX�� < > M L Ta b c dQ 2� �� �

SIQ kg m s Ka b c d  ¯ �¡ °¢ ±� �

JGMH�VX�RVQRYQH�IL]LNDOQH�YHOLþLQH��þLMH�VX�GLPHQ]LMH�RVQRYQH��X�PHKDQLFL�IOXLGD�9HOLþLQD� 2]QDND� 'LPHQ]LMD� -HGLQLFD�X�6,�VXVWDYX�GXOMLQD�YULMHPH�PDVD�WHPSHUDWXUD�

/�W�P�7�

/�7�0�Θ�

P�V�NJ�.�D� HNVSRQHQWL�D��E��F� L�G� WLSLþQL� ]D� IL]LNDOQX�NDWHJRULMX�4��1LVX�XYLMHN�SRWUHEQH�VYH�þHWLUL�RVQRYQH� GLPHQ]LMH�� 7DNR� VH� GLPHQ]LMH� VYLK� IL]LNDOQLK� YHOLþLQD� X� NLQHPDWLFL� IOXLGD�PRJX�RSLVDWL� V� GYLMH�GLPHQ]LMH��GXOMLQH� L�YUHPHQD��8�GLQDPLFL�QHVWODþLYRJ�VWUXMDQMD� IOXLGD�JGMH�WHPSHUDWXUD�IOXLGD�QH�LJUD�XORJX�GRYROMQH�VX�WUL�GLPHQ]LMH��GXOMLQH��YUHPHQD�L�PDVH��D�WHN�X�GLQDPLFL�VWODþLYRJ�VWUXMDQMD�WDM�VNXS�VH�SURãLUXMH�GLPHQ]LMRP�WHPSHUDWXUH���2]QDNH��GLPHQ]LMH�L�MHGLQLFH�QHNLK�L]YHGHQLK�IL]LNDOQLK�YHOLþLQD�X�PHKDQLFL�IOXLGD�)L]LNDOQD�YHOLþLQD� 2]QDND� 'LPHQ]LMD� -HGLQLFD�X�6,�VXVWDYX�EU]LQD��EU]LQD�]YXND� Y��F� /7��� P�V�VLOD� )� 0/7��� 1�JUDYLWDFLMD� J� /7��� P�V��WHåLQVNL�SURWRN� G� � 0/7��� 1�V�YROXPHQVNL�PRGXO�HODVWLþQRVWL� .� 0/��7��� 3D�PDVHQL�SURWRN� �m � 07��� NJ�V�PRPHQW�VLOH� 0� 0/�7��� 1P�VQDJD� 3� 0/�7��� :�WODN� S� 0/��7��� 3D�YROXPHQVNL�SURWRN� 4� /�7��� P��V�SOLQVND�NRQVWDQWD� 5� /�7��Θ��� -�NJ.�SRWHQFLMDO�PDVHQH�VLOH� 8� /�7��� P��V��VSHFLILþQD�XQXWUDãQMD�HQHUJLMD� X� /�7��� -�NJ�UDG�VLOH��HQHUJLMD� :��(� 0/�7��� -�

Page 47: 892’ - Fakultet strojarstva i brodogradnje · 0(+$1,.$)/8,’$ 892’ 892’ 0hkdqlndioxlgdmhglril]lnhnrmlvhedyljledqmhpioxlgdlvlodpdnrmhgmhoxmxqdioxlg 0hkdqlndioxlgdvhglmholqdvwdwlnxioxlgdnrmdsurxþdydudyqrwhåxioxlgdxplurydqmx

0(+$1,.$�)/8,'$��–�',0(1=,-6.$�$1$/,=$� ��

JXVWRüD�IOXLGD� ρ� 0/��� NJ�P��NLQHPDWLþND�YLVNR]QRVW� ν� /�7��� P��V�GLQDPLþND�YLVNR]QRVW� µ� 0/��7��� 3DV�.XWQD�EU]LQD� ω� 7��� UDG�V�QDSUH]DQMH� τ�� iT �� ijT � 0/��7��� 1�P��NXW� α� �� UDG�površinska napetost� σ� 07��� 1�P������'LPHQ]LRQR�QH]DYLVDQ�VNXS��7UHED�QDJODVLWL�GD�MH�L]ERU�VNXSD�RVQRYQLK�IL]LNDOQLK�YHOLþLQD�X�SULQFLSX�SURL]YROMDQ��WH�VH�PRåH�NRULVWLWL�ELOR�NRML�VNXS�RG�þHWLUL�GLPHQ]LRQR�QH]DYLVQH�IL]LNDOQH�YHOLþLQH��'LPHQ]LRQD�QH]DYLVQRVW�RVQRYQRJ�VNXSD� IL]LNDOQLK�YHOLþLQD�SRGUD]XPLMHYD�GD� VH�GLPHQ]LMD�QLWL� MHGQH�RG�IL]LNDOQLK�YHOLþLQD�L]DEUDQRJ�VNXSD�QH�PRåH�SULND]DWL�GLPHQ]LMDPD�SUHRVWDOLK�IL]LNDOQLK�YHOLþLQD� X� WRP� VNXSX�� ãWR� MH� VDGUåDQR� X� WHRUHPX� R� GLPHQ]LRQR� QH]DYLVQRP� VNXSX� NRML�JODVL���Ako samo trivijalno rješenje a� D� � ��� DQ ��� þLQL� SURGXNW� SRWHQFLMD� Q Q Q1

a2a

na1 2 n⋅ ⋅ �

EH]GLPHQ]LMVNLP��RQGD�MH�VNXS�Q�IL]LNDOQLK�YHOLþLQD� Q ,Q , .. . ,Q1 2 n �GLPHQ]LRQR�QH]DYLVDQ��$NR� MH� Q!N�� JGMH� MH� N� EURM� RVQRYQLK� GLPHQ]LMD� �PMHUQLK� MHGLQLFD�� X� VNXSX�� WDGD� VNXS� Q�IL]LNDOQLK�YHOLþLQD�QH�PRåH�ELWL�GLPHQ]LRQR�QH]DYLVDQ���3ULPMHUL�����6LOD��PDVD�L�XEU]DQMH�VX�GLPHQ]LRQR�]DYLVQH�YHOLþLQH��MHU�VX�YH]DQH�GUXJLP�1HZWRQRYLP�]DNRQRP�����6NXS�RG�Q ��YHOLþLQH��EU]LQD��XEU]DQMH�L�NXWQD�EU]LQD�þLMH�VX�GLPHQ]LMH�RSLVDQH�V�GYLMH�RVQRYQH�GLPHQ]LMH�GXOMLQH�L�YUHPHQD��N ����]ERJ�Q!N�QH�PRJX�ELWL�GLPHQ]LRQR�QH]DYLVQH�����,VSLWDWL�GLPHQ]LRQX�QH]DYLVQRVW�VNXSD�YHOLþLQD�ρ��Y��/��3UHPD�WHRUHPX�R�GLPHQ]LRQR�QH]DYLVQRP�VNXSX�WUDåL�VH�UMHãHQMH�]D�HNVSRQHQWH�D��E��F��NRML�þLQH�SURGXNW�SRWHQFLMD�YHOLþLQD�EH]GLPHQ]LP��WM���� < > < > < > 0 0 0M L T

a b cv Lρ � �

$NR�VH�GLPHQ]LMH�ρ��Y��/�L]UD]H�SRPRüX�0��/��7��JRUQMD�MHGQDGåED�SUHOD]L�X�REOLN��� < >-3 -1 0 0 0ML LT L M L T

a b c  ¯   ¯ �¡ ° ¡ °¢ ± ¢ ±�

,]MHGQDþDYDQMHP� HNVSRQHQDWD� QDG� LVWLP� ED]DPD� OLMHYH� L� GHVQH� VWUDQH� JRUQMH� MHGQDGåEH��VOLMHGL�VXVWDY�OLQHDUQLK�DOJHEDUVNLK�MHGQDGåEL��

M: =0

L: -3 + + =0

T: - 0

a

a b c

b �

�kojeg je rješenje trivijalno (D E F ����ãWR�]QDþL�GD�MH�VNXS�YHOLþLQD�ρ��Y��/�GLPHQ]LRQDOQR�QH]DYLVDQ�������%DFNLQJKDPRY�WHRUHP��3L�WHRUHP���.OMXþQR�]QDþHQMH�X�GLPHQ]LMVNRM�DQDOL]L�LPD�3L�WHRUHP�NRML�JODVL��6YDNL� IL]LNDOQL� ]DNRQ� L]PH X� Q� IL]LNDOQLK� YHOLþLQD� Q ,Q , .. . ,Q1 2 n �� L]UDåHQ� IXQNFLMRP�*�Q Q Q n1 2, , ..., � ��� QHRYLVQRP� R� SURPMHQL� PMHULOD� �YHOLþLQVND� MHGQDGåED��� PRåH� VH�L]UD]LWL�NDR�IXQNFLMD�Q�N�EH]GLPHQ]LMVNLK�YDULMDEOL�X�REOLNX� � �* Π Π Π1 2, , . . . n- k 0 ��JGMH�MH�N�

Page 48: 892’ - Fakultet strojarstva i brodogradnje · 0(+$1,.$)/8,’$ 892’ 892’ 0hkdqlndioxlgdmhglril]lnhnrmlvhedyljledqmhpioxlgdlvlodpdnrmhgmhoxmxqdioxlg 0hkdqlndioxlgdvhglmholqdvwdwlnxioxlgdnrmdsurxþdydudyqrwhåxioxlgdxplurydqmx

0(+$1,.$�)/8,'$��–�',0(1=,-6.$�$1$/,=$� ��EURM� RVQRYQLK� YHOLþLQD� þLMLP� VH� GLPHQ]LMDPD� PRJX� RSLVDWL� GLPHQ]LMH� þLWDYRJ� VNXSD� Q�IL]LNDOQLK�YHOLþLQD���,OXVWUDFLMD��3URPMHQD�SXWD�SUL�SUDYRFUWQRP�JLEDQMX�NRQVWDQWQLP�XEU]DQMHP� 2

01

2s v t at� � �MH�

IL]LNDOQL�]DNRQ�L]PH X�Q ��IL]LNDOQH�YHOLþLQH�X�þLMLP�VH�GLPHQ]LMDPD�SRMDYOMXMH�VDPR�SXW�L�YULMHPH� �N ���� SD� VH� ]DNRQ� PRåH� SULND]DWL� SRPRüX� GYLMH� EH]GLPHQ]LMVNH� YDULMDEOH��'LMHOMHQMHP�JRUQMH�MHGQDGåEH�V� 0v t �GRELMH�VH�

0 0

11

2

s at

v t v� � �LOL� 1 2

1�2

� � ��JGMH�VX� 10

s

v t� �

L� 20

at

v� ��

�3ULPMHQRP�3L�WHRUHPD�VH�VPDQMXMH�EURM�YDULMDEOL�X�SRMDYL��þLPH�VH�VPDQMXMH�SRWUHEQL�EURM�mjerenja i olakšava analiza rezultata. Pi�WHRUHP�VH�RSüHQLWR�UHDOL]LUD�NUR]�VOMHGHüH�NRUDNH������3UHWSRVWDYOMD� VH� VNXS� Q� IL]LNDOQLK� YHOLþLQD� ]D� NRML� VH� VPDWUD� GD� XSUDYOMD� IL]LNDOQRP�SRMDYRP�� WH� VH� VDVWDYL� WDEOLFD� V� QMLKRYLP� VLPEROLPD� L� GLPHQ]LMDPD� LOL� PMHUQLP�MHGLQLFDPD��L]�NRMH�VH�RGUHGL�EURM�N��GLPHQ]LRQR�QH]DYLVQLK�YHOLþLQD����� ,]�VNXSD�RG�Q� IL]LNDOQLK�YHOLþLQD�L]DEHUH�VH�N�GLPHQ]LRQR�QH]DYLVQLK�YHOLþLQD�L�GRNDåH�GLPHQ]LRQX�QH]DYLVQRVW�L]DEUDQRJ�VNXSD�SUHPD�GDQRP�WHRUHPX�����2G� VYDNH� IL]LNDOQH� YHOLþLQH� L]YDQ� VNXSD� GLPHQ]LRQR� QH]DYLVQLK� YHOLþLQD� IRUPLUD� VH�

EH]GLPHQ]LMVNL� Π� SDUDPHWDU� QD� QDþLQ� GD� VH� QMHQD� GLPHQ]LMD� SULNDåH� GLPHQ]LMDPD�IL]LNDOQLK�YHOLþLQD�L]�GLPHQ]LRQR�QH]DYLVQRJ�VNXSD��X�REOLNX�

� � � + 1 2k1 2 aa a

k i k i kQ Q Q ... Q� � ¸ ¸ ¸ ¸ ������L ���Q�N��1D�WDM�QDþLQ�VNXS�RG�Q�IL]LNDOQLK�YHOLþLQD�]DPLMHQMHQ�MH�VNXSRP�RG�Q�N�EH]GLPHQ]LMVNLK�Π�SDUDPHWDUD��3UL�WRPH�YULMHGH�VOMHGHüD�SUDYLOD��D�� DNR�MH�Q�N�≤����ãWR�]QDþL�GD�VH�QH�PRåH�IRUPLUDWL�QLWL�MHGDQ�Π�SDUDPHWDU��XND]XMH�GD�MH�

VNXS�RG�Q�XWMHFDMQLK�YHOLþLQD�QHSRWSXQ��E��DNR�MH�Q�N ���PRJXüH�MH�VDþLQLWL�VDPR�MHGDQ�Π�SDUDPHWDU��D�SUREOHP�VH�VYRGL�QD�Γ�Π� ��

LOL� Π=konst, što znDþL� GD� MH� SUREOHP� SULQFLSLMHOQR� PRJXüH� ULMHãLWL� VDPR� MHGQLP�PMHUHQMHP��

F�� )XQNFLMD� PH X� EH]GLPHQ]LMVNLP�Π� SDUDPHWULPD�� LGHQWLþQRJ� MH� REOLND� ]D� EHVNRQDþQX�RELWHOM� JHRPHWULMVNL�� NLQHPDWLþNL� L� GLQDPLþNL� VOLþQLK� SRMDYD�� 6OLþQRVW� GYDMX� SRMDYD�SRGUD]XPLMHYD�GD�VH�L]�UH]XOWDWD�GRELYHQLK�QD�MHGQRM�SRMDYL�PRJX�RGUHGLWL�UH]XOWDWL�QD�GUXJRM� SRMDYL� MHGQRVWDYQLP� PQRåHQMHP� UH]XOWDWD� SUYH� SRMDYH� V� NRQVWDQWQLP�NRHILFLMHQWRP� �NRHILFLMHQWRP� VOLþQRVWL��� 3RVHEQR�� RQD� MH� MHGQD� L� LVWD� IXQNFLMD� ]D�PRGHOVNX�L�SURWRWLSQX�SRMDYX��

G��%H]GLPHQ]LMVND� YHOLþLQD� �QSU�� NXW�� YHü� MH� VDPD� SR� VHEL�Π� SDUDPHWDU� L� QH� PRåH� ELWL�XNOMXþHQD�X�VNXS�GLPHQ]LRQR�QH]DYLVQLK�IL]LNDOQLK�YHOLþLQD��H�� 3RVWRML�YLãH�PRJXüQRVWL�L]ERUD�VNXSD�GLPHQ]LRQR�QH]DYLVQLK�YHOLþLQD��D�X�WDM�VNXS�VH�QH�VWDYOMDMX�IL]LNDOQH�YHOLþLQH�þLML�VH�XWMHFDM�åHOL�SURPDWUDWL�L]ROLUDQR��FLOM�MH�GD�VH�SRMDYOMXMH�X�VDPR�MHGQRP�Π�SDUDPHWUX���

I�� 6YDNL�Π�SDUDPHWDU�VH�VPLMH�SRWHQFLUDWL�L�PQRåLWL�SURL]YROMQRP�NRQVWDQWRP��J��8NROLNR�MH�QHND�RG�XWMHFDMQLK�IL]LNDOQLK�YHOLþLQD�LVSXãWHQD�L]�SROD]QRJ�VNXSD��UH]XOWDWL�PMHUHQMD�QHüH�OHåDWL�QD�NULYXOML�QHJR�üH�ELWL�UDVXWL�SR�þLWDYRP�GLMDJUDPX���

Page 49: 892’ - Fakultet strojarstva i brodogradnje · 0(+$1,.$)/8,’$ 892’ 892’ 0hkdqlndioxlgdmhglril]lnhnrmlvhedyljledqmhpioxlgdlvlodpdnrmhgmhoxmxqdioxlg 0hkdqlndioxlgdvhglmholqdvwdwlnxioxlgdnrmdsurxþdydudyqrwhåxioxlgdxplurydqmx

0(+$1,.$�)/8,'$��–�',0(1=,-6.$�$1$/,=$� ��3ULPMHU�� 7UHED� LVWUDåLWL� ]DYLVQRVW� VLOH� RWSRUD� 5� KLGUDXOLþNL� JODWNH� NXJOH� SURPMHUD� '�

SRWRSOMHQH�X� IOXLG� �JXVWRüH�ρ��NRHILFLMHQWD�GLQDPLþNH�YLVNR]QRVWL�µ��NUR]�NRML�VH� WD�NXJOD�JLED�VWDOQRP�EU]LQRP�Y�X�KRUL]RQWDOQRM�UDYQLQL��

3UHWSRVWDYOMD�VH�GDNOH�GD�MH�VLOD�RWSRUD�GHILQLUDQD�QHNRP�IXQNFLMRP� , , , , 0R D vS N �G �PH X�Q ��IL]LNDOQLK�YHOLþLQD���3UYL�NRUDN�MH�IRUPLUDQMH�WDEOLFH�V�GLPHQ]LMDPD�VYLK�IL]LNDOQLK�YHOLþLQD�X�SRMDYL��9HOLþLQD� '� Y� ρ� 5� µ�'LPHQ]LMD� /� /7��� 0/��� 0/7��� 0/��7���,]� WDEOLFH� MH� YLGOMLYR� GD� VH� RG� RVQRYQLK� GLPHQ]LMD� SRMDYOMXMX� 0�� /�� 7�� GDNOH� N=3, što RPRJXüXMH� L]ERU�VNXSD�RG� WUL�GLPHQ]LRQDOQR�QH]DYLVQH�IL]LNDOQH�YHOLþLQH��RGQRVQR�PRJX�VH�IRUPLUDWL�GYD�Π�SDUDPHWUD���'UXJL�NRUDN�MH�L]ERU�VNXSD�GLPHQ]LRQR�QH]DYLVQLK�YHOLþLQD��]D�ãWR�SRVWRML�YLãH�PRJXüQRVWL��$NR�VH�WUDåL�]DYLVQRVW�VLOH�RWSRUD�5��RQD�VH�QHüH�XNOMXþLWL�X�WDM�VNXS��D�RE]LURP�GD�MH�RQD�SRVOMHGLFD� YLVNR]QRVWL� þLML� VH� XWMHFDM� åHOL� SRVHEQR� DQDOL]LUDWL�� NRHILFLMHQW� GLQDPLþNH�YLVNR]QRVWL� WDNR HU� QHüH� XüL� X� WDM� VNXS�� WH� RVWDMH� VNXS� ρ�� Y�� '� þLMD� MH� GLPHQ]LRQD�QH]DYLVQRVW�YHü�GRND]DQD�X�SUHWKRGQRP�SULPMHUX��JGMH�MH�X]HWR�/�XPMHVWR�'���8� WUHüHP� NRUDNX� IRUPLUDMX� VH� EH]GLPHQ]LMVNL�Π� SDUDPHWUL�� MHGDQ� RG� VLOH�)�� D� GUXJL� RG�NRHILFLMHQWD�GLQDPLþNH�YLVNR]QRVWL�µ�X�REOLNX�� a b cR v DS�1 �LOL�SRPRüX�GLPHQ]LMD�� � < >0 0 0 -2 -3 -1M L T MLT ML LT L

a b c  ¯   ¯� ¡ ° ¡ °¢ ± ¢ ±�

1DNRQ� L]MHGQDþDYDQMD� HNVSRQHQDWD� QDG� LVWLP� ED]DPD� QD� OLMHYRM� L� GHVQRM� VWUDQL� JRUQMH�MHGQDGåEH�VOLMHGL�VXVWDY�WUL�OLQHDUQH�DOJHEDUVNH�MHGQDGåEH��

M: 0 1

L: 0 1 -3

T: 0 -2 -

a

a b c

b

� �

� � �

�kojeg je rješenje D ���� E ���� F ���� ãWR� XYUãWHQR� X� GHILQLFLMVNX� MHGQDGåEX� ]D� SDUDPHWDU�

a b cR v DS�1 �GDMH�� �

2 2

R

v DS�1 �

3R]LYDMXüL�VH�QD�SUDYR�PQRåHQMD�Π�SDUDPHWDUD�SURL]YROMQRP�NRQVWDQWRP�SDUDPHWDU�Π��VH�PRåH�SUHXUHGLWL�X�REOLN�NRHILFLMHQWD�VLOH��NRHILFLMHQWD�RWSRUD���

221

2 4

DR

CD

vQ

S� �

JGMH�MH� 21 2 vS �GLQDPLþNL�WODN��D� 2 4D Q površina presjeka kugle suprostavljenog sWUXMDQMX�IOXLGD���$QDORJQR�VH�GHILQLUD�GUXJL�Π�SDUDPHWDU�X�REOLNX�� � a b cv DNS�2 �L]�NRMHJ�VOLMHGL�VXVWDY�WUL�OLQHDUQH�DOJHEDUVNH�MHGQDGåEH���

M: 0 1

L: 0 1 -3

T: 0 -1 -

a

a b c

b

� �

� � �

Page 50: 892’ - Fakultet strojarstva i brodogradnje · 0(+$1,.$)/8,’$ 892’ 892’ 0hkdqlndioxlgdmhglril]lnhnrmlvhedyljledqmhpioxlgdlvlodpdnrmhgmhoxmxqdioxlg 0hkdqlndioxlgdvhglmholqdvwdwlnxioxlgdnrmdsurxþdydudyqrwhåxioxlgdxplurydqmx

0(+$1,.$�)/8,'$��–�',0(1=,-6.$�$1$/,=$� ��

Rješenje kojega je D ����E ����F ���� ãWR�XYUãWHQR�X�GHILQLFLMVNX� MHGQDGåEX� a b cv DNS�2 �GDMH�� �

vD

NS

�2 �NRML�R]QDþXMH�UHFLSURþQX�YULMHGQRVW�5H\QROGVRYD�EURMD�� 1 vDS

N�Re

2= �

�3UHPD� WRPH�� IXQNFLMD�*�PH X�SHW� IL]LNDOQLK�YHOLþLQD��SUHYRGL� VH�X� IXQNFLMX�PH X�GYD�Π�SDUDPHWUD�REOLND��

22

D1

ili2 4

D vDC R v

Q SS

N

� ¬­� ­� � � ­� ­� ­� ®ReΦ Φ �

�-HGQRP�RGUH HQD�EH]GLPHQ]LMVND�IXQNFLMD�Φ�5H��PRåH�SRVOXåLWL�]D�RGUH LYDQMH�VLOH�RWSRUD�5�SUL�JLEDQMX�NXJOH�ELOR�NRMHJ�SURPMHUD��ELOR�NRMRP�EU]LQRP�X�ELOR�NRMHP�IOXLGX��$NR�EL�VH�UDVSRODJDOR� VDPR� MHGQLP� PMHUHQMHP�&'� Φ�5H���� RQR� EL� MRã� XYLMHN� PRJOR� SRVOXåLWL� ]D�RGUH LYDQMH�VLOH�5�X�YHOLNRP�EURMX�VLWXDFLMD�NRMLPD�MH�]DMHGQLþND�YULMHGQRVW�5H\QROGVRYD�EURMD�5H�� L]�NRMH� VOLMHGL� MHGQD� WH� LVWD�YULMHGQRVW�NRHILFLMHQWD� VLOH�&'���GDNOH�SDGDMX�X� LVWX�WRþNX� SURVWRUD� EH]GLPHQ]LMVNLK� YDULMDEOL�� 2SüHQLWR� ]D� SRMDYH� NRMH� VX� RSLVDQH� LVWLP�IL]LNDOQLP�MHGQDGåEDPD�L�NRMH�VX�NDUDNWHUL]LUDQH�LVWRP�WRþNRP�X�SURVWRUX�EH]GLPHQ]LMVNLK�YDULMDEOL� NDåH� VH�GD� VX� IL]LNDOQR� VOLþQH��7DNR�EL� VH�QSU�� L]� VOMHGHüLK�SRGDWDND�PMHUHQLK�X�YRGL�� 999,8S� � NJ�P��� 31,03 10N �� ¸ � 3DV�� 300D � � PP�� 0.142v � � P�V�� 0,3R � � 1�� PRJDR�L]UDþXQDWL� NRHILFLMHQW� RWSRUD�

221

2 4

DR

CD

vQ

S� ������� SUL� 5H\QROGVRYRP� EURMX�

44.135 10vDSN

� � ¸Re �� ,]� WLK� VH�SRGDWDND�PRåH� L]UDþXQDWL� VLOD�QD�NXJOX�SURPMHUD� 1 50D � �PP�� NRMD� VH� JLED� X� XOMX� JXVWRüH� 1 820S � � NJ�P��� YLVNR]QRVWL� 1 0.08N � � 3DV�� EU]LQRP�

11

1 1

80.7 m/svD

NS

� �Re �SUL�NRMRM�üH�VLOD�RWSRUD�ELWL� 2

2 11 1 1

12207 N

2 4 DD

R v CQ

S� � ���%H]GLPHQ]LMVNL�Π�SDUDPHWUL�X�PHKDQLFL�IOXLGD�VH�PRJX�VYUVWDWL�X�QH]DYLVQH�SDUDPHWUH�LOL�NULWHULMH� VOLþQRVWL� NRML� SRWMHþX� RG� QH]DYLVQLK� YHOLþLQD� X� SRMDYL�� WH� ]DYLVQH� SDUDPHWUH� NRML�SRWMHþX�RG�UH]XOWDWD��7DNR�QD�SULPMHU�VLOD�RWSRUD�X�SUHWKRGQRP�SULPMHUX�]DYLVL�RG�YHOLþLQD�V� NRMLPD� VH� IRUPLUD�5H\QROGVRY� EURM� NDR� QH]DYLVQL� SDUDPHWDU�� D� VLOD�RWSRUD� VH�SULND]XMH�SRPRüX�EH]GLPHQ]LMVNRJ�NRHILFLMHQWD�VLOH��NRML�MH�]DYLVQL�SDUDPHWDU��6OMHGHüH�WDEOLFH�GDMX�SUHJOHG� QDMþHãüLK� QH]DYLVQLK� L� ]DYLVQLK� EH]GLPHQ]LMVNLK� SDUDPHWDUD� X� QHVWODþLYRP�VWUXMDQMX�IOXLGD��� 1H]DYLVQL�EH]GLPHQ]LMVNL�SDUDPHWUL��NULWHULML�VOLþQRVWL��

1D]LY� 2]QDND� 'HILQLFLMD�5H\QROGVRY�EURM� ,Re Rn � vLS

N�� vDS

N�� 4 Q

D

SQ N

)URXGHRY�EURM� ,Fr Fn � v

gL�� 2v

gL�

6WURXKDORY�EURM� Sh � L

vt�� D

v

X �

Page 51: 892’ - Fakultet strojarstva i brodogradnje · 0(+$1,.$)/8,’$ 892’ 892’ 0hkdqlndioxlgdmhglril]lnhnrmlvhedyljledqmhpioxlgdlvlodpdnrmhgmhoxmxqdioxlg 0hkdqlndioxlgdvhglmholqdvwdwlnxioxlgdnrmdsurxþdydudyqrwhåxioxlgdxplurydqmx

0(+$1,.$�)/8,'$��–�',0(1=,-6.$�$1$/,=$� ��

(XOHURY�EURM� Eu � 0

21

2

p

vS�

� 1HNL�]DYLVQL�EH]GLPHQ]LMVNL�SDUDPHWUL�1D]LY� 2]QDND� 'HILQLFLMD�

.RHILFLMHQW�WODND��WUHQMD� pC � ,fC CU � 21

2

p p

vS

d� ��21

2v

U

S��

21

2

w

v

U

S�

.RHILFLMHQW�VLOH��RWSRUD��X]JRQD�� , ,F D LC C C � 21

2

F

v AS�� D

21

2

F

v AS�� L

21

2

F

v AS�

.RHILFLMHQW�PRPHQWD� MC � 21

2

M

v ALS�

)DNWRU�WUHQMD�]D�VWUXMDQMH�X�FLMHYLPD� M � 21

2

fp

Lv

DS

��2

2

fh

v L

g D

�'RSULQRV�GLPHQ]LMVNH�DQDOL]H�MH�X�VPDQMHQMX�EURMD�YDULMDEOL�NRMLPD�MH�RSLVDQD�QHND�SRMDYD��D� RQD� QH� PRåH� GDWL� RGJRYRU� R� IXQNFLML� NRMD� SRYH]XMH� EH]GLPHQ]LMVNH� SDUDPHWUH�� 8�SRMHGLQLP� VOXþDMHYLPD� VH� UD]PLãOMDQMHP� PRåH� GRüL� GR� QHNLK� ]DNOMXþDND� R� QHSR]QDWRM�IXQNFLML�� NRMD� SRYH]XMH� EH]GLPHQ]LMVNH� SDUDPHWUH�� 7DNR� EL� VH� X� SUHWKRGQRP� SULPMHUX�PRJOR� SUHWSRVWDYLWL�� GD� VX� SUL� JLEDQMX� NXJOH� PDORP� EU]LQRP� Y� �WRþQLMH� SUL� QLVNLP�YULMHGQRVWLPD�5H\QROGVRYD�EURMD 1Reb ���LQHUFLMVNH�VLOH�]DQHPDULYH�X�RGQRVX�QD�YLVNR]QH�VLOH�� SD� X� WRP� VOXþDMX� VLOD� RWSRUD� QHüH� ]DYLVLWL� RG� JXVWRüH�ρ� IOXLGD� �NRMD� MH� SUHGVWDYQLN�PDVH�� RGQRVQR� LQHUFLMVNLK� VLOD���8]LPDMXüL� X� RE]LU� GD� MH� VLOD� RWSRUD� SUL�PLURYDQMX� NXJOH�MHGQDND�QXOL��]DNOMXþXMH�VH�GD�]DYLVQRVW�NRHILFLMHQWD�RWSRUD�R�5H\QROGVRYX�EURMX�PRUD�ELWL�REOLND� D

konstC �

Re�� RGQRVQR� VLOD� RWSRUD� MH� konstR vDN� ¸ �� 2YR� MH� SRWYU HQR�

HNVSHULPHQWLPD�L�YULMHGL�QH�VDPR�]D�NXJOX�QHJR�]D�RSWMHFDQMH�ELOR�NRMHJ�WLMHOD��YULMHGQRVW�NRQVWDQWH�RYLVL�R�REOLNX�WLMHOD���D�GRQMD�WDEOLFD�GDMH�QHNH�SULPMHUH���

Page 52: 892’ - Fakultet strojarstva i brodogradnje · 0(+$1,.$)/8,’$ 892’ 892’ 0hkdqlndioxlgdmhglril]lnhnrmlvhedyljledqmhpioxlgdlvlodpdnrmhgmhoxmxqdioxlg 0hkdqlndioxlgdvhglmholqdvwdwlnxioxlgdnrmdsurxþdydudyqrwhåxioxlgdxplurydqmx

0(+$1,.$�)/8,'$��–�',0(1=,-6.$�$1$/,=$� ��

��Gornja slika prikazuje u logaritamskom mjerilu zavisnost koeficijenta otpora kugle u širem UDVSRQX�5H\QROGVRYD�EURMD��8RþDYD�VH�SRGUXþMH�5H\QROGVRYD�EURMD�X�NRMHP�MH�NRHILFLMHQW�RWSRUD�SULEOLåQR�NRQVWDQWDQ��L�SRGUXþMH�X�NRMHP�GROD]L�GR�SDGD�NRHILFLMHQWD�RWSRUD�]ERJ�SURPMHQH�VOLNH�VWUXMDQMD��SRPLFDQMD�WRþNH�RGYDMDQMD�SUHPD�VWUDåQMHP�GLMHOX�NXJOH��ãWR�üH�VH�REMDVQLWL�X�0)�,,����

Page 53: 892’ - Fakultet strojarstva i brodogradnje · 0(+$1,.$)/8,’$ 892’ 892’ 0hkdqlndioxlgdmhglril]lnhnrmlvhedyljledqmhpioxlgdlvlodpdnrmhgmhoxmxqdioxlg 0hkdqlndioxlgdvhglmholqdvwdwlnxioxlgdnrmdsurxþdydudyqrwhåxioxlgdxplurydqmx

0(+$1,.$�)/8,'$��–�',0(1=,-6.$�$1$/,=$� ��7LMHOD� GUXJDþLMHJ� REOLND� LPDMX� VOLþQH� IXQNFLRQDOQH� ]DYLVQRVWL� NRHILFLMHQWD� RWSRUD� R�5H\QROGVRYX� EURMX�� 6OLMHGHüD� VOLND� SULND]XMH� GLMDJUDP� NRHILFLMHQWD� RWSRUD� ]D� UD]OLþLWH�SURILOH� �RG� UDYQH� SORþH� GR� NUXåQRJ� FLOLQGUD�� L� WR� SUL� YLãLP� YULMHGQRVWLPD� 5H\QROGVRYD�EURMD�� NRMH� VH� X� SUDNVL� þHãüH� SRMDYOMXMX�� 8� VYLP� VOXþDMHYLPD� VH� PRåH� XRþLWL� SRGUXþMH�NRQVWDQWQH� YULMHGQRVWL� NRHILFLMHQWD� RWSRUD�� D� ãWR� MH� SURILO� WDQML� WR� MH� SRGUXþMH� SRPDNQXWR�prema višim vrijednostima Reynoldsova broja. �

�6OMHGHüH�WDEOLFH�GDMX�YULMHGQRVWL�NRHILFLMHQDWD�RWSRUD�]D�UD]OLþLWH�REOLNH�SURILOD�L�WLMHOD�L�WR�X�SRGUXþMX� 5H\QROGVRYD� EURMD� JGMH� VX� RQH� SULEOLåQR� NRQVWDQWQH�� 3UL� YLãLP� YULMHGQRVWLPD�5H\QROGVRYD� EURMD� VWUXMDQMH� SUHOD]L� L]� UHåLPD� ODPLQDUQRJ� VWUXMDQMD� X� UHåLP� WXUEXOHQWQRJ�VWUXMDQMD��8�UHåLPX�ODPLQDUQRJ�VWUXMDQMD�þHVWLFH�VH�JLEDMX�SUDYLOQR�X�VORMHYLPD��D�SXWDQMH�þHVWLFD�VX�JODWNH�NULYXOMH��7XUEXOHQWQR�VWUXMDQMH�MH�L]UD]LWR�QHVWDFLRQDUQR�VWUXMDQMH�X�NRMHP�VH�þHVWLFH� IOXLGD�JLEDMX�SR�YUOR�QHSUDYLOQLP�SXWDQMDPD�L� LQWHQ]LYQR�VH�PLMHãDMX��D�EU]LQD��WODN� L�RVWDOH�YHOLþLQH�SRND]XMX�VOXþDMQH�SXO]DFLMH��8� WXUEXOHQWQRP�VWUXMDQMX�VLOD�RWSRUD� MH�IXQNFLMD�L�KUDSDYRVWL�SRYUãLQH�SUL�þHPX�V�SRYHüDQMHP�KUDSDYRVWL�X�SUDYLOX�VLOD�RWSRUD�UDVWH��3RYHüDQMHP� KUDSDYRVWL� VH� VPDQMXMH� L� YULMHGQRVW� 5H\QROGVRYD� EURMD� SUL� NRMRM� ODPLQDUQR�VWUXMDQMH�SUHOD]L�X�WXUEXOHQWQR��

Page 54: 892’ - Fakultet strojarstva i brodogradnje · 0(+$1,.$)/8,’$ 892’ 892’ 0hkdqlndioxlgdmhglril]lnhnrmlvhedyljledqmhpioxlgdlvlodpdnrmhgmhoxmxqdioxlg 0hkdqlndioxlgdvhglmholqdvwdwlnxioxlgdnrmdsurxþdydudyqrwhåxioxlgdxplurydqmx

0(+$1,.$�)/8,'$��–�',0(1=,-6.$�$1$/,=$� ��

��

Page 55: 892’ - Fakultet strojarstva i brodogradnje · 0(+$1,.$)/8,’$ 892’ 892’ 0hkdqlndioxlgdmhglril]lnhnrmlvhedyljledqmhpioxlgdlvlodpdnrmhgmhoxmxqdioxlg 0hkdqlndioxlgdvhglmholqdvwdwlnxioxlgdnrmdsurxþdydudyqrwhåxioxlgdxplurydqmx

0(+$1,.$�)/8,'$��–�',0(1=,-6.$�$1$/,=$� ��

Page 56: 892’ - Fakultet strojarstva i brodogradnje · 0(+$1,.$)/8,’$ 892’ 892’ 0hkdqlndioxlgdmhglril]lnhnrmlvhedyljledqmhpioxlgdlvlodpdnrmhgmhoxmxqdioxlg 0hkdqlndioxlgdvhglmholqdvwdwlnxioxlgdnrmdsurxþdydudyqrwhåxioxlgdxplurydqmx

0(+$1,.$�)/8,'$��–�',0(1=,-6.$�$1$/,=$� ��3UL� RSVWUXMDYDQMX�QHVLPHWULþQLK� WLMHOD�� LOL� SUL� RSVWUXMDYDQMX� VLPHWULþQLK� WLMHOD�NRG�NRMLK� VH�SUDYDF�YHNWRUD�EU]LQH�QH�SRNODSD�V�RVL�VLPHWULMH�SRMDYOMXMH�VH�L�VLOD�X�RNRPLWRP�VPMHUX�QD�VPMHU� EU]LQH�� NRMD� VH� QD]LYD� VLORP� X]JRQD� �]D� UD]OLNX� RG� VLOH� RWSRUD� þLML� VH� SUDYDF�GMHORYDQMD� SRNODSD� V� SUDYFHP� EU]LQH��� $YLRQL� OHWH� XSUDYR� ]DKYDOMXMXüL� VLOL� X]JRQD�� 6LOD�X]JRQD�SRYHüDYD�VH�SRYHüDQMHP�NXWD�L]PH X�YHNWRUD�EU]LQH�L�VNHOHWQH�OLQLMH�NULOD��QDSDGQL�NXW�B ���D�VOMHGHüH�VOLNH�SULND]XMX�]DYLVQRVW�VLOH�X]JRQD�L�VLOH�RWSRUD�R�QDSDGQRP�NXWX��3UL�RGUH HQRM� YULMHGQRVWL� NXWD� B � NRHILFLMHQW� X]JRQD� GRåLYOMDYD� PDNVLPXP�� QDNRQ� þHJD� V�SRYHüDQMHP� QDSDGQRJ� NXWD� VLOD� X]JRQD� QDJOR� RSDGD� MHU� VH� QD� JRUQMRM� VWUDQL� SURILOD�SRMDYOMXMH�RGYDMDQMH�VWUXMDQMD��

� �2ED� VH� GLMDJUDPD� PRJX� SULND]DWL� QD� MHGQRM� VOLFL� V� NXWRP� B kao parametrom, kao što SULND]XMH�VOMHGHüD�VOLND�]D�SRGUXþMH�NXWD�B �GR�����2SWLPDOQD�WRþND�MH�RQD�X�NRMRM�MH�RGQRV�VLOH�X]JRQD�VSUDP�VLOH�RWSRUD�PDNVLPDODQ��FUWNDQL�SUDYDF�V�QDMYHüLP�QDJLERP�NRML�WDQJLUD�NULYXOMX���

Page 57: 892’ - Fakultet strojarstva i brodogradnje · 0(+$1,.$)/8,’$ 892’ 892’ 0hkdqlndioxlgdmhglril]lnhnrmlvhedyljledqmhpioxlgdlvlodpdnrmhgmhoxmxqdioxlg 0hkdqlndioxlgdvhglmholqdvwdwlnxioxlgdnrmdsurxþdydudyqrwhåxioxlgdxplurydqmx

0(+$1,.$�)/8,'$��–�35,0-(1$�8�+,'5$8/,ý.,0�6752-(9,0$� ������ 35,0-(1$� 261291,+� =$.21$� ',1$0,.(� )/8,'$� 1$� 6758-$1-(� 8�+,'5$8/,ý.,0�6752-(9,0$������2VQRYQL�]DNRQL�X�NRRUGLQDWQRP�VXVWDYX�NRML�VH�JLED�SUDYRFUWQR�EU]LQRP�X�.RRUGLQDWQL�VXVWDY�NRML�VH�JLED�NRQVWDQWQRP�EU]LQRP�uG ��NRQVWDQWQRP�SR�YHOLþLQL�L�VPMHUX��MH� LQHUFLMVNL� NRRUGLQDWQL� VXVWDY�� 3URPDWUDþ� L]� DSVROXWQR�PLUXMXüHJ� NRRUGLQDWQRJ� VXVWDYD�PMHUL�DSVROXWQX�EU]LQX� vG ��D�SURPDWUDþ�NRML�VH�JLED�]DMHGQR�V�NRRUGLQDWQLP�VXVWDYRP�PMHUL�UHODWLYQX� EU]LQX� w

G �� SUL� þHPX� YULMHGL� v u w= +G G G �� 6YL� ]DNRQL� PHKDQLNH� IOXLGD� X�NRRUGLQDWQRP�VXVWDYX�NRML�VH�JLED�NRQVWDQWQRP�EU]LQRP�VX�LVWRJ�REOLND�NDR�L�]D�DSVROXWQR�PLUXMXüL�NRRUGLQDWQL�VXVWDY�X]�XYMHW�GD�VH�XPMHVWR�DSVROXWQH�EU]LQH�X]PH�UHODWLYQD�EU]LQD���3ULPMHU��VLOD�POD]D�QD�SRPLþQX�ORSDWLFX��

y

x

C

pa

v

A

S

nepomi namlaznica

þ

pomi na lopaticaþ

uO

z=konst.

/RSDWLFD� VH� JLED� NRQVWDQWQRP�EU]LQRP�X, a fluid u mlazu površine $�SRSUHþQRJ� SUHVMHND� VWUXML� EU]LQRP� Y��=D� SURPDWUDþD� L]� NRRUGLQDWQRJ�VXVWDYD� 2[\� NRML� VH� JLED� ]DMHGQR� V�ORSDWLFRP�� IOXLG� QDLOD]L� QD� ORSDWLFX�UHODWLYQRP� EU]LQRP� w v u� � � �VYH� VX�EU]LQH� KRUL]RQWDOQH� SD� YULMHGL�DOJHEDUVNL�]EURM���

Cpa

y

x

AV Sw

w1

wSa

A1

6OLND� OLMHYR� SULND]XMH� NRQWUROQL�YROXPHQ�NRML�REXKYDüD�IOXLG�NRML�MH�X�GRGLUX� V� ORSDWLFRP�� V� XFUWDQLP�UHODWLYQLP� EU]LQDPD�� .RQWUROQD�površina VH� VDVWRML� RG� XOD]QRJ� GLMHOD�$�� L]OD]QRJ� $��� UXED� POD]D� 6D�� WH�površine 6Z� QD�NRMRM� VH�RVWYDUXMH� VLOD�GRGLUD�POD]D�L�ORSDWLFH��%HUQRXOOLMHYD�MHGQDGåED�RG�XOD]QRJ�GR�L]OD]QRJ�SUHVMHND�X]�SUHWSRVWDYNX�GD�MH�ORSDWLFD�X�KRUL]RQWDOQRM�UDYQLQL�JODVL�

22a a 1

2 2

p p wwz z

g g g gS S� � � � � �L]�NRMH�MH�MDVQR�GD�YULMHGL� 1w w� �

-HGQDGåED� NRQWLQXLWHWD� JODVL� � �rel 1 1Q wA w A � L]� NRMH� MH� MDVQR� GD� MH� 1A A� �� 7UHED�QDJODVLWL� GD� MH� relQ � UHODWLYQL� SURWRN� NRMLP� IOXLG� VWUXML� SUHNR� ORSDWLFH� L� GD� MH� WDM�PDQML� RG�DSVROXWQRJ�SURWRND�Q vA� �NRMLP�IOXLG�L]OD]L�L]�POD]QLFH��MHU�VH�UD]PDN�L]PH X�POD]QLFH�L�ORSDWLFH� VWDOQR� SRYHüDYD�� WH� VH� GLR� DSVROXWQRJ� SURWRND� WURãL� QD� SRSXQMDYDQMH� POD]D� X�SURVWRUX�L]PH X�POD]QLFH�L�ORSDWLFH��

Cpa

Sw

Sa

x

A

V

I

I

A

6LOD� IOXLGD� QD� ORSDWLFX� GHILQLUDQD�MH� MHGQDGåERP� NROLþLQH� JLEDQMD��9HOLþLQH� LPSXOVQLK� IXQNFLMD� QD�XOD]QRP� L� L]OD]QRP� SUHVMHNX� VX�MHGQDNH�� WM� YULMHGL�2

relI w A Q wS S� � �� GRN� MH� QD�površini 6D� LPSXOVQD� IXQNFLMD�MHGQDND� QXOL�� 6LOD� IOXLGD� QD�ORSDWLFX� MH� MHGQDND� YHNWRUVNRP�]EURMX�LPSXOVQLK�IXQNFLMD��.RPSRQHQWD�VLOH�X�VPMHUX�RVL�[�GHILQLUDQD�MH�L]UD]RP��

Page 58: 892’ - Fakultet strojarstva i brodogradnje · 0(+$1,.$)/8,’$ 892’ 892’ 0hkdqlndioxlgdmhglril]lnhnrmlvhedyljledqmhpioxlgdlvlodpdnrmhgmhoxmxqdioxlg 0hkdqlndioxlgdvhglmholqdvwdwlnxioxlgdnrmdsurxþdydudyqrwhåxioxlgdxplurydqmx

0(+$1,.$�)/8,'$��–�35,0-(1$�8�+,'5$8/,ý.,0�6752-(9,0$� ��

2rel1 1xF I I cos w A cos wQ cosC S C S C� � � � � � �

.DG� EL� WR� ELOD� MHGLQD� VLOD� QD� ORSDWLFX� X� VPMHUX� RVL� [�� ORSDWLFD� EL� VH� SR� ,,��1HZWRQRYRP�]DNRQX�XErzavala, što je protivno pretpostavci da se lopatica giba konstantnom brzinom X��'DNOH�]DNOMXþXMH�VH�GD�]D�RGUåDYDQMH�NRQVWDQWQH�EU]LQH�QD�ORSDWLFX�PRUD�L]YDQD�GMHORYDWL�VLOD� xF� �NRMD�üH�MX�NRþLWL��XUDYQRWHåLWL�VLOX�IOXLGD�QD�ORSDWLFX���7LP�NRþHQMHP�VH�GRELYD�UDG��ORSDWLFD�GMHOXMH�SRSXW�WXUELQH���D�VQDJD�WRJ�NRþHQMD�MH�GHILQLUDQD�L]UD]RP��

S C� ¸ � �T rel 1 cosxP F u uwQ �-DVQR�MH�GD�üH�VQDJD�ELWL�PDNVLPDOQD�]D� 180C � D �L�MHGQDND�QXOL�]D� 0C � D �

�180C � D �

��IOXLG�MH�QHYLVNR]DQ�SD�QHPD�VLOH�X�[�VPMHUX��WM��VQDJD�MH�MHGQDND�QXOL��0C � D �

3DG�YLVLQH�HQHUJLMH�GHILQLUDQ�MH�L]UD]RP� T

Trel

1 cosP uw

hgQ g

CS

� � � �'R� LVWRJ� VH� UH]XOWDWD� PRJOR� GRüL� SURPDWUDQMHP� SUREOHPD� L]� DSVROXWQRJ� NRRUGLQDWQRJ�VXVWDYD��6OLND�SULND]XMH�NRQWUROQL�YROXPHQ�NRML�PLUXMH��WH�WURNXW�EU]LQD�QD�L]OD]X�NRMLP�VH�GHILQLUD�DSVROXWQD�EU]LQD� 1v

G ���

C

v

wv =u+w1

u

PT �%XGXüL�GD�VH�NUR]�NRQWUROQX�SRYUãLQX�L]PMHQMXMH�VQDJD�V�RNROLQRP��RGYRGL�VH�VQDJD� TP ���VQDJD� QD� L]OD]X� üH� ELWL� PDQMD� RG� VQDJH� QD� XOD]X� ]D� RGYHGHQX� VQDJX�� WM�� SUHPD�%HUQRXOOLMHYRM�MHGQDGåEL��X]LPDMXüL�X�RE]LU�GD�VX�WODN�L�YLVLQD�NRQVWDQWQL��YULMHGL�

2 21

T2 2

v vh

g g� � ���

D�EXGXüL�GD�YULMHGL� 2 2 21 1 1 2 2 cosv v v w w w u u u w wu uC� ¸ � ¸ � ¸ � ¸ � � �

G G G G G G G G �L� v u w� � � što uvršteno

X�%HUQRXOOLMHYX�MHGQDGåEX�GDMH� T 1 cosuw

hg

C� � �������%HUQRXOOLMHYD�MHGQDGåED�]D�URWLUDMXüX�VWUXMQX�FLMHY�����������

3UHWSRVWDYNH�����)OXLG�MH�QHVWODþLY� .constρ = ����)OXLG�MH�QHYLVNR]DQ� 0µ = �� pnσ = −G G ����� .u const= ����� 0q =G ����6WUXMDQMH�MH�VWDFLRQDUQR� 0

t

∂ =∂

�6Z�$��

$��

$�G9 $GV�

w wn= −G G

w wn=G G

2rreρω G �

Usds se=G G

Page 59: 892’ - Fakultet strojarstva i brodogradnje · 0(+$1,.$)/8,’$ 892’ 892’ 0hkdqlndioxlgdmhglril]lnhnrmlvhedyljledqmhpioxlgdlvlodpdnrmhgmhoxmxqdioxlg 0hkdqlndioxlgdvhglmholqdvwdwlnxioxlgdnrmdsurxþdydudyqrwhåxioxlgdxplurydqmx

0(+$1,.$�)/8,'$��–�35,0-(1$�8�+,'5$8/,ý.,0�6752-(9,0$� ����

M M M M M

MM M

M

2

Brzina promjene energije snaga masenih snaga vanjskih brzina dovo HQMD�sila na površinskih topline na

sila na

D Dd d d d d

D D 2V V V S S

VV V

V

we V u V f w V w S q n S

t tρ ρ ρ σ

= + = ⋅ + ⋅ − ⋅

∫ ∫ ∫ ∫ ∫G G G G G G

��� ��� ��� ���

N NM M M M M

M

22

0 snaga vanjskih 0površinskihsila na

D Dd d 2 d d d

D D 2 r

KoriolisV V V S Sgravitacija centrifugaln a

V

wu V V gk re w w V w S q n S

t tρ ρ ρ ω ω σ

= =

+ = − + + × ⋅ + ⋅ − ⋅

∫ ∫ ∫ ∫ ∫G GG G G G G G G

���� ��� ���

,QWHJUDFLMRP�MHGQDGåEH�RþXYDQMD�HQHUJLMH��SR�NRQWUROQRP�YROXPHQX�SUHPD�VOLFL��GRELMH�VH�( )

( ) ( ) ( )

( )( )2 2 2 2 2 2 2 2 12 1

2 1 2 2 1 1

1

2 2 2

1 1d 0 2 2

d 1 1d d d d d

d 2 2

w r r

r

KV KP KV KV KP

Q p pQg z zw Q w QQ ss

w V w w n S gk w V re w V pn w St

ω ωρρ ρρ

ρ ρ ρ ρω

− −− −∂ − −=∂

+ ⋅ = ⋅ + ⋅ + − ⋅

∫ ∫ ∫ ∫ ∫GG G G G G G G

������� ���������� ����� ������

JGMH�VX� 1v �L� 2v �SURVMHþQH�EU]LQH�QD�SUHVMHFLPD� 1A �L� 2A ��D�Q �SURWRN�NUR]�FLMHY��3ULPMHQRP�]DNRQD� RþXYDQMD� HQHUJLMH� ]D� MHGQRGLPHQ]LMVNR� VWUXMDQMH� X� URWLUDMXüRM� FMHYþLFL� L]ERGL� VH�%HUQRXOOLMHYD�MHGQDGåED�]D�URWLUDMXüX�VWUXMQX�FLMHY�

2 2 2 2 2 2

2 1

snaga na izlazu iz cijevi snaga na ulazu u cijev

2 2

w r w rp gz Q p gz Q

ω ωρ ρ ρ ρ − −+ + = + + ����������� �����������

2ERGQD�EU]LQD�X�GHILQLUDQD�MH�L]UD]RP�u rω= ���2 2 2 22 2 2 1 1 1

2 12 2

w u p w u pz z

g g g gS S� �

� � � � � ������(XOHURYD�MHGQDGåED�]D�WXUERVWURMHYH��3URPDWUDþ�L]�NRRUGLQDWQRJ�VXVWDYD�NRML�URWLUD�]DMHGQR�V�URWRURP�PMHUL�UHODWLYQX�EU]LQX� w

G �NRMD�MH�WDQJHQFLMDOQD�QD�ORSDWLFH��2ERGQD�EU]LQD� u rX� q

G GG �MH�QD�XOD]X�X�URWRU�SR�YHOLþLQL�MHGQDND� 1 1u RX� �� D� QD� L]OD]X� 2 2u RX� � L� RNRPLWD� MH� QD� UDGLMXV��$SVROXWQD� EU]LQD� v

G � MH�]EURM�RERGQH�L�UHODWLYQH�EU]LQH� v u w� �

G G G, što se SULND]XMH�WURNXWRP�EU]LQD��.XW� C �MH�NXW�

ORSDWLFH�� D� R]QDþXMH� NXW� L]PH X� YHNWRUD� UHODWLYQH� EU]LQH� wG � �WDQJHQWD� QD� ORSDWLFX�� L�

QHJDWLYQH�RERGQH�EU]LQH� u�G ��6OMHGHüH�VOLND�SULND]XMH�SULPMHU�WURNXWD�EU]LQD��QD�NRMHP�MH�R]QDþHQ�L�NXW�B �GHILQLUDQ�NDR�NXW�L]PH X�RERGQH�L�DSVROXWQH�EU]LQH���

�-DVQR�MH�GD�YULMHGH�UHODFLMH��2 2 2 2 cosv u w uw C� � � �

cosv u wR C� � �n sinv w C� �

$SVROXWQD�EU]LQD�VH�PRåH�UDVWDYLWL�X�UDGLMDOQL�L�RERGQL�VPMHU��JGMH�MH�UDGLMDOQD�NRPSRQHQWD�RNRPLWD�QD�XOD]QL� L� L]OD]QL�SUHVMHN��SD�VH�R]QDþXMH�V� nv ��D�RERGQD�V� vR ��$NR�VH�MHGLQLþQL�

uG

wG

vG

nv

vR

CB

Page 60: 892’ - Fakultet strojarstva i brodogradnje · 0(+$1,.$)/8,’$ 892’ 892’ 0hkdqlndioxlgdmhglril]lnhnrmlvhedyljledqmhpioxlgdlvlodpdnrmhgmhoxmxqdioxlg 0hkdqlndioxlgdvhglmholqdvwdwlnxioxlgdnrmdsurxþdydudyqrwhåxioxlgdxplurydqmx

0(+$1,.$�)/8,'$��–�35,0-(1$�8�+,'5$8/,ý.,0�6752-(9,0$� ��

YHNWRUL� X� UDGLMDOQRP� L� RERGQRP� VPMHUX� R]QDþH� V� reG � L� eR

G � �QHND� XYLMHN� JOHGD� X� VPMHUX�RERGQH�EU]LQH��WDGD�YULMHGL� n rv v e v eR R� �

G G G ��%HUQRXOOLMHYD�MHGQDGåED�GXå�VWUXMQLFH�RG�XOD]D�GR�L]OD]D�L]�SXPSH�NDåH�GD�üH�VH�HQHUJLMD�QD�L]OD]X�SRYHüDWL�]D�YLVLQX�GREDYH�SXPSH��WM��YULMHGL�� 2 2

2 2 1 12 1 P2 2

v p v pz z h

g g g gS S� � � � � � �

$NR�VH�X�%HUQRXOOLMHYRM�MHGQDGåEL�DSVROXWQH�EU]LQH�SULNDåX�SUHNR�RERGQH�L�UHODWLYQH�EU]LQH�� 2 2 2 2 cosv u w uw C� � � ���WH�RG�QMH�RGX]PH�%HUQRXOOLMHYD�MHGQDGåED�]D�URWLUDMXüX�VWUXMQX�FLMHY�

2 2 2 22 2 2 1 1 1

2 12 2

w u p w u pz z

g g g gS S� �

� � � � � �,]YRGL�VH�(XOHURYD�MHGQDGåED�]D�WXUERVWURMHYH�

P 2 2 2 2 1 1 1 1 2 12 1

1 1cos cosh u u w u u w u v u v

g g R RC C  ¯� � � � � �¢ ± ������3ULPMHQD�QD�URWLUDMXüX�FMHYþLFX�2VQRYQD� (XOHURYD� MHGQDGåED� ]D� WXUERVWURMHYH� L� %HUQRXOOLMHYD� MHGQDGåED� ]D� URWLUDMXüX�VWUXMQLFX� VH�PRJX�SULPLMHQLWL� L�QD� VWUXMDQMH�X� URWLUDMXüRM�FMHYþLFL��1DUDYQR� L�GDOMH�YULMHGH�SUHWSRVWDYNH�R�QHYLVNR]QRP�QHVWODþLYRP�VWUXMDQMX��X]�GRGDWQX�SUHWSRVWDYNX�GD�MH�SURPMHU�FMHYþLFH�PDOL� X� RGQRVX� QD� QMHQX� GXOMLQX�� 6YLQXWD� FMHYþLFD�PRåH� UDGLWL� SRSXW� SULPLWLYQH�SXPSH�� NRMD� SRGLåH� IOXLG� QD� YLVLQX� H � �VOLND� OLMHYR�� LOL� SRSXW� WXUELQH�� NRMD� SUHWYDUD�UDVSRORåLYX�YLVLQX� H �X�PHKDQLþNL�UDG��VOLND�GHVQR���$NR�VH�FMHYþLFL�VQDJD�QLWL�GRYRGL�QLWL�RGYRGL�JRYRUL�VH�R�VORERGQRURWLUDMXüRM�FLMHYL��DNR�VH�]DQHPDUH�XþLQFL�WUHQMD�WR�EL�ELR�VOXþDM�SROMHYDþD� WUDYH��� -DVQR� MH� GD� ]D� VOXþDM� SXPSH� FMHYþLFD� SULMH� SRþHWND� URWDFLMH� PRUD� ELWL�LVSXQMHQD�IOXLGRP��LQDþH�VH�VWUXMDQMH�QH�EL�XVSRVWDYLOR���

H

S

pa

pa

R

X=konst.

X

g

w

vu

C=90o

D

1

2

S

X=konst.

X

C

H

Dg

pa

pa

1

vTR

w

v

u

2

3ULPLWLYQD�SXPSD�%HUQRXOOLMHYD�MHGQDGåED�]D�URWLUDMXüX�VWUXMQLFX�� 3ULPLWLYQD�WXUELQD�%HUQRXOOLMHYD�MHGQDGåED�]D�URWLUDMXüX�VWUXMQLFX��

Page 61: 892’ - Fakultet strojarstva i brodogradnje · 0(+$1,.$)/8,’$ 892’ 892’ 0hkdqlndioxlgdmhglril]lnhnrmlvhedyljledqmhpioxlgdlvlodpdnrmhgmhoxmxqdioxlg 0hkdqlndioxlgdvhglmholqdvwdwlnxioxlgdnrmdsurxþdydudyqrwhåxioxlgdxplurydqmx

0(+$1,.$�)/8,'$��–�35,0-(1$�8�+,'5$8/,ý.,0�6752-(9,0$� ��

P2

22

02

u

w RH

g

X�� � �

,]�WURNXWD�EU]LQD�MH�� v u RR X� � ��EU]LQD� vR �MH�XYLMHN�SR]LWLYQD�WM��JOHGD�X�VPMHUX�EU]LQH� u

G � SD� üH�PRPHQW� L� VQDJD� ELWL� SR]LWLYQL��RGQRVQR�UDGL�VH�R�SXPSL���

P2

22

2

u

w RH

g

X�� �

,]�WURNXWD�EU]LQD�MH�� cosv u wR C� � ��DNR�MH�EU]LQD� vR �SR]LWLYQD�UDGL�VH�R�SXPSL��DNR�MH�QHJDWLYQD� UDGL� VH�R� WXUELQL�� D�DNR� MH� MHGQDND�QXOL�R�VORERGQRURWLUDMXüRM�FLMHYL��

-HGQDGåED�NRQWLQXLWHWD�� 2

konst.4

DQ w

Q� � �

9LVLQD�GREDYH�SXPSH�� P

uvh

gR� �NRG�WXUELQH�üH�VH�GRELWL�QHJDWLYQD�YLVLQD�GREDYH��

6QDJD�NRMD�VH�SUHGDMH�IOXLGX�� P PP gQhS� ��]D�WXUELQX�QHJDWLYQR��0RPHQW�VLOH�NRMRP�FMHYþLFD�GMHOXMH�QD�IOXLG�� P

PM

X� �

������3ULPMHQD�QD�KLGUDXOLþNH�VWURMHYH�� 3ULPLWLYQD�WHRULMD�SURSHOHUD��2YD�VH�WHRULMD�WHPHOML�QD�LGHDOL]LUDQRM�VOLFL�VWUXMDQMD�X]�SUHWSRVWDYNX�QHYLVNR]QRJ�VWUXMDQMD�IOXLGD� L� GHILQLUD� VDPR� RNYLUQH� RGQRVH� PH X� LQWHJUDOQLP� YHOLþLQDPD� NDUDNWHULVWLþQLP� ]D�SURSHOHU�� SD� VH� RYRP� WHRULMRP� SURSHOHUL� QH� PRJX� SURMHNWLUDWL�� $QDOL]LUDW� üH� VH� VOXþDM�DYLRQVNRJ� SURSHOHUD� NRML� VH� JLED� NRQVWDQWQRP� EU]LQRP� vd � X� PLUXMXüHP� ]UDNX�� ,]�NRRUGLQDWQRJ� VXVWDYD� YH]DQRJ� QD� SURSHOHU� L]JOHGDW� üH� NDR� GD� IOXLG� QDLOD]L� QD� SURSHOHU�EU]LQRP� vd ��6OMHGHüD�VOLND�VKHPDWVNL�SULND]XMH�SURSHOHU�L�RGDEUDQL�NRQWUROQL�YROXPHQ��

� ��Površina 1A NUR]�NRMX�IOXLG�XOD]L�X�NRQWUROQL�YROXPHQ�MH�GRYROMQR�GDOHNR�LVSUHG�SURSHOHUD��WDNR� GD� MH� SURILO� EU]LQH� MHGQROLN�� D� X� WRP� SUHVMHNX� YODGD� QHSRUHPHüHQL� WODN� pd ��1HSRVUHGQR� LVSUHG� L� QHSRVUHGQR� L]D� SURSHOHUD� �SUHVMHFL� 2A � L� 3A ) površine su jednake

2 3 PA A A� � �� D� SUHPD� MHGQDGåEL� NRQWLQXLWHWD� L� EU]LQH� VX� MHGQDNH� 2 3 Pv v v� � �� D� ]ERJ�VQDJH� NRMX� SURSHOHU� SUHGDMH� IOXLGX� WODN� 3p � L]D� SURSHOHUD� üH� ELWL� YHüL� RG� WODND� 2p � LVSUHG�SURSHOHUD��'RYROMQR�GDOHNR�L]D�SURSHOHUD�WODN�üH�VH�VPDQMLWL�QD�YULMHGQRVW�QHSRUHPHüHQRJ�

Page 62: 892’ - Fakultet strojarstva i brodogradnje · 0(+$1,.$)/8,’$ 892’ 892’ 0hkdqlndioxlgdmhglril]lnhnrmlvhedyljledqmhpioxlgdlvlodpdnrmhgmhoxmxqdioxlg 0hkdqlndioxlgdvhglmholqdvwdwlnxioxlgdnrmdsurxþdydudyqrwhåxioxlgdxplurydqmx

0(+$1,.$�)/8,'$��–�35,0-(1$�8�+,'5$8/,ý.,0�6752-(9,0$� ��

WODND� pd �� D� EU]LQD� üH� QDUDVWL� QD� YULMHGQRVW� 4v v vd� � �� 8� L]OD]QRP� SUHVMHNX� 4A �SUHWSRVWDYOMD� VH� MHGQROLNL� SURILO� EU]LQH�� 3UHRVWDOL� GLR� NRQWUROQH� SRYUãLQH� þLQL� VWUXMQD�SRYUãLQD�NUR]�NRMX�QHPD�SURWRND�L�QD�NRMRM�VH�SUHWSRVWDYOMD�QHSRUHPHüHQL�WODN� pd ��3UHPD�MHGQDGåEL�NRQWLQXLWHWD�SURWRN�Q �NUR]�SURSHOHU�MH�� 1 P P 4 4 4Q v A v A v A v v Ad d� � � � � �.RPSRQHQWD� VLOH� IOXLGD� QD� SURSHOHU� X� SUDYFX� JLEDQMD� SURSHOHUD� MH� 3 2 PF p p A� � � L�GMHOXMH�VXSURWQR�RG�YHNWRUD�EU]LQH� vdG ��3UHPD�MHGQDGåEL�NROLþLQH�JLEDQMD��WD�MH�VLOD�MHGQDND�zbroju impulsnih funkcija na ulaznoj i izlaznoj površini, gdje se impulsne funkcije UDþXQDMX� V� SUHWODNRP� X� RGQRVX� QD� WODN� pd �� WH� YULMHGL�� 2

4 4 4 4I v A QvS S� � � L�2

1 1I v A QvS Sd d� � , dok je impulsna funkcija po plaštu kontrolnog volumena jednaka nuli

�YLGMHWL�VOLNX�JRUH�GHVQR���6LOD�F MH�GDNOH�SR�YHOLþLQL�MHGQDND��� 3 2 P 4F p p A Q v vS d� � � � �%HUQRXOOLMHYH�MHGQDGåEH�L]PH X�SUHVMHND���L����RGQRVQR�SUHVMHND���L���JODVH�� 2 2

2 P

1 1

2 2p v p vS Sd d� � � ���L��� 2 2

3 P 4

1 1

2 2p v p vS Sd� � � �

þLMRP�NRPELQDFLMRP�VH�GRELMH�� 2 2

3 2 4

1

2p p v vS d� � � , što uvršteno u izraz za silu F �GDMH�UHODFLMX�

� 4

2 2P

v v vv vd

d

�� � � �

6QDJD� P � NRMX� SURSHOHU� SUHGDMH� IOXLGX� MH� GHILQLUDQD�%HUQRXOOLMHYRP� MHGQDGåERP� L]PH X�SUHVMHND���L���L]PH X�NRMLK�MH�SURSHOHU��WH�YULMHGL�� 2 2

4

2 2

v v P

g g gQSd� � �

RGDNOH�MH� 2 24 P

1

2P Q v v Qv vS Sd� � � ��

Korisna snaga propelera je ona snaga koja se troši na potisak aviona, a definirana je L]UD]RP�� k 4P Fv Q v v v Qv vS Sd d d d� � � � ��

D�IDNWRU�NRULVQRVWL�SURSHOHUD��

4k

2 2 P4

2 11 2 12 2

Q v v vP v vvP v v vQ v v

v

SI

S

d d d d

dd

d

�� � � � �

�� ��

��3ULPMHU��$YLRQ� OHWL�EU]LQRP� 320vd � NP�K�NUR]�PLUXMXüL�]UDN�JXVWRüH� 1,22S� �NJ�P���SUL�þHPX� NUR]� QMHJRYD� GYD� SURSHOHUD� SURPMHUD� 800D � � PP� SURWMHþH� 100Q � P��V� ]UDND��2GUHGLWH�WHRULMVNL�IDNWRU�NRULVQRVWL�SURSHOHUD��SRWLVQX�VLOX��VNRN�WODND�NUR]�SURSHOHU�L�VQDJX�SRWUHEQX�]D�SRJRQ�SURSHOHUD���Rješenje: Brzina aviona je 88.9vd � �P�V��D�EU]LQD�]UDND�NUR]�SURSHOHU�NUR]�NRML�MH�SURWRN�

/ 2Q �

Page 63: 892’ - Fakultet strojarstva i brodogradnje · 0(+$1,.$)/8,’$ 892’ 892’ 0hkdqlndioxlgdmhglril]lnhnrmlvhedyljledqmhpioxlgdlvlodpdnrmhgmhoxmxqdioxlg 0hkdqlndioxlgdvhglmholqdvwdwlnxioxlgdnrmdsurxþdydudyqrwhåxioxlgdxplurydqmx

0(+$1,.$�)/8,'$��–�35,0-(1$�8�+,'5$8/,ý.,0�6752-(9,0$� ��

� P 2

42 99,5

Q

vD Q

� � �P�V�7HRULMVNL�IDNWRU�NRULVQRVWL�SURSHOHUD�MH�

P

0.894v

vI d� � ���

D�EU]LQD� 4v �L]D�SURSHOHUD�MH� 4 P2 110.1v v vd� � � �P�V���3RWLVQD�VLOD�RG�RED�SURSHOHUD�MH� 4 2,58F Q v vS d� � � �N1��$�VNRN�WODND�NUR]�SURSHOHU� 3 2 2

2 2.57

4

Fp p

D Q� � � N3D��

6QDJD�NRMX�SURSHOHU�SUHGDMH�IOXLGX�MH� 2 24

1256.8

2P Q v vS d� � � �N:��

�� 3ULPMHQD�QD�FHQWULIXJDOQL�VWURM���z

b1

b2

Q

g

rotor

NXüLãWH

lopatica

R2

R 1

lopaticaX=konst.

w2

w1

v2

v1

u2

u1

n2

n1

C2

C1

3UHWSRVWDYNH����� 5RWRU� VH� RNUHüH� NRQVWDQWQRP� NXWQRP�EU]LQRP�X ����� 6WUXMDQMH� L]PH X� ORSDWLFD� MH� X�UDGLMDOQRP� VPMHUX� �YLVLQD� ORSDWLFH� QD�XOD]X�MH 1b ��D�QD�L]OD]X�L]�URWRUD� 2b ������ .RQWUROQL� YROXPHQ� REXKYDüD� SURVWRU�L]PH X� ORSDWLFD�� .RQWUROQD� SRYUãLQD� VH�VDVWRML� RG� XOD]QRJ� GLMHOD� YHOLþLQH�

1 1 12S R bQ� �� L]OD]QRJ� GLMHOD� YHOLþLQH�2 2 22S R bQ� te plašta, kroz kojeg nema

SURWRND����� 1D� URWRUX� VH� SUHWSRVWDYOMD� EHVNRQDþQR�SXQR�EHVNRQDþQR�WDQNLK�ORSDWLFD��ãWR�]QDþL�GD� üH� VH� REOLN� VWUXMQLFD� JOHGDQR� L]�NRRUGLQDWQRJ�VXVWDYD�NRML� URWLUD�]DMHGQR�V�URWRURP�SRNODSDWL�V�REOLNRP�ORSDWLFD��D�GD�üH�VWUXMDQMH�IOXLGD�ELWL�SXQLP�SUHVMHNRP�����6WUXMDQMH�MH�QHYLVNR]QR�L�QHVWODþLYR����8WMHFDM�VLOH�WHåLQH�VH�]DQHPDUXMH�

3URPDWUDþ�L]�NRRUGLQDWQRJ�VXVWDYD�NRML�URWLUD�]DMHGQR�V�URWRURP�PMHUL�UHODWLYQX�EU]LQX� wG �

NRMD�MH�WDQJHQFLMDOQD�QD�ORSDWLFH��2ERGQD�EU]LQD� u rX� qG GG �MH�QD�XOD]X�X�URWRU�SR�YHOLþLQL�

MHGQDND� 1 1u RX� �� D� QD� L]OD]X� 2 2u RX� � L� RNRPLWD� MH� QD� UDGLMXV��$SVROXWQD� EU]LQD� vG � MH�

]EURM�RERGQH�L�UHODWLYQH�EU]LQH� v u w� �G G G

, što se prikazuje trokutom brzina. Kut C �MH�NXW�ORSDWLFH�� D� R]QDþXMH� NXW� L]PH X� YHNWRUD� UHODWLYQH� EU]LQH� w

G � �WDQJHQWD� QD� ORSDWLFX�� L�QHJDWLYQH�RERGQH�EU]LQH� u�

G ��6OMHGHüH�VOLND�SULND]XMH�SULPMHU�WURNXWD�EU]LQD��QD�NRMHP�MH�R]QDþHQ�L�NXW�B �GHILQLUDQ�NDR�NXW�L]PH X�RERGQH�L�DSVROXWQH�EU]LQH���

Page 64: 892’ - Fakultet strojarstva i brodogradnje · 0(+$1,.$)/8,’$ 892’ 892’ 0hkdqlndioxlgdmhglril]lnhnrmlvhedyljledqmhpioxlgdlvlodpdnrmhgmhoxmxqdioxlg 0hkdqlndioxlgdvhglmholqdvwdwlnxioxlgdnrmdsurxþdydudyqrwhåxioxlgdxplurydqmx

0(+$1,.$�)/8,'$��–�35,0-(1$�8�+,'5$8/,ý.,0�6752-(9,0$� ��

�-DVQR�MH�GD�YULMHGH�UHODFLMH��2 2 2 2 cosv u w uw C� � � �

cosv u wR C� � �n sinv w C� �

$SVROXWQD�EU]LQD�VH�PRåH�UDVWDYLWL�X�UDGLMDOQL�L�RERGQL�VPMHU��JGMH�MH�UDGLMDOQD�NRPSRQHQWD�RNRPLWD�QD�XOD]QL� L� L]OD]QL�SUHVMHN��SD�VH�R]QDþXMH�V� nv ��D�RERGQD�V� vR ��$NR�VH�MHGLQLþQL�YHNWRUL� X� UDGLMDOQRP� L� RERGQRP� VPMHUX� R]QDþH� V� re

G � L� eRG � �QHND� XYLMHN� JOHGD� X� VPMHUX�

RERGQH�EU]LQH��WDGD�YULMHGL� n rv v e v eR R� �G G G ��

-HGQDGåED� NRQWLQXLWHWD� ]D� NRQWUROQL� YROXPHQ� NDåH� GD� MH� SURWRN� NUR]� XOD]QX� L� L]OD]QX�površinu jednak�� 1 1 n1 2 2 n22 2Q R b v R b vQ Q� � �3ULPMHQRP� ]DNRQD� PRPHQWD� NROLþLQH� JLEDQMD� ]D� NRPSRQHQWX� PRPHQWD� VLOH� IOXLGD� X�RGQRVX�QD�RV�URWDFLMH��NRMRM�QH�GRSULQRVH�VLOH�WODND��D�YLVNR]QH�VLOH�VX�]DQHPDUHQH��VOLMHGL�izraz za moment kojim fluid djeluje na rotor, što je po definiciji moment turbine TM �� T 2 2 1 1M Q R v R vθ θS�� � �0RPHQW�SXPSH�MH�GDNDNR�VXSURWQRJ�SUHG]QDND��SD�YULMHGL�� P 2 2 1 1M Q R v R vθ θS� � �8RELþDMHQR� MH� XYLMHN� UDGLWL� V� L]UD]LPD� ]D� SXPSX�� D� DNR� VH� GRELMH� QHJDWLYDQ� UH]XOWDW�� WR�XND]XMH�GD�VH�UDGL�R�WXUELQL��6QDJD�SXPSH�MH�GHILQLUDQD�L]UD]RP�� P P 2 2 1 1P M Q u v u vθ θX S� � � �9LVLQD�GREDYH�SXPSH�MH�� P

P 2 2 1 1 2 2 2 2 1 1 1 1

1 1cos cos

Ph u v u v u u w u u w

gQ g gR R C CS

  ¯� � � � � � �¢ ± �što se naziva osnovnom (XOHURYRP�MHGQDGåERP�]D�WXUERVWURMHYH��,]�WH�MH�MHGQDGåEH�MDVQR�GD�üH�YLVLQD�GREDYH�SXPSH�ELWL�PDNVLPDOQD�SUL� 1 0vR � ��ãWR�]QDþL�GD�MH�DSVROXWQD�EU]LQD�QD�XOD]X�X�ORSDWLFH�RNRPLWD�QD�RERGQX�EU]LQX���D�SDG�YLVLQH�HQHUJLMH�X�WXUELQL�SUL� 2 0vR � ���%HUQRXOOLMHYD�MHGQDGåED�GXå�VWUXMQLFH�RG�XOD]D�GR�L]OD]D�L]�SXPSH�NDåH�GD�üH�VH�HQHUJLMD�QD�L]OD]X�SRYHüDWL�]D�YLVLQX�GREDYH�SXPSH��WM��YULMHGL�� 2 2

2 2 1 12 1 P2 2

v p v pz z h

g g g gS S� � � � � � �

$NR�VH�X�%HUQRXOOLMHYRM�MHGQDGåEL�DSVROXWQH�EU]LQH�SULNDåX�SUHNR�RERGQH�L�UHODWLYQH�EU]LQH�� 2 2 2 2 cosv u w uw C� � � ��� WH� XYUVWL� L]UD]� ]D� YLVLQX� GREDYH� SXPSH�� VOLMHGL�%HUQRXOOLMHYD�MHGQDGåED�]D�URWLUDMXüX�VWUXMQLFX�� 2 2 2 2

2 2 2 1 1 12 12 2

w u p w u pz z

g g g gS S� �

� � � � � ��� 3ULPMHQD�QD�3HOWRQ�WXUELQX�� � �

uG

wG

vG

nv

vR

CB

Page 65: 892’ - Fakultet strojarstva i brodogradnje · 0(+$1,.$)/8,’$ 892’ 892’ 0hkdqlndioxlgdmhglril]lnhnrmlvhedyljledqmhpioxlgdlvlodpdnrmhgmhoxmxqdioxlg 0hkdqlndioxlgdvhglmholqdvwdwlnxioxlgdnrmdsurxþdydudyqrwhåxioxlgdxplurydqmx

0(+$1,.$�)/8,'$��–�35,0-(1$�8�+,'5$8/,ý.,0�6752-(9,0$� ��

u= RX

R

vA

w

� ��

w

w

w=v-u

Cu

� �.DGD�VH�XVSRUHGL�VOXþDM�ORSDWLFH�3HOWRQ�WXUELQH��NRMD�VH�JLED�RERGQRP�EU]LQRP� u RX� �V�SRPLþQRP�ORSDWLFRP�X�SUHWKRGQRP�SULPMHUX��MDVQR�MH�GD�üH�WURNXW�EU]LQD�QD�L]OD]X�POD]D�V�ORSDWLFH�ELWL� LVWL�� SD� üH� L� L]UD]� ]D�SDG�YLVLQH� HQHUJLMH�ELWL� LVWL��0H XWLP�ELWQD� MH� UD]OLND�X�WRPH�ãWR�MH�UD]PDN�L]PH X�PLUXMXüH�POD]QLFH�L�SRNUHWQLK�ORSDWLFD�VWDOQR�MHGDQ�WH�LVWL��SD�VH�DSVROXWQL� SURWRN� QH� JXEL� QD� SRSXQMDYDQMH� SURVWRUD� L]PH X� POD]D� L� ORSDWLFD�� QHJR� VDY�SURWRN� IOXLGD�SULMH H�SUHNR� ORSDWLFD� �SUL� þHPX�POD]�PRåH�ELWL�X zahvatu s više lopatica), WDNR�GD�MH�VQDJD�WXUELQH�MHGQDND� T T 1 cosP gQh QuwS S C� � � ��8]LPDMXüL�X�RE]LU�GD�QD�3HOWRQ�WXUELQX�PRåH�GMHORYDWL�YLãH�POD]RYD��QSU��Q�POD]RYD���WH�GD�MH� w v u v RX� � � � ��WHRULMVNL�L]UD]�]D�VQDJX�3HOWRQ�WXUELQH�MH�� N N T 1 cos

Q w

P n u vA v uS C� ¬­�� � �­� ­� ­�� ®

�0DNVLPDOQD�VQDJD� T maxP � WXUELQH� MH�SUL� 180C � D � L�SUL�RERGQRM�EU]LQL�

2

vu � ��SUL�þHPX�MH�

3T max

12

P v AS� �� ãWR� RGJRYDUD� UDVSRORåLYRM� VQD]L� POD]D�� WDNR� GD� MH� WHRULMVNL� IDNWRU�NRULVQRVWL� MHGQDN� MHGLQLFL��8� WRP�EL� VOXþDMX� DSVROXWQD�EU]LQD�POD]D�QD� L]ODVNX� V� ORSDWLFH�ELOD�MHGQDND�QXOL��GDNOH�VYD�VQDJD�POD]D�EL�ELOD�SUHWYRUHQD�X�VQDJX�WXUELQH����