research article unsteady hydromagnetic rotating flow...

Hindawi Publishing CorporationMathematical Problems in EngineeringVolume 2013 Article ID 508431 8 pageshttpdxdoiorg1011552013508431

Research ArticleUnsteady Hydromagnetic Rotating Flow through an OscillatingPorous Plate Embedded in a Porous Medium

I Khan12 A Khan1 A Farhad1 M Qasim3 and S Sharidan1

1 Department of Mathematical Sciences Faculty of Science Universiti Teknologi Malaysia 81310 Skudai Malaysia2 College of Engineering Majmaah University PO Box 66 Majmaah 11952 Saudi Arabia3 Department of Mathematics COMSATS Institute of Information Technology Park Road Chak Shahzad Islamabad 44000 Pakistan

Correspondence should be addressed to S Sharidan sharidanutmmy

Received 23 July 2013 Revised 2 October 2013 Accepted 9 October 2013

Academic Editor Tirivanhu Chinyoka

Copyright copy 2013 I Khan et al This is an open access article distributed under the Creative Commons Attribution License whichpermits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

This paper investigates unsteady hydromagnetic flow of a viscous fluid in a rotating frame The fluid is bounded by an oscillatingporous plate embedded in a porous mediumThe Laplace transform and Fourier sine transform methods are employed to find theexact solutionsThey satisfy all imposed initial and boundary conditions and as special cases are reduced to some published resultsfrom the literature The graphical results are plotted for different values of pertinent parameters and some interesting conclusionsare made

1 Introduction

Stokesrsquo problem for the flow of an incompressible viscousfluid over an oscillating plane serves as a benchmark in theliterature of fluid dynamics [1] It admits an exact analyticalsolution The Stokesrsquo problem not only is of fundamentaltheoretical interest but also occurs in many applied problems[2]The transient solutions in terms of tabulated functions forthe flowof viscous fluid due to oscillating boundary have beenexpressed by Panton [3] Later in 2000 Erdogan [4] studiedthe unsteady flow of Newtonian fluid past an oscillating hor-izontal plane wall and obtained the exact solutions Fetecauet al [5] presented new exact solutions corresponding toStokesrsquo second problem for both small and large timesErdogan and Imrak [6] made comparative study for the solu-tion of Stokesrsquo second problem by employing two differenttransform methods They expressed transient solutions interms of tabulated functions

On the other hand flow through porous media is veryprevalent in nature and therefore has become a principalinterest of researchers in many scientific and engineeringstudies For example one can refer to the books of Pop andIngham [7] Ingham et al [8] Ingham and Pop [9] Vafai [10]and Nield and Bejan [11] for the detailed literature on thistopic Moreover the rotating flow through porous media has

been the subject ofmany studies in the last few decades due totheir wide range of applications in cosmological and geophys-ical fluid dynamics astrophysics meteorology petroleumand hydrology to study the movement of underground water(Jana et al [12]) According to Hayat et al [13] the analysisof the effects of rotation and magnetic field in fluid flows hasbeen an active area of research because of its geophysical andtechnological applications such as theMHD power generatorand boundary layer flow control Based on this motivationDas et al studied in [14] the simultaneous effects of rotationandmagnetic field on the flowof a second grade fluid betweentwo parallel plates Bearing in mind the importance of MHDand porosity in a rotating fluid Hayat et al [15 16] Abelmanet al [17] Sahoo et al [18] Seth et al [19] and Farhad et al[20] studied the hydromagnetic flow of rotating fluids inporous media Recently Jana et al [21] studied the unsteadyflow of viscous fluid through a porous medium bounded by aporous plate in a rotating system

In order to further discuss the work of Jana et al [21]and to make closer relations between this study and practicalengineering we study in this work hydromagnetic rotatingflow of a viscous fluid bounded by an oscillating porous plateembedded in a porousmediumThe governingmathematicalproblem is solved by using the Laplace transform and Fouriersine transformmethodsThe expressions for velocity and skin

2 Mathematical Problems in Engineering

friction in case of cosine and sine oscillations of the plate areobtainedThe analytical results are plotted and discussedTheresults of Jana et al [21] are recovered as a special case fromour obtained solutions The present study is useful in astron-omy to study the rotating motion of astrophysical objectssuch as magnetic stars

2 Mathematical Formulation of the Problem

Consider the unsteady flow of an incompressible viscousfluid occupying the upper porous half-space of the (119909 119910)plane The fluid is bounded by a porous plate at 119911 = 0 suchthat the positive 119911-axis is taken normal to the plate and the 119909-axis is taken parallel to the plate We consider the hydromag-netic flow induced in the fluid in the presence of a uniformmagnetic field of strength 119861

0applied in a direction normal to

the plate by means of the plate oscillations in its plane (Hayatet al [13]) The electric field due to polarization of charges isneglected Both of the fluid and plate are in a state of rigidbody rotation with constant angular velocity Ω = Ωk k isa unit vector in the 119911-directionThe geometry of the problemis shown in Figure 1 Initially for 119905 le 0 both of the fluid andplate are at rest At time 119905 = 0+ the lower plate suddenlystarts to move in its own plane with oscillating velocity inthe flow direction Under these assumptions the equations ofmomentum in a rotating frame along the 119909- and 119910-directionsare [18ndash20]

120597119906

120597119905minus 1199080

120597119906

120597119911minus 2ΩV = ]

1205972119906

1205971199112minus]120601119896119906 minus1205901198612

0

120588119906

119911 119905 gt 0

120597V120597119905minus 1199080

120597V120597119911+ 2Ω119906 = ]

1205972V1205971199112minus]120601119896V minus1205901198612

0

120588V

119911 119905 gt 0

(1)

where 119906 and V are the velocity components in 119909- and 119910-directions 119908

0is the suction velocity at the plate ] is the kine-

matic viscosity 119905 is the time 120590 is the electrical conductivity1198610is the applied magnetic field 120588 is the fluid density 120601 is the

porosity and 119896 is the permeability of the porous mediumThe corresponding initial and boundary conditions are

119906 = V = 0 119905 le 0 (2)

119906 = 119880119867 (119905) cos (1205960119905) or 119880 sin (120596

0119905)

V = 0 at 119911 = 0 119905 gt 0(3)

119906 997888rarr 0 V 997888rarr 0 as 119911 997888rarr infin 119905 gt 0 (4)

in which 119880 is the constant plate velocity 1205960is the frequency

of oscillation of the plate and119867(119905) is the Heaviside functionDefining119865 = 119906+119894V the above systemof equations reduces

to

120597119865

120597119905minus 1199080

120597119865

120597119911+ 2119894Ω119865 = ]

1205972119865

1205971199112minus]120601119896119865 minus1205901198612

0

120588119865

119911 119905 gt 0

Ox

B0

Magnetic fieldΩ

z

Porous medium z gt 0

y

u(0 t) = UH(t) cos(1205960t)

u(0 t) = Usin(1205960t)

(0 t) = 0 t gt 0

Figure 1 Geometry of the problem

119865 (119911 0) = 0 119905 le 0

119865 (0 119905) = 119880119867 (119905) cos (1205960119905) or 119880 sin (1205960119905) 119905 gt 0

119865 997888rarr 0 as 119911 997888rarr infin 119905 gt 0(5)

Introducing the following dimensionless variables (see Janaet al [21])

119866 =119865

119880 120585 =

119911119880

] 120591 =

1198802119905

] 120596 =

1205960]1198802 (6)

the dimensionless problem takes the following form

120597119866

120597120591minus 119878120597119866

120597120585+ 2119894120578119866 =

1205972119866

1205971205852minus1198722119866 minus

1

119870119866 (7)

119866 (120585 0) = 0 120591 le 0 (8)

119866 (0 120591) = 119867 (120591) cos (120596120591) or sin (120596120591) 120591 gt 0 (9)

119866 (infin 120591) = 0 120591 gt 0 (10)

1198722=12059011986120]

1205881198802

1

119870=

]21206011198961198802 120578 =

Ω]1198802 119878 =

1199080

119880

(11)

where 119872 119870 120578 and 119878 are the magnetic permeability rota-tion and suction parameters respectively

3 Solution of the Problem byUsing Laplace Transform

Taking the Laplace transform of (7) and using the initialcondition (8) we get the following transformed differentialequation

1198892119866 (120585 119902)

1198891205852+ 119878119889119866 (120585 119902)

119889120585

minus [119902 +1198722+1

119870+ 2119894120578]119866 (120585 119902) = 0

(12)

Mathematical Problems in Engineering 3

where 119902 is the transform parameter In view of the boundaryconditions (9) and (10) the Laplace transform 119866(120585 119902) of119866(120585 120591) has to satisfy the following conditions

119866 (0 119902) =119902

1199022 + 1205962or 119866 (0 119902) =

120596

1199022 + 1205962 (13)

119866 (0 119902) = 0 (14)

Solution of (12) under the boundary conditions (13) and (14)yields

119866119888(120585 119902) =

1

2119890minus12058511987821

119902 + 119894120596119890minus120585radic119902+119886

+1

119902 minus 119894120596119890minus120585radic119902+119886

(15)

respectively

119866119904(120585 119902) =

1

2119894119890minus1205851198782minus

1

119902 + 119894120596119890minus120585radic119902+119886

+1


(16)

where the subscripts 119888 and 119904 denote the cosine and sine oscil-lations of the plate and

119886 =1198782

4+1

119870+1198722+ 2119894120578 (17)

Now in order to find the inverse Laplace transform of (15)and (16) we use formula (A4) from the Appendix of Farhadet al [20] hence we get the following solutions

119866119888(120585 120591)

=119867 (120591) 119890

minus1205851198782

4

times [119890minus119894120596120591119890minus120585radic119886minus119894120596 erf 119888 ( 120585

2radic120591minus radic(119886 minus 119894120596) 120591)

+ 119890120585radic119886minus119894120596 erf 119888 ( 120585

2radic120591+ radic(119886 minus 119894120596) 120591)]

+119867 (120591) 119890

minus1205851198782

4

times [119890119894120596120591119890minus120585radic119886+119894120596 erf 119888 ( 120585

2radic120591minus radic(119886 + 119894120596) 120591)

+ 119890minus120585radic119886+119894120596 erf 119888 ( 120585

2radic120591+ radic(119886 + 119894120596) 120591)]

(18)

respectively

119866119904(120585 120591)

=119890minus1205851198782

4119894

times [minus119890minus119894120596120591119890minus120585radic119886minus119894120596 erf 119888 ( 120585


+ 119890120585radic119886minus119894120596 erf 119888 ( 120585


+119890minus1205851198782

4119894[119890119894120596120591119890minus120585radic119886+119894120596 erf 119888 ( 120585


+ 119890120585radic119886+119894120596 erf 119888 ( 120585

2radic120591+radic(119886+ 119894120596) 120591)]

(19)

4 Solution of the Problem byUsing Fourier Sine Transform

Now applying Fourier sine transform to (12) and usingboundary conditions (13) and (14) we obtain

119866119888(120577 119902) =

radic2 (120577 + 120577119878) 119902

(1205772 + 1205772119878 + 119902 + 119886) (1199022 + 1205962) (20)

respectively

119866119904(120577 119902) =

radic2 (120577 + 120577119878) 120596

(1205772 + 1205772119878 + 119902 + 119886) (1199022 + 1205962) (21)

where 119866(120577 119902) denotes Fourier sine transform of 119866(120585 119902)Now we take the inverse Fourier sine transform of (20)

which yields

119866119888(120585 119902) =

2radic2119902

120587 (1199022 + 1205962)intinfin

0

(120577 + 120577119878)

(1205772 + 1205772119878 + 119902 + 119886)sin (120577120585) 119889120577

(22)

Taking the inverse Laplace transform of (22) we obtain

119866119888(120585 120591) =

2

120587intinfin

0

120577 (1205772 + 1205772119878 + 119886) 119890minus(1205772+1205772119878+119886)120591

((1205772 + 1205772119878 + 119886)2+ 1205962)

times sin (120577120585) 119889120577

minus2

120587intinfin

0

120577119878 (1205772 + 1205772119878 + 119886) 119890minus(1205772+1205772119878+119886)120591

((1205772 + 1205772119878 + 119886)2+ 1205962)

times sin (120577120585) 119889120577


1

08

06

04

02

01080604020 12

G

120585

M = 0

M = 3M = 6

M = 9

120591 = 09 K = 08 120578 = 03 S = 2 m = 05 120596 = 02 120596120591 = 0

(a)

01

008

006

004

002

01080604020 12

120585

M = 0

M = 3M = 6

M = 9

120591 = 09 K = 08 120578 = 03 S = 2 m = 05 120596 = 02 120596120591 = 0

minusiG

(b)

Figure 2 Profiles of velocity for different values of119872

+2

120587intinfin

0

120577 (1205772 + 1205772119878 + 119886)

((1205772 + 1205772119878 + 119886)2+ 1205962)

times cos (120596120591) sin (120577120585) 119889120577

+2

120587intinfin

0

120577120596

((1205772 + 1205772119878 + 119886)2+ 1205962)

times sin (120596120591) sin (120577120585) 119889120577

+2

120587intinfin

0

120577119878 (1205772+ 1205772119878 + 119886)

((1205772 + 1205772119878 + 119886)2+ 1205962)

times cos (120596120591) sin (120577120585) 119889120577

+2

120587intinfin

0

120577119878120596

((1205772 + 1205772119878 + 119886)2+ 1205962)

times sin (120596120591) sin (120577120585) 119889120577(23)

Similarly for the sine oscillation of the boundary we get from(21) the following

119866119904 (120585 120591) =

2

120587intinfin

0

120577120596119890minus(1205772+1205772119878+119886)120591

((1205772 + 1205772119878 + 119886)2+ 1205962)

sin (120577120585) 119889120577

minus2

120587intinfin

0

120577120596

((1205772+1205772119878+119886)2+ 1205962)

cos (120596120591) sin (120577120585) 119889120577

+2

120587intinfin

0

120577 (1205772 + 1205772119878 + 119886)120596

((1205772+1205772119878+119886)2+1205962)

sin (120596120591) sin (120577120585) 119889120577

+2

120587intinfin

0

120577119878120596119890minus(1205772+1205772119878+119886)120591

((1205772 + 1205772119878 + 119886)2+ 1205962)

sin (120577120585) 119889120577

minus2

120587intinfin

0

120577119878120596

((1205772+1205772119878+119886)2+1205962)

cos (120596120591) sin (120577120585) 119889120577

+2

120587intinfin

0

120577119878 (1205772 + 1205772119878 + 119886)

((1205772+1205772119878+119886)2+1205962)

sin (120596120591) sin (120577120585) 119889120577

(24)

The nondimensional shear stresses 120591119888and 120591119904at the plate 120585 =

0 due to cosine and sine oscillations of the plate are obtainedfrom (18) and (19) as follows

120591119888= minus (120591

119909+ 119894120591119910)

= minus119867 (120591) 119890

minus119886119905minus119894120596120591

4radic120587120591

times [(minus4119890119894120596120591minus 119878radic120587120591119890

119886120591minus 119878radic120587120591119890

119886120591+2119894120596120591

minus 2119890119886120591radic(119886 minus 119894120596) 120587120591 minus 2119890

119886120591+2119894120596120591radic(119886 + 119894120596) 120587120591

+ 2119890119886120591radic(119886 minus 119894120596) 120587120591 erf 119888 (radic(119886 minus 119894120596) 120591)

+ 2radic(119886 + 119894120596) 120587120591119890119886120591+2119894120596120591 erf 119888 (radic(119886 + 119894120596) 120591))]


1

08

06

04

02

0151050 2

G

120585

K = 02

K = 04

K = 06

K = 08

120591 = 09M = 02 120578 = 03 S = 2 m = 05 120596 = 02 120596120591 = 0

(a)

15 251050 32120585

K = 02

K = 04

K = 06

K = 08

120591 = 09M = 02 120578 = 03 S = 2 m = 05 120596 = 02 120596120591 = 0

0035

003

0025

002

0015

001

0005

0

minusiG

(b)

Figure 3 Profiles of velocity for different values of 119870

120591119904= minus (120591

119909+ 119894120591119910)

=119894119890minus119886119905minus119894120596120591

4radic120587120591


119886120591minus 119878radic120587120591119890

119886120591+2119894120596120591


119886120591+2119894120596120591radic(119886 + 119894120596) 120587120591


+ 2radic(119886 + 119894120596) 120587120591119890119886120591+2119894120596120591 erf 119888 (radic(119886 + 119894120596) 120591))]

(25)

5 Special Cases

In order to verify the correctness of our obtained solutionsit is important to note that (18) and (19) satisfy the governingequation and the imposed initial and boundary conditionsFurther these solutions are more general comparing to theexisting solutions in the literature In this section we derivesome special cases of these solutions The solution given by(18) for hydrodynamic fluid (119872 = 0) over an impulsivelymoved plate (120596 = 0) reduces to the following form

119866 (120585 120591) =119867 (120591) 119890

minus1205851198782

2

times[[[

[

119890minus120585radic119878

24+1119870+2119894120578

times erf 119888( 120585

2radic120591minus radic(

1198782

4+1

119870+ 2119894120578) 120591)

+ 119890120585radic11987824+1119870+2119894120578

times erf 119888( 120585

2radic120591+ radic(

1198782

4+1

119870+ 2119894120578) 120591)

]]]

]

(26)

which is identical to the solution given by (13) obtained byJana et al [21] Further the present solution given by (18) forsuddenly moved plate (120596 = 0) reduces to the solution (19)obtained by Farhad et al [20] in the absence of Hall currentand slip boundary condition Hence this provides a usefulmathematical check to our calculi

6 Results and Discussion

The exact solutions for the unsteady hydromagnetic flow ofviscous fluid bounded by a porous plate in a porous mediumare obtained The analytical results are displayed for variousvalues of emerging parameters such as Hartmann number119872 permeability parameter 119870 rotation parameter 120578 suctionparameter 119878 and phase angle 120596120591 Figures 2ndash6 are plotted forthe cosine velocity given by (18) In these figures (a) and (b)show the real and imaginary parts of velocity

Figure 2 is prepared to show the variation of velocity fordifferent values of Hartmann number119872 It is found that thereal part of velocity and boundary layer thickness decreaseswith increasing values of Hartmann number It is due to thefact that the application of transverse magnetic field results aresistive type force (called Lorentz force) similar to drag forceand upon increasing the values of119872 the drag force increaseswhich leads to the deceleration of the flow However it isobserved that this behavior is quite opposite for the imaginarypart of velocity Figure 3 reveals that the presence of porousmedium increases the resistance to flow thus reducing its


0075

01

005

0025

0

151050 2

G

120585

120591 = 09 K = 08M = 03 S = 2 m = 05 120596 = 02 120596120591 = 0

120578 = 0

120578 = 2

120578 = 4

120578 = 6

minus0025

minus005

minus0075

(a)

151050 2120585

120591 = 09 K = 08M = 03 S = 2 m = 05 120596 = 02 120596120591 = 0

120578 = 0

120578 = 2

120578 = 4

120578 = 6

03

025

02

015

01

005

0

minusiG

(b)


08

08

06

06

04

04

02

02

1

1 120

0

G

120585

S = 00

S = 03

S = 06

S = 09

120591 = 09 K = 08 120578 = 03M = 2 m = 05 120596 = 02 120596120591 = 0

(a)

05 151 20

0120585

120591 = 09 K = 08 120578 = 03M = 2 m = 05 120596 = 02 120596120591 = 0

S = 00

S = 03

S = 06

S = 09

006

005

004

003

002

001

minusiG

(b)


velocity Itmay also be expected due to the fact that increasingvalues of119870 reduces the friction forces which assists the fluidconsiderably to move fast and increases the boundary layerthicknessThe graphs for the rotation parameter are plotted inFigure 4 It is observed that the real part of velocity increaseswhen 120578 is increased However when observed carefully it isfound from Figure 4(b) that the behavior of imaginary part ofvelocity is more oscillatory in nature The velocity increasesfirst and then decreases when 120578 is increased

Figure 5 shows the effect of suction parameter 119878 on theflow through a porous medium in a rotating frame Clearlythe absolute value of velocity and boundary layer thicknessfor both real and imaginary parts of velocity decrease with

an increase in suction parameterThe variation of velocity fordifferent values of phase angle 120596120591 is shown in Figure 6 It isfound that the real part of velocity decreases with increasingphase angle 120596120591 Two different values of the phase angle 120596120591 =0120587 and 1205872 are chosen It is interesting to note that when120596120591 = 0120587 the real part of velocity is 1 which corresponds tothe impulsivemotion of the plateMoreover for120596120591 = 1205872 thereal part of velocity is 0 The absolute value of the imaginarypart of velocity is increasing with increasing phase angle 120596120591Furthermore for large values of 120585 that is 120585 rarr infin both ofthe real and imaginary parts of velocity are approaching zeroThe graphical behavior in this figure is in accordance withthe imposed boundary conditions (3) and (4) Moreover it


08

08

06

06

04

04

02

02

1

10

0

G

120585

120591 = 09 K = 08 120578 = 03 S = 2 m = 05 120596 = 02M = 0

120596120591 = 0120587

120596120591 =120587

2

(a)

107505 17515025 1250120585

120591 = 09 K = 08 120578 = 03 S = 2 m = 05 120596 = 02M = 0

120596120591 = 0120587

120596120591 =120587

2

0

minusiG

005

004

003

002

001

(b)


is interesting to note from Figure 6 that for a hydrodynamicfluid and impulsive motion of the plate that is when119872 = 0and 120596 = 0 the graph for velocity matches with that of Janaet al [21] which ensures the accuracy of the obtainedanalytical results

7 Conclusions

The unsteady hydromagnetic rotating flow of viscous fluidbounded by an oscillating porous plate embedded in a porousmedium is studied The Laplace transform and Fourier sinetransform methods are used for finding the solutions of theproblem The analytical results for nondimensional velocityand skin friction are obtained The graphical results are

plottedThe results show that velocity increases with increas-ing rotation parameter and permeability parameter whereasit decreases with increasing Hartmann number suctionparameter and phase angle Moreover as the permeabilityof the medium increases the velocity field increases in theboundary layer Thus we can control the velocity field byintroducing porous medium in a rotating system

References

[1] H Schlichting Grenzchicht Theorie Braun Karlsruhe Ger-many 8th ed edition 1982

[2] N Tokuda ldquoOn the impulsive motion of a flat plate in a viscousfluidrdquo Journal of Fluid Mechanics vol 33 no 4 pp 657ndash6721968

[3] R Panton ldquoThe transient solution for Stokesrsquos oscillating planea solution in terms of tabulated functionsrdquo Journal of FluidMechanics vol 31 no 4 pp 819ndash825 1968

[4] M E Erdogan ldquoNote on an unsteady flow of a viscous fluid dueto an oscillating plane wallrdquo International Journal of Non-LinearMechanics vol 35 no 1 pp 1ndash6 2000

[5] C Fetecau D Vieru and C Fetecau ldquoA note on the secondproblem of Stokes for Newtonian fluidsrdquo International Journalof Non-Linear Mechanics vol 43 no 5 pp 451ndash457 2008

[6] M E Erdogan andC E Imrak ldquoOn the comparison of the solu-tions obtained by using two different transformmethods for thesecond problem of Stokes for Newtonian fluidsrdquo InternationalJournal of Non-Linear Mechanics vol 44 no 1 pp 27ndash30 2009

[7] I Pop and D B Ingham Convective Heat Transfer Mathemat-ical and Computational Modeling of Viscous Fluids and PorousMedia Pergamon Oxford UK 2001

[8] D B Ingham A Bejan E Mamut and I Pop Emerging Tech-nologies and Techniques in Porous Media Kluwer AcademicDodrecht The Netherlands 2004

[9] D B Ingham and I PopTransport Phenomena in PorousMediaPergamon Oxford UK 2005

[10] K Vafai Handbook of Porous Media Taylor amp Francis NewYork NY USA 2005

[11] D A Nield and A BejanConvection in PorousMedia SpringerNew York NY USA 3rd edition 2006

[12] M Jana S Das and R N Jana ldquoUnsteady Couette flow througha porous medium in a rotating systemrdquo Open Journal of FluidDynamics vol 2 pp 149ndash158 2012

[13] T Hayat S Nadeem A M Siddiqui and S Asghar ldquoAn oscil-lating hydromagnetic non-Newtonian flow in a rotating sys-temrdquo Applied Mathematics Letters vol 17 no 6 pp 609ndash6142004

[14] S Das S L Maji M Guria and R N Jana ldquoUnsteady MHDCouette flow in a rotating systemrdquoMathematical and ComputerModelling vol 50 no 7-8 pp 1211ndash1217 2009

[15] T Hayat C Fetecau and M Sajid ldquoAnalytic solution for MHDTransient rotating flow of a second grade fluid in a porousspacerdquo Nonlinear Analysis Real World Applications vol 9 no4 pp 1619ndash1627 2008

[16] T Hayat C Fetecau andM Sajid ldquoOnMHD transient flow of aMaxwell fluid in a porous medium and rotating framerdquo PhysicsLetters A vol 372 no 10 pp 1639ndash1644 2008

[17] S Abelman E Momoniat and T Hayat ldquoSteady MHD flowof a third grade fluid in a rotating frame and porous spacerdquoNonlinear Analysis Real World Applications vol 10 no 6 pp3322ndash3328 2009


[18] S N Sahoo J P Panda and G C Dash ldquoHydromagneticoscillatory flow and heat transfer of a viscous liquid past avertical porous plate in a rotating mediumrdquo Indian Journal ofScience and Technology vol 3 pp 817ndash821 2010

[19] G S Seth M S Ansari and R Nandkeolyar ldquoUnsteady hydro-magnetic Couette flow within porous plates in a rotating sys-temrdquo Advances in Applied Mathematics and Mechanics vol 2no 3 pp 286ndash302 2010

[20] A Farhad M Norzieha S Sharidan I Khan and SamiulhaqldquoHydromagnetic rotating flow in a porous medium with slipcondition and Hall currentrdquo International Journal of PhysicalSciences vol 7 no 10 pp 1540ndash1548 2012

[21] M Jana S L Maji S Das and R N Jana ldquoUnsteady flowofviscous fluid through a porous medium bounded bya porousplate in a rotating systemrdquo Journal of Porous Media vol 13 no7 pp 645ndash653 2010

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of


Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of


Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of


Mathematical PhysicsAdvances in

Complex AnalysisJournal of


OptimizationJournal of


CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of


Operations ResearchAdvances in

Journal of


Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences


The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014


Algebra

Discrete Dynamics in Nature and Society



Decision SciencesAdvances in

Discrete MathematicsJournal of


Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of


friction in case of cosine and sine oscillations of the plate areobtainedThe analytical results are plotted and discussedTheresults of Jana et al [21] are recovered as a special case fromour obtained solutions The present study is useful in astron-omy to study the rotating motion of astrophysical objectssuch as magnetic stars

2 Mathematical Formulation of the Problem

Consider the unsteady flow of an incompressible viscousfluid occupying the upper porous half-space of the (119909 119910)plane The fluid is bounded by a porous plate at 119911 = 0 suchthat the positive 119911-axis is taken normal to the plate and the 119909-axis is taken parallel to the plate We consider the hydromag-netic flow induced in the fluid in the presence of a uniformmagnetic field of strength 119861

0applied in a direction normal to

the plate by means of the plate oscillations in its plane (Hayatet al [13]) The electric field due to polarization of charges isneglected Both of the fluid and plate are in a state of rigidbody rotation with constant angular velocity Ω = Ωk k isa unit vector in the 119911-directionThe geometry of the problemis shown in Figure 1 Initially for 119905 le 0 both of the fluid andplate are at rest At time 119905 = 0+ the lower plate suddenlystarts to move in its own plane with oscillating velocity inthe flow direction Under these assumptions the equations ofmomentum in a rotating frame along the 119909- and 119910-directionsare [18ndash20]

120597119906

120597119905minus 1199080

120597119906

120597119911minus 2ΩV = ]

1205972119906

1205971199112minus]120601119896119906 minus1205901198612

0

120588119906

119911 119905 gt 0

120597V120597119905minus 1199080

120597V120597119911+ 2Ω119906 = ]

1205972V1205971199112minus]120601119896V minus1205901198612

0

120588V

119911 119905 gt 0

(1)

where 119906 and V are the velocity components in 119909- and 119910-directions 119908

0is the suction velocity at the plate ] is the kine-

matic viscosity 119905 is the time 120590 is the electrical conductivity1198610is the applied magnetic field 120588 is the fluid density 120601 is the

porosity and 119896 is the permeability of the porous mediumThe corresponding initial and boundary conditions are

119906 = V = 0 119905 le 0 (2)

119906 = 119880119867 (119905) cos (1205960119905) or 119880 sin (120596

0119905)

V = 0 at 119911 = 0 119905 gt 0(3)

119906 997888rarr 0 V 997888rarr 0 as 119911 997888rarr infin 119905 gt 0 (4)

in which 119880 is the constant plate velocity 1205960is the frequency

of oscillation of the plate and119867(119905) is the Heaviside functionDefining119865 = 119906+119894V the above systemof equations reduces

to

120597119865

120597119905minus 1199080

120597119865

120597119911+ 2119894Ω119865 = ]

1205972119865

1205971199112minus]120601119896119865 minus1205901198612

0

120588119865

119911 119905 gt 0

Ox

B0

Magnetic fieldΩ

z

Porous medium z gt 0

y

u(0 t) = UH(t) cos(1205960t)

u(0 t) = Usin(1205960t)

(0 t) = 0 t gt 0

Figure 1 Geometry of the problem

119865 (119911 0) = 0 119905 le 0

119865 (0 119905) = 119880119867 (119905) cos (1205960119905) or 119880 sin (1205960119905) 119905 gt 0

119865 997888rarr 0 as 119911 997888rarr infin 119905 gt 0(5)

Introducing the following dimensionless variables (see Janaet al [21])

119866 =119865

119880 120585 =

119911119880

] 120591 =

1198802119905

] 120596 =

1205960]1198802 (6)

the dimensionless problem takes the following form

120597119866

120597120591minus 119878120597119866

120597120585+ 2119894120578119866 =

1205972119866

1205971205852minus1198722119866 minus

1

119870119866 (7)

119866 (120585 0) = 0 120591 le 0 (8)

119866 (0 120591) = 119867 (120591) cos (120596120591) or sin (120596120591) 120591 gt 0 (9)

119866 (infin 120591) = 0 120591 gt 0 (10)

1198722=12059011986120]

1205881198802

1

119870=

]21206011198961198802 120578 =

Ω]1198802 119878 =

1199080

119880

(11)

where 119872 119870 120578 and 119878 are the magnetic permeability rota-tion and suction parameters respectively

3 Solution of the Problem byUsing Laplace Transform

Taking the Laplace transform of (7) and using the initialcondition (8) we get the following transformed differentialequation

1198892119866 (120585 119902)

1198891205852+ 119878119889119866 (120585 119902)

119889120585

minus [119902 +1198722+1

119870+ 2119894120578]119866 (120585 119902) = 0

(12)



119866 (0 119902) =119902

1199022 + 1205962or 119866 (0 119902) =

120596

1199022 + 1205962 (13)

119866 (0 119902) = 0 (14)


119866119888(120585 119902) =

1

2119890minus12058511987821

119902 + 119894120596119890minus120585radic119902+119886

+1


(15)

respectively

119866119904(120585 119902) =

1

2119894119890minus1205851198782minus

1

119902 + 119894120596119890minus120585radic119902+119886

+1


(16)


119886 =1198782

4+1

119870+1198722+ 2119894120578 (17)


119866119888(120585 120591)

=119867 (120591) 119890

minus1205851198782

4



+ 119890120585radic119886minus119894120596 erf 119888 ( 120585


+119867 (120591) 119890

minus1205851198782

4



+ 119890minus120585radic119886+119894120596 erf 119888 ( 120585

2radic120591+ radic(119886 + 119894120596) 120591)]

(18)

respectively

119866119904(120585 120591)

=119890minus1205851198782

4119894



+ 119890120585radic119886minus119894120596 erf 119888 ( 120585


+119890minus1205851198782

4119894[119890119894120596120591119890minus120585radic119886+119894120596 erf 119888 ( 120585


+ 119890120585radic119886+119894120596 erf 119888 ( 120585

2radic120591+radic(119886+ 119894120596) 120591)]

(19)



119866119888(120577 119902) =

radic2 (120577 + 120577119878) 119902

(1205772 + 1205772119878 + 119902 + 119886) (1199022 + 1205962) (20)

respectively

119866119904(120577 119902) =

radic2 (120577 + 120577119878) 120596

(1205772 + 1205772119878 + 119902 + 119886) (1199022 + 1205962) (21)


which yields

119866119888(120585 119902) =

2radic2119902

120587 (1199022 + 1205962)intinfin

0

(120577 + 120577119878)

(1205772 + 1205772119878 + 119902 + 119886)sin (120577120585) 119889120577

(22)


119866119888(120585 120591) =

2

120587intinfin

0

120577 (1205772 + 1205772119878 + 119886) 119890minus(1205772+1205772119878+119886)120591

((1205772 + 1205772119878 + 119886)2+ 1205962)

times sin (120577120585) 119889120577

minus2

120587intinfin

0

120577119878 (1205772 + 1205772119878 + 119886) 119890minus(1205772+1205772119878+119886)120591

((1205772 + 1205772119878 + 119886)2+ 1205962)

times sin (120577120585) 119889120577


1

08

06

04

02

01080604020 12

G

120585

M = 0

M = 3M = 6

M = 9

120591 = 09 K = 08 120578 = 03 S = 2 m = 05 120596 = 02 120596120591 = 0

(a)

01

008

006

004

002

01080604020 12

120585

M = 0

M = 3M = 6

M = 9

120591 = 09 K = 08 120578 = 03 S = 2 m = 05 120596 = 02 120596120591 = 0

minusiG

(b)


+2

120587intinfin

0

120577 (1205772 + 1205772119878 + 119886)

((1205772 + 1205772119878 + 119886)2+ 1205962)

times cos (120596120591) sin (120577120585) 119889120577

+2

120587intinfin

0

120577120596

((1205772 + 1205772119878 + 119886)2+ 1205962)

times sin (120596120591) sin (120577120585) 119889120577

+2

120587intinfin

0

120577119878 (1205772+ 1205772119878 + 119886)

((1205772 + 1205772119878 + 119886)2+ 1205962)

times cos (120596120591) sin (120577120585) 119889120577

+2

120587intinfin

0

120577119878120596

((1205772 + 1205772119878 + 119886)2+ 1205962)

times sin (120596120591) sin (120577120585) 119889120577(23)


119866119904 (120585 120591) =

2

120587intinfin

0

120577120596119890minus(1205772+1205772119878+119886)120591

((1205772 + 1205772119878 + 119886)2+ 1205962)

sin (120577120585) 119889120577

minus2

120587intinfin

0

120577120596

((1205772+1205772119878+119886)2+ 1205962)

cos (120596120591) sin (120577120585) 119889120577

+2

120587intinfin

0

120577 (1205772 + 1205772119878 + 119886)120596

((1205772+1205772119878+119886)2+1205962)

sin (120596120591) sin (120577120585) 119889120577

+2

120587intinfin

0

120577119878120596119890minus(1205772+1205772119878+119886)120591

((1205772 + 1205772119878 + 119886)2+ 1205962)

sin (120577120585) 119889120577

minus2

120587intinfin

0

120577119878120596

((1205772+1205772119878+119886)2+1205962)

cos (120596120591) sin (120577120585) 119889120577

+2

120587intinfin

0

120577119878 (1205772 + 1205772119878 + 119886)

((1205772+1205772119878+119886)2+1205962)

sin (120596120591) sin (120577120585) 119889120577

(24)



120591119888= minus (120591

119909+ 119894120591119910)

= minus119867 (120591) 119890

minus119886119905minus119894120596120591

4radic120587120591


119886120591minus 119878radic120587120591119890

119886120591+2119894120596120591


119886120591+2119894120596120591radic(119886 + 119894120596) 120587120591


+ 2radic(119886 + 119894120596) 120587120591119890119886120591+2119894120596120591 erf 119888 (radic(119886 + 119894120596) 120591))]


1

08

06

04

02

0151050 2

G

120585

K = 02

K = 04

K = 06

K = 08

120591 = 09M = 02 120578 = 03 S = 2 m = 05 120596 = 02 120596120591 = 0

(a)

15 251050 32120585

K = 02

K = 04

K = 06

K = 08

120591 = 09M = 02 120578 = 03 S = 2 m = 05 120596 = 02 120596120591 = 0

0035

003

0025

002

0015

001

0005

0

minusiG

(b)


120591119904= minus (120591

119909+ 119894120591119910)

=119894119890minus119886119905minus119894120596120591

4radic120587120591


119886120591minus 119878radic120587120591119890

119886120591+2119894120596120591


119886120591+2119894120596120591radic(119886 + 119894120596) 120587120591


+ 2radic(119886 + 119894120596) 120587120591119890119886120591+2119894120596120591 erf 119888 (radic(119886 + 119894120596) 120591))]

(25)

5 Special Cases


119866 (120585 120591) =119867 (120591) 119890

minus1205851198782

2

times[[[

[

119890minus120585radic119878

24+1119870+2119894120578

times erf 119888( 120585


1198782

4+1

119870+ 2119894120578) 120591)

+ 119890120585radic11987824+1119870+2119894120578

times erf 119888( 120585


1198782

4+1

119870+ 2119894120578) 120591)

]]]

]

(26)






0075

01

005

0025

0

151050 2

G

120585

120591 = 09 K = 08M = 03 S = 2 m = 05 120596 = 02 120596120591 = 0

120578 = 0

120578 = 2

120578 = 4

120578 = 6

minus0025

minus005

minus0075

(a)

151050 2120585

120591 = 09 K = 08M = 03 S = 2 m = 05 120596 = 02 120596120591 = 0

120578 = 0

120578 = 2

120578 = 4

120578 = 6

03

025

02

015

01

005

0

minusiG

(b)


08

08

06

06

04

04

02

02

1

1 120

0

G

120585

S = 00

S = 03

S = 06

S = 09

120591 = 09 K = 08 120578 = 03M = 2 m = 05 120596 = 02 120596120591 = 0

(a)

05 151 20

0120585

120591 = 09 K = 08 120578 = 03M = 2 m = 05 120596 = 02 120596120591 = 0

S = 00

S = 03

S = 06

S = 09

006

005

004

003

002

001

minusiG

(b)






08

08

06

06

04

04

02

02

1

10

0

G

120585

120591 = 09 K = 08 120578 = 03 S = 2 m = 05 120596 = 02M = 0

120596120591 = 0120587

120596120591 =120587

2

(a)

107505 17515025 1250120585

120591 = 09 K = 08 120578 = 03 S = 2 m = 05 120596 = 02M = 0

120596120591 = 0120587

120596120591 =120587

2

0

minusiG

005

004

003

002

001

(b)



7 Conclusions



References






























Volume 2014




Journal of











Journal of


Function Spaces






Algebra











119866 (0 119902) =119902

1199022 + 1205962or 119866 (0 119902) =

120596

1199022 + 1205962 (13)

119866 (0 119902) = 0 (14)


119866119888(120585 119902) =

1

2119890minus12058511987821

119902 + 119894120596119890minus120585radic119902+119886

+1


(15)

respectively

119866119904(120585 119902) =

1

2119894119890minus1205851198782minus

1

119902 + 119894120596119890minus120585radic119902+119886

+1


(16)


119886 =1198782

4+1

119870+1198722+ 2119894120578 (17)


119866119888(120585 120591)

=119867 (120591) 119890

minus1205851198782

4



+ 119890120585radic119886minus119894120596 erf 119888 ( 120585


+119867 (120591) 119890

minus1205851198782

4



+ 119890minus120585radic119886+119894120596 erf 119888 ( 120585

2radic120591+ radic(119886 + 119894120596) 120591)]

(18)

respectively

119866119904(120585 120591)

=119890minus1205851198782

4119894



+ 119890120585radic119886minus119894120596 erf 119888 ( 120585


+119890minus1205851198782

4119894[119890119894120596120591119890minus120585radic119886+119894120596 erf 119888 ( 120585


+ 119890120585radic119886+119894120596 erf 119888 ( 120585

2radic120591+radic(119886+ 119894120596) 120591)]

(19)



119866119888(120577 119902) =

radic2 (120577 + 120577119878) 119902

(1205772 + 1205772119878 + 119902 + 119886) (1199022 + 1205962) (20)

respectively

119866119904(120577 119902) =

radic2 (120577 + 120577119878) 120596

(1205772 + 1205772119878 + 119902 + 119886) (1199022 + 1205962) (21)


which yields

119866119888(120585 119902) =

2radic2119902

120587 (1199022 + 1205962)intinfin

0

(120577 + 120577119878)

(1205772 + 1205772119878 + 119902 + 119886)sin (120577120585) 119889120577

(22)


119866119888(120585 120591) =

2

120587intinfin

0

120577 (1205772 + 1205772119878 + 119886) 119890minus(1205772+1205772119878+119886)120591

((1205772 + 1205772119878 + 119886)2+ 1205962)

times sin (120577120585) 119889120577

minus2

120587intinfin

0

120577119878 (1205772 + 1205772119878 + 119886) 119890minus(1205772+1205772119878+119886)120591

((1205772 + 1205772119878 + 119886)2+ 1205962)

times sin (120577120585) 119889120577


1

08

06

04

02

01080604020 12

G

120585

M = 0

M = 3M = 6

M = 9

120591 = 09 K = 08 120578 = 03 S = 2 m = 05 120596 = 02 120596120591 = 0

(a)

01

008

006

004

002

01080604020 12

120585

M = 0

M = 3M = 6

M = 9

120591 = 09 K = 08 120578 = 03 S = 2 m = 05 120596 = 02 120596120591 = 0

minusiG

(b)


+2

120587intinfin

0

120577 (1205772 + 1205772119878 + 119886)

((1205772 + 1205772119878 + 119886)2+ 1205962)

times cos (120596120591) sin (120577120585) 119889120577

+2

120587intinfin

0

120577120596

((1205772 + 1205772119878 + 119886)2+ 1205962)

times sin (120596120591) sin (120577120585) 119889120577

+2

120587intinfin

0

120577119878 (1205772+ 1205772119878 + 119886)

((1205772 + 1205772119878 + 119886)2+ 1205962)

times cos (120596120591) sin (120577120585) 119889120577

+2

120587intinfin

0

120577119878120596

((1205772 + 1205772119878 + 119886)2+ 1205962)

times sin (120596120591) sin (120577120585) 119889120577(23)


119866119904 (120585 120591) =

2

120587intinfin

0

120577120596119890minus(1205772+1205772119878+119886)120591

((1205772 + 1205772119878 + 119886)2+ 1205962)

sin (120577120585) 119889120577

minus2

120587intinfin

0

120577120596

((1205772+1205772119878+119886)2+ 1205962)

cos (120596120591) sin (120577120585) 119889120577

+2

120587intinfin

0

120577 (1205772 + 1205772119878 + 119886)120596

((1205772+1205772119878+119886)2+1205962)

sin (120596120591) sin (120577120585) 119889120577

+2

120587intinfin

0

120577119878120596119890minus(1205772+1205772119878+119886)120591

((1205772 + 1205772119878 + 119886)2+ 1205962)

sin (120577120585) 119889120577

minus2

120587intinfin

0

120577119878120596

((1205772+1205772119878+119886)2+1205962)

cos (120596120591) sin (120577120585) 119889120577

+2

120587intinfin

0

120577119878 (1205772 + 1205772119878 + 119886)

((1205772+1205772119878+119886)2+1205962)

sin (120596120591) sin (120577120585) 119889120577

(24)



120591119888= minus (120591

119909+ 119894120591119910)

= minus119867 (120591) 119890

minus119886119905minus119894120596120591

4radic120587120591


119886120591minus 119878radic120587120591119890

119886120591+2119894120596120591


119886120591+2119894120596120591radic(119886 + 119894120596) 120587120591


+ 2radic(119886 + 119894120596) 120587120591119890119886120591+2119894120596120591 erf 119888 (radic(119886 + 119894120596) 120591))]


1

08

06

04

02

0151050 2

G

120585

K = 02

K = 04

K = 06

K = 08

120591 = 09M = 02 120578 = 03 S = 2 m = 05 120596 = 02 120596120591 = 0

(a)

15 251050 32120585

K = 02

K = 04

K = 06

K = 08

120591 = 09M = 02 120578 = 03 S = 2 m = 05 120596 = 02 120596120591 = 0

0035

003

0025

002

0015

001

0005

0

minusiG

(b)


120591119904= minus (120591

119909+ 119894120591119910)

=119894119890minus119886119905minus119894120596120591

4radic120587120591


119886120591minus 119878radic120587120591119890

119886120591+2119894120596120591


119886120591+2119894120596120591radic(119886 + 119894120596) 120587120591


+ 2radic(119886 + 119894120596) 120587120591119890119886120591+2119894120596120591 erf 119888 (radic(119886 + 119894120596) 120591))]

(25)

5 Special Cases


119866 (120585 120591) =119867 (120591) 119890

minus1205851198782

2

times[[[

[

119890minus120585radic119878

24+1119870+2119894120578

times erf 119888( 120585


1198782

4+1

119870+ 2119894120578) 120591)

+ 119890120585radic11987824+1119870+2119894120578

times erf 119888( 120585


1198782

4+1

119870+ 2119894120578) 120591)

]]]

]

(26)






0075

01

005

0025

0

151050 2

G

120585

120591 = 09 K = 08M = 03 S = 2 m = 05 120596 = 02 120596120591 = 0

120578 = 0

120578 = 2

120578 = 4

120578 = 6

minus0025

minus005

minus0075

(a)

151050 2120585

120591 = 09 K = 08M = 03 S = 2 m = 05 120596 = 02 120596120591 = 0

120578 = 0

120578 = 2

120578 = 4

120578 = 6

03

025

02

015

01

005

0

minusiG

(b)


08

08

06

06

04

04

02

02

1

1 120

0

G

120585

S = 00

S = 03

S = 06

S = 09

120591 = 09 K = 08 120578 = 03M = 2 m = 05 120596 = 02 120596120591 = 0

(a)

05 151 20

0120585

120591 = 09 K = 08 120578 = 03M = 2 m = 05 120596 = 02 120596120591 = 0

S = 00

S = 03

S = 06

S = 09

006

005

004

003

002

001

minusiG

(b)






08

08

06

06

04

04

02

02

1

10

0

G

120585

120591 = 09 K = 08 120578 = 03 S = 2 m = 05 120596 = 02M = 0

120596120591 = 0120587

120596120591 =120587

2

(a)

107505 17515025 1250120585

120591 = 09 K = 08 120578 = 03 S = 2 m = 05 120596 = 02M = 0

120596120591 = 0120587

120596120591 =120587

2

0

minusiG

005

004

003

002

001

(b)



7 Conclusions



References






























Volume 2014




Journal of











Journal of


Function Spaces






Algebra










1

08

06

04

02

01080604020 12

G

120585

M = 0

M = 3M = 6

M = 9

120591 = 09 K = 08 120578 = 03 S = 2 m = 05 120596 = 02 120596120591 = 0

(a)

01

008

006

004

002

01080604020 12

120585

M = 0

M = 3M = 6

M = 9

120591 = 09 K = 08 120578 = 03 S = 2 m = 05 120596 = 02 120596120591 = 0

minusiG

(b)


+2

120587intinfin

0

120577 (1205772 + 1205772119878 + 119886)

((1205772 + 1205772119878 + 119886)2+ 1205962)

times cos (120596120591) sin (120577120585) 119889120577

+2

120587intinfin

0

120577120596

((1205772 + 1205772119878 + 119886)2+ 1205962)

times sin (120596120591) sin (120577120585) 119889120577

+2

120587intinfin

0

120577119878 (1205772+ 1205772119878 + 119886)

((1205772 + 1205772119878 + 119886)2+ 1205962)

times cos (120596120591) sin (120577120585) 119889120577

+2

120587intinfin

0

120577119878120596

((1205772 + 1205772119878 + 119886)2+ 1205962)

times sin (120596120591) sin (120577120585) 119889120577(23)


119866119904 (120585 120591) =

2

120587intinfin

0

120577120596119890minus(1205772+1205772119878+119886)120591

((1205772 + 1205772119878 + 119886)2+ 1205962)

sin (120577120585) 119889120577

minus2

120587intinfin

0

120577120596

((1205772+1205772119878+119886)2+ 1205962)

cos (120596120591) sin (120577120585) 119889120577

+2

120587intinfin

0

120577 (1205772 + 1205772119878 + 119886)120596

((1205772+1205772119878+119886)2+1205962)

sin (120596120591) sin (120577120585) 119889120577

+2

120587intinfin

0

120577119878120596119890minus(1205772+1205772119878+119886)120591

((1205772 + 1205772119878 + 119886)2+ 1205962)

sin (120577120585) 119889120577

minus2

120587intinfin

0

120577119878120596

((1205772+1205772119878+119886)2+1205962)

cos (120596120591) sin (120577120585) 119889120577

+2

120587intinfin

0

120577119878 (1205772 + 1205772119878 + 119886)

((1205772+1205772119878+119886)2+1205962)

sin (120596120591) sin (120577120585) 119889120577

(24)



120591119888= minus (120591

119909+ 119894120591119910)

= minus119867 (120591) 119890

minus119886119905minus119894120596120591

4radic120587120591


119886120591minus 119878radic120587120591119890

119886120591+2119894120596120591


119886120591+2119894120596120591radic(119886 + 119894120596) 120587120591


+ 2radic(119886 + 119894120596) 120587120591119890119886120591+2119894120596120591 erf 119888 (radic(119886 + 119894120596) 120591))]


1

08

06

04

02

0151050 2

G

120585

K = 02

K = 04

K = 06

K = 08

120591 = 09M = 02 120578 = 03 S = 2 m = 05 120596 = 02 120596120591 = 0

(a)

15 251050 32120585

K = 02

K = 04

K = 06

K = 08

120591 = 09M = 02 120578 = 03 S = 2 m = 05 120596 = 02 120596120591 = 0

0035

003

0025

002

0015

001

0005

0

minusiG

(b)


120591119904= minus (120591

119909+ 119894120591119910)

=119894119890minus119886119905minus119894120596120591

4radic120587120591


119886120591minus 119878radic120587120591119890

119886120591+2119894120596120591


119886120591+2119894120596120591radic(119886 + 119894120596) 120587120591


+ 2radic(119886 + 119894120596) 120587120591119890119886120591+2119894120596120591 erf 119888 (radic(119886 + 119894120596) 120591))]

(25)

5 Special Cases


119866 (120585 120591) =119867 (120591) 119890

minus1205851198782

2

times[[[

[

119890minus120585radic119878

24+1119870+2119894120578

times erf 119888( 120585


1198782

4+1

119870+ 2119894120578) 120591)

+ 119890120585radic11987824+1119870+2119894120578

times erf 119888( 120585


1198782

4+1

119870+ 2119894120578) 120591)

]]]

]

(26)






0075

01

005

0025

0

151050 2

G

120585

120591 = 09 K = 08M = 03 S = 2 m = 05 120596 = 02 120596120591 = 0

120578 = 0

120578 = 2

120578 = 4

120578 = 6

minus0025

minus005

minus0075

(a)

151050 2120585

120591 = 09 K = 08M = 03 S = 2 m = 05 120596 = 02 120596120591 = 0

120578 = 0

120578 = 2

120578 = 4

120578 = 6

03

025

02

015

01

005

0

minusiG

(b)


08

08

06

06

04

04

02

02

1

1 120

0

G

120585

S = 00

S = 03

S = 06

S = 09

120591 = 09 K = 08 120578 = 03M = 2 m = 05 120596 = 02 120596120591 = 0

(a)

05 151 20

0120585

120591 = 09 K = 08 120578 = 03M = 2 m = 05 120596 = 02 120596120591 = 0

S = 00

S = 03

S = 06

S = 09

006

005

004

003

002

001

minusiG

(b)






08

08

06

06

04

04

02

02

1

10

0

G

120585

120591 = 09 K = 08 120578 = 03 S = 2 m = 05 120596 = 02M = 0

120596120591 = 0120587

120596120591 =120587

2

(a)

107505 17515025 1250120585

120591 = 09 K = 08 120578 = 03 S = 2 m = 05 120596 = 02M = 0

120596120591 = 0120587

120596120591 =120587

2

0

minusiG

005

004

003

002

001

(b)



7 Conclusions



References






























Volume 2014




Journal of











Journal of


Function Spaces






Algebra










1

08

06

04

02

0151050 2

G

120585

K = 02

K = 04

K = 06

K = 08

120591 = 09M = 02 120578 = 03 S = 2 m = 05 120596 = 02 120596120591 = 0

(a)

15 251050 32120585

K = 02

K = 04

K = 06

K = 08

120591 = 09M = 02 120578 = 03 S = 2 m = 05 120596 = 02 120596120591 = 0

0035

003

0025

002

0015

001

0005

0

minusiG

(b)


120591119904= minus (120591

119909+ 119894120591119910)

=119894119890minus119886119905minus119894120596120591

4radic120587120591


119886120591minus 119878radic120587120591119890

119886120591+2119894120596120591


119886120591+2119894120596120591radic(119886 + 119894120596) 120587120591


+ 2radic(119886 + 119894120596) 120587120591119890119886120591+2119894120596120591 erf 119888 (radic(119886 + 119894120596) 120591))]

(25)

5 Special Cases


119866 (120585 120591) =119867 (120591) 119890

minus1205851198782

2

times[[[

[

119890minus120585radic119878

24+1119870+2119894120578

times erf 119888( 120585


1198782

4+1

119870+ 2119894120578) 120591)

+ 119890120585radic11987824+1119870+2119894120578

times erf 119888( 120585


1198782

4+1

119870+ 2119894120578) 120591)

]]]

]

(26)






0075

01

005

0025

0

151050 2

G

120585

120591 = 09 K = 08M = 03 S = 2 m = 05 120596 = 02 120596120591 = 0

120578 = 0

120578 = 2

120578 = 4

120578 = 6

minus0025

minus005

minus0075

(a)

151050 2120585

120591 = 09 K = 08M = 03 S = 2 m = 05 120596 = 02 120596120591 = 0

120578 = 0

120578 = 2

120578 = 4

120578 = 6

03

025

02

015

01

005

0

minusiG

(b)


08

08

06

06

04

04

02

02

1

1 120

0

G

120585

S = 00

S = 03

S = 06

S = 09

120591 = 09 K = 08 120578 = 03M = 2 m = 05 120596 = 02 120596120591 = 0

(a)

05 151 20

0120585

120591 = 09 K = 08 120578 = 03M = 2 m = 05 120596 = 02 120596120591 = 0

S = 00

S = 03

S = 06

S = 09

006

005

004

003

002

001

minusiG

(b)






08

08

06

06

04

04

02

02

1

10

0

G

120585

120591 = 09 K = 08 120578 = 03 S = 2 m = 05 120596 = 02M = 0

120596120591 = 0120587

120596120591 =120587

2

(a)

107505 17515025 1250120585

120591 = 09 K = 08 120578 = 03 S = 2 m = 05 120596 = 02M = 0

120596120591 = 0120587

120596120591 =120587

2

0

minusiG

005

004

003

002

001

(b)



7 Conclusions



References






























Volume 2014




Journal of











Journal of


Function Spaces






Algebra










0075

01

005

0025

0

151050 2

G

120585

120591 = 09 K = 08M = 03 S = 2 m = 05 120596 = 02 120596120591 = 0

120578 = 0

120578 = 2

120578 = 4

120578 = 6

minus0025

minus005

minus0075

(a)

151050 2120585

120591 = 09 K = 08M = 03 S = 2 m = 05 120596 = 02 120596120591 = 0

120578 = 0

120578 = 2

120578 = 4

120578 = 6

03

025

02

015

01

005

0

minusiG

(b)


08

08

06

06

04

04

02

02

1

1 120

0

G

120585

S = 00

S = 03

S = 06

S = 09

120591 = 09 K = 08 120578 = 03M = 2 m = 05 120596 = 02 120596120591 = 0

(a)

05 151 20

0120585

120591 = 09 K = 08 120578 = 03M = 2 m = 05 120596 = 02 120596120591 = 0

S = 00

S = 03

S = 06

S = 09

006

005

004

003

002

001

minusiG

(b)






08

08

06

06

04

04

02

02

1

10

0

G

120585

120591 = 09 K = 08 120578 = 03 S = 2 m = 05 120596 = 02M = 0

120596120591 = 0120587

120596120591 =120587

2

(a)

107505 17515025 1250120585

120591 = 09 K = 08 120578 = 03 S = 2 m = 05 120596 = 02M = 0

120596120591 = 0120587

120596120591 =120587

2

0

minusiG

005

004

003

002

001

(b)



7 Conclusions



References






























Volume 2014




Journal of











Journal of


Function Spaces






Algebra










08

08

06

06

04

04

02

02

1

10

0

G

120585

120591 = 09 K = 08 120578 = 03 S = 2 m = 05 120596 = 02M = 0

120596120591 = 0120587

120596120591 =120587

2

(a)

107505 17515025 1250120585

120591 = 09 K = 08 120578 = 03 S = 2 m = 05 120596 = 02M = 0

120596120591 = 0120587

120596120591 =120587

2

0

minusiG

005

004

003

002

001

(b)



7 Conclusions



References






























Volume 2014




Journal of











Journal of


Function Spaces






Algebra





















Volume 2014




Journal of











Journal of


Function Spaces






Algebra









research article unsteady hydromagnetic rotating flow...

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