Momentum differential equation: Navier-Stokes equation

CHEE 3363Spring 2014Handout 15

Reading: Fox 5.25.4

1

Learning objectives for lecture

1. State the most general form of the Navier-Stokes equations.2. State the version of the N-S equations valid for incompressible A3;7;-8,'3278%28:-7'37-8=

3. !8%8)8,)90)6)59%8-32%2(8,)'32(-8-32792()6;,-',-8-7valid.

2

Conservation of linear momentum 1);832?72(

Write:

dx

dy

dz

x

y

z

u

v

w

))(83@2()

Conservation of linear momentum 2

dx

dy

dz

x

y

z

u

v

w

7979%0)

Conservation of linear momentum 320=&3(=*36')-7+6%:-8=73)

Conservation of linear momentum 4

6

Write differential momentum equation (equate to acceleration):

gx +xx

x+

yx

y+

zx

z=

(u

t+ u

u

x+ v

u

y+ w

u

z

)

gy +xy

x+

yy

y+

zy

z=

(v

t+ u

v

x+ v

v

y+ w

v

z

)

gz +xz

x+

yz

y+

zz

z=

(w

t+ u

w

x+ v

w

y+ w

w

z

)

T = pnn+

Navier-Stokes equation 1We need a three-dimensional version of LMB:

#7-2+8,) )=230(7"6%274368",)36)1%2(

7

dP

dt

sys

= F =

CS

n TdA+

CV

gdVF =dP

dt

sys

stresses +6%:-8=

t

CV

vdV +

CS

vv dA =

CS

pdA+

CS

dA+

CV

gdV

(Recall: is a tensor)

Navier-Stokes equation 2

8

#7-2+%977?78,)36)1CS

vv dA =

CV

(vv)dV

CS

pdA =

CV

pdV

CS

dA =

CV

dV

CV

[

t(v) + (vv) +p g

]dV = 0

Navier-Stokes equation 3

-2%00=97)'327)6:%8-323*1%77837-140-*=

",-7-78,)Navier-Stokes equation:> $-8,'328-29-8=%&3:)*361*396'3940)(2320-2)%67*36v, >

xy = yx =

(v

x+

u

y

)

yz = zy =

(w

y+

v

z

)

zx = xz =

(u

z+

w

x

)

xx = p2

3

v + 2u

x

yy = p2

3

v + 2v

y

zz = p2

3

v + 2w

z

Stresses for N-S equations (rectilinear)

10

Shear stresses:

Normal stresses:

!7-140-@'%8-32For ;-8,constant viscosity:

)6-:%8-32&)=32(7'34)3*'3967)

11

(v

t+ v v

)= p+

2v + g

9278)%(=acceleration

convective acceleration

inertia

&3(=*36')78=4-'%00=+6%:-8=:-7'37-8=pressure

divergence of stress

90)6?7)59%8-32779148-327*6-'8-320)77A3; = 0 ):

Used for:

12