Chapter 3

Low Reynolds number flows ( Lubrication theory, stoke’s flow, Oseen's flow)

3-1

21' ' : motion pressureV V p V p

Limiting cases for the Navier-Stokes equation

(a) Low Re creeping flow slow motion, larege viscosity

Consider steady, incompressible, Constant flow

Navier-Stokes eq.

0

1

0 1

Make non-dimensconal

Assume single scale velocity =

length =

dimensionle

ss quantities

pressure =

, , ,

V

l

p

V p xV p x y

V p l

2

2 2

, z

Note: ,

y z

l l

l l

3-2

2

0 1

0

substitute

V l p lV V p V

v

inertiaCharacteristic Reynolds number = Re

viscous

2

cm1 mm V=1.4 sec

Re 1 cm in air =0.14

sec

d

Limiting cases for the Navier-Stokes equation

3-3

At small Re, to a first approximation, neglect the inertia

terms in the equation.

2 linear P.D.E. like pa1

rallel flo1 N-S eq.

2 continuit

0 '

y eq.

w

0

p V

V

4 eqations 4 unknows ,v, ,u w p

Limiting cases for the Navier-Stokes equation

3-4

2

2

0

1 '

'

=

contin

= -

uity

p V

p V

V V

0

2 p' is a potentia' 0 l function p

Limiting cases for the Navier-Stokes equation

3-5

2a vorticity eq. 0

0V

2 eqations

2 unknow u,v

(recover p from N-S eq.)

recover p from N-S eq.

4

b vorticity eq. in terms of , stream function

0

4th order linear P.D.E. eq.

known as the bi-harmonic eq.

Limiting cases for the Navier-Stokes equation

2 2 D flow

3-6

x

Φ

r

y

z

R

r

The oldest known solution for a creeping motion

, ,

Stokes flow 1851 (Sphere)

U

3-7

2

2

2

V

vorticit

spherical coordinate

solve using vorticity eq. in terms of

y

sin1 10

sin

1

sin

r r

r

r

r

V V V

rV VV

r r

r V VV

r r r

Vr

1

sinV

r r

Stokes flow 1851 (Sphere)

0 (symmetry)

3-8

2

2 2

2

2 2

2

2 2

2 2

wher

sin 1

sin sin

1or E

sin

sin 1 E

vorticity

= sin

0

E1 sin

e

=-sin

eq.

r r r

r

r r

r r

E1

sin

4 0

0

Stokes flow 1851 (Sphere)

3-9

2

vorticity eq. in terms of

E 0

r 1 , 0 0

Boundary Condit

2 , 0 0

3 , uniform x-directed flow

4

ions

?

r R V

r R Vr

r

linear 4th order P.D.E

Stokes flow 1851 (Sphere)

3-10

θ

θ U

rVV

2

2 2

1sin

sin

1cos

sin

Integrate

= sin2

& compare

r

V Ur r

V Ur

Ur

---------------- B.C. (3)

Stokes flow 1851 (Sphere)

3-11

2

2

22

2 2

2 22 2

2 2 2 2

22

2 2

Assume =

since the B.C. at r holds for all

let g sin

= sin

2sin

2 2sin 0

Must hold for all

2 0

f r g

f r

E fr r

E fr r r r

df

dr r

Stokes flow 1851 (Sphere)

3-12

4 2

4 2 2 3 4

2 4

2 2

4 8 8 0

1, 1, 2, 4

expanding

solution

substitute

Boundary condition r = s in2

n

d f d f dff

dr r dr r dr r

f r

n

Af r Br Cr Dr

r

Ur

2 2

D=0 ; C=2

sin2

U

A UBr r

r

Stokes flow 1851 (Sphere)

3-13

2

3

3

3

1r=R, 0

sin

= cos 2 cos

1r=R, 0 0

sin

sin cos

=4solving

3

4

1 3 11

2 2 2

rVr

A BU

R R

Vr r

A BU

R R

U RA

B U R

R RU

r r

3

2 2sinr

B.C. (1)

B.C. (2)

Stokes flow 1851 (Sphere)

3-14

D

2 2

d

d

2 skin friction

3 D=6

1 pressure

3

6D

1frontal area dynamic head

2

24

Re

d=2R

Drag :

Drag c

oefficie

Re

nt :

va

D

RU

RUC

R U

C

U d

lid Re 1

Drag and Drag coefficient

3-15

dA

pdA

r dAU

drag D

lift L

Drag and Drag coefficient

3-16

3/13/2014

a circular disk perpendicular t

special

o the s

cases

treamdisk

U

R = radius

D 16 RU

b circular disk parallel to the stream

32

D = 3

RU

Ellipsoid in a parallel stream

R = radius

U

disk

3-18

inertia terms

' ' '' ' '

v ' v ' v ' ' ' ' + ' v ' ' j + U ' ' ' k

U u U u U uV V u u v w i

x y z

w w wU u w u v w

x y z x y z

Oseen’s approximation

'

v v ' '

let

'

u U u

V U i V

w w

U u',v',w'

'

As

' ', '

sume

wher ,e 'V

U V u v wx

3-19

2

D

0 '

Hence '

' 1momentum eq

Note:Approximation must fail near the surface, sin

. U ' '

continuity eq. ' 0

24Drag coefficient

R

e

e

c

Cd

u u U

u U

Vp V

x

V

dd

4.5

where Re hold for Re up t 5 oU d

Oseen’s approximation

3-20

NCKU Department of Mechanical Engineering