UNSTEADY VISCOUS FLOW

upgDt

uD

2

2

21

y

u

x

p

t

u

Viscous effects confined to within some finite area near the boundary → boundary layer

In unsteady viscous flows at low Re the boundary layer thickness δ grows with time; but in periodic flows, it remains constant

If the pressure gradient is zero, Navier-Stokes equation (in x) reduces to:

2

2

y

u

t

u

Assume linear, horizontal motion

2

2

y

u

t

u

Heat Equation– parabolic partial differential equation - linear

Requires one initial condition and two boundary conditions

U

y

Uuy 0@

0@ uy

00,, tyxu

Total of three conditions

Impulsively started plate –

Stokes first problem

2

2

y

u

t

u

Heat Equation– parabolic partial differential equation

Can be solved by “Separation of Variables”

Suppose we have a solution: tTyYtyu ,

Substituting in the diff eq: tTyYy

tTyYt 2

2

May also be written as: tTyYtTyY

Moving variables to same side:

yY

yY

tT

tT

The two sides have to be equal for any choice of y and t ,

kyY

yY

tT

tT

The minus sign in front of k is for convenience

This equation contains a pair of ordinary differential equations:

kyY

yY

tT

tT

0

0

TkT

kYY

0

02

2

Tkt

T

kYy

Y

Uuy 0@

0@ uy

00,, tyxu tTyYtyu ,

0YU

Y0

00 T

0

Tkt

T

tkT

T

tkT

T AtkT ln tkAeT

02

2

kYy

Y ykCykBY sincos

t

y

Ueu 4

2

ykCykBY sincos tTyYtyu ,

increasing time

2

L

nkn

Applying B.C.,B = 0; C =1;

L

yneAu

tL

n

n

sin

2

2

2

y

u

t

u

Uuy 0@

0@ uy

00,, tyxu

New independent variable:t

y

2

η is used to transform heat equation:

d

d

td

d

tt 2

d

d

td

d

yy 2

1

2

2

2

2

4

1

d

d

ty

Substituting into heat equation:

2

2

42

d

ud

td

du

t

Alternative solution to“Separation of Variables” – “Similarity Solution”

from:2y

u

t

u

022

2

d

du

d

ud

022

2

d

du

d

ud

Uu 0@

0@ u

asu 0

To transform second order into first order: d

duf

2 BC turn into 1

02 fd

df

With solution:2Aef

Integrating to obtain u:

BdeAu

0

2

Or in terms of the error function:

deerf0

22 erfUu 1

df

df2

erf

2e

For η > 2 the error function is nearly 1, so that u → 0

erfUu 1 For η > 2 the error function is nearly 1, so that u → 0

Then, viscous effects are confined to the region η < 2

This is the boundary layer δ

t

y

2

t

22

t 4

δ grows as the squared root of time

increasing time

erfUu 1

t

y

2

2

2

y

u

t

u

UNSTEADY VISCOUS FLOW

Oscillating Plate – Stokes’ second problem

tUuy cos0@

boundeduy @

Ucos(ωt)

y

Look for a solution of the form: tieyYtyYu Recos

tite ti sincos Euler’s formula

Fourier’s transform in the time domain: tieyYu Re 0YU

0Y

B.C. in Y

2

2

y

u

t

u

Substitution into:

titi YeiYet

2

2

y

YYi

2

2

2

2

y

Ye

y

u ti

02

2

Yi

y

Y

2

1 ii

yiByiAY

21exp

21exp

00 BY UAUY 0

yiUY

2

1exp

yiUY

2

1exp

y

tUeYeuy

ti cosRe

Most of the motion is confined to region within:

2

Ucos(ωt)

y

UUeUe

yy

37.0

@

1

UUeUeUe

yy

06.0

/4@

24

y

tUeuy

cos

2

2

y

u

t

u

UNSTEADY VISCOUS FLOW

Oscillating Plate

tUuy cos0@

0@ uWy

Look for a solution of the form: tieyYtyYu Recos

tite ti sincos Euler’s formula

Ucos(ωt)

y

W

Fourier’s transform in the time domain: tieyYu Re UY 0

0WY

B.C. in Y

2

2

y

u

t

u

Substitution into:

titi YeiYet

2

2

y

YYi

2

2

2

2

y

Ye

y

u ti

02

2

Yi

y

Y

2

1 ii

yiByiAY

21exp

21exp

UBA

WiBWiA

21exp

21exp0

2

UBA

W

iW

i

BeAe11

0 sinh2 ee

sinh2

Ue

B

Wi 1

sinh21

eUA

yy BeAeY

Wi

yWiUY

)1(sinh

))(1(sinh

Wi

WiW)1(

)1(sinh@

BeeBU 0

W

yU

W

yWUY 1

tW

yUu cos1

Wi

eWiW)1(

21)1(sinh@

yi

UY)1(

exp

Wi

yWiUY

)1(sinh

))(1(sinh

sinh2 ee Same solution as for unbounded oscillating plate

tW

yUu cos1