175

Some Exact Solutions of Unsteady Boundary Layer Equations - II

By MICHAEL HALL and LOKENATH DEBNATH 1)

Summary - Some exact solutions are presented for the unsteady boundary layer flows of a homo- geneous, viscous, incompressible fluid bounded by (i) an infinite rigid oscillating flat plate or (ii) two parallel rigid oscillating flat plates. An explicit representation of the velocity fields for both the con- figurations has been given. The structures of the associated periodic boundary layers are determined with physical interpretations. Several results of interest have been recovered as special cases of this general theory. The Heaviside operational calculus along with the theory of residues of analytic func- tions is adopted in finding the solutions.

1. Introduction

In an earlier paper [1 ]2), the authors have investigated unsteady boundary layer flows generated in a homogeneous, incompressible viscous fluid by moving the boundary of the fluid impulsively with a prescribed velocity. The Laplace transform method has been used to obtain exact solutions of the unsteady boundary layer equations in some general configurations. The structures of the unsteady velocity field and the associated boundary layers have been determined with physical interpretations. In addition to the generalizations of the earlier results, some new results of interest have been found.

In this paper, we are primarily interested in the study of the unsteady periodic boundary layer phenomena in a homogeneous incompressible viscous nonrotating fluid. Some general and exact solutions of the unsteady boundary layer equation are presented for two geometric configurations. It is shown that the ultimate steady solu-

tion consists of double Stokes layers of thicknesses of the orders x / v ~ , ( r= 1, 2) where v is the kinematic viscosity of the fluid and co r represents the forcing frequency of the boundary of the fluid or the fluid in the inviscid region. The results of the earlier investigators are recovered as special cases of the present general result. In addition to the extensions of the earlier results, some new results of interest are obtained with their physical significance explored. The Heaviside operational calculus along with the theory of residues of analytic functions is employed for the investigation of the problems.

2. Constitutive equations and general solution

We consider the periodic unsteady boundary layer flows engendered in a semi-

t) Department of Mathematics, East Carolina University, Greenville, N.C. 27834, U.S.A. 2) Numbers in brackets refer to References, page 181.

176 Michael Hall and Lokenath Debnath (Pageoph,

infinite expanse of a homogeneous incompressible viscous fluid bounded by (i) an infi- nite rigid flat plate or (ii) two infinite parallel rigid flat plates. The flow is generated in configuration (i) by the oscillations of the plate and the oscillations of the main flow outside the boundary layer. In configuration (ii), both the boundary plates perform elliptic harmonic oscillations in their own planes so that an unsteady flow is set up in the fluid.

The Cartesian coordinate system is employed with the origin, x-axis and y-axis in the plate at z= 0 and the z-axis vertically normal to the plate. With this coordinate system, the unsteady flow is governed by the Navier-Stokes equation and the conti- nuity equation in the forms

au 1 + ( u . V ) u = - - V p + v v 2 u , (2 .2)

div u = 0, (2.2)

where u = (u,v, w) is the velocity field, 0 is the density, andp is the pressure. In view of the symmetry of the configurations, all the quantities depend only on z

and time t. Based on the usual assumptions of the unsteady two dimensional flow, it can readily be seen that the non-linear term (u. V) u disappears exactly from (2.1). Con- sequently, it follows from (2.1) and (2.2) that the equations of the boundary layer flow take the form

Olg ~2U ~ t - P + v ~z 2, (2.3)

where P---- (1/O)(Op/Ox) is the pressure gradient. The boundary conditions for configuration (i) are taken to be

u(z,t)=f(t) o n z = 0 , t > 0 , (2 .4)

u ( z , t ) - + g ( 0 as z - + o o , t > 0 , (2 .5)

where f (t) and g(t) are arbitrary functions of t and their particular forms specified later.

For the flow in the inviscid region to be consistent with the basic equations of motion, we require

~u - P , ( 2 . 6 )

0t

where u is given by (2.5). The initial condition is

u(z,t)=O at t < 0 , for all z. (2.7)

Vol. 102(1), 1973) Solutions of Unsteady Boundary Layer Equations 177

This initial value problem can be readily solved by introducing the Laplace trans- form

co

~(z, s) = f e -s' u(z , t) d t . (2.8)

0

The inverse Laplace transform is given by

C+ico

1 f e s t f t ( z , s ) ds , C > O . (2.9) u(z, t) - 2 zc i C - loo

Using the Laplace transform method and the initial condition (2.7), the solution for u(z, t ) , subject to the required boundary conditions, can be represented by the Laplace inversion integral

where

C + i m

u ( z , t ) - 2 ~ i C - io~

I f ( s ) e -xz + ~(s) (1 - e-)'z)] e st ds , (2.10)

By virtue of the convolution theorem for the Laplace transform, solution (2.10) reduces to

t (J) u(z, t) = g( t ) + { f ( t - z) - g( t - "c)} z -3/2 exp - ~ dz . (2.12)

0

This is the most general as well as exact solution of the velocity distribution. To investigate the principal features of the flow and the structures of the associated boundary layers, it is of interest to consider some special cases.

We consider the following particular case

f ( t ) = a e i~ + b e -'~ and g( t ) = c e '~ + d e -'~ (2.13a, b)

where ~1, o)2 are the given frequency of oscillations, a, b, c, and d are complex con- stants of order one.

This case includes several other special cases of interest which arise when

(a) b = 0 and c = d= O, (b) c=d=O, (c) f ( t ) = a + b e i~ and e=d--O,

12 P A G E O P H 102(I), (1973)

178 Michael Hall and Lokenath Debnath (Pageoph,

(d) a = b = 0 and g(t ) = c + d e i~ (e) a = b = O a n d g ( t ) = c ei'~ + d e -~2~, ( f ) f (t) = a e u~ and g(t ) = c; and ( g ) 0)1 = ~o~.

Substituting (2.13a, b) into (2.10) and using the tables of CAMPBELL and FOSTER [3], the solution u(z, t ) takes the form

a I u(z , t) = ~ ei~ ( - 21 z) erfc (( - J i oq t) + exp (21 z)

b x erfc (~ + j i o91 t)] + ~ e-U~ ( - 2; z)

x erfc (( - J ~ i 0)2 t) + exp (2 i z) erfc (~ + x / - i 0) 1 t)]

_ c e~,2,fexp ( _ 22 z) erfc (( - x / i 0)2 t) + exp (22 z) 2

d eUO=,[exp ( - 21 z) x erfc (( + x / i 092 t)] -

x erfc (ff - x / - - i 0)2 t) + exp (2 i z) erfc (~ + x / ~ i m 2 t)]

+ c e u~ + d e-i~

(2.14)

where ~ =-z/2 x/-vtt is the well-known similarity variable in viscous boundary layer

theory,

/ i cot\ 1/2 - cor 2 r - \ ] [ ~ - } , 2 : = ~ , r = l , 2; (2.15a, b)

and erfc (x) represents the complementary error function. The unsteady flow fields associated with all cases (a) - (g) can be directly obtained

from the general result (2.14). It may be confirmed that result (2.14) describes all the salient features of periodic

boundary layer solutions and may be regarded as an extension of the corresponding results available in the existing literature.

We now use the asymptotic behavior of the complementary error function [4] as

e r f c ( ( + x / i 0 ) , t ) + O and e r f c ( ~ - x / i 0 ) ~ t ) ~ 2 , as t ~ o o

to obtain the steady-state structure of the flow field. It can easily be seen that in the limit t ~ Qo, (2.14) approaches

u(z, t) ~ a exp( io) 1 t - 21 z) + b e x p ( - i 0)1 t - 21z) + c e"~ - e -a==) + d e - ' ' = ' ( 1 - e-Z'=~).

(2.16)

Vol. 102(1), 1973) Solutions of Unsteady Boundary Layer Equations 179

We concluse that the ultimate steady state is reached in the limit t ~ oe and lepre- sents an extension of the well-known Stokes solution [5]. This steady solution consists of double Stokes layers on the plate of thicknesses of the orders (v/o31) ~/2 and (v/032)1/2 which coalesce into a single layer (v/co) ~/2 as co 1 = e92 = co. Physically, these boundary layers represent the depths of penetration of vorticity due to the imposed oscillations.

3. Boundary layers in a vertically bounded v&cous f luid

We next turn to the problem of unsteady periodic boundary layer flows generated in a homogeneous incompressible viscous fluid between two infinitely long parallel plates at z = 0 and z = D. In the absence of an imposed pressure gradient, the unsteady boundary layer equation becomes

Ou ~2u - v - - - 0 < z < D ( 3 . 1 )

3t Oz z ' - - "

This equation has to be solved subject to the imposed boundary conditions

u ( z , t ) = f ( t ) on z = O , t > O , (3.2)

u ( z , t ) = g ( t ) on z = D , t > 0 , (3.3)

where f (t) and g(t) have been specified earlier. The initial condition is

u ( z , t ) = O at t < 0 for all z, (3.4)

In view of the Laplace transform (2.8), equation (3.1) subject to the boundary and initial conditions (3.2)-(3.4) admits the solution

C+i~

1 f sinh2(D - z) u ( z , t ) - 2 r c i f ( s ) s inh2D e s 'ds

C - ico

+ - -

C + i ~

1 f sinh2 z st 2 h i ~ ( S ) ~ m h 2 D e ds.

(3.5)

Invoking (2.13a, b) into (3.5), this integral can be exactly and readily evaluated by using the theory of residues. The solution of the problem is then given by

180 Michael Hall and Lokenath Debnath (Pageoph,

where

u(z, t) = a e ' ' ' 1 ' sinh2,(D - z) sinh).l D

+ b e -i~''' sinh 2'1(D - z)

sinh).~ D

" k C e i~~ sinh ).2 Z

sinh).2 D ~- d e-io~t sinh ).; z

sinh).2 D

+

co ) [( ( - - 1 ) " 2 n n v a s i n nn 1 - e -~t

*' ( n 2 7g 2 v -b i (.01 D 2) n = i

+

oo

) ' ( - 1 ) " 2 n rt v b sin[n r c ( 1 - D ) l e -~'

' (n 2 n 2 v -- i o~, D 2) n = l

+

oo

( - 1 ) "2n n v c s i n ~ - )

(n Ere 2 v -k i09 2 D 2) n = l

+

oo

. f n ( -1)n 2 n z~ v d sln ~ , ~ - j e -~t

(n 2 ~ 2 v - i % D 2) n = l

(3.6)

n 2 7/: 2 1/

~ D 2 > 0 . (3.7)

It is evident from (3.6) that the velocity field consists of both steady-state and transient components. In the limit t ~ 0% the last four infinite series representing the transient solution decay exponentially to zero. Consequently, the ultimate steady- state is attained in the limit. The first four terms in (3.6) represent the steady-state solution which consists of the Stokes boundary layers on both the plates. These layers have depths of penetration of oscillations of the order (v/cot) 1/2 on z = 0 and (v/m2) 1/2 on z = D. Solution (3.6) includes several special cases of interest.

In particular, when f ( t ) - a + b e u~ and g(t)=-c+de *o'2t, the velocity field u(z, t) can be found in the form

u(z, t) = a 1 - + b sinh), t D e

z de i~~ sinh).2 z + c ~ + sinh 22 D

+ the same transient terms as those in (3.6).

Vol. 102(1), 1973) Solutions of Unsteady Boundary Layer Equations 181

In the limit t--. oo, the transient terms die out and the solution approaches the

ultimate steady-state represented by the first four terms in (3.8). The first and the third terms of (3.8) corresponds to the steady Couette flow in which the shear is uniform in z. The second and the fourtl~ terms represent the Stokes layers of thick-

nesses of the order (v/o)r) 1/2 (r = 1, 2) on the plates.

Acknowledgement

The authors express their sincere thanks to Dr. W. M. WHYBURN and Dr. T. J. PrGNANI for their interest in this work. MICHAEL HALL is grateful to the Mathematics Department of East Carolina University for a graduate assistantship. This work was partially supported by the Research Council of East Carolina University.

REFERENCES

[1 ] MICHAEL HALL and LOKENATH DEBNATH, Some exact solutions of unsteady boundary layer equa- tions I, Pure and Applied Geophysics (1972) (to appear).

[2] C. J. TRANTER, Integral Transforms in Mathematical Physics (Metheum Monographs, London, 1966).

[31 G. K. CAMWELL and R. M. FOSTER, Fourier Integrals for PractieaI Applications (D. van Nostrand, 1948).

[4] N. N. LE~EDEV, Speeial Functions and Their Applieations, (Prentice Hall, 1965). [5] L. ROSENHEAD (ed.), Laminar Boundary Layers (Clarendon Press, Oxford, 1963).

(Received 24th April 1972)