An analysis of the boundary element method for the heat conduction equation with periodic heating

Engineering Analysis with Boundary Elements 9 (1992) 137-144

~

An analysis of the boundary element method for the heat conduction equation with periodic heating D.B. Ingham Department of Applied Mathematical Studies, The University of Leeds, Leeds, West Yorkshire, LS2 9JT, UK

&

H. Han Department of Applied Mathematics, Tsinghua University, Beijing 100084, People's Republic of China In this paper a formulation of the unsteady heat transfer equation in both two and three dimensions with periodic heating using the boundary element method is presented. In the case of problems involving Dirichlet and Neumann boundary conditions error estimates are obtained.

1 INTRODUCTION

In this paper we consider the unsteady heat transfer equation,

The mathematical formulation of problems involving the steady-state heat transfer in bodies of various shapes with variable conductivity and differing boundary conditions usually leads to the need to solve one or more partial differential equation. Consequently a variety of numerical schemes have been developed with the most widely used being the finite difference and finite element 'space discretisation' techniques, in which the governing partial differential equations are approximated by a set of discretised equations, whose solution is subsequently obtained numerically at a finite number of prespecified points in the solution domain. An alternative approach is to employ the boundary element method (BEM) so that the governing equations are solved only on the boundary of the required solution domain. An excellent general review of the BEM may be found in Brebbia et al., ~ whilst a review of the application of the method to steadystate heat transfer problems may be found in Ingham & Kelmanson. 2 A very important application of the solution of the heat transfer equation is when considering the steady-state heat transfer in extended surfaces and this has been reviewed by Manzoor. 3 However, the performance of fins under unsteady ~onditions has received little attention and this situation occurs frequently in practice, e.g. in electrical systems where fins are often subjected to periodic heating. Engineering Analysis with Boundary Elements 0955-7997/92/$05.00 © 1992ElsevierSciencePublishers Ltd.

~T t~t

-

AT

(1)

where T is the temperature, under the assumption that the heat flow is generated by some periodic heating. Such problems may be solved analytically if the geometry is simple, but invariably one has to employ numerical techniques. Normally this would involve using either finite difference or finite element methods. However, because of the substantial advantages gained by using the BEM, in that only the surface of the body has to be discretised, then we describe how this problem may be formulated using this technique. Typically, the boundary within which we seek a solution is not simple and the boundary conditions are of a mixed type, e.g. Fig. 1 represents the kind of boundary and boundary conditions present in finned problems. However, the mathematical formulation of the BEM is independent of the boundary shape and boundary conditions and is only dependent on the governing equations. This analysis is presented in Section 2. The question of the existence and/or uniqueness now arises and for general mixed boundary conditions this is not always easy to resolve. However, for Dirichlet and Neumann boundary conditions this may be determined and the results of these investigations are presented in Sections 3-6, along with the error estimates. 137

138

D.B. Ingham, H. Han ~~T o-o

and the boundary conditions reduce to

r,

]

(i)

on F~ and 1~3 OUo

To Tm÷%

~ ~t

P2

~

= k(T

Ouo On

- TO)

r,

(iii)

. . . . ~

Ou~

0

0--g

=

0

Uo

k(uo-

0U 1

To), ~-~n = k u ~ , -

=

T~t,

uI

=

To,

0,

in t~

OF PROBLEM

0n

Let t2 be a bounded domain in R2 with boundary F = Fl w F2 w Fa w F~, as shown in Fig. 1. We study the following heat conduction problem with t2 with the periodic boundary conditions

0T _ 0t

Au,

inO

0,

onF~andF 3

Ot

T =

= ku2

On

u2

=

0

x

(-o~,o~)

-

0,

on F~ ~ 1"3

0u0 0n Uo and

=

k ( u o - To), on F~ Tu,

on F4

t

Au~ + u2 = 0,

int~

Au~-

inf~

u~

=

0,

Ou~ 0--~- = 0,

Ou2 On -

Ou~ On

0u--2 = kuz On

O,

onF~ L) F~

(3)

(4)

(2)

OT -

0//2

]

0u0

=

0,

Hence the original problem (2), which is time dependent, is separated into two problems which are independent of the time, namely, AUo =

0T -0t

=

on F4

= o

Fig. 1.

2 DESCRIPTION

Ou2

(ii) on F2 ~

~

=

k(T

-

To)

T~ + T c s i n t

on Fz onG

where To, T~, T~ are given functions which are independent of t. It is well known that the problem specified by (2) has a unique periodic solution and the aim of this paper is to investigate the BEM formulation of this problem. We let

ku~,

on F2

u~ = T~, uz = 0 onF4 The problem (3) is a mixed boundary value, Laplace equation, problem which has a unique solution and may be solved by using the finite difference, finite element or boundry element methods numerically. In this paper we investigate boundary value problems of the equations in the problem (4) numerically by the boundary element method.

T(x~, x~, t) = Uo(X~, x~) + u~(x~, x~)sint "1- U2(X1,

X2)COSt

and then

3 THE FUNDAMENTAL SOLUTION By definition, the distribution matrix

0T 0--~- =

u~cos t - u2 sin t

AT =

Au0 + Au, sin t + Au2cos t

Thus equations (2) may be written Auo = 0,

in fl

G = F GI' LG~

G,2] G~ J

is a fundamental solution of the equations A u , + u 2 = 0, i n f , ) Au2 - ut = 0, in

ol

if and only if

=

0,

infl

AGII "+- GI2 = ~

Au2 - u~ =

0,

in f~

AG~2 - Gt~

AUl+U2

0J

(6a)

An analysis of the boundary element method and

139

~ is Euler's constant, 7 = 0.577 215 6649 . . . .

AG2, +

G22

--

0

~

(6b)

AG22 - G2~ = 6 ~

Ker(r) and Kei(r) have a singularity at r = 0, and

d2

"~

1 d

where 6 denotes the Dirac measure at the parameter point y. Eliminating G. from eqn (6a) gives A2G~2 + G12 = 6

dr--~ Ker(r) + -r ~ Ker(r) = -Kei(r)/~ d2 1 d dr--~ Kei(r) + -r~rr Kei(r) Ker(r) )

where G1, = AG~2

K e r ( r ) ~ l n / ~ ),

and similarly we obtain from eqns (6b) A2G~I + G2~ = - 6

Kei(r) ~ ¼r 2In

G22

--

/1\

Ker(r)

- - AG21

_G~ 2)

"~

(7)

We now look for a fundamental solution of the equation A2 W + W = 0, which depends only on the distance

r =

X/~'

7~rrJ

W+

e-/4

W

=

0,

in~2,

(5

, r --* ÷ ~ ,

+~

, r ~ + o~,

solution of the eqns (4), namely,

G,2(x,y)) G2~(x,y)

(11)

where 1 = ~ ker(Ix - Yl)

G~(x,y)

(8)

and

sin

÷ ~n

(see Ref. 4). Hence we know that 1/2~ Ker(r) is a fundamental two-dimensional solution of the equation A2 W + W = 0. By using (7) we obtain a fundamental

(G,,(x,y) G(x,y) = \G2~(x,y)

(xi - yi) 2

from the point x to the parameter point y. By transforming the Laplacian operator to polar coordinates we obtain (d 2 1 d'~ 2 ~+

, r ~ 0,

e -r/42 COS

£r

Kei(r).-~ -

G~ = AG~2 I G22 = AGI2

0,

r~

(!)

Hence G2 is a fundamental solution of the equation G2~ =

(10)

1

G~2(x, y) = ~ kei([x - Yl) (12)

(d~ 2d) ~ ~ +-r~rr W+

W = 0,

inl~ ~,

(9)

G~(x,y)

1 - - - 2 r t k e i ( I x - Yl)

=

We know that the eqn (8) has four independent solutions (the Kelvin Functions): ber(r), bei(r), Ker(r), Kei(r)

Gz2(X,

where

In three dimensions, the equation (9) has solutions

ber(r) =

1

(¼ r2)2 (¼ r2)~ (2!) 2 + ~(4!)

bei(r) = ¼ r 2 - 0 r2)3

r

1

+ - f ( _ 1)~ ~b(2k((2k)!) ~ ' +1) (½ r~)~~

Further (d2 2d) ~-~ + 7~rr

- l n ( ½ r ) b e i ( r ) - ¼nber(r) E ( - - l ) k ~h(2k + 2) (½ r2)2,+ , k=0 ((2k + l)[) 2

-?

d2 2 d ~r~ ÷ 7 ~r

W~ =

- y + ~ k-' k~l

1

(n >~ 1)

--

e -~r

4n - -r

W~

W2 "~- -- WI

and n-~

~O(n) =

W2(r) =

(¼ r2)~ -

~=o

where ~h(l) =

W~(r) = ~ s i n a r ,

with~ = - x/~"

÷

1 ~ k e r ( I x - y])

- e- ~'

"''

~ + ~ "'" - I n (½ r) ber(r) + ¼ n bei(r)

Ker(r) =

Kei(r) =

y) =

sin ~r

e-" ,, cos~r F

140

D.B. Ingham, H. Han

is a fundamental solution of the equation N W + W = 0 in three dimensions. Hence we let

where ( 1,

x ~ fL

1 e -~r --

G~I(X, y)

4~t - r-

COS ~ r

1 e ~

~

/¢

sin ~r (13)

G~(x, y)

-

4n

r

G~(x, y)

-

4~ ~ r

G22(x' Y)

-

4n - -rc o s ~ r

l e~r

sin ~r

=

~ ~-~,

x e F,

where ~ is the angle included between the tangents of F on either side of x. Similarly, we have xuz(x)

:

le="

( OG2, 0u,

fv u~ ~ . - fr

where

(

)

~ . ~ G2, ds~..

aG22 au2 u ~ + ~ .G , ~

)

ds,.~

(16)

Combining the formulas (1 5) and (1 6), we have

~ _

~

~

=

Ix - ,I

{u,{x), u~{x))

=

f~

;~u,,~u,) ~&,. ~ / ~ , , .

~

.

.

"

-- ~ ~,, u , ~dG s,,,

an~

~x ~ ,

G{x, y) = ~G~ G=/

07)

is a fundamental solution of the eqns (4).

and

(~u, au2)

x(<(x), u2(x)) = ~ v an~' &,.J Gds.~. 4 GREEN'S

FORMULA

aG - ~ (u~, u2) ~ ds,.,

Suppose ~ e R2 with bounded boundary F, (u~, u2) is a solution of the eqns (4), i.e. A u , ( y ) + u2(y) = 0, Au~(y) - u~(y) = 0,

in~} in

(14)

for any point x e ~, let & denote the disc with centre x a n ~ r a ~ i u s e a n ~ a ~ = n~K~. By Green's formula we have ff(--UlGl

2 +

~,:

(<(x),u=(x))

=

ff(&' ~u~) ~ ~ dn.,.' On.~.J G ds.~.

OG ff (<, .p ~ d<,,

-

(

~

F°rthethree-dimensi°nalcase, supp ° s e O e u ~ w i t h bounded boundary F. For any solution (u,, u2) satisfying the equations (4), we have

u2Gll)dy

(U2Gll

-

F

)

= fv~ u, ~

any G,, dsy

k On,.' an,./G

} (u, (x), u2(x)) =

Vxea

.

u~Gi2)dy

ds.,,

8G

.

= ~: u2 ~

Vx e ~, (18)

-

)

~

ff(u~ F

G~2 dsy

u2)~ds,. ~

~*

Vx6F '

where G is given by (13) and the boundary F is smooth.

Moreover we may obtain 5 DIRICHLET ~

~

~

- fz, ( u~ dGI, ~

In this section we consider the boundary element approximations of the following equations

~aul G , ~ ) dsy = 0

On letting e + 0, we obtain ~GI2

xu~(x) = ~v uz ~ - ~,~

PROBLEM

-

~U2

any G~2 + ~

~,, ~s~

AUl + u: = 0,

in~}

A u 2 - u~ = 0,

infl

(19)

with the Dirichlet boundary conditions 05)

u,

one}

u~ = f2,

on r

~20)

An analysis of the boundary element method where f~ c ~: is a bounded domain with smooth boundary F and f,, f~ are two given functions on F. We know that the fundamental solution of the eqns (19) is

( G~,(x, y) G,z(x, y) )

at x = y and the result for the Laplace equation Au = 0 in Ref. 6, then the inequality (25) follows immediately. We now only need to prove the inequality (26). For any p ~ H ~/2(G),let

~r p(y)G(x, y)ds,.,

~b(x) =

G(x, y) = \ Gz~(x, y) G~ (x, y)

141

(27)

then ~(x) = ( ~ (x), ~ (x)) satisfies the eqn (19) on ~ ~F and

where ,

G~(x, y) = ~ Ker (Ix - y[) a~(x,

y)

=

a~ (x, y) =

~

K e i (lx -

-a~(x,

=

yl)

y)

O

,

Ixl ~ + ~

i =

Ixl ~

i,j

+ ~

1, 2 =

I, 2

Hence

G~,(x, y) = G,~(x, y)

~i(x) s W~(~2),

Let the solution of the problem (19)-(20) be

~(x) = f~ p(y)6(x, y)~s~, where u(x) = (u~(x), u~(x)) and to be determined on F. Introduce the spaces IHI'/2(F) =

=

0x~

,

O

where

(2~)

Vx ~ ~

i = 1, 2,

w ' ( ~ :) = {u ~ e ' ( ~ ) ; (~ + ?)-'/~

p(x) = (p,(x), p2(x)) are

x (1 + log ~

+ 1)-'u ~ L~(~);

grad u ~ (LZ(~ ~))2}

(v = (v,, vz)~ v,, vz e H~/2(F))

with n o ~ U2

IHI-'/z(F) = (v = (v" ~2)' ~" z~ e H - ' ~ ( F ) ) where H ' ( F ) denotes the usual Sobolev space on boundary F (see Ref. 5). By the boundary condition (20), we obtain the following integral equations to d e t e ~ i n e the density p(x):

Ilu[l~,(~)

=

~ dx ~ (fl + 1)(1 + L o g ~ r Z + 1)~ + ~ Igrad ul~ dx

~ by the Hardy inequality we know that the seminorm

~r p(y)G(x, y)ds.~, = f(x), ~x ~ r

(22)

furthermore, the integral eqn (22) is equivalent to the following variational problem: Find (p~, p~) = p ~ H-~/Z(F), such t h a t )

f p( y)G(x, y)r(x)~ ds~.ds~ = 2rffrr ds~' ~ Jr ~r

=

(fi,~l)

6H

(~

lU]w~n~) =

lgrad

ulZdx)~/~

is equivalent to the n o ~ Ilullw ~ in the quotient space W~(~Z)/Po, where P0 = {u, u is a constant on ~ } (see (23)

~/~(F)

Ref. 7). By the following equations A ~ + ~ = 0, inRZ~F A~ - ~

wheref = (~,fz). We now let

=

0,

inR:~F

We have

Ao(p, ~) = fr fr p(y)G(x, y)~(x)rds~ds~

(24)

~f(Igrad~,~ ~ + ~grad~l ~)dx R2

and f o forl this l bilinear o w i f on~ gAo(p, L ~), e we m have m athe :

{[~] =

Lemma 5.1: Ao(O, ~) is a bounded and coercive bilinear f o ~ on H-'/~(F) x H--'~(F), i.e. there M0 > 0, eo > 0 such that

are two constants

(i) IA0(0, ~)1 ~ m~ll01~-~.r~lrll-~.r, V0, ~ e H-~/~(F) (25) ~ H ~ ( F ) (26) (ii) A~(p, p) ~ e~llPll~¢~.r, Vp ~

fr

4~ +

[ 0 }4 ~~]J n ~

ds

Hence [. = ~.~(Igrad~'l ~ + Igrad~l~)dx ~ ~ll~ll~ze.r

Ao(p,O) = f r 4 " p d s

(28)

with a positive constant ~. The last inequality follows from the trace theorem, s On the other hand, from (27) we obtain an operator ~ : H - ~ ( F ) ~ H~/~(F), i.e.

Proof" From the singularity of the fundamental solution G(x, y)

~P

~

~(x) ~

~r O(Y)G(x, y) ds,~.. Vx e F

(29)

142

D.B. Ingham, H. Han

It is straightforward to check 3ff is bounded and, furthermore, 3f" is a bijective mapping. In fact if we suppose that there are two elements p, z • H-I/2(F) such that 3ffp =

3fir

holds lip - Phll-i/z,r < Mo inf lip - Phil l/2,F ~0 ~h~Vh

(33)

where p is the solution of problem (23).

i.e. ~(x) = fr (p(y) - z(y))G(x, y) dsy =- O, Yx • R 2,

then the equations

Proof." From the problems (23) and (32), we obtain Ao(p - Ph, rh) = O, VZh • Vh

A~I -- ~2 = 0,

inR E

A~2 + ~1 =

inR 2

0,

and hence Ao(p -- ph,p -- ph) = Ao(p -- Ph, P -- rh), V~h•Vh from an application of Lemma 5.1. Then the inequality (33) follows immediately. We now assume that the boundary F offl is represented as

for any v • WI(R 2) x WI(R2), we have 0 = ff (grad ~1" gradvl + grad ~2 ° gradv2 R2

+ ~2 v, - ~ , v z ) d x =

[_(p-

dl

z)'vds

"~)o ~ ) d s

0,

Vv

= xl(s)l

X2

X 2 (s) )

0 <-_ s <= L

and xj(0) = xj(L), j = 1, 2. Furthermore, F is divided into some segments {T} by the points x i = (xl(s~), x:(si)), i = 1, 2, 3 . . . . . N with s~ = 0, Sn+ 1 = L. Let

i.e. fr(p

x,

HI/~(F)

h =

Hence p -- r in H-I/~(F) Thus we have proved that ~ is a bijective mapping. Moreover the Banach theorem implies the continuity of the inverse o,~ -~' HI/~(F) -~ H l/~(F). Namely,

max Is~+l - s~l

I<~i<~N

The partition of F is denoted oCh. Let S~ = {Ph, Phlr is a constant, V T • o~h}

~< I1~ -t 114~ll~/zr (30)

V = Sh X Sh The Vh is a subspace of H-~/2(F) and a regular finite element space in the sence by Babuska & Aziz 8 that satisfy the following approximation property:

where ~b is given by (29). Combining results (28) and (30), then the inequality follows immediately with P ~0 = i i ~ _ t I-------~> 0

inf lip - Oh II-i/~,r <~ CohllPlll/~,r (34) ~h~v~ where Co (later C1, C~ . . . . ) denotes a constant which is independent of h. Combining (33) and (34) we obtain:

By applying Lemma 5.1 and the Lax-Milgram theorem we obtain the following theorem:

Corollary 5.1 Suppose the solution of the problem (23), p • H-~/~(F) then the following error estimate holds

IIp(x)ll ~/2,r --

II~-~ll-~/~,r

Theorem 5.1 For any f • HI/2(F), the variational problem (23) has a

unique solution p s H ~/2(F) and (31)

Suppose that Vh is a subspace of H-~/2(F) with dimension 2N. Consider the discrete problem:

Ao(ph, zh) = ;r f ' z ~ d s ,

~0

'

(35)

From the approximate solution Ph, we can get the ap-

Ilpll-1/2,r <~ Mo Ilflll/z,r ~0

Find 0h ~ Vh, such that

__

lip - Ph 11-1/2 ~< C0 M0 Ilpll-~/2 rh

1

(32)

Vzh • Vh

For the problem (32) we have the following estimate: Theorem 5.2 For any f • H~/2(F), the variational problem (32) has a unique solution Ph ~ Vh and the following inequality

proximation of the original problem (19)-(20). Letting uh(x) = fr ph(y)G(x, y)ds~,

Vx • F

we have u(x) - uh(x) -- fr (P(Y) - ph(y))G(X, y ) d s and

[lu(x) - uAx)lll,~ <~ C, Ilu(x) - u~(x)lll/2,r ~< C, I1~11

lip - mll-~/=,r

<~ CoC, II~¢ll M0~0[Ipll ~/:,rh

(36)

An analysis of the boundary element method By the equations

Thus we have:

Theorem 5.3 Uh(X) is an approximation of the original problem (19)-(20) and the following error estimate holds

A~GI1 4- Giz =

x # y,

A~GI2 - Gij = 0, x V= y, x # y,

AxGz2 - Gzl = 0, x # y,

with

the equality (41) follows immediately. Inserting (42) into (39) and integrating by parts, we have

CoC, II~ll M° Ilpll ,/2,r

C =

~0

6 NEUMANN

Ou(x) d dp(y) = dc fr G(x, y)ds v Onx

PROBLEM

- f~ n~" nyp(y)G*(x, y)dsy.

We consider the following boundary value problem: Au~ + uz =

0,

inf~ J

A u 2 - ul =

0,

0U 1

infl~

l

0U 2 =

(37)

=

Al(p, z) = -- 0i1 . g ' z d s , l ' j r p ( y ) OG(x, y) dsy,

Vx • £~

(38)

Vx • t~

From the boundary conditions in the problem (37), we obtain the following variational problem for determining the density p(x) on F: Find p(x) • H~/2(F), such that

d-'-n- = gz" on F / J

gl,

We now let

u(x)

0,

A~G21 + G22 = 0,

Ilu(x) - uh(x)ll,,~ ~ Ch

0--n--

143

] (42)

Vz • H~/Z(F)

where At (r, z) = fr fJr

dp(y)

dz(x) r

G(x, y)

ds*'

be the solution of the problem (37), where p(x) = (Pl (x), P2 (x)) is to be determined by the boundary conditions in the problem (37), and ny denotes the unit outward normal vector on the boundary F. For x • f2 and an arbitrary unit vector nx = (n~, n~), we have

and the nx, ny denote outward unit normal vectors at the point x, y • F. For the bijective form A~(p, z), we have:

i_ P(y) 02G(x, y) dsy, VX • ~ J~ OnxOny

Lemma 6.1 A~(r. z) is a bounded and coercive bilinear form on

du(x) dn~

(39)

For the derivative of the double-layer (39), we have:

Lemma 6.1 Ou(x)

On~

1

dt~ fr

G(x, y)dsy

(ii) A,(p, p) >1 oq Ilpll~/2.r

- fr n~" nyp(y)G*(x, y)dsy Vx • ~ (40) where ty = ( - n 2, n~), t~ = ( - n x2, nl) are two unit vectors perpendicular to ny and nx respectively, and G*(x, y) =

G,' I G2t/

- G22

Theorem 6.1

First we prove the following equality

02G(x, y) OtxOty

G*(x, y)nx" ny.

y)

Ilpllm.r ~< M--2 Ilgll-,/2.r

S~ =

Ot~Oty

[ O~G(x_,y) = L OxlOyl

(41)

4-

(43)

Supposed that Jh is a partition of F described in Section t 5. If we let

A computation shows that

o 6(x, y) + dnxdny

The proof of Lemma 6.1 is similar to the proofs given for Lemma 4.1 and Lemma 4.2 in Ref. 9 and hence are not presented here. By the Lemma 6.1, an application of the Lax-Milgram theorem yields the following theorem:

For any g e H-~/2(F), the variational problem (42) has a unique solution p • H-I/2(F), and

Proof"

02G(x, y) = OnxOny

H~/2(F) x Hm(F), i.e. there are two constants Mj > 0 and cq > 0, such that (i) IA,(r, r)l <~ M~ Ilpll~/2.rllrll~/2,r

dp(y)

( -G"

+ fr fr nx "nyp(y)G*(x, y)z(x)rdsxdsv

O~G(x,y) ] ~-x2~ .J nx'ny

= -n~'ny(A~G~/),

i,j = 1, 2.

w

=

{Ph;Ph• C°(F), PhlTis a linear function VT• ~ },, ×

then W is a subspace of H~/2(F) and a regular finitce element space in the sence by Bakuska & Aziz s thatt

144

D.B. Ingham, H. Han

satisfy the following approximation property:

governing equations reduce to

inf lip - rhll~/2,r ~< C2hllPll3/2.r (44) ~w~ We now consider the discrete approximation of the variational problem (6), namely Find ph(x) ~ W~, such that fr A~(ph, Vh) = -g " vhds, and then we have,

) (45) ~1~ ~ W~

Theorem 6.2

For any g ~ H-~/~(F), the variational problem (45) has a unique solution, furthermore the following error estimate holds lip -- ~h II~/=,f ~< C2 M---L~Ilpll3/2,Fh ~ By the Lemma 6.1 and the Lax-Milgram theorem the conclusion follows immediately,

7 DISCUSSION In this paper the general formulation of the unsteady heat transfer equation in two and three dimensions with periodic heating has been presented. In the cases of the situations where the boundary conditions are either Neumann or Dirichlet the error estimates have been obtained. However, in most practical applications of this work the boundary conditions are of the mixed type and an error analysis of this situation is at present in progress. Although in this work we have concentrated on the heat conduction equation, the same general conclusions may be drawn for other applications where the governing equation is the same, i.e. the mass diffusion equation, etc. It is hoped that in a later paper that we will extend the above analysis to some oscillatory fluid flow problems where the inertial terms are very small, i.e. when the Reynolds number of the flow is small. In this case the

0~o - V2co 0t VZ~ = ~o

(46) (47)

where ~o is the vorticity ~ and the streamfunction. Although eqn (46) is identical to the governing equation considered in this paper, eqn (47) is different. Further, the two equations are normally linked through the boundary conditions which are usually either on ~ or the normal derivative of q~. Thus the work presented in this paper is not directly applicable. REFERENCES

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