Acta Mechaniea 77, 121--129 (1989) A C T A M E C H A N I C A �9 by Springer-Verlag 1989

A Note on the Porous-Wavemaker Problem

By

A. Chakrabarti, Bangalore, India

(Received ~February 10, 1988; revised July 19, 1988)

Summary

A complete analytical solution is obtained, by using an integral transform method, for the porous-wavemaker problem, when the effect of surface tension is taken into account on the free surface of water of finite-depth in which surface waves are produced by small horizontal oscillations of a porous vertical plate. The final results are expressed in the form of convergent integrals as well as series and known results ~re reproduced when surface tension is neglected.

1. Introduction

A porous-wavemaker theory has been worked out by Chwang [2], by way of analysing the small-amplitude surface waves on water of finite depth which are produced by small horizontal oscillations of a porous vertical plate. By employing an impedence type of boundary condition on the porous wavemaker, derived by using Taylor 's idea [8], and by neglecting the effect of surface tension on the free upper surface of the fluid, Chwang has reduced the wave-maker problem to a mixed boundary value problem for Laplacc 's equation and has obtained its solution by using an eigen-function-expansion procedure.

In the present note we have considered the wave-maker problem of Chwang with the extra effect of surface tension on the upper surface of the fluid in question, as studied by Packham [6] and Evans [3], in connection with other water wave problems. Introducing the effect of surface tension into the picture amounts, mathematical ly speaking, to solving a more complicated mixed boundary value problem in the sense tha t third-order derivatives also occur in the form of the boundary condition on the upper surface. This new mathemat ical problem, as well as the problem of Chwang, are shown to be amenable to an integral t ransform technique in which the Kernel of the defining transform consists of general trigonometrical functions of a type described in Sneddon's book [7]. Two different representations of the velocity potential describing the irrotational motion of the fluid are derived, one of which is in the form of an infinite integral

122 A. Chakrabarti:

and the other, an infinite series. The infinite-series-form of the solution fully agrees with the one obtained by Chwang, in the case when surface tension is

neglected.

2. The Boundary Value Problem

In terms of the notations of Chwang [2] and Packham [6], the physical problem under consideration can be translated into the following mixed boundary value

problem for the determination of the Velocity potential 6(x, y), describing the irrotational motion of the fluid, in the region x > O, 0 < y < h:

~2q~ 82r (i) ~x--- 7 + - - - ay 2 - - 0 , in ( x > O , O < y < h ) ,

(ii) . . . . . . . ~r 2i be~~ qb = io)d, on x -~ 0, 3x /~

(iii) - - = 0 , on y = O , ~y

~3~ ~ (iv) - - n ~ + ~Y m~b 0, on y h~.

and

(V) ~ ~ e--lk~ as x -+ 0~.

We have dropped above the t ime-dependent factor e i~t all through, and have

associated ourselves with the following meaning of the various symbols:

(a) m = ~o~/g, (dpae ~Jt, eo = circular frequency),

(b) n = T/eft, (T = surface tension, ~ = density, g = acceleration due to gravity),

(c) k0 is a positive root of the transcendental equation

ko(nko 2 + 1) sinh (koh) - - m cosh (koh) = O, (1)

(d) s0 = de i~t, denotes the small horizontM displacement (d ~ h) of the porous vertical plate at x ---- O, producing the corresponding velocity and acceleration

as given by

Uo = i(~de i ' t , ao -~ --w2de r

respectively.

(e) /~ is the dynamic viscosity of the fluid, and

(f) b is a constant having the dimension of length.

In addition to the above boundary and infinity conditions, appropriate edge conditions must also be imposed at the edge x = 0, y = h, where the upper

On the Porous-Wavemaker Problem 123

surface of the fluid meets the wave maker (see Murray [5]), in order to ensure uniqueness of the solution of the above boundary value problem. The edge condition that we need in our approach is tha t

(~) - = - - 2 i ~ = o = 2 i - - ~ , tt ~x 8y # ~y

at x = 0, y = h, compatible with the condition (ii), on the boundary x = 0. The physical meaning of this edge behaviour is tha t the free surface is assumed

to be in fixed contact with the wave maker and that this surface also remains perpendicular to the wave maker for all time. This observation is supported by the free surface condition

(vii) ~--~-~ = ~-~r (d = h), ~t ~y

where ~(x, t) represents the surface elevation. More complicated water-wave problems using more general edge-constraints

have been investigated by Benjamin and Scot [1]. In the present note we have utilized the simplified edge-constraints as given by (vi).

3. The Method of S o l u t i o n

We first observe that the function

~o(X, y) = Ao cosh (/Coy) e -ik~ (2)

in which A0 is a constant, and the positive real number/co satisfies the Eq. (1), is a special solution of the boundary value problem posed by the relations (i) to (v), minus the condition (ii), on x = 0. As a mat ter of fact, the function ~b0 represents a surface wave on the surface of water of finite depth, when surface-tensional forces are also taken into account.

We then split the unkown function ~(x, y), in question, into the following form:

~(x, y) -- ~0(x, y) -~ ~(x, y), (3)

where the function ~b(x, y) and the constant A0, will have to be determined satisfying the following requirements:

~2~ ~2~ (i)' - - + - -

~x 2 ~y2 = 0 , in ( x > O , O < y < h ) ,

~p b~o (ii)' - - -- 2i

5x /t = iwd -~ iAo(ko -~ 2b~(o/#) eosh (/COY), on x ~ 0,

124 A. Chakrabarti:

@ (iii)' - - = 0 , on y = O ,

Oy

(iv)' - ~ ~y-~z~ + ~-~ -

(v) ' ~ - + 0 , as x - > ~ ,

m) ~b = 0 , on y = h ,

and

) (vi)' ~ ~ x -- 2ibQw/#~ = iAlko(ko ~- 2b~o~/#) sinh (koh), at x = O, y = h.

In what follows we shall describe an integral transform technique, similar to the one explained in Sneddon's book [7], to attack the problem posed through the relations (i)' to (iv)'. We define an integral transform T($, y) of the function ~b(x, y), as given by

o o

0

Multiplying both sides of the partial differential equation (i)' by the Kernel

(~ cos ~x-~ 2 i b ~ sin~x) and integrating with respeet to x, we easily

when the boundary condition (ii)' is also utilized, the following ordinary differen- tial equation for the function T($, y) :

d2T

dy~ -- $2~V = $[ia~d + iAo(ko + 2b~w/#) cosh (k0y)]. (5)

The solution of Eq. (5), satisfying the transformed boundary conditions (iii)' to (iv)' is next obtained in the form:

with

T(~, y) = C1 cosh @ -- - - iwd iAo~(ko +2b~(o/#)

+ cosh (koy) (6) (~o~--~ ~)

C1 : --[imwd]/(~[~(n~ 2 A- 1) sinh th -- m cosh ~h]). (7)

Our next job is to determine an appropriate inversion formula for the transform defined by the relation (4). We observe that

I ] T($, y) = F~ - - - z - A- 2i ?; x - ~ ~ , (8)

where F~[/(x); x---~] represents the Fourier sine transform (see Sneddon [7]),

On the Porous-Wavemaker Problem 125

as defined by oo

F~[/(x) ; x --+ ~] ----- f / ( x ) sin ~x dx. (9) 0

Using Fourier sine inversion formula to the relation (8), we then arrive at the relation:

2 ~ 8r 2i b@o# ~ -- z ] T(~, y) sin (~x) d~, (10)

0

and, then solving the differential equation (10) we obtain

2 / ~P(~, y) d~ /~ @(x, y) -= Co(y) e 2ibe~''x/~ -~ --~ (~ _ 4b,@2o~,/#2)/r co~ ~x ~-

0

2ib@o~ ) sin~x , (11)

~u

where Co(y) is an arbitrary function of y, to be determined and where singular integrals are understood as their Cauchy principal values.

The formula (11), with Co(y) fully determined, will therefore serve as an inversion formula for the integral transform defined by the relation (4), and this, in turn, will determine the complete solution of the boundary value problem (i)' to (vi)', if the constant A0 is also determined in the process. In order to achieve this goal, we proceed as follows.

(A) As observed earlier already, the function ~P(#, y) can be looked at as a

Fourier sine transform of the function ~b - - - - + 2 ~ @ - r with ~ -+ 0 as x -+ ~ , ~x #

satisfying Laplace's equation (i)'. Then, from the theory of Fourier sine trans- forms, we notice that we cannot have any singularity of the function T(~, y), treated as a function of the complex variable $, on the real axis. But, as is apparent from the expressions (6) and (7), we may have singularities of the function T at the points ~ = ~k0, which must not happen. This fact immediately suggests tha t the constant Ao in (6) must satisfy the requirement that

lim (~ -- ko) [iAo~(ko q- 2b@o~/#) ) ,-~k. [" ko 2 ~ ~ cosh (/Coy)

e-~k0 ~(~(n~ 2 Jr 1) sinh ~h -- m cosh ~h) cosh (~y) ,

i.e.,

/Ao (ko + 2b@~o/#) -=- 2

im~od

k o - ~ (~(n~ 2 + 1) sinh ~h -- m cosh ~h)j~=ko

126 A. Chakrabarti:

giving

2o~dPo A~ = ko2(1 + Go)[(1 + C~o~(3nko ~ + 1)) hi ' (12)

after a straightforward algebra, with

1 Po : sinh (koh), m = -~, 2b~o)//~ : Goko (see Chwang [2]). (13)

Thus the constant Ao gets completely determined and this agrees with the one obtained by Chwang, in the case when n = 0.

(B) The expression on the right of Eq. (11) must tend to zero as x -~ cr This fact, along with a standard contour integration procedure employed to evaluate the integral in Eq. (11) suggests that we must have that

i.e.,

Co(y) : - - ~ i - - lim ~(~, y)

Co(y) = --i~(2b~oJ//~, y) = --iT(Goko, y). (14)

The relations (12) and (14), along with the expressions (6) and (7) solve the problem completely, and we thus obtain the form of the function ~b(x, y), from Eq. (11) as given by:

O;9

O(z , y) = z J (8 3 - Go~ko ~) 0

with

(~ cosSx ~- iGoko sin ~x) d~, (15)

T($, y) = --io~d 1 @ (~(n~2 + 1) sinh Sh -- m eosh ~h

2Po cosh koy ~ ]

+ k0h(1 + CP02(3nko 2 + 1)) (ko ? -- ~2) "

(16)

An alternative representation of the function p(x, y), can be derived by evaluating the integral on the right-hand side of Eq. (15) with the help of a standard contour integration procedure and Cauchy's residue theorem. We easily find that

tp(x, y) = --2me)d

(x)

[

j : 1 I~j2 ~-~ ] -~- (~(n~ ~ ~- 1) sinh ~h -- m cosh ~h)j(~=ikj '

On the Porous-Wavemaker Problem 127

where k/s (j = 1, 2 . . . . ) are the positive roots of the equation

k(1 -- nk 2) sin (kh) + m cos (kh) -- O. (18)

Simplifying the denominator of the right hand side of Eq. (17) by using Chwang's notations,

( 1 tx

we find that

2~d ~ Pi cos (kjy) e-~ ~ ~b(x, y) -- h i ~ ki2[1 -- C(1 -- 3n]ci 2) pis] (G) -- i) ' (19)

with P~ = sin (kjh). In the particular case, when surface tension is neglected, i.e., when n = 0, we

get back the known results of Chwang. With ~b(x, y), as given by Eq. (19), and the constant A0, as given by Eq. (12),

we finally obtain the solution of our boundary value problem in the form of a convergent series and the computational aspects of the present general problem can be carried out in the lines similar to those of Chwang's work.

4. S o m e I m p o r t a n t R e s u l t s

Using Chwang's notations we find, for the present problem, that the pressure distribution on the wave maker surface is given by (taking real part only)

where

and

R e [ _ . . d~ - - R e [ ~ dh] = ~o(y) = C. ~ ~t + C~ sin ~,t.

oo 2Pj cos kiy v~ v "

=i~=1 k~h2[1 -- C(1 -- 3nki z) pj2] (Gi~ +4:- 1)'

2Po cosh koy Cq = kj2h2( 1 + Go)(1 + CPo2(3nko 2 + 1))

2P i cos kjy +i~= iv, kJ~h~[ 1 _ C(1 -- 3nkj ~) Pi ~] (Gj -4- Gi-1)"

(20)

(21)

(22)

The total hydrodynamic pressure distribution on the wavemakex', normalized with respect to the factor 0h2(--~o~d), as given by

P = C'~ cos wt + Cz sin cot, (23)

9 Aot~ Mech. 77]1--2

128 A. Chakrabarti :

with h h

CF = h - l f C~ody and Cz = h-1 f C q d y o o

(24)

can be derived easily f rom Eqs. (21) and (22). The surface elevat ion f rom the undis turbed level y = h, ~(x, t) is obta ined

( taking real p a r t only) in the form:

where

and

1 }) dg - ~ ---- Eo sin (]~0x - - cot) + ~ (Ej cos wt + Fj. sin o)t) e -~Jx,

j=l (25)

2/:)0 2

Eo : ko(1 -~- G0 ) (1 4- CPoe(3nko u + 1)) h ' (26)

2pi2(1 -- nkj 2)

Ej ~- hki[ 1 _ C(1 -- 3n]~ 2) p 2 ] (Gj2 + 1)' (27)

2pj2(1 - - nk~ 2)

F j ~- hki[ 1 _ C(1 - - 3 n k i 2) P i 2] (G i + Gs-* )" (2s)

The effect of surface-tension, i.e. the presence of the cons tant n in the forms of the various coefficients obta ined above is very much p rominen t and all the

results reduce to those of Chwang if we s imply subst i tu te n ~ 0.

Table 1. The w~riation o/CF and C L with G,/or m = .1, 1, 2.6, and n ~ 0.075. (The numbers within the parentheses correspond to the case n ~ 0). h = 1

m = 0 . 1 m = 1 m ~ 2 . 6

c~ eL cF eL c~ c,

0 0.0007174 3.1030498 0.0405391 0.7668046 0.1776196 0.286667

(0.0014724) (3.5640477) (0.0115162) (0.804375) (0.1162758) (0.271781)

1 0.0003587 1.5744575 0.0202695 0.2209365 0.0888098 0.143 672 (0.0007362) (1.8464718) (0.0057581) (0.1902478) (0.0581379) (0.096726)

2 0.0001435 1.0548003 0.0081078 0.1316542 0.0355239 0.101381 (0.0002945) (1.2459206) (0.0023032) (0.1088002) (0.0232552) (0.067279)

5 0.0000276 0.5299768 0.0015592 0.0585573 0.0068315 0.047113 (0.0000566) (0.6305508) (0.000443) (0.0474042) (0,0044721) (0.031067)

10 0.0000071 0.2897188 0.0004014 0.0302629 0.0017586 0.024213 (0.0000146) (0.3458465) (0.000114) (0.024388) (0.0011512) (0.015936)

On the P o r o u s - W a v e m a k e r P r o b l e m 129

The numerical evaluation of the various quantities can be carried out in the

lines similar to those of Chwang and we have presented just one set of results

here in Table 1, showing the variations of CF and CL with G, for h --=- 1, and for three different choices of the parameter m, after fixing n ~ 0.075 which is a representative of the water-air interface at the free surface (see Packham [6]). The results for n ~ 0 are shown within parentheses in the same table.

5. Conclusions

The integral t ransform as defined by Eq. (4), applied in a straightforward manner to the mixed boundary value problem under consideration, has produced the results in neat computable forms for the general porous wave-maker problem when the effect of surface-tension is also taken into account. The known results of Chwang [2], when surface-tension is neglected, have been reproduced as a particular case of the general situation considered here.

References

[1] Benjamin , T. B., Scot t , J . C. : Grav i ty -cap i l l a ry waves wi th edge cons t ra in t s . J . F lu id Mech. 92, 241- -267 (1979).

[2] Chwang , A. T. : A p o r o u s - w a v e m a k e r theory . J . F lu id Mech. 132, 395- -406 (1983). [3] E vans , D. V. : The effect of surface t ens ion on t h e waves p r o d u c e d b y a heav ing circular

cyl inder . Proc . Camb. Phil . Soc. 64, 833- -847 (1968). [4] Jones , D. S.: Me thods in e l ec t romagne t i c wave p ropaga t ion , p. 409. Oxford :

C la rendon 1979. [5] Muray, J . C. : On the capi l la ry-gravi ty wave-maker problem. Ac ta l~[echanica 24,

289--295 (1976). [6] P a c k h a m , B. A. : Capi l l a ry-grav i ty waves aga ins t a ver t ica l cliff. Proc . Camb. Phi l .

Soc. 64, 827- -832 (1968). [7] Sneddon , I . N . : The use of in tegra l t r ans fo rms , pp . 70- -75 . N e w Delhi : Tara McGraw-

Hill 1974. [8] Taylor , G. I . : F lu id f low in regions b o u n d e d b y porous surfaces. Proc . R . Soc. Lond .

A 234, 456- -475 (1956).

A. Chakrabarti Department o/Applied Mathematics

Indian Institute o/Science Bangalore 560 012

India

Vertagsort: Wien. - IIerstellungsort: Alteuburg, DDR. -- Printed in the German Democratic Republic. Verleger: Springer-Verlag KG, MSlkerbastei 5, A-1010 Wien. -- Herausgeber: Prof. Dipl.-Ing. Dr. Hans Troger und Prof. Dipl.-Ing, Dr. Franz Ziegler, beide Technische Universit~t, Karlsplatz 13, A- 1040 Wien. -- Redaktion: Karlsplatz :13, A-1040 Wien. -- ttersteller: VEB Druckhaus ,,Maxim Gorki", DDR-7400 Altenburg 5001/006/89

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