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Reflection of capillary-gravity waves
WAVE MOTION 9 (1987) 217-226 NORTH-HOLLAND
R E F L E C T I O N OF C A P I L L A R Y - G R A V I T Y
217
WAVES
L.M. H O C K I N G Deportment of Mathematics, University College London, London WCIE 6BT, United Kingdom Received 18 August 1986, Revised 5 December 1986
The reflection of a capillary-gravity wave by a partially immersed obstacle depends on the conditions applied at the contact line between the free surface of the fluid and the boundary of the obstacle. The contact angle varies with the speed of the contact line relative to the boundary and a simple model of this variation is used to determine the reflection and transmission coefficients for a vertical plane barrier of finite depth. The incident plane wave has its crests parallel to the plate and the fluid is of infinite depth. The varying contact angle requires that energy be dissipated at the edge, and the proportion of the incident energy that is reflected, transmitted and dissipated is also calculated.
1. Introduction The scattering o f a gravity wave on the surface o f a fluid by a solid obstacle is one o f the classic problems o f wave motion. A natural extension is to examine what effect surface tension has on the scattering, but very little attention seems to have been devoted to this topic. O f course, in m a n y realistic situations surface tension effects are negligible, but precise experimental studies on a laboratory scale do require t h e m to be taken into account. N o r m a l l y the restoring force o f gravity greatly exceeds that o f capillarity, except when very short waves are present or when the effective gravity is reduced, as, for example, at the interface between two immiscible fluids o f nearly equal density. The effect o f surface tension on the p r o p a g a t i o n o f surface waves in u n b o u n d e d regions has been studied extensively [1-3], and its primary effect is to change the dispersion relation. In the presence o f lateral boundaries, however, there is an a d d e d complication. The presence o f surface tension adds a term to the d y n a m i c free-surface condition which is p r o p o r t i o n a l to the m e a n curvature. The differential order o f this condition is therefore increased and it is necessary to specify conditions at the line o f contact between the fluid and the b o u n d a r y to
close the problem. For pure gravity waves such a condition c a n n o t be specified and the free surface intersects a vertical b o u n d a r y orthogonally. But there is no reason to suppose that the same condition should hold when surface tension is present; even if a 90 ° contact angle exists u n d e r static conditions, this angle will change when the surface is moving. In fact, it m a y well be that the contact line does not move relative to the b o u n d a r y . Such a possibility was examined by Benjamin and Scott [4], who were the first to m y knowledge to consider capillary effects on the p r o p a g a t i o n o f surface waves in the presence o f boundaries. They dealt with the progression o f a wave along a rimfull channel and applied the condition that the edge of the free surface remain fixed along the rims o f the side walls. As well as the two extreme cases when the contact line can move freely up and d o w n the b o u n dary, maintaining a 90 ° contact angle, or remain fixed, with a contact angle that varies t h r o u g h o u t the motion, there are intermediate possibilities. The details o f fluid d y n a m i c s in the vicinity o f moving contact lines contain m a n y obscure points and difficulties [5]. Without considering the microscopic-scale physics o f the processes involved near the contact line, it is justifiable, in some contexts,
0165-2125/87/$3.50 O 1987, Elsevier Science Publishers B.V. (North-Holland)
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L.M. Hocking / Capillary-gravity waves
to postulate that their macroscopic effect can be accounted for by a contact angle that exhibits both a dynamic variation with the speed of the contact line and contact-angle hysteresis. The typical behaviour of the contact angle for an air-fluid-solid contact is sketched in Fig. 1. The contact angle
corlt~ic t m;;le
Pet Peat in :
Y speed
a d w ~ n c J rlr"
Fig. 1. Dynamic contact angle with hysteresis.
measured in the fluid increases with the speed of advance of the fluid into the air and decreases when the fluid retreats. This dynamic variation of the contact angle is accompanied by a discontinuity between the minimum advancing angle and the m a x i m u m retreating angle. This hysteresis implies that there is no single static contact angle, but a range of possible static angles. With this model for the behaviour of the contact line, it follows that an oscillatory motion of the free surface will produce a so-called "stick-slip" motion of the contact line. The contact line will remain at rest until the slope increases sufficiently far for it to move up the boundary. As the speed decreases, the contact line will stop moving and remain at rest until the slope has decreased far enough for it to begin to move down the boundary. This motion will again be followed by an interval during which the contact line remains at rest. If the hysteresis is sufficiently large, as it will always be for waves of infinitesimally small amplitude, the contact line will remain at rest throughout the whole of the oscillation, as in [4]. An important feature of the contact-angle variation is that the physical
processes in the vicinity of the contact line involve a dissipation of energy there [6]. The edge condition enables this dissipation to be calculated without reference to the details of the motion on the microscopic scale near the contact line. An edge condition that incorporates the dynamic variation, but not the hysteresis, was used in [6] to calculate the frequency and damping of standing waves between two parallel vertical walls. The contribution to the damping from the edge condition is sometimes much larger than that produced by viscosity. Young and Davis [7] studied the motion of a vertical oscillating plate partially immersed in fluid with both the dynamic variation and the hysteresis of the contact angle included in their analysis. In the parameter range studied, there was no coupling between the induced fluid motion and the position of the contact line, to leading order. With an increase in the size of the surface tension parameter, the fluid motion and the contact-line motion have to be determined simultaneously and this problem has been solved in [8]. When the contact line can move freely along the plate, there is, of course, no resulting fluid motion, but the stick-slip motion resulting from the contactangle behaviour generates a radiated wave, as well as dissipating some of the energy introduced by the forced motion of the plate, and the amplitude of the radiated wave was calculated in [8]. It is natural to follow the determination of the radiated wave from an oscillating plate with a study of the scattering of an incident wave by a fixed plate. This problem forms the subject of this paper; the dynamic variation of the contact angle is included, but not the hysteresis. If the dynamic variation is assumed to be linear, and there is no hysteresis, the edge condition is linear and the solution becomes much simpler. Although hysteresis of the contact angle is a nearly universal feature of real materials, the results obtained here should be applicable to cases when the hysteresis is relatively weak and also when it is very strong. For in the latter case, the contact line will not move and this possibility is included as a limiting case of the dynamic-variation model. The particular
L.M. Hocking / Capillary-gravity waves problem studied here is for a plate partially immersed in fluid of infinite depth. A plane wave of given amplitude is progressing along the surface of the fluid in a direction normal to the plate, so that the motion is two-dimensional. The mean static contact angle is taken to be 90 °, for simplicity, so that the free surface is horizontal in equilibrium. The linear dynamic variation of the contact angle, for small amplitude waves, has the form
Oil'~at'= A' o~'lox',
(1)
where x' is measured horizontally away from the plate, t' is the time and 71'(x', t') is the elevation of the free surface. The constant A' measures the strength of the dynamic variation; A ' = 0 corresponds to a fixed contact line (and so to large hysteresis) and with A' infinite the contact line can move freely along the plate with the contact angle fixed at 90 °. For a wave of given frequency, the parameters of the problem are the depth of the immersed portion of the plate, the relative strengths of capillarity and gravity and the edgecondition parameter A'. The quantities to be determined are the reflection and transmission coefficients, from which the radiated, transmitted and dissipated energies can be calculated. The waves are of small amplitude and the fluid motion satisfies linearized inviscid equations. Viscosity will be important in a Stokes layer on the wall and near the edge, but it was shown in [6] that the associated dissipation is of a smaller order of magnitude than that produced by the edge condition.
2. Formulation
Consider a fixed thin vertical plate immersed to a depth d' below the free surface of fluid of infinite depth. Choose horizontal and vertical coordinate axes with origin at the intersection of the plate and the free surface. If the incident wave has a wave length equal to 2~r/k, we can scale all lengths with k -1, so that the horizontal and vertical coordinates of a point in the fluid are k-l(x, y); the free surface
219
is given by y = 0 and the immersed portion of the plate by x = 0, y > - d , where d = kd'. The velocity components are taken to be (g/k)l/2(u, v), the time is (gk)-l/2t, and the pressure is (pg/k)p, where p is the density of the fluid and g is the gravitational acceleration. The linearized equations of inviscid fluid motion then become
Ou/Ot = -Op/Sx, av/~t = - a p / a y ,
8u/Ox + av/Oy = 0, which have to be solved subject to the conditions u,/9-->0 u=0
as y--> -o0, on
x=O,
0>y>-d.
If the elevation of the free surface above its equilibrium position is denoted by ~?(x, t), the kinematic and dynamic conditions at the free surface are given by
O~7/Ot=v o n y = 0 , -KoE~/0x 2=p
(2) ony=0,
(3)
where
K = yk2/pg and y is the surface tension. The edge conditions to be applied to ~7 where the free surface meets the plate are, as discussed in the introduction,
871/0t = +hO~l/Ox at x = 0 ± ,
(4)
where A = (k/g)l/EA'. The free-surface elevation for the incident wave is taken to be 7~i : e i ( ° t + x )
and the corresponding velocity and pressure are given by U : --O.~I eY,
v = io.~h e y, p : O'27~i e y,
where o.2 = 1 + K. At large distances from the plate
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L.M. Hocking / Capillary-gravity waves
the elevation has the forms ~7--r/l+~R
and for the o d d solution that
asx~+oo,
(5) 77 ~
T~T
as x ~ - ~ ,
as x ~ +oo,
(9)
where a and /3 are c o m p l e x constants that have to be determined. By taking a linear c o m b i n a t i o n o f these two solutions, we can express the freesurface elevation in the form
where ~TR= R e i(~t-x),
f o - - + c o s x +/3 sin x
"OT= T e i(~rt+x).
R and T are the c o m p l e x reflection and transmission coefficients, respectively, and, if ER and E r are the p r o p o r t i o n s o f the incident energy that are reflected and transmitted,
~7~ ei~'{A(cos x + a sin x) + B ( c o s x + /3 sin x)}
asx~+ce,
r / ~ ei"t{A(cos x - a sin x)
ER =IRI:,
E T = I T I 2. + B ( - c o s x + /3 sin x)}
In general, En + E r < 1, since the edge condition implies that energy is dissipated there. N o energy is dissipated when the contact line remains fixed ()t = 0), n o r w h e n it can m o v e freely along the plate (A = oo) [6]. In terms o f a velocity potential ~b(x, y), we can express the velocity c o m p o n e n t s and pressure in the forms
asx-->-o%
where A and B are constants. This form for the solution can then be written in terms o f progressive waves, p r o p o r t i o n a l to exp{i(crt±x)}, and we obtain a solution with the required asymptotic b e h a v i o u r at infinity p r o v i d e d that R = (1 + a/3)/{(1 - i a ) ( 1 -i/3)},
(lo)
U = ei~tocb/Ox,
T = i ( a - / 3 ) / { ( 1 - i a ) ( 1 - i/3)},
V = ei°~tO~/Oy,
where a a n d / 3 d e p e n d on the values o f the three parameters d, K and ~.
p = -krei~'d~" I f we define two functions f ( x ) and h ( x ) by f ( x ) = ~b(x, 0),
h(x) =
o6(x, o) Oy
3. Even solution
,
the free-surface elevation is given by r / = ei~'h/io -, and the d y n a m i c b o u n d a r y condition (3) b e c o m e s h - K d 2 h / d x 2 -- (1 + K)f.
(6)
W h e n ~b is an even function o f x, 04a/Ox = 0 at x = 0 for all negative y, and not just on the plate. Hence the solution is i n d e p e n d e n t o f d and can be f o u n d by a procedure similar to that used in [8]. We can take a Fourier cosine transform o f the even part o f ~b and write it in the form
The edge conditions are then given by h = :t:(iA/tr) d h / d x
at x = 0+.
(7)
The solution o f the p r o b l e m so p o s e d can be f o u n d by considering two standing-wave problems, in which ~b, f and h are respectively even a n d o d d functions o f x. If we write f = f e + f o , then, for the even solution we require that fe ~ cos x ± a sin x
as x -->+oe,
(8)
4~ =
fo
F ( k ) cos kx e ky dk,
so that fe(x) =
;o o F ( k )
cos kx dk.
(11)
The value o f the even part o f h can then be f o u n d from the d y n a m i c free-surface condition (6), which
LM. Hocking / Capillary-gravity waves
221
the singularities at k = 1 and we find that
can be solved to give
f ~ - cos x -2Ce{K1/2/(1 +3K)} sin x,
he(x) = (1 + K) Io'~ 1F(k) +--~2 cos kx dk
so that (see (8))
+ Ce e-XlK'/~/io', where Ce is a complex constant. The edge condition (7) then shows that Ce-
I+K 1-iJK-1/2(1
(12)
From the kinematic free-surface condition (2) we obtain an alternative expression for he, namely
;o o kF(k) cos kx dk,
and equating these two representations of he, we find that {k(1 + Kk 2) - 1 - K } F = 2K1/2Cd~r, so that
2K1/2 F - ~(k - 1)(Kk2 + Kk+ K + 1) Ce+ ~(k - 1), where the delta-function is introduced to provide the correct form for 7/at infinity. When this value of F is substituted in (12), we obtain
Ce{JI(K) -iA (1 + K) -1/2} = - K 1/2, where Jr(K) =--2Kfo~
~r
k 1) dk (k-1)(Kk2+Kk+K+
The boundary condition at x = 0 on the odd part of $ has the form
aqb/Ox=O
{
K+I ½log K
for y < - d ,
and a Fourier transform method is not appropriate. Instead, we can use a conformal mapping in which the equilibrium free surface and the two sides of the plate form the real axis in the transformed plane. Since the normal derivative of ~bo is known on the plate from (14), it is possible to express the solution in terms of a source distribution on the transformed free surface, with a density given by ho(x). The problem can, therefore, be reduced to the solution of an integral equation for fo(X). A similar method could, of course, be used for the even part of the solution, but the Fourier transform method gives the solution explicitly. The dynamic boundary condition (6) can be solved for ho in terms of fo, and the solution obtained by variation of parameters simplifies to the form
ho(x)
I +K 2K1/2
ioo e-lx-x'l/r'/2fo(xt)
+ Co e -x/'°',
xtan-1 (3K~-+4) 1/2} -
dxl (15)
where Co is a complex constant. The required mapping to the (X, Y)-plane is defined by
3K+2 K1/E(3K +4) 1/2 -
for 0 > y > - d , (14)
=0
-
2K =l+3K
(13)
4. Odd solution
+ K ) -1/2
× Io 1 F(k) ~-K~- dk.
he(x) =
2K (1 +3K){JI(K) -iA(1 + K)-~/2}"
X 2_ y2=x2_y2+d2" .
X Y = xy, For large values of x, the most important contributions to the asymptotic value of fe come from
the free surface being then given by Y = 0, Ixl > d, and the plate by Y = 0, Ixl < d. The value of ho(x)
222
L.M. Hocking / Capillary-gravity waves
in the new coordinates gives the density of the source distribution on Y = 0, from which the velocity potential can be found. The value of ~bo on the free surface can then be determined and, when we revert to the original coordinates, we obtain the equation
The two integrals can be reduced to forms more suited to numerical evaluation by standard methods, and we find that 11 = - 2 ' r r - l ( x 2+ d2) 1/2
X
fo(X) = _ -1
S(x, Xl)ho(Xl) dXl,
x>O,
_acos _ 2 s e_,,oo h ds
X2q- d 2 c o s h
and (16)
12 = - - ' r r - l S ( x , 0 ) --
/2
where ×
S(x, y)
=
logJ(y2 + d2) '/2 - (x2+ d2)1/2 I _log](ya + d 2) 1/2 + (x 2 + d2)1/2];
we have used the fact that ho is an odd function to write the integral over a semi-infinite range. The solution of (15) and (16) has to be found, subject to the edge condition (7) and the required form at infinity, given by (9). If we write fo = cos x+/3 sin x+fo(x), ho = cos x+/3 sin x + ho(x), then we can solve for fo and ho, provided these functions are o(1) as x ~ o o . The form of (15) is unchanged, except for the value of the arbitrary constant, and we obtain
ho(x )
I0
I+K -2K1/2
;ooe-lX-x,I/K'/2fo(x 0 dx,
+ C o e -x/K'/2.
(17)
In terms of the new variables, (16) has the form
fo(x) = -'rr-'
S(x, xl)ho(x,) dxl
I/
+11+[312,
(18)
where 11 and I2 are real functions of x and d, defined by
S(x, xl) e ix' d x l - e ix.
I1 +iI2 = _ - 1 o
I;
2"rr-l(x 2 + d2) 1/2
d cos s -a cos x2+d2cos2s e Sds.
A n alternative form for the second integral is needed if x = 0. The integral operators in (17) and (18) can be written in finite-difference form using the trapezium rule, and the equations become two matrix equations for the vectors containing the unknown values of fp and hp at the mesh points. The solution can be written as the sum of three parts, one of which is proportional to/3 and one to C O. The value of C o can then be chosen so that the whole solution has the correct asymptotic form. The edge condition (7) has the form
gp(O)+l=-iA(l+K)-l/2(g'o(O)+/3 )
(19)
in the new variables and from this equation the value of/3 can be determined. Some care is needed in the numerical work to deal adequately with the singularity of the operator in (18). Tests showed that it was sufficient to truncate the integrals at x = 14, and quadratic extrapolation was used, after it bad been verified as appropriate for the numerical approximations employed. Most of the calculations were done with the integrals truncated at xl = 14 and with quadratic extrapolation from the results obtained using 35 and 70 mesh points. Sample calculations were done with more mesh points and with truncation at a larger value of xl and these tests indicated that the results presented here are accurate to within 0.1 per cent. Some care was needed to deal adequately with the singularity of the operator in (18). In order
223
L M . Hocking/Capillary-gravity waves
infinite depth, there is no transmission and so T = 0; the value of the reflection coefficient is then given by
to maintain the overall accuracy of the numerical scheme, the integral was evaluated analytically on either side of xl = x with a linear approximation to hp(xl) there.
R=(l+ia)/(1-ia), with ot given by (13). For A equal to zero or infinity, a is real and ]R] = 1; for other values of A there is some dissipation of energy and [R] < 1. For d -- 0, the limiting solution is not the same as if there were no obstruction to the incident wave, since the conditions at the origin still have to be satisfied. Without the control exerted on the motion at this point we would, of course, have R = 0 and T = 1. The odd part of the solution can be found by a Fourier sine transform, since the condition to be applied at x = 0 now has the same form for all y. I f we proceed in a manner analogous to the method used for the even solution in Section 3, we find that
5. Results
The values of the two quantities ot a n d / 3 were found for a range of values of the parameters d, K and A. The calculation of a from (13) is straightforward. To find /3, the numerical procedure described in the previous section was used; the unknown terms appearing in (19) could be found independently of A, and then is was an easy task to find/3 for any value of A. Once a a n d / 3 were found, it was possible to calculate R and T and the associated energies from (10). Certain limiting cases required a separate treatment, or could be treated more expeditiously. For
K = 10 /
0.8
K)I/2/2KA,
f. / i - ~
-//"
J,'
\;/
O.G
/3 = J 2 ( K ) + i ( 1 + 3 K ) ( 1 +
j//
./'/
///
// /
IRI
/
0.4
K=o.i
/
~ /
/
/
/ /
0.2
/ ,.,
/
/
/
/
/
/
/
I{
=
0
/
/
/ /
//
/ /
00--
/
/
/
/
/
/
/
8.?5
8,'5
e°!75
1'
1 ~75
! ,'5
1 o~75
d
Fig. 2. Modulus of the reflection coefficient as a function of the depth of the plate for different values of the capillary parameter K. The solid curves are for fixed contact lines (A = 0), and the dashed curves for free contact lines (A = oo).
L.M. Hocking / Capillary-gravity waves
224
where J2(K) =~--~
K+I -½K log-K 3K2+6K+2 K I / 2 ( 3 K + 4) 1/2
Both a and fl are now given explicitly and the values of R and T are easily calculated. A third limiting case is when there is no surface tension, and we set K = 0. The dynamic free-surface condition (6) now reduces to h = f and we no longer have the freedom to apply an edge condition. The value o f / 3 can be found by an adaptation of the method used in the general case and the reflection coefficient is now given by
R= l / ( 1 - i f l ) . There is no dissipation of energy in this case and the solution is just that for the reflection of a pure gravity wave by a plate of depth d. Some of the results obtained are shown in Figs. 2 and 3. In Fig. 2 the dependence of the modulus of the reflection coefficient on the depth of the plate is shown for three different values of K and for the extreme values of h. There is no dissipation in these cases, so that the transmission coefficient can be found from the equation IRI2+IT[ 2= 1. Fixing the position of the contact line on the plate increases the amount of the incident energy reflected c o m p a r e d with that reflected when the contact line can move freely along the plate. For values of d greater than 2 there is almost complete reflection. In dimensional terms, this means that there is very little transmission of energy past a plate that extends to a distance below the free surface greater than about one-third of the wave length of the incident wave. There is greater reflection of energy, for moderate values of d, when the contact line is fixed than when it is free to move, because of the increased constraint on the surface near the edge. For values of h that are finite and nonzero, the reflection coefficients are reduced because of the dissipation of energy at the edge. In Fig. 3 the
reflected, transmitted and dissipated energies are shown as functions of the edge-condition parameter h for small and moderate depths and for three different values of K. The lower curve gives the value of the reflected energy ER and the upper curve the value of 1 - ET, so that the gap between the two curves represents the dissipated energy. It can be seen that a large proportion of the incident energy can be dissipated by the processes occurring in the vicinity of the contact line. For small depths, as much as 90 per cent of the incident energy may be dissipated, and the plate can act as an efficient absorber of the wave energy. When d is infinite, the plate forms a complete barrier and there is no transmission of energy past it. The dissipation in this case is similar in its dependence on h and K to that for d -- 1, with values that are less than 10 per cent lower.
6. Conclusions
The results presented in this paper have shown that the problem of reflection of a plane gravity wave by an obstacle, even in the simplest case of normal incidence and a plane barrier, is made much more complicated when surface tension is not negligible. Not only is an extra parameter added to the problem, but it is also necessary to postulate a condition at the line of contact between the fluid and the plate. The form of this condition employed here contains some of the features of the known behaviour of fluid near moving contact lines. It has included the important special case in which the contact line remains fixed in its position on the plate. There are obvious extensions of the problem discussed here. The scattering of a wave by a cylinder is one possibility, in which a threedimensional solution would be required. Another is the inclusion of moderate hysteresis in which a "stick-slip" motion of the contact line would be present. These and other wave problems involving fixed and moving contact lines are currently under investigation.
LM. Hocking I Capillary-gravity waves
225
0.8 ~
+/ /
\ \ \ \ \ \ /"
i
-
~ ~"-..
ET ,
\
~
\
/ ~
/
ER
8.4
\\
0°2
'
//
'
// \
/ ,,
x\\
0.01
0.1
I
,
l I/
,
A
10
100
0.8
0.6 i-
ET, ER
\\.•
~// \
i
(J.01
)
0.1
i
/
, ' ,
,
i
I
i
X
i
10
i
100
Fig. 3. Energy partition as a function of A. The lower curves in each pair give the values of the reflected energy, the distance between the upper curves and the upper boundary represents the transmitted energy and the gap between them gives the dissipated energy. The solid curves are for K =0.1, the long-dashed curves for K = 1 and the short-dashed curves for K = 10. (a) d =0.1; (b) d = 1.
226
L.M. Hocking / Capillary-gravity waves
References [1] Sir W. Thomson, "Hydrokinetic solutions and observations", Phil. Mag. (4) xlii, 374 (1871). [2] S.J. Hogan, "Some effects of surface tension on steep water waves", J. Fluid Mech. 91, 167-180 (1979). [3] J.-M. Vanden-Broeck, "Nonlinear gravity-capillary standing waves in water of arbitrary uniform depth", J. Fluid Mech. 139, 97-104 (1984). [4] T.B. Benjamin and J.C. Scott, "Gravity-capillary waves with edge constraints", J. Fluid Mech. 92, 241-267 (1979).
[5] E.B. Dussan V., "On the spreading of liquids on solid surfaces: Static and dynamic contact lines", Ann. Rev. Fluid Mech. 11, 371-400 (1979). [6] L.M. Hocking, "The damping of gravity-capillary waves at a rigid boundary", J. Fluid Mech. 174, 327-356 (1987). [7] G.W. Young and S.H. Davis, "A plate oscillating across a liquid interface: Effects of contact-angle hysteresis", J. Fluid Mech., to appear. [8] L.M. Hocking, "Waves produced by a vertically oscillating plate", J. Fluid Mech., to appear.
R E F L E C T I O N OF C A P I L L A R Y - G R A V I T Y
217
WAVES
L.M. H O C K I N G Deportment of Mathematics, University College London, London WCIE 6BT, United Kingdom Received 18 August 1986, Revised 5 December 1986
The reflection of a capillary-gravity wave by a partially immersed obstacle depends on the conditions applied at the contact line between the free surface of the fluid and the boundary of the obstacle. The contact angle varies with the speed of the contact line relative to the boundary and a simple model of this variation is used to determine the reflection and transmission coefficients for a vertical plane barrier of finite depth. The incident plane wave has its crests parallel to the plate and the fluid is of infinite depth. The varying contact angle requires that energy be dissipated at the edge, and the proportion of the incident energy that is reflected, transmitted and dissipated is also calculated.
1. Introduction The scattering o f a gravity wave on the surface o f a fluid by a solid obstacle is one o f the classic problems o f wave motion. A natural extension is to examine what effect surface tension has on the scattering, but very little attention seems to have been devoted to this topic. O f course, in m a n y realistic situations surface tension effects are negligible, but precise experimental studies on a laboratory scale do require t h e m to be taken into account. N o r m a l l y the restoring force o f gravity greatly exceeds that o f capillarity, except when very short waves are present or when the effective gravity is reduced, as, for example, at the interface between two immiscible fluids o f nearly equal density. The effect o f surface tension on the p r o p a g a t i o n o f surface waves in u n b o u n d e d regions has been studied extensively [1-3], and its primary effect is to change the dispersion relation. In the presence o f lateral boundaries, however, there is an a d d e d complication. The presence o f surface tension adds a term to the d y n a m i c free-surface condition which is p r o p o r t i o n a l to the m e a n curvature. The differential order o f this condition is therefore increased and it is necessary to specify conditions at the line o f contact between the fluid and the b o u n d a r y to
close the problem. For pure gravity waves such a condition c a n n o t be specified and the free surface intersects a vertical b o u n d a r y orthogonally. But there is no reason to suppose that the same condition should hold when surface tension is present; even if a 90 ° contact angle exists u n d e r static conditions, this angle will change when the surface is moving. In fact, it m a y well be that the contact line does not move relative to the b o u n d a r y . Such a possibility was examined by Benjamin and Scott [4], who were the first to m y knowledge to consider capillary effects on the p r o p a g a t i o n o f surface waves in the presence o f boundaries. They dealt with the progression o f a wave along a rimfull channel and applied the condition that the edge of the free surface remain fixed along the rims o f the side walls. As well as the two extreme cases when the contact line can move freely up and d o w n the b o u n dary, maintaining a 90 ° contact angle, or remain fixed, with a contact angle that varies t h r o u g h o u t the motion, there are intermediate possibilities. The details o f fluid d y n a m i c s in the vicinity o f moving contact lines contain m a n y obscure points and difficulties [5]. Without considering the microscopic-scale physics o f the processes involved near the contact line, it is justifiable, in some contexts,
0165-2125/87/$3.50 O 1987, Elsevier Science Publishers B.V. (North-Holland)
218
L.M. Hocking / Capillary-gravity waves
to postulate that their macroscopic effect can be accounted for by a contact angle that exhibits both a dynamic variation with the speed of the contact line and contact-angle hysteresis. The typical behaviour of the contact angle for an air-fluid-solid contact is sketched in Fig. 1. The contact angle
corlt~ic t m;;le
Pet Peat in :
Y speed
a d w ~ n c J rlr"
Fig. 1. Dynamic contact angle with hysteresis.
measured in the fluid increases with the speed of advance of the fluid into the air and decreases when the fluid retreats. This dynamic variation of the contact angle is accompanied by a discontinuity between the minimum advancing angle and the m a x i m u m retreating angle. This hysteresis implies that there is no single static contact angle, but a range of possible static angles. With this model for the behaviour of the contact line, it follows that an oscillatory motion of the free surface will produce a so-called "stick-slip" motion of the contact line. The contact line will remain at rest until the slope increases sufficiently far for it to move up the boundary. As the speed decreases, the contact line will stop moving and remain at rest until the slope has decreased far enough for it to begin to move down the boundary. This motion will again be followed by an interval during which the contact line remains at rest. If the hysteresis is sufficiently large, as it will always be for waves of infinitesimally small amplitude, the contact line will remain at rest throughout the whole of the oscillation, as in [4]. An important feature of the contact-angle variation is that the physical
processes in the vicinity of the contact line involve a dissipation of energy there [6]. The edge condition enables this dissipation to be calculated without reference to the details of the motion on the microscopic scale near the contact line. An edge condition that incorporates the dynamic variation, but not the hysteresis, was used in [6] to calculate the frequency and damping of standing waves between two parallel vertical walls. The contribution to the damping from the edge condition is sometimes much larger than that produced by viscosity. Young and Davis [7] studied the motion of a vertical oscillating plate partially immersed in fluid with both the dynamic variation and the hysteresis of the contact angle included in their analysis. In the parameter range studied, there was no coupling between the induced fluid motion and the position of the contact line, to leading order. With an increase in the size of the surface tension parameter, the fluid motion and the contact-line motion have to be determined simultaneously and this problem has been solved in [8]. When the contact line can move freely along the plate, there is, of course, no resulting fluid motion, but the stick-slip motion resulting from the contactangle behaviour generates a radiated wave, as well as dissipating some of the energy introduced by the forced motion of the plate, and the amplitude of the radiated wave was calculated in [8]. It is natural to follow the determination of the radiated wave from an oscillating plate with a study of the scattering of an incident wave by a fixed plate. This problem forms the subject of this paper; the dynamic variation of the contact angle is included, but not the hysteresis. If the dynamic variation is assumed to be linear, and there is no hysteresis, the edge condition is linear and the solution becomes much simpler. Although hysteresis of the contact angle is a nearly universal feature of real materials, the results obtained here should be applicable to cases when the hysteresis is relatively weak and also when it is very strong. For in the latter case, the contact line will not move and this possibility is included as a limiting case of the dynamic-variation model. The particular
L.M. Hocking / Capillary-gravity waves problem studied here is for a plate partially immersed in fluid of infinite depth. A plane wave of given amplitude is progressing along the surface of the fluid in a direction normal to the plate, so that the motion is two-dimensional. The mean static contact angle is taken to be 90 °, for simplicity, so that the free surface is horizontal in equilibrium. The linear dynamic variation of the contact angle, for small amplitude waves, has the form
Oil'~at'= A' o~'lox',
(1)
where x' is measured horizontally away from the plate, t' is the time and 71'(x', t') is the elevation of the free surface. The constant A' measures the strength of the dynamic variation; A ' = 0 corresponds to a fixed contact line (and so to large hysteresis) and with A' infinite the contact line can move freely along the plate with the contact angle fixed at 90 °. For a wave of given frequency, the parameters of the problem are the depth of the immersed portion of the plate, the relative strengths of capillarity and gravity and the edgecondition parameter A'. The quantities to be determined are the reflection and transmission coefficients, from which the radiated, transmitted and dissipated energies can be calculated. The waves are of small amplitude and the fluid motion satisfies linearized inviscid equations. Viscosity will be important in a Stokes layer on the wall and near the edge, but it was shown in [6] that the associated dissipation is of a smaller order of magnitude than that produced by the edge condition.
2. Formulation
Consider a fixed thin vertical plate immersed to a depth d' below the free surface of fluid of infinite depth. Choose horizontal and vertical coordinate axes with origin at the intersection of the plate and the free surface. If the incident wave has a wave length equal to 2~r/k, we can scale all lengths with k -1, so that the horizontal and vertical coordinates of a point in the fluid are k-l(x, y); the free surface
219
is given by y = 0 and the immersed portion of the plate by x = 0, y > - d , where d = kd'. The velocity components are taken to be (g/k)l/2(u, v), the time is (gk)-l/2t, and the pressure is (pg/k)p, where p is the density of the fluid and g is the gravitational acceleration. The linearized equations of inviscid fluid motion then become
Ou/Ot = -Op/Sx, av/~t = - a p / a y ,
8u/Ox + av/Oy = 0, which have to be solved subject to the conditions u,/9-->0 u=0
as y--> -o0, on
x=O,
0>y>-d.
If the elevation of the free surface above its equilibrium position is denoted by ~?(x, t), the kinematic and dynamic conditions at the free surface are given by
O~7/Ot=v o n y = 0 , -KoE~/0x 2=p
(2) ony=0,
(3)
where
K = yk2/pg and y is the surface tension. The edge conditions to be applied to ~7 where the free surface meets the plate are, as discussed in the introduction,
871/0t = +hO~l/Ox at x = 0 ± ,
(4)
where A = (k/g)l/EA'. The free-surface elevation for the incident wave is taken to be 7~i : e i ( ° t + x )
and the corresponding velocity and pressure are given by U : --O.~I eY,
v = io.~h e y, p : O'27~i e y,
where o.2 = 1 + K. At large distances from the plate
220
L.M. Hocking / Capillary-gravity waves
the elevation has the forms ~7--r/l+~R
and for the o d d solution that
asx~+oo,
(5) 77 ~
T~T
as x ~ - ~ ,
as x ~ +oo,
(9)
where a and /3 are c o m p l e x constants that have to be determined. By taking a linear c o m b i n a t i o n o f these two solutions, we can express the freesurface elevation in the form
where ~TR= R e i(~t-x),
f o - - + c o s x +/3 sin x
"OT= T e i(~rt+x).
R and T are the c o m p l e x reflection and transmission coefficients, respectively, and, if ER and E r are the p r o p o r t i o n s o f the incident energy that are reflected and transmitted,
~7~ ei~'{A(cos x + a sin x) + B ( c o s x + /3 sin x)}
asx~+ce,
r / ~ ei"t{A(cos x - a sin x)
ER =IRI:,
E T = I T I 2. + B ( - c o s x + /3 sin x)}
In general, En + E r < 1, since the edge condition implies that energy is dissipated there. N o energy is dissipated when the contact line remains fixed ()t = 0), n o r w h e n it can m o v e freely along the plate (A = oo) [6]. In terms o f a velocity potential ~b(x, y), we can express the velocity c o m p o n e n t s and pressure in the forms
asx-->-o%
where A and B are constants. This form for the solution can then be written in terms o f progressive waves, p r o p o r t i o n a l to exp{i(crt±x)}, and we obtain a solution with the required asymptotic b e h a v i o u r at infinity p r o v i d e d that R = (1 + a/3)/{(1 - i a ) ( 1 -i/3)},
(lo)
U = ei~tocb/Ox,
T = i ( a - / 3 ) / { ( 1 - i a ) ( 1 - i/3)},
V = ei°~tO~/Oy,
where a a n d / 3 d e p e n d on the values o f the three parameters d, K and ~.
p = -krei~'d~" I f we define two functions f ( x ) and h ( x ) by f ( x ) = ~b(x, 0),
h(x) =
o6(x, o) Oy
3. Even solution
,
the free-surface elevation is given by r / = ei~'h/io -, and the d y n a m i c b o u n d a r y condition (3) b e c o m e s h - K d 2 h / d x 2 -- (1 + K)f.
(6)
W h e n ~b is an even function o f x, 04a/Ox = 0 at x = 0 for all negative y, and not just on the plate. Hence the solution is i n d e p e n d e n t o f d and can be f o u n d by a procedure similar to that used in [8]. We can take a Fourier cosine transform o f the even part o f ~b and write it in the form
The edge conditions are then given by h = :t:(iA/tr) d h / d x
at x = 0+.
(7)
The solution o f the p r o b l e m so p o s e d can be f o u n d by considering two standing-wave problems, in which ~b, f and h are respectively even a n d o d d functions o f x. If we write f = f e + f o , then, for the even solution we require that fe ~ cos x ± a sin x
as x -->+oe,
(8)
4~ =
fo
F ( k ) cos kx e ky dk,
so that fe(x) =
;o o F ( k )
cos kx dk.
(11)
The value o f the even part o f h can then be f o u n d from the d y n a m i c free-surface condition (6), which
LM. Hocking / Capillary-gravity waves
221
the singularities at k = 1 and we find that
can be solved to give
f ~ - cos x -2Ce{K1/2/(1 +3K)} sin x,
he(x) = (1 + K) Io'~ 1F(k) +--~2 cos kx dk
so that (see (8))
+ Ce e-XlK'/~/io', where Ce is a complex constant. The edge condition (7) then shows that Ce-
I+K 1-iJK-1/2(1
(12)
From the kinematic free-surface condition (2) we obtain an alternative expression for he, namely
;o o kF(k) cos kx dk,
and equating these two representations of he, we find that {k(1 + Kk 2) - 1 - K } F = 2K1/2Cd~r, so that
2K1/2 F - ~(k - 1)(Kk2 + Kk+ K + 1) Ce+ ~(k - 1), where the delta-function is introduced to provide the correct form for 7/at infinity. When this value of F is substituted in (12), we obtain
Ce{JI(K) -iA (1 + K) -1/2} = - K 1/2, where Jr(K) =--2Kfo~
~r
k 1) dk (k-1)(Kk2+Kk+K+
The boundary condition at x = 0 on the odd part of $ has the form
aqb/Ox=O
{
K+I ½log K
for y < - d ,
and a Fourier transform method is not appropriate. Instead, we can use a conformal mapping in which the equilibrium free surface and the two sides of the plate form the real axis in the transformed plane. Since the normal derivative of ~bo is known on the plate from (14), it is possible to express the solution in terms of a source distribution on the transformed free surface, with a density given by ho(x). The problem can, therefore, be reduced to the solution of an integral equation for fo(X). A similar method could, of course, be used for the even part of the solution, but the Fourier transform method gives the solution explicitly. The dynamic boundary condition (6) can be solved for ho in terms of fo, and the solution obtained by variation of parameters simplifies to the form
ho(x)
I +K 2K1/2
ioo e-lx-x'l/r'/2fo(xt)
+ Co e -x/'°',
xtan-1 (3K~-+4) 1/2} -
dxl (15)
where Co is a complex constant. The required mapping to the (X, Y)-plane is defined by
3K+2 K1/E(3K +4) 1/2 -
for 0 > y > - d , (14)
=0
-
2K =l+3K
(13)
4. Odd solution
+ K ) -1/2
× Io 1 F(k) ~-K~- dk.
he(x) =
2K (1 +3K){JI(K) -iA(1 + K)-~/2}"
X 2_ y2=x2_y2+d2" .
X Y = xy, For large values of x, the most important contributions to the asymptotic value of fe come from
the free surface being then given by Y = 0, Ixl > d, and the plate by Y = 0, Ixl < d. The value of ho(x)
222
L.M. Hocking / Capillary-gravity waves
in the new coordinates gives the density of the source distribution on Y = 0, from which the velocity potential can be found. The value of ~bo on the free surface can then be determined and, when we revert to the original coordinates, we obtain the equation
The two integrals can be reduced to forms more suited to numerical evaluation by standard methods, and we find that 11 = - 2 ' r r - l ( x 2+ d2) 1/2
X
fo(X) = _ -1
S(x, Xl)ho(Xl) dXl,
x>O,
_acos _ 2 s e_,,oo h ds
X2q- d 2 c o s h
and (16)
12 = - - ' r r - l S ( x , 0 ) --
/2
where ×
S(x, y)
=
logJ(y2 + d2) '/2 - (x2+ d2)1/2 I _log](ya + d 2) 1/2 + (x 2 + d2)1/2];
we have used the fact that ho is an odd function to write the integral over a semi-infinite range. The solution of (15) and (16) has to be found, subject to the edge condition (7) and the required form at infinity, given by (9). If we write fo = cos x+/3 sin x+fo(x), ho = cos x+/3 sin x + ho(x), then we can solve for fo and ho, provided these functions are o(1) as x ~ o o . The form of (15) is unchanged, except for the value of the arbitrary constant, and we obtain
ho(x )
I0
I+K -2K1/2
;ooe-lX-x,I/K'/2fo(x 0 dx,
+ C o e -x/K'/2.
(17)
In terms of the new variables, (16) has the form
fo(x) = -'rr-'
S(x, xl)ho(x,) dxl
I/
+11+[312,
(18)
where 11 and I2 are real functions of x and d, defined by
S(x, xl) e ix' d x l - e ix.
I1 +iI2 = _ - 1 o
I;
2"rr-l(x 2 + d2) 1/2
d cos s -a cos x2+d2cos2s e Sds.
A n alternative form for the second integral is needed if x = 0. The integral operators in (17) and (18) can be written in finite-difference form using the trapezium rule, and the equations become two matrix equations for the vectors containing the unknown values of fp and hp at the mesh points. The solution can be written as the sum of three parts, one of which is proportional to/3 and one to C O. The value of C o can then be chosen so that the whole solution has the correct asymptotic form. The edge condition (7) has the form
gp(O)+l=-iA(l+K)-l/2(g'o(O)+/3 )
(19)
in the new variables and from this equation the value of/3 can be determined. Some care is needed in the numerical work to deal adequately with the singularity of the operator in (18). Tests showed that it was sufficient to truncate the integrals at x = 14, and quadratic extrapolation was used, after it bad been verified as appropriate for the numerical approximations employed. Most of the calculations were done with the integrals truncated at xl = 14 and with quadratic extrapolation from the results obtained using 35 and 70 mesh points. Sample calculations were done with more mesh points and with truncation at a larger value of xl and these tests indicated that the results presented here are accurate to within 0.1 per cent. Some care was needed to deal adequately with the singularity of the operator in (18). In order
223
L M . Hocking/Capillary-gravity waves
infinite depth, there is no transmission and so T = 0; the value of the reflection coefficient is then given by
to maintain the overall accuracy of the numerical scheme, the integral was evaluated analytically on either side of xl = x with a linear approximation to hp(xl) there.
R=(l+ia)/(1-ia), with ot given by (13). For A equal to zero or infinity, a is real and ]R] = 1; for other values of A there is some dissipation of energy and [R] < 1. For d -- 0, the limiting solution is not the same as if there were no obstruction to the incident wave, since the conditions at the origin still have to be satisfied. Without the control exerted on the motion at this point we would, of course, have R = 0 and T = 1. The odd part of the solution can be found by a Fourier sine transform, since the condition to be applied at x = 0 now has the same form for all y. I f we proceed in a manner analogous to the method used for the even solution in Section 3, we find that
5. Results
The values of the two quantities ot a n d / 3 were found for a range of values of the parameters d, K and A. The calculation of a from (13) is straightforward. To find /3, the numerical procedure described in the previous section was used; the unknown terms appearing in (19) could be found independently of A, and then is was an easy task to find/3 for any value of A. Once a a n d / 3 were found, it was possible to calculate R and T and the associated energies from (10). Certain limiting cases required a separate treatment, or could be treated more expeditiously. For
K = 10 /
0.8
K)I/2/2KA,
f. / i - ~
-//"
J,'
\;/
O.G
/3 = J 2 ( K ) + i ( 1 + 3 K ) ( 1 +
j//
./'/
///
// /
IRI
/
0.4
K=o.i
/
~ /
/
/
/ /
0.2
/ ,.,
/
/
/
/
/
/
/
I{
=
0
/
/
/ /
//
/ /
00--
/
/
/
/
/
/
/
8.?5
8,'5
e°!75
1'
1 ~75
! ,'5
1 o~75
d
Fig. 2. Modulus of the reflection coefficient as a function of the depth of the plate for different values of the capillary parameter K. The solid curves are for fixed contact lines (A = 0), and the dashed curves for free contact lines (A = oo).
L.M. Hocking / Capillary-gravity waves
224
where J2(K) =~--~
K+I -½K log-K 3K2+6K+2 K I / 2 ( 3 K + 4) 1/2
Both a and fl are now given explicitly and the values of R and T are easily calculated. A third limiting case is when there is no surface tension, and we set K = 0. The dynamic free-surface condition (6) now reduces to h = f and we no longer have the freedom to apply an edge condition. The value o f / 3 can be found by an adaptation of the method used in the general case and the reflection coefficient is now given by
R= l / ( 1 - i f l ) . There is no dissipation of energy in this case and the solution is just that for the reflection of a pure gravity wave by a plate of depth d. Some of the results obtained are shown in Figs. 2 and 3. In Fig. 2 the dependence of the modulus of the reflection coefficient on the depth of the plate is shown for three different values of K and for the extreme values of h. There is no dissipation in these cases, so that the transmission coefficient can be found from the equation IRI2+IT[ 2= 1. Fixing the position of the contact line on the plate increases the amount of the incident energy reflected c o m p a r e d with that reflected when the contact line can move freely along the plate. For values of d greater than 2 there is almost complete reflection. In dimensional terms, this means that there is very little transmission of energy past a plate that extends to a distance below the free surface greater than about one-third of the wave length of the incident wave. There is greater reflection of energy, for moderate values of d, when the contact line is fixed than when it is free to move, because of the increased constraint on the surface near the edge. For values of h that are finite and nonzero, the reflection coefficients are reduced because of the dissipation of energy at the edge. In Fig. 3 the
reflected, transmitted and dissipated energies are shown as functions of the edge-condition parameter h for small and moderate depths and for three different values of K. The lower curve gives the value of the reflected energy ER and the upper curve the value of 1 - ET, so that the gap between the two curves represents the dissipated energy. It can be seen that a large proportion of the incident energy can be dissipated by the processes occurring in the vicinity of the contact line. For small depths, as much as 90 per cent of the incident energy may be dissipated, and the plate can act as an efficient absorber of the wave energy. When d is infinite, the plate forms a complete barrier and there is no transmission of energy past it. The dissipation in this case is similar in its dependence on h and K to that for d -- 1, with values that are less than 10 per cent lower.
6. Conclusions
The results presented in this paper have shown that the problem of reflection of a plane gravity wave by an obstacle, even in the simplest case of normal incidence and a plane barrier, is made much more complicated when surface tension is not negligible. Not only is an extra parameter added to the problem, but it is also necessary to postulate a condition at the line of contact between the fluid and the plate. The form of this condition employed here contains some of the features of the known behaviour of fluid near moving contact lines. It has included the important special case in which the contact line remains fixed in its position on the plate. There are obvious extensions of the problem discussed here. The scattering of a wave by a cylinder is one possibility, in which a threedimensional solution would be required. Another is the inclusion of moderate hysteresis in which a "stick-slip" motion of the contact line would be present. These and other wave problems involving fixed and moving contact lines are currently under investigation.
LM. Hocking I Capillary-gravity waves
225
0.8 ~
+/ /
\ \ \ \ \ \ /"
i
-
~ ~"-..
ET ,
\
~
\
/ ~
/
ER
8.4
\\
0°2
'
//
'
// \
/ ,,
x\\
0.01
0.1
I
,
l I/
,
A
10
100
0.8
0.6 i-
ET, ER
\\.•
~// \
i
(J.01
)
0.1
i
/
, ' ,
,
i
I
i
X
i
10
i
100
Fig. 3. Energy partition as a function of A. The lower curves in each pair give the values of the reflected energy, the distance between the upper curves and the upper boundary represents the transmitted energy and the gap between them gives the dissipated energy. The solid curves are for K =0.1, the long-dashed curves for K = 1 and the short-dashed curves for K = 10. (a) d =0.1; (b) d = 1.
226
L.M. Hocking / Capillary-gravity waves
References [1] Sir W. Thomson, "Hydrokinetic solutions and observations", Phil. Mag. (4) xlii, 374 (1871). [2] S.J. Hogan, "Some effects of surface tension on steep water waves", J. Fluid Mech. 91, 167-180 (1979). [3] J.-M. Vanden-Broeck, "Nonlinear gravity-capillary standing waves in water of arbitrary uniform depth", J. Fluid Mech. 139, 97-104 (1984). [4] T.B. Benjamin and J.C. Scott, "Gravity-capillary waves with edge constraints", J. Fluid Mech. 92, 241-267 (1979).
[5] E.B. Dussan V., "On the spreading of liquids on solid surfaces: Static and dynamic contact lines", Ann. Rev. Fluid Mech. 11, 371-400 (1979). [6] L.M. Hocking, "The damping of gravity-capillary waves at a rigid boundary", J. Fluid Mech. 174, 327-356 (1987). [7] G.W. Young and S.H. Davis, "A plate oscillating across a liquid interface: Effects of contact-angle hysteresis", J. Fluid Mech., to appear. [8] L.M. Hocking, "Waves produced by a vertically oscillating plate", J. Fluid Mech., to appear.