An exact solution to the one-phase zero-surface-tension Hele-Shaw free-boundary problem

Computers Math. Applic. Vol. 29, No. 10, pp. 45-50, 1995 Copyright(~1995 Elsevier Science Ltd Printed in Great Britain. All rights reserved 0898-1221/95 $9.50 + 0.00

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0898-1221(95)00044-5

A n E x a c t S o l u t i o n to t h e O n e - P h a s e Zero-Surface-Tension Hele-Shaw Free-Boundary Problem C. HUNTINGFORD* Mathematical Institute, Oxford University 24-29 St. Giles', Oxford, OX1 3LB, U.K.

(Received and accepted October 1994) A b s t r a c t - - A family of explicit solutions are presented to the one-phase zero-surface-tension HeleShaw problem with a single source/sink. The free boundary is described as a function of three parameters, whose dependence on time is evaluated through two different yet complementary methods. Possible mechanisms for nonexistence beyond a critical time are found in the case of both a contracting and expanding fluid region. K e y w o r d s - - F r e e - b o u n d a r y problem, Hele-Shaw cell, Schwartz function, Galin equation, Complex variable methods.

1. I N T R O D U C T I O N A Hele-Shaw cell [1] consists of two narrowly spaced, parallel plates between which two immiscible fluids of different viscosities, /zl,2, are trapped. T h e fluids are forced to move by either sources/sinks within them, or by injection/extraction at the cell edges. For slow flow within this gap, and with the z-direction normal to the plates, the mean horizontal fluid velocities are given by (e.g., [21) b2 ,

u(x, y, t) = (u, v) -

--vp(x, 12#i

y, t).

Here, p is the pressure within the cell and b the plate spacing. Assuming incompressibility, the m e a n pressure field for p(x, y, t) is Laplacian. T h e different fluids meet at a sharp interface Of~(t). Continuity of normal velocity, Vn, on c0fl(t) gives Vn = -(b2/12#1~°-~j on = - ( b 2 / 1 2 # 2 ) ~ n 2" T h e free b o u n d a r y requires a second condition: surface tension creates a pressure j u m p on Off(t), which is often modelled by P2 - Pl = - a n , where a is the surface tension coefficient and ~ the curvature of O~(t), measured in the x, y plane (see [3] for discussion of this b o u n d a r y condition). This note is restricted to a particular solution of the zero-surface-tension one-phase problem with a single source/sink: only one fluid region, fl(t), has viscosity sufficiently large to sustain a pressure gradient and a is negligible. O n O~(t), the b o u n d a r y conditions reduce to V,~ = -(b2/12#)o°-~n and p = 0. Rescaling by a typical length scale 1 and pressure P0 within the cell, and time by to = 12#12/b21pol, the following problem results:

V2p(x,y,t) = 0,

(x,y) e ~(t),

(1)

*Now at Institute of Hydrology, Wallingford, OX10 8BB, U.K. I should like to thank my doctoral supervisors, J.R. Ockendon and S.D. Howison for their help. Typeset by Ajk,bS-TEX 45

46

C. HUNTINGFORD

where

Op

Vn =

On'

p = O,

(2)

( z , y ) E Oft(t).

A single source/sink of (nondimensional) strength Q (Q > 0 : source) is specified at (x, y) = (0,0) e 12(t). That is

Q

P ~ -2-~r In V / ~ + y2

as

tx 2 + y2] __, 0.

(3)

A linear stability analysis of (1) and (2) with an initial planar free boundary 0ft(0) : x = 0 (but with sink p ~ V x as x ~ - o o ) is known to be unstable to perturbations of all wavelengths [4]. In addition, many explicit solutions to (1)-(3) exist (e.g., [5]), and for all 0~(0) that are not circles centered at (0,0), finite-time blow-up occurs when Q < 0. Here, a class of exact solutions is presented that exhibits blow-up for both Q < 0 (suction) and Q > 0 (blowing).

2. P O L U B A R I N O V A - G A L I N E Q U A T I O N Polubarinova-Kochina [6,7] and Calin [8] developed a complex variable method solutions to (1)-(3), where Oft(t) is simply connected. A time-dependent conformal from a unit disk ]¢1 <- 1 in the ~-plane, to ft(t) in the physical plane; that is z = x where O~(t) is given by the image of ]~[ -- 1. The complex potential w = - p w --- (Q/27r)in ~, and, with boundary conditions (2) (in complex form: Rl(w) = - ] ow 12), leads to the Polubarinova-Galin equation for the evolution of f(~, t) Re ;

-~

=

on

[¢[ = 1.

to find exact map f is used + iy = f(~, t) + i¢ satisfies 0, Rl ( - ~ ) =

(4)

Certain restrictions are placed on f(¢, t). For the pressure field to be well-defined, f(¢, t) must be univalent (analytic and one-to-one) for [¢] < 1. To guarantee the source/sink is at z = 0, f(0, t) - 0. Two invariance properties exist: for a map f(¢, t) satisfying (4), then f(~, k ( t ) ) also satisfies with a source/sink Q d k / d t , while for a constant k, k f ( ~ , t) satisfies (4) with Q replaced by k2Q. Solutions to (4) can be found by inspection. The example considered here is the map z -- f(~, t) = a l (t)~ 4- a 2 ( t ) ~ 2 -t- a 3 ( t ) ~ 3,

ai E ~,

(5)

satisfying (4) if

Q

(6)

aid1 + 2a2d2 + 3a363 =~-~,

(7) (s)

ala2 + 2a2a3 + 2a2al + 3a3a2 = 0, 3a3&l + a l a 3 = 0.

It is interesting to note that (6)-(8) can be derived in a more convenient form by appealing to the Schwartz function, g(z, t) [9] near z = O. Ogl(t) is the solution to -2 = g(z, t), and the relationship between f and g is g(z, t) = f(1/~(z, t), t). Hence, the Schwartz function corresponding to (5) is given by g ( z , t ) -- a l ( t ) + a2(t) a3(t)

-~- +

¢--~-

(9)

Inverting (5) for small z, ~(z, t) = z / a l - a 2 z 2 / a 3 + (2a2/a~ - a 3 / a 4) z 3 + O(z4), which upon substitution in (9) gives the Laurent expansion g(z, t) = 7

+

z~

+

z

+

o(11.

(10)

Free-BoundaryProblem

47

For the one-phase zero-surface-tension Hele-Shaw cell, the complex potential satisfies [10],

dw 1 09 d---~ = 2 0 t "

(11)

Near z = 0, ~z N Q/27rz, and so from (10), (11) d(a3a3)d~ = 0,

d(a2a2 +dt3aia2a3) = O,

d(a21 + 2a22dt+ 3a32) = -~.Q

(12)

Equations (12) satisfy (6)-(8), and so the Schwartz function has led to a trivial determination of the global time-evolution of the unknown coefficients al,a2 and a3. This is because it has been possible to use analytic continuation to relate the evolution near z = 0 to that of the free boundary. Setting A2 = a2/al, A3 = a3/al, dt/dT = a l / a l , (6)-(8) become

1 + 2A2A~ + 2A~ + 3A3A~ + 3A 2 =

Q 27raldl '

(13)

A'2 + 3A2 + 2A2A~ + 5A2A3 + 3A3A'2 = O,

(14)

4A3 + A~ = 0,

(15)

where P --- d/dT. (14) and (15) are independent of al, with solution A3(T) = A3(0)e -4T,

(3A (O) + 1 )

A2(T) ----A2(0)e r \3A3(0) + e4r " .

(16)

As al(t) is now only a multiplicative factor in the formula for f(~, t), the solution structure is determined by A2(T), A3(T). Indeed, (16) represents a one-dimensional family of trajectories in (a2/al, a3/al): this reduction from three to one in the order of the problem is not surprising in view of the two invariance arguments above. 0.4-

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A2 Figure 1. Bounds of univalency, which are symmetrical about the Aa axis, for the

function f(~) = ~+A2~2+A3~3 (continuous line). These bounds are given by straight lines between A, (0, -1/3) and B, (4/5, 1/5) and between C, (2v~/3, 1/3) and D, (0, 1/3) while the elliptical segment, BC, satisfies A2 = 1 - 4(A3 - 1/2) 2. Also plotted is a trajectory OBE (dashed line) satisfying (16) with A2(0) = 4/5, A3(0) = 1/5, that is A2 ----(4/5) -45V~318A3/(3A3 + 1)]. At B, Ol-~(t)passes through a 5/2-power cusp; the point E is given by (16 ~f~/15, 1/3).

1.0

48

C. HUNTINGFORD 0.35-

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A= Figure 2. An enlarged plot of region BCE in Figure 1. In this region of trajectory space, solutions to the blowing problem eventually fail by overlapping on BC. In the suction case, solutions fail by the free boundary developing two cusps when the trajectory reaches CE.

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(a) Figure 3. The image of I¢l = 1 under the conformal map z = f(¢) = ~ + 0.92~ 2 + (0.5 - ~ ) ~ 3 . This is a possible final state for O~(t) in the blowing problem with overlap. As an initial condition to the suction problem, the free boundary evolves such that the curves part at A, although the solution eventually fails due to the formation of 3/2-power cusps near B and B': this corresponds to a trajectory between curve BC and CE in Figure 1.

3,5

Free-Boundary Problem

49

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0.04"

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(b) Figure 3. (cont.)

Bounds of univalency for a cubic map with real coefficients are presented in [11] and trajectories of A2(~-), A3(v) must lie within these bounds (see Figure 1 and [Figure 4, 12]). It is found that in the small closed region of trajectory space 1/5 < A3(0) < 1/3 and 4/515A3(O)l-1/418Aa(O)/ (3A3(0) + 1)] < IA2(0)I < [1 - 4(Aa(0) - 1/2)~] 1/2 (see Figure 2), the solution will cease to exist for the blowing problem due to the loss of univalency by overlapping at ~- = ~-*. Such solutions satisfy A2(~-*) 2 + 4(A3(r*) - 1/2) 2 = 1 and 4/5 < IA2(r*)l < 2v~/3 (Figure 1,2; curve BC): an example is presented in Figure 3. For A2(0), A3(0) that lie outside this region but still within the bounds of univalency, for the blowing problem, the solution continues for all time, tending to a circle of increasing radius (that is A2, A3 --~ 0). The trajectory dividing the blowing solutions that lose univalency from those that continue for all time has the property that the solution develops a 5/2-power cusp, beyond which the solution may continue (either to the expanding circle in the blowing case, or to a solution with two cusps in the suction case). The ability for Hele-Shaw solutions to continue beyond 5/2-power cusps for the suction problem has already been observed [5]. For the suction problem, blow-up always occurs, with the formation of either one (Figure 1; line AB) or two (Figure 1; line CD) 3/2-power cusps. The cusp tips are images of points ~ = ~0 satisfying of(;o,t) = 0 and I~01 = 1. Further continuation in time of suction solutions results in o¢ unphysical mappings, as zeros of the first derivative move into I~l < 1, and hence, the map is no longer conformal in the fluid region.

3. C O N C L U S I O N S A particular class of solutions to the one-phase zero-surface-tension Hele-Shaw problem has been found by appealing both to the Polubarinova-Galin equation and the idea of the Schwartz function. Certain solutions can cease to exist for the blowing problem by overlapping; blow-up through the formation of cusps in the free boundary is inevitable in the suction case. 29:10-£

50

C. HUNTINGFORD

It would be i n t e r e s t i n g to classify blowing solutions for p o l y n o m i a l maps into those t h a t cease to exist due to loss of u n i v a l e n c y a n d those t h a t continue for all time. T h i s m a y p e r h a p s be done by p a r t i t i o n i n g t r a j e c t o r y space into regions b o u n d e d by trajectories t h a t can "pass t h r o u g h " cusps.

REFERENCES 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12.

H.S. Hele-Shaw, The flow of water, Nature 58, 34-35 (1898). G.K. Batchelor, A n Introduction to Fluid Dynamics, Cambridge University Press, Cambridge, (1967). P.G. Saffman, Viscous fingering in Hele-Shaw cells, J. Fluid Mech. 173, 73-94 (1986). P.G. Saffman and G.I. Taylor, The penetration of a fluid into a porous medium or Hele-Shaw cell containing a more viscous fluid, Proc. R. Soc. London A245, 312-329 (1958). S.D. Howison, Cusp development in Hele-Shaw flow with a free surface, S I A M J. Appl. Math. 46, 20-26 (1986). P.Ya Polubarinova-Kochina, Dokl. Akad. Nauk SSSR (Russian) 47 (1945a). P.Ya. Polubarinova-Kochina, Prikl. Matem. Mech. (Russian) 9, 79-90 (1945b). L.A. Galin, Dokl. Akad. Nauk SSSR (Russian) 47, 246-249 (1945). P.J. Davis, The Schwartz Function and its Applications, Math. Assoc. of America, (1974). S.D. Howison, Complex variable methods in Hele-Shaw moving boundary problems, Europ. J. Appl. Math. 3, 209-224 (1992). V.F. Cowling and W.C. Royster, Domains of variability for univalent polynomials, Proc. Amer. Math. Soc. 19, 767-772 (1968). Yu.E. Hohlov, S.D. Howison, C. Huntingford, J.R. Ockendon and A.A. Lacey, A model for non-smooth free boundaries in Hele-Shaw flows, Q. J. Mech. Appl. Math. 47, 107-128. (1994).