ON CONVEXITY OF HELE- SHAW CELLS Ibrahim W. Rabha Institute of Mathematical Sciences, University Malaya, Kuala Lumpur, Malaysia

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1 ROMAI J., v.8, no.1(2012), ON CONVEXITY OF HELE- SHAW CELLS Ibrahim W. Rabha Institute of Mathematical Sciences, University Malaya, Kuala Lumpur, Malaysia Abstract The main aim of this paper is to apply methods of the theory of univalent functions and geometric functions to some problems of fluid mechanics. We study the time evolution of the free boundary of a viscous fluid for planar flows in Hele- Shaw cells under injection; we prove the invariance in time of convexity of complex order for two basic cases: the inner problem and the outer problem. Keywords: free boundary problem, conformal map, complex analysis, univalent functions, unit disk, Hele-Shaw cells, Hadamard product, convex function, convex of complex order MSC: 30C45,76S05, 76D INTRODUCTION One of the basic models is a fluid flow in a Hele-Shaw cell. Hele-Shaw [1] first described his cell, which was an experimental device for studying fluid flow by pumping a viscous liquid into the gap between two closely-separated glass plates. Using dye-lines, he was able to observe the flow patterns generated when the flow was impeded by various kinds of obstacles, such as aerofoil sections, placed between the plates. If the displacing fluid has lower viscosity than the displaced fluid the interface will develop hydrodynamical instability which results in highly ramified patterns [2-4]. This phenomenon is known as viscous fingering, and it may also occur when elasticity of the fluids acts as another driving mechanism [5-7]. Pattern formation of similar type has been observed in a variety of non equilibrium systems besides viscous fingering, such as crystal growth [8], electrodeposition [9] and solidification [10]. The canonical mathematical model for Hele-Shaw flows is Darcy s law, where the flow velocity is proportional to the pressure gradient: Darcy Law : v n (z) H(z); z γ(t), where v n is the velocity of the curve, γ(t) is a simple planar curve - boundary of a simply-connected domain and H(z) is the normal gradient of a solution of the Dirichlet problem in a fluid domain D with a source (a sink) at a distant location (often at infinity). Under the assumption that the flow is incompressible, the pressure field satisfies a Laplace s equation; therefore, such an evolution of the free interface is also called Laplacian growth process. The method of conformal mapping resides its 93

2 94 Ibrahim W. Rabha strength of transforming the generally difficult task of solving a moving-free boundary problem into finding solutions to a single differential equation of an analytic function on a fixed domain, usually the half plane or the interior of the unit disk. Indeed, the simply connected domains are preserved in rotating cell [11]. Recently, the stability of this method was considered in [12]. The time evolution of the free boundary of a viscous fluid for planar flows in Hele- Shaw cells under injection was studied by many authors. By using methods of univalent functions theory, they proved that certain geometric properties (such as starlikeness, directional convexity Φ like and E family) are preserved in time [13-20]. In fact, as any zero-surface tension problem is timer-eversible, blow-up solutions can be generated by injecting with a non-smooth free boundary and then reversing the sequence of solutions so obtained. This procedure can reveal unexpected features, an example being the waiting time that can occur when injection takes place into an initial domain with a corner [21]. The analysis of this situation again involves the Baiocchi transform of the pressure, and this device is also instrumental in the analysis of allowable cusps in injection problems [22]. In this paper, we continue their study by proving the invariance in time of another geometric property. We study the time evolution of the free boundary of a viscous fluid for planar flows in Hele- Shaw cells under injection. Applying methods from the theories of univalent functions and geometric functions; we prove the invariance in time of convexity of complex order property for two basic cases: the inner problem and the outer problem. 2. HELE-SHAW CELLS PROBLEM We study the flow of a viscous fluid in a planar Hele-Shaw cell under injection through a source (of constant strength Q, Q < 0 in case of injection) which is situated at the origin. Suppose that for the initial time, the phase domain Ω 0 occupied by the fluid is simply connected and bounded by a smooth curve Γ 0. The evolution of the phase domains Ω(t), Ω(0) = Ω 0 is described by an auxiliary conformal mapping f (ζ, t), f (ζ, 0) = f 0 (ζ) on the unit disk U := {z C : z < 1} onto Ω(t), Γ(t) = Ω(t) normalized by f (0, t) = 0, f (0, t) > 0. We denote the derivatives by f (z, t) = f z and f (z, t) = f t, where t is the time parameter. The function f (ζ, 0) = f 0(ζ) poses a parametrization of Γ 0 = { f 0 (e iθ ), θ [0, 2π)}, while the moving boundary is parameterized by Γ(t) = { f (e iθ, t), θ [0, 2π)}. This mapping satisfies the equation (see [23-27]) R[ f (ζ, t)ζ f (ζ, t)] = Q 2π, ζ = eiθ. (1) In the case of the problem of injection Q < 0 of the fluid into a bounded simply connected with small surface tension γ > 0, the Polubarinova-Galin equation [18] is of the form:

3 On convexity of Hele- Shaw cells 95 where R[ f (ζ, t)ζ f (ζ, t)] = Q 2π + γh[ik θ (eiθ, t)](θ), ζ = e iθ, (2) k(e iθ, t) = 1 f (e iθ, t) R( 1 + eiθ f (e iθ, t) ), θ [0, 2π), f (e iθ, t) k θ (eiθ, t) = I( e2iθ S f (e iθ, t) f (e iθ ),, t) S f = ( f f ) 1 2 ( f f )2 is the Schwarzian derivative, and H[Φ](θ) is the Hilbert transform given by H[Φ](θ) = 1 π P.V. θ 2π (PV denotes the Cauchy principal value) satisfying where 0 Φ(e iθ ) 1 e i(θ θ ) dθ θ [ik θ (eiθ, t)](θ) = H[iA](θ), A(ζ) = ( 1 )[ ( R 2ζ 2 f S f (ζ)+ζ( f (ζ) (ζ) f (ζ) ) ζ( f (ζ) f (ζ) )( f (ζ) f (ζ) ) ) +I ( ζ f (ζ)) Iζ 2 f S f (ζ) ]. (ζ) The case of unbounded domain with bounded complement virtues as the dynamics of a contracting bubble in a Hele-Shaw cell since the fluid occupies a neighborhood of infinity and injection (of constant strength Q < 0) is supposed to take place at infinity. Again, we denote by Ω(t) the domain occupied by the fluid at the moment t, Γ(t) = Ω(t). By using the Riemann mapping theorem, the domain Ω(t) can be described by a univalent function F(ζ, t) from the exterior of the unit disk U = {ζ : ζ > 1} onto Ω(t). The equation satisfied by the free boundary is [16,18] for the zero tension surface model and R[Ḟ(ζ, t)ζf (ζ, t)] = Q 2π, ζ = eiθ, (3) R[Ḟ(ζ, t)ζf (ζ, t)] = Q 2π γh[ik θ (eiθ, t)](θ), ζ = e iθ, (4) for the small surface tension model.

4 96 Ibrahim W. Rabha 3. THE INNER PROBLEM Let us define the classes of convex functions which will parameterize our phase domains. If a function f (z) maps U onto a convex domain and f (0) = 0, then we say that f (z) is a convex function. A necessary and sufficient condition for a function f to be convex is the inequality R(1 + z f (z) f (z) ) > 0. In this section, we impose the invariance in time of convex of complex order for the inner problem. Starting with an initial bounded domain Ω(0) which is convex of complex order, we prove that at each moment t [0, T) the domain Ω(t) is convex of complex order. Definition 3.1. Let f be a holomorphic function on U such that f (0) = 0, f (0) 0. A function f is called convex of complex order b (b C\{0}) if it satisfies the following inequality R{1 + 1 b (z f (z) f 1)} > 0, (z U). (z) We denoted this class C(b). Definition 3.2. A simply connected domain Ω C is convex in the direction of the real axis R if each line parallel to R either misses Ω or the intersection with Ω is a connected set. If a function f (ζ) maps U(the unit disk) onto a domain which is convex in the direction of the real axis, f (0) = 0, then we say that f (z) is a convex function in the direction of the real axis and denote the class of such function by C R and by C R (b) for convex of complex order in the direction of the real axis. Theorem 3.1. Let Q < 0 and f 0 C R (b) on U and univalent in U. Let f (ζ, t) be the classical solution of the Polubarinova-Galin equation (1) with the initial condition f (ζ, 0) = f 0 (ζ). Moreover, let Ω := t [0,T) Ω(t) = t [0,T) f (U, t), where T is the blow-up time. If Rb = 2 and 1 < Ib < 2 then f (ζ, t) C R (b) in the direction of the real axis R for ζ [ɛ, 1), 0 < ɛ < 1. Proof. Assume that there exists a complex number such that which satisfies the equality ζ 0 := (e iθ 0 ) b, ζ0 1 ( arg ζ 0 b f (ζ 0, t 0 ) ) b = π 2 (or π ), b 0 2 and for ɛ > 0 there is t > t 0 and θ (θ 0 ɛ, θ 0 + ɛ) such that arg ζ ( b f (ζ, t) ) b π 2 (or π 2 ).

5 Then for ζ = (e iθ ) b, which yields Since f (e i( b )θ, t) 0 we obtain ( θ arg(eiθ ) b b f (e On convexity of Hele- Shaw cells 97 i( b R { b [ζ f (ζ, t 0 ) f (ζ, t 0 ) )θ, t) ) b 0, 1] } > 0. Iζ 0 (b f (ζ 0, t 0 )) b > 0. Since ζ 0 is a critical point then we deduce that and We put r θ arg ei( )θ( b b f (e arg ei( )θ( b b f (re i( b i( b )θ, t) ) b )θ, t) ) b R ( b [ζ 0 f (ζ 0, t 0 ) f (ζ 0, t 0 ) I ( b [ζ 0 f (ζ 0, t 0 ) f (ζ 0, t 0 ) = 0 θ=θ0,t=t 0 0. θ=θ0,t=t 0,r=1 1] ) = 0, 1] ) > 0. Equality does not hold for the last assertion because the level lines preserve the property of being convex in the direction of R. By the assumptions Rb = 2 and Ib < 2, we obtain R ( 1 + ζ 0 f (ζ 0, t 0 )) = 0, (5) f (ζ 0, t 0 ) Then we derive I ( 1 + ζ 0 f (ζ 0, t 0 ) f (ζ 0, t 0 ) ) > 0. (6) t arg ζ(b f (ζ, t)) b 1 = I t f (ζ, t) b 1 f (ζ, t). (7) By differentiating the Polubarinova-Galin equation with respect to θ for ζ = 1, we obtain or I ( f (ζ, t) t f (ζ, t) ζ f (ζ, t) f (ζ, t) ζ 2 f (ζ, t) f (ζ, t) ) = 0 f (ζ, t) 2 I ( t f (ζ, t) f (ζ, t) ζ f (ζ, t) f (ζ, t) ζ 2 f (ζ, t) f (ζ, t) ) = 0

6 98 Ibrahim W. Rabha and consequently, we have f (ζ, t) 2 I 1 t f (ζ, t) b 1 f = I 1 (ζ, t) b 1 ζ f (ζ, t) f (ζ, t) ( ζ f (ζ, t) f 1 ) (ζ, t) Substituting (1) and (5) in last equation, we have t arg ζ( b f (ζ, t) ) b = ζ=ζ0,t=t 0 Q 2π f (ζ 0, t 0 ) 2 I 1 (ζ f (ζ 0, t 0 ) b 1 f + 1 ). (ζ 0, t 0 ) The right-hand side of this equality is strictly negative because of (6) and Ib > 1. Hence arg ζ ( b f (ζ, t) ) b < π 2 for t close to t 0 and in a neighborhood of θ 0 and this is a contradiction. 4. THE OUTER PROBLEM In this section, we obtain the invariance in time of the same geometric property (denoted by C(b)) for the outer problem (injection at infinity). Here we prove that our problem preserves geometric properties of domains with C(b) complements. Let us suppose that the complement of the fluid domain contains the origin and is in the class C(b) with respect to the origin at the initial instant, we have the following definition: Definition 4.1. Let F be a holomorphic function on U such that F(ζ) = aζ + a 0 + a 1 ζ +..., a 0. The C(b) is defined so that R{1 + 1 b (zf (z) F (z) 1)} > 0, ζ U. (8) Straightforward computations give the following facts: Remark 4.1. (a) If the function F belongs to the C(b) in U then the function f : U C given by is a member in C(b) on U. f (z) = 1 F( 1 z 0, f (0) = 0 z ),

7 On convexity of Hele- Shaw cells 99 (b) If the function f belongs to the C(b) in U then the function F : U C given by F(ζ) = 1 f ( 1 ζ 0, ζ ), is a member in C(b) on U. (c) If F C(b), then it is univalent on U. Theorem 4.1. Let Q < 0 and F 0 be a function which is in the class C(b) on U and univalent in U. Let F(ζ, t) be the classical solution of the Polubarinova-Galin equation (3) with the initial condition F(ζ, 0) = F 0 (ζ). Moreover, let Ω := t [0,T) Ω(t) = t [0,T) F(U, t), where T is the blow-up time. If If Rb = 2 and 1 < Ib < 2 then F(ζ, t) is in C(b). Proof. By assuming the function f (ζ, t) = 1/F( 1 ζ ), the Polubarinova-Galin equation (3) can be rewritten in terms of f as follows: R[ f (ζ, t)ζ f Q f (ζ, t) 4 (ζ, t)] =, ζ = 1. (9) 2π According to the previous remark, the function F(ζ, t), ζ U is in C(b) if and only if f (ζ, t), ζ U is in C(b). Thus, it suffices to prove that the functions f (ζ, t), ζ U, t [0, T), is in C(b). Suppose by contrary that the previous statement is not true. Then there exist t 0 0 and ζ 0 = e iθ 0 such that equations (5-7) are held. Calculation gives t arg ζ( b f (ζ, t) ) b = Q f (ζ 0, t 0 ) 2 I 1 (ζ f (ζ 0, t 0 ) ζ=ζ0,t=t 0 2π b 1 f + 1 ). (ζ 0, t 0 ) The right-hand side of this equality is strictly negative because of (6) and Ib > 1. Hence arg ζ ( b f (ζ, t) ) b < π 2 for t close to t 0 and in a neighborhood of θ 0 or arg[1 + 1 b (eiθ f (e iθ, t) f (e iθ, t) 1)] < π 2 and this is a contradiction. Hence f (ζ, t), ζ U is in the C(b) and consequently F(ζ, t), ζ U is in C(b).

8 100 Ibrahim W. Rabha 5. CONCLUSION We imposed sufficient conditions for convexity of Hele- Shaw Cells in complex domain. This problem was studied by Gustafsson, Friedman, Hohlov, Markina, Prokhorov, Vasil ev and others. We considered a modification of a logarithmic antiderivative of the convex of complex order. The functional that assumed here taking the form ζ ( b f (ζ, t) ) b such that Rb = 2 and 1 < Ib < 2. This functional satisfied ( arg ζ 0 b f (ζ 0, t 0 ) ) b = π 2 (or π ), b 0 2 where ζ 0 := (e iθ 0 ) b, ζ0 1. References [1] H.S. Hele-Shaw, The flow of water, Nature, 58(1898), [2] P. Saffman, 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, Ser. A, 245(1958), [3] L. Paterson, Radial fingering in a Hele-Shaw cell, J. Fluid Mech. 113(1981), [4] D. Bensimon, Stability of viscous fingering, Phys. Rev., A 33(1986), [5] E. Lemaire, P. Levitz, G. Daccord, H. Damme, From viscous fingering to viscoelastic fracturing, Phys. Rev. Lett., 67(1991), [6] T. Podgorski, M. C. Sostarecz, S. Zorman, A. Belmonte, Fingering instabilities of a reactive micellar interface, Physical Review E, 76( )(2007), 1-6. [7] S. Mora, M. Manna, Saffman-Taylor instability of viscoelastic uids: From viscous fingering to elastic fractures, Physical Review E, 81(026305) (2010), [8] J. Langer, Instabilities and pattern formation in crystal growth, Rev. Mod. Phys., 52(1980), [9] M. Matsushita, M. Sano, Y. Hayakawa, H. Honjo, Y. Sawada, Fractal structures of zinc metal leaves grown by electrodeposition, Phys. Rev. Lett., 53(1984), [10] J. Hunt, Pattern formation in solidification, Materials Science & Technology, 15(1999), [11] V.M. Entov, P.I. Etingof, D. Ya, Kleinbock, On nonlinear interface dynamics in Hele-Shaw flows, Eur. J. Appl. Math., 6(1995), [12] A. He, A. Belmonte, Inertial effects on viscous fingering in the complex plane, J. Fluid Mech., 668(2011), [13] Y. E. Hohlov, D. V. Prokhorov, A. J. Vasil ev, On geometrical properties of free boundaries in the Hele-Shaw flows moving boundary problem, Lobachevskii Journal of Mathematics, 1(1998), [14] K. Kornev, A. Vasil ev, Geometric properties of the solutions of a Hele-Shaw type equation, Proceedings of the American Mathematical Society, 128, 9(2000), [15] D. Prokhorov, A. Vasil ev, Convex dynamics in Hele-Shaw cells, International Journal of Mathematics and Mathematical Sciences, 31, 11(2002), [16] A. Vasil ev, Univalent functions in the dynamics of viscous flows, Computational Methods and Function Theory, 1, 2(2001),

9 On convexity of Hele- Shaw cells 101 [17] A. Vasil ev, Univalent functions in two-dimensional free boundary problems, Acta Applicandae Mathematicae, 79, 3(2003), [18] A. Vasil ev, I. Markina, On the geometry of Hele-Shaw flows with small surface tension, Interfaces and Free Boundaries, 5, 2(2003), [19] P. Curt, D. Fericean, A Special class of univalent functions in Hele-Shaw flow problems, Abstract and Applied Analysis Volume 2011, Article ID , [20] R.W. Ibrahim, On Geometric Properties for Hele- Shaw Cells, Complex Variables and Elliptic Equations. To appear. [21] J.R. King,, A.A. Lacey, J.L. Vazquez, Persistence of corners in free boundaries in Hele-Shaw flow, Europ. J. Appl. Math., 6(1995), [22] S.D Howison, Cusp development in Hele-Shaw flow with a free surface, SIAM J. Appl. Math., 46(1986), [23] L. A. Galin, Unsteady filtration with a free surface, Doklady Akademii Nauk USSR, 47(1945), , (Russian). [24] P. Y. Polubarinova-Kochina, On a problem of the motion of the contour of a petroleum shell, Doklady Akademii Nauk USSR, 47(1945), , (Russian). [25] P. J. Poloubarinova-Kochina, Concerning unsteady motions in the theory of filtration, 9, 1(1945), 79-90, (in Russian). [26] L. Fejer, Mechanische Quadraturen mit positiven Cotesschen Zahlen, Math. Zeitschrift, 37, 2 (1933), [27] B. Gustafsson, A. Vasil ev, Conformal and Potential Analysis in Hele-Shaw Cells, Advances in Mathematical Fluid Mechanics, Birkhäuser, Basel, Switzerland, 2006.