Interface Dynamics in Two-dimensional Viscous Flows.

Abstract

The subject of this thesis is viscous fingering in Hele-Shaw cells, or Hele-Shaw flows. We look for insights into the fundamental mechanisms<br/>underlying the physics of interface dynamics, which we hope will exhibit some degree of universality. The aim is twofold: on the one hand we focus on the role of surface tension and viscosity contrast in the dynamics of fingering patterns. On the other hand we introduce a modification of the original problem and study the effects of a inhomogeneous gap between the<br/>plates of a Hele-Shaw cell.<br/><br/>A dynamical systems approach to competition of Saffman-Taylor (ST) fingers in a Hele-Shaw channel is developed. This is based on global analysis of the phase space flow of the ODE sets associated to the exact solutions of the problem without surface tension. A general proof of the existence of finite-time singularities for broad classes of solutions is given. The existence of a continuum of multifinger fixed points and its dynamical implications are discussed. We conclude that exact zero-surface tension solutions taken in a global sense as families of trajectories in phase space are unphysical because the multifinger fixed points are nonhyperbolic. Hyperbolicity (saddle-point structure) of multifinger fixed points is argued to be essential to the physically correct qualitative<br/>description of finger competition. The restoring of hyperbolicity by surface tension is proposed as the key point to formulate a generic Dynamical Solvability Scenario for interfacial pattern selection.<br/><br/>We study the singular effects of vanishingly small surface tension on the dynamics of finger competition in the Saffman-Taylor problem, using the asymptotic techniques developed by Tanveer and Siegel, and numerical computation, following the numerical scheme of Hou, Lowengrub, and<br/>Shelley. We demonstrate the dramatic effects of small surface tension on the late time evolution of two-finger configurations with respect to exact<br/>(non-singular) zero-surface tension solutions. The effect is present even when the zero surface tension solution has asymptotic behavior consistent with selection theory. Such singular effects therefore cannot be traced back to steady state selection theory, and imply a drastic global change in the structure of phase-space flow.<br/><br/>Finger competition with arbitrary viscosity contrast (or Atwood ratio) is<br/>studied by means of numerical computation. Two different types of dynamics<br/>are observed, depending on the value of the viscosity contrast an the<br/>initial condition. One of them exhibits finger competition and ends up in<br/>the ST finger. In opposition, the second dynamics does not exhibit finger<br/>competition and the long time dynamics seems attracted to bubble shaped<br/>solutions. An initial condition appropriate to study the ST finger basin<br/>of attraction is identified, and used to characterize its dependence on<br/>the viscosity contrast, obtaining that its size decreases for decreasing<br/>viscosity contrast, being very small for zero viscosity contrast. The ST<br/>finger is not the universal attractor for arbitrary viscosity contrast. An<br/>alternative class of attractors is identified as the set of Taylor-Saffman<br/>bubble solutions, and one important implication of this result is that the<br/>interface is strongly attracted to finite time pinchoff.<br/><br/>A nonlocal interface equation is derived for two-phase fluid flow, with<br/>arbitrary wettability and viscosity contrast c, in a model porous medium defined as a Hele-Shaw cell with random gap b. Fluctuations of both capillary and viscous pressure are explicitly related to the microscopic quenched disorder, yielding conserved, non-conserved and power-law correlated noise terms. Two length scales are identified that control the possible scaling regimes and which depend on capillary number Ca as l sub 1 = b sub zero (c Ca)(superindex -1/2) and l sub 2 = b sub zero/Ca. Exponents for forced fluid invasion are obtained from numerical simulation and compared with recent experiments,obtaining good partial agreement.