In this project, we will be considering three problems in fluid dynamics whose common features involve free surface dynamics. In a two-dimensional space, these problems can be conveniently formulated by conformal mapping methods. The history and status quo of the application of conformal mapping methods to fluid dynamics are briefly discussed in Chapter 1. In Chapter 2 we study theoretically and experimentally the deformation of a free surface between two fluids in a gravitational field, due to a jet in the lighter fluid impinging at right angles to the surface. A mathematical model is built using the method of conformal mapping. The strength of our method lies in its general applicability to analytically study the interface between two fluids in a gravitational field, one of which has an arbitrary potential velocity field, while the other is assumed to be motionless. An asymptotic solution is derived for the cavity shape with the density ratio of fluids as the small expansion parameter. We present in Chapter 3 an unsteady nonlinear Darcy’s equation which includes inertial effects for flows in a Hele-Shaw cell, and discuss the conditions under which it reduces to the classical Darcy’s law. In the absence of surface tension we derive a generalized Polubarinova-Galin equation in a circular geometry, using the method of conformal mapping. The linear stability of the base-flow state is examined by perturbing the corresponding conformal map. We show that inertia always tends to stabilize the interface, regardless of whether a less viscous fluid is displacing a more viscous fluid or vice versa. In Chapter 4 a mathematical model of reactive Hele-Shaw flows when two immiscible fluids meet, chemically react and form an elastic interface is considered. This reaction brings about significant changes in the interfacial tension, which is crucial in determining the stability of such a system. We model this by treating the interface as an elastic membrane whose bending stiffness depends on the local curvature. We derive from energy variation a dynamic boundary condition at the interface. An analysis of the roles that several parameters play in affecting stability is performed. We are able to qualitatively account for the anomalous fingering instabilities that have been seen experimentally.

Conformal mapping and variational methods for interfacial dynamics in fluids.