Conformal and Potential Analysis in Hele-Shaw cells

Preface One of the most influential works in Fluid Dynamics at the edge of the 19-th century was a short paper [130] written by Henry Selby Hele-Shaw (1854–1941). There Hele-Shaw first described his famous cell that became a subject of deep investigation only more than 50 years later. A Hele-Shaw cell is a device for investigating two-dimensional flow of a viscous fluid in a narrow gap between two parallel plates. This cell is the simplest system in which multi-dimensional convection is present. Probably the most important characteristic of flows in such a cell is that when the Reynolds number based on gap width is sufficiently small, the Navier-Stokes equations averaged over the gap reduce to a linear relation similar to Darcy's law and then to a Laplace equation for pressure. Different driving mechanisms can be considered , such as surface tension or external forces (suction, injection). Through the similarity in the governing equations, Hele-Shaw flows are particularly useful for visualization of saturated flows in porous media, assuming they are slow enough to be governed by Darcy's low. Nowadays, the Hele-Shaw cell is used as a powerful tool in several fields of natural sciences and engineering, in particular, matter physics, material science, crystal growth and, of course, fluid mechanics. who developed a complex variable method to deal with non-gravity Hele-Shaw flows neglecting surface tension. The main idea was to apply the Riemann mapping from an appropriate canonical domain (the unit disk in most situations) onto the phase domain to parameterize the free boundary. The equation for this map, named after its creators, allows to construct many explicit solutions and to apply methods of conformal analysis and geometric function theory to investigate Hele-Shaw flows. In particular, solutions to this equation in the case of advancing fluid give subordination chains of simply connected domains which have been studied for a long time in the theory of univalent functions. The Polubarinova-Galin equation and the Löwner-Kufarev one, having some evident geometric connections, are VI PREFACE not closely related analytically. The Polubarinova-Galin equation is essentially non-linear and the corresponding subordination chains are of rather complicated nature. Among other remarkable contributions we distinguish the discovery of the viscous fingering phenomenon by Sir Geoffrey Ingram Taylor (1886–1975) and Philip Geoffrey Saffman [224], [225], and the first modern description of the complex variable approach and the study of the complex moments made by Stanley Richardson [215]. Contributions made by …