Saffman-Taylor fingers in deformed Hele-Shaw cells

NOTE: Text or symbols not renderable in plain ASCII are indicated by [...]. Abstract is included in .pdf document. Viscous fingering occurs when an essentially inviscid fluid is used to displace a viscous fluid in a porous medium or a Hele-Shaw cell. Long finger-shaped protrusions of the non-viscous fluid are found to advance into the viscous fluid. The importance of viscous fingering was first realised when attempts were made to retrieve oil from underground reservoirs by water injection. When one displaces oil in a Hele-Shaw cell of width a with air, an initially plane interface between the two fluids becomes unstable. This interface eventually develops into a finger shaped protrusion of width [...] moving at constant speed. Experimentally, in the presence of small surface tension at the air-oil interface the constant [...] is found to be near 1/2. An exact solution of the model equations was found by Saffman and Taylor in the absence of surface tension. Until now the fingering problem has only been studied in Hele-Shaw of constant gap corresponding to a porous medium with constant permeability. In order to understand the effect of non-homogenous permeability we look at the problem of fingering in a Hele-Shaw cell of varying gap. We consider variations in the direction perpendicular to the motion of the finger allowing steady fingers to exist. We extend previous work on the constant gap case to show how the presence of surface tension interacts with variations in the gap to change the solution structure in a non-trivial way. For example values of [...] less than a half are possible. It is even found that solutions cease to exist for small surface tension in some cases. We solve the problem using both numerical methods based on conformal mapping and analytical methods involving asymptotics beyond all orders