Avtandil Tsitskishvili

Solution of Some Plane Filtration Problems with Partially Unknown Boundaries

Plane problems of the stationary filtration theory with partially unknown boundaries are considered. The porous medium is assumed to be homogeneous, isotropic and non-deformable. The motion of the fluid obeys the Darcy law. The simply connected domain occupied by the moving fluid is bounded by a simple sectionally analytic contour consisting of unknown depression curves, line segments, half-lines and straight lines. The paper describes mathematical methods of finding the unknown parts of the boundary of the fluid motion domain, as well as of determining geometric, cinematic and physical characteristics of the moving fluid. In solving the corresponding mathematical problem, the use is made of the general solution of the non-linear Schwarz differential equation. The general solution is constructed in the paper.

Mathematics Subject Classification: 34A20, 34B15

Key words and phrases: Filtration, analytic functions, conformal mapping, differential equation.