A. Tsitskishvili

Solution of Spatial Axially Symmetric Problems of the Theory of Filtration with Partially Unknown Boundaries

abstract:
In the present work we consider spatial axially symmetric stationary motions of incompressible liquid in a porous medium with partially unknown
boundaries. The domain of liquid motion is bounded by an unknown depression curve and by known segments of lines, half-lines and lines. The liquid motion is subjected to the Darcy law. The porous medium is assumed to be undeformable, isotropic and homogeneous.
First, we prove that to the domain of the liquid motion, on the plane of complex velocity there corresponds a circular polygon of particular type. Then we construct an algorithm for solution of spatial axially symmetric problems of filtration with partially unknown boundaries. We construct an algorithm for finding three analytic functions, by means of which the half-plane is conformally mapped on a circular polygon, on the domain of liquid motion and on the domain of complex potential.
Finally, the construction of the solutions is reduced to the construction of solutions of integral and integro-differential equations, which are solved by the method of successive approximations. Here, the use is made of ordinary and generalized analytic functions. The systems of equations are set up for determination of unknown parameters of the problem of filtration, and equations are found for determination of unknown segments of boundaries.

Mathematics Subject Classification: 35J55, 76S05

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