A. Tsitskishvili
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