A. Tsitskishvili

The Exact Solution of Spatial Axisymmetric Problems of the Theory of Filtration with Partially Unknown Boundaries

abstract:
In the present work we suggest a general method of solution of spatial axisymmetric problems of steady liquid motion in a porous medium with partially unknown boundaries. The liquid motion of ground waters in a porous medium is subjected to the Darcy law. The porous medium is undeformable, isotropic and homogeneous. The velocity potential $\varphi(z,\rho)$ and the flow function $\psi(z,\rho)$ are mutually connected and separately they satisfy different equations of elliptic type, where $z$ is the coordinate of the axis of
symmetry, and $\rho$ is the distance to that axis.
To the domain $S(\sigma)$ of the liquid motion on the plane of complex velocity there corresponds a circular polygon. The mapping $\omega=\varphi+i\psi$ belongs to the class of quasi-conformal mappings. Using the functions $\omega_0(\zeta)=\varphi_0(\xi,\eta)+i\psi_0(\xi,\eta)$, $\sigma(\zeta)=z(\xi,\eta)+i\rho(\xi,\eta)$ we map conformally the half-plane $\mbox{\rm Im\,}(\zeta)>0$ onto the domains $S(\sigma)$, $S(\omega_0)$ and $S(\omega_0'(\zeta)/\sigma'(\zeta))$. These functions satisfy all the boundary conditions, and the functions $\varphi_1(\xi,\eta)=\varphi(\xi,\eta)-\varphi_0(\xi,\eta)$, $\psi_1(\xi,\eta)=\psi(\xi,\eta)-\psi_0(\xi,\eta)$ satisfy the system of differential equations and also zero boundary conditions. The solution of these equations is reduced to a system of Fredholm integral equations of second kind which are solved uniquely by rapidly converging series.

Mathematics Subject Classification: 34A20,34B15

Key words and phrases: Filtration, analytic functions, generalized analytic functions, quasi-conformal mappings, differential equation