A. Tsitskishvili and R. Tsitskishvili

A. Razmadze Mathematical Institute, Georgian Academy of Sciences Tbilisi, Georgia

ON THE CONSTRUCTION OF SOLUTIONS OF CERTAIN SPATIAL AXISYMMETRIC MIXED PROBLEMS OF

FILTRATION WITH PARTIALLY UNKNOWN BOUNDARIES

The axis of symmetry is assumed to be the x-axis directed downwards, the distance to the x-axis is denoted by y, and the velocity vector is expressed as follows: V⃗ (u, v) = grad φ. The conditions for incompressibility and potentiality of a moving liquid are of the form div (yV⃗) = 0 (1), \frac{∂u}{∂y} − \frac{∂v}{∂x} = 0 (2). The streamline equation vdx − udy = 0 multiplied by y becomes the exact differential of the stream function ψ(x, y), where u = \frac{∂φ}{∂x} = y^{−1} \frac{∂ψ}{∂y} (3), v = \frac{∂φ}{∂y} = −y^{−1} \frac{∂ψ}{∂x} (4). The functions φ(x, y) and ψ(x, y) with respect to y are even functions (surface of rotation). Therefore y^{−1} \frac{∂φ}{∂y} and y^{−1} \frac{∂ψ}{∂y} tend to finite limits, as y → ∞. The domain S(z) with the boundary ℓ(z), occupied by a moving liquid we combine with the plane z = x + iy. The boundary ℓ(z) consists of an unknown curve and known segments, lines and of their portions. We seek for the functions ω(z) = φ(x, y)+iψ(x, y), w(z) = u(x, y)+iv(x, y), where φ(x, y) and ψ(x, y) must satisfy equations (3),(4) and also the following boundary conditions: ψ(x, y) = const along nonpermeable boundaries; φ(x, y) = const along water boundaries; φ(x, y)− kx = const, ψ(x, y) = Q, k = const, Q = const along an unknown curve; φ(x, y)− kx = const along the leaking interval; ψ(x, y) = 0, (x, y) ∈ ℓ(z), along the axis of symmetry. To the above-mentioned conditions are added to the corresponding equations of curves.

Let us consider the right half of the plane axisymmetric domain S_{0}(z) with the boundary ℓ_{0}(z) which coincides with the domain S(z) with the boundary ℓ(z) and respectively with the boundary conditions. The equation of depression curve in the plane problem must be the function y^{2}. Assume that to the domain S_{0}(z) there correspond the domains S(ω_{0}) and S(w_{0}) of a complex potential ω_{0}(z) = φ_{0}(x, y) + iψ_{0}(x, y) and of a complex velocity w_{0}(z) = \frac{dω_{0}(z)}{dz}. Geometrical characteristics respectively of the domains S(z) = S(z_{0}), S(ω) = S(ω_{0}), S(w) = S(w_{0}) and boundaries ℓ(z) = ℓ(z_{0}), ℓ(ω) = ℓ(ω_{0}), ℓ(w) = ℓ(w_{0}) coincide. The boundary conditions are likewise coincide. Here we emphasize that the functions ω_{0}(z) and w_{0}(z) are holomorphic, while the functions ω(z) and w(z) are analytic ones.

The half-plane Im (ζ) > 0 of the plane ζ = ξ + iη is conformally mapped onto the domains S(z_{0}), S(ω_{0}) and S(w_{0}). Conformally mapping functions we denote by z_{0}(ζ), ω_{0}(ζ) and w_{0}(ζ). Generalized functions φ(ξ, η) and ψ(ξ, η) must satisfy the equations \frac{∂φ}{∂ξ} = y^{−1} \frac{∂ψ}{∂η} (5), \frac{∂φ}{∂η} = y^{−1} \frac{∂ψ}{∂ξ} (6), or Δφ(ξ, η) + y^{−1}( \frac{∂y}{∂ξ} \frac{∂φ}{∂ξ} + \frac{∂y}{∂η} \frac{∂φ}{∂η}) = 0 (7), Δψ(ξ, η) − y^{−1}( \frac{∂y}{∂ξ} \frac{∂ψ}{∂ξ} + \frac{∂y}{∂η} \frac{∂ψ}{∂η}) = 0 (8). We consider it possible to construct holomorphic functions z_{0}(ζ), ω_{0}(ζ), w_{0}(ζ) efficiently. A solution of

(7) (analogously, of (8)) is sought in the form φ(ξ, η) = ∑_{k=0}^{∞} φ_{k}(ξ, η) (9), where

φ_{0}(ξ, η) (analogously, ψ_{0}(ξ, η)) is known, φ_{k}(ξ, η) is defined by the Poisson formula φ_{k}(ξ, η) = ∫∫_{Im(ζ)≥0} G(ξ, η; ξ_{1}, η_{1})φ_{k−1}(ξ,_{1} , η_{1})dξ_{1}dη_{1}, k = \overline{1, ∞} (10), where G is Green’s function for the half-plane with a coefficient. It remains to prove the convergence of series (9). Equation (8) is treated analogously.