Abstract
Abstract We consider axial-symmetric stationary flows of the ideal incompressible fluid as an important case of potential solenoid fields. We use an integral expression of the Stokes flow function via the corresponding complex analytic function for solving a boundary value problem with respect to a steady streamline of the ideal incompressible fluid along an axial-symmetric body. We describe the solvability of the problem in terms of the singularities of the mentioned complex analytic function. The obtained results are illustrated by concrete examples of modelling of steady axial-symmetric flows.
Highlights
It is well-known that a three-dimensional potential solenoid field symmetric with respect to the axis Ox with the Cartesian coordinates x, y, z is described in a meridian plane xOr, where r := √︀y2 + z2, in terms of the axial-symmetric potential φ and the Stokes flow function ψ, satisfying the following system of equations: r ∂φ(x, r) ∂x = ∂ψ(x, r) ∂r, r ∂φ(x, r) ∂r − ∂ψ(x, ∂x r) (1)While the function φ(x, √︀y2 + z2) is a spatial harmonic function in the variables x, y, z, the same is not true for the function ψ(x, √︀y2 + z2).Under the condition that there exist continuous second-order partial derivatives of the function ψ(x, r), system (1) implies the following equation for the Stokes flow function:
We use an integral expression of the Stokes flow function via the corresponding complex analytic function for solving a boundary value problem with respect to a steady streamline of the ideal incompressible fluid along an axial-symmetric body
We describe the solvability of the problem in terms of the singularities of the mentioned complex analytic function
Summary
It is well-known that a three-dimensional potential solenoid field symmetric with respect to the axis Ox with the Cartesian coordinates x, y, z is described in a meridian plane xOr, where r := √︀y2 + z2, in terms of the axial-symmetric potential φ and the Stokes flow function ψ, satisfying the following system of equations: r. We established a relation between the solutions of system (1) in the form of integral expressions via complex analytic functions and principal extensions of these functions into the mentioned topological vector space In such a way we developed a method for explicit construction of axial-symmetric potentials and Stokes flow functions in an arbitrary connected domain symmetric with respect to the axis Ox by means of components of the mentioned principal extensions. We develop a constructive functionally-analytic method for modelling steady streamlines of the ideal incompressible fluid along axial-symmetric bodies Such a method is essentially based on using integral expressions of the Stokes flow functions via complex analytic functions. We shall consider a boundary value problem for the Stokes flow function in unbounded domains only
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