Exact solutions of the problem of free-boundary unsteady flows

TL;DR

This paper introduces a method for solving boundary value problems in wedge-shaped domains, reducing them to finite ODE systems when the wedge angle ratio is rational, and provides exact solutions for self-similar free-boundary flows.

Contribution

It presents a novel approach to boundary value problems in wedge geometries, enabling exact solutions for certain free-boundary unsteady flows.

Findings

Reduced boundary value problems to finite ODE systems for rational angle ratios.

Derived four exact self-similar flow solutions with free boundaries.

Enhanced understanding of free-boundary unsteady flow behaviors.

Abstract

Some approach to the solution of boundary value problems for finding functions, which are analytical in a wedge, is proposed. If the ratio of the angle at the wedge vertex to a number \pi is rational, then the boundary value problem is reduced to the finite system of ordinary differential equations. Such approach, applied the problem of inertial motion of a liquid wedge, made it possible to sum series with small denominators arising in the problem and find four exact examples of self-similar flows with a free boundary.