On steady motions of an ideal fibre-reinforced fluid in a curved stratum. Geometry and integrability

TL;DR

This paper explores the geometric and integrability properties of steady motions of an ideal fibre-reinforced fluid in curved layers, revealing connections to classical differential equations and special coordinate systems on surfaces.

Contribution

It expresses the governing equations in terms of the intrinsic Gauss equation, identifies integrable cases, and relates them to known equations like the Tzitzéica equation, extending geometric fluid mechanics.

Findings

Derived a third-order PDE for the surface of the stratum.

Identified integrable cases linked to the Tzitzéica equation.

Applied the formalism to flat, spherical, and pseudospherical geometries.

Abstract

It is shown that the kinematic equations governing steady motions of an ideal fibre-reinforced fluid in a curved stratum may be expressed entirely in terms of the intrinsic Gauss equation, which assumes the form of a partial differential equation of third order, for the surface representing the stratum. In particular, the approach adopted here leads to natural non-classical orthogonal coordinate systems on surfaces of constant Gaussian curvature with one family of coordinate lines representing the fibres. Integrable cases are isolated by requiring that the Gauss equation be compatible with another third-order hyperbolic differential equation. In particular, a variant of the integrable Tzitz\'eica equation is derived which encodes orthogonal coordinate systems on pseudospherical surfaces. This third-order equation is related to the Tzitz\'eica equation by an analogue of the Miura transformation for the (modified) Korteweg-de Vries equation. Finally, the formalism developed in this paper is illustrated by focussing on the simplest ``fluid sheets'' of constant Gaussian curvature, namely the plane, sphere and pseudosphere.