On non-steady planar motions of fibre-reinforced fluids. Geometry and integrable structure

TL;DR

This paper demonstrates that non-steady planar motions of ideal fibre-reinforced fluids can be modeled by a single integrable PDE, linking steady and non-steady states and connecting to the mKdV hierarchy.

Contribution

It introduces a reduction of the kinematic system to a single PDE and establishes a method to derive non-steady motions from steady solutions, revealing their integrable structure.

Findings

Reduction to a single third-order PDE

Mapping steady motions to non-steady motions

Connection with the mKdV hierarchy

Abstract

It is shown that the kinematic system describing planar non-steady motions of ideal fibre-reinforced fluids may be reduced to a single two-dimensional third-order partial differential equation in which time enters parametrically. A procedure is established which maps steady motions to non-steady motions. The resulting motions inherit their hidden integrable structure from the steady case. The formalism presented here also readily recovers the connection with the scattering problem of the modified Korteweg-de Vries hierarchy established in previous work.