TY - JOUR

T1 - On steady motions of an ideal fibre-reinforced fluid in a curved stratum

T2 - Geometry and integrability

AU - Demskoi, Dmitry

AU - Schief, Wolfgang

PY - 2021/11/26

Y1 - 2021/11/26

N2 - 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 Tzitzéica equation is derived which encodes orthogonal coordinate systems on pseudospherical surfaces. This third-order equation is related to the Tzitzéica 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.

AB - 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 Tzitzéica equation is derived which encodes orthogonal coordinate systems on pseudospherical surfaces. This third-order equation is related to the Tzitzéica 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.

KW - fibre-reinforced fluid

KW - geometry

KW - integrability

U2 - 10.1088/1751-8121/ac38eb

DO - 10.1088/1751-8121/ac38eb

M3 - Article

SN - 1751-8121

VL - 54

JO - Journal of Physics A: Mathematical and Theoretical

JF - Journal of Physics A: Mathematical and Theoretical

IS - 50

M1 - 505205

ER -