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

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