The Kinematics of the Planar Motion of Ideal Fiber-Reinforced Fluids: An Integrable Reduction and Bдcklund Transformation

We establish that the kinematic constraints on the steady planar motion of an ideal fiber-reinforced fluid can be consolidated in a single third-order nonlinear equation. Remarkably, this equation admits a solitonic reduction related to the classical sine-Gordon equation. The kinematic conditions in this case admit a novel duality property and a Bäcklund transformation.