The conduit equation: Hyperbolic approximation and generalized Riemann problem

Abstract

The conduit equation is a dispersive non-integrable scalar equation modeling the flow of a low-viscous buoyant fluid embedded in a highly viscous fluid matrix. This equation can be written in a particular form reminiscent of the famous Godunov form proposed in 1961 for the Euler equations of compressible fluids. We propose a hyperbolic approximation of the conduit equation by retaining the Godunov-type structure. The comparison of solutions to the conduit equation and those to the approximate hyperbolic system is performed: the wave fission of a large initial perturbation of a rectangular or Gaussian form. The results are in good agreement. New generalized solutions to the conduit equation composed of a finite set of waves of the same period and linked with a constant solution by generalized Rankine-Hugoniot relations are discovered. Such multi-hump structures interact with each other almost as solitary waves: they collide, merge, and reconstruct after the interaction. This partly indicates the stability of such multi-hump solutions under small perturbations. The exact and approximate hyperbolic system describes such an interaction with good accuracy.