Optimal systems of Lie subalgebras for a two-phase mass flow

Abstract

We apply the Lie symmetry method to a two-phase mass flow model (Pudasaini, 2012 [18]) and construct one-, two- and three-dimensional optimal systems of Lie subalgebras corresponding to the non-linear PDEs. As an optimal system contains structurally important information about different types of invariant solutions, it provides precise insights into all possible invariant solutions emerging from infinitesimal symmetries. We use the optimal system of one-dimensional Lie subalgebras to reduce the two-phase mass flow model to other systems of PDEs. Using the fact that the Lie bracket contains information about further reduction, we further reduce to systems of ODEs and PDEs. We solve a system numerically and present results for different physical and Lie parameters. Simulations reveal fluid and solid dynamics are distinctly sensitive to different Lie parameters, whereas both phases are influenced by the solid and the fluid pressure parameters. Higher pressure gradients result in higher flow velocities and lower flow heights. Fluid velocities dominate solid velocities, but the solid heights are higher than the fluid heights. Results provide an overall picture of the physical process, and the coupled dynamics of the solid and fluid phase velocities and the flow heights. These are physically meaningful results in sheared inclined channel flow of coupled two-phase mixture. This confirms the consistency of the obtained similarity solutions and potential applicability of the models and the constructed optimal systems.