A non-hydrostatic multi-phase mass flow model

Modeling mass flows is classically based on the hydrostatic, depth-averaged balance equations. However, if the momentum transfers scale similarly in the slope parallel and the flow depth directions, then the gravity and the acceleration can have the same order of magnitude effects. This urges for a non-hydrostatic model formulation. Here, I extend existing single-phase Boussinesq-type gravity wave models by developing a new non-hydrostatic model for multi-phase mass flows consisting of the solid and fine-solid particles, and viscous fluid (Pudasaini and Mergili, 2019 [1]). The new model includes enhanced gravity and dispersion effects taking into account the interfacial momentum transfers due to the multi-phase nature of the mass flow. I outline the fundamentally new contributions in the non-hydrostatic Boussinesq-type multi-phase gravity waves emerging from the phase-interactions including buoyancy, drag, virtual mass and Newtonian as well as non-Newtonian viscous effects. So, this contribution presents a more general, well-structured framework of the multi-phase flows with enhanced gravity and dispersion effects, setting a foundation for a comprehensive simulation of such flows. I discuss some particular situations where the non-hydrostatic and dispersive effects are more pronounced for multi-phase mass flows. Even the reduced models demonstrate the importance of non-hydrostatic contributions for both the solid and fine-solid particles, and the viscous fluid. Analytical solutions are presented for some simple situations demonstrating how the new dispersive model can be reduced to non-dispersive motions, yet largely generalizing the existing non-dispersive models. I postulate a novel, spatially varying dissipative force, called the prime-force, which physically controls the dynamics, run-out and the deposition of the mass flow in a precise way. The practitioners and engineers may find this force very useful in relevant technical applications. This illuminates the need of formally including the prime-force in the momentum balance equation. A simple dispersion equation is derived. I highlight the essence of dispersion on the mass flow dynamics. Dispersion consistently produces a wavy velocity field about the reference state without dispersion. Emergence of such a dispersive wave is the first of this kind for the avalanching debris mass. It is revealed that the dispersion intensity increases energetically as the solid volume fraction or the friction decreases.