Dispersion and Group Analysis of Dusty Burgers Equations

We investigate the system of non-stationary one-dimensional equations consisting of a parabolic Burgers equation for the velocity of a viscous gas and a hyperbolic Hopf equation for the velocity of solid particles. The Burgers and Hopf equations are connected into a system due to relaxation terms simulating the momentum transfer between the carrier phase (gas) and the dispersed phase (particles). The momentum transfer intensity is inversely proportional to the relaxation time of the particle velocity to the gas velocity (stopping time).

A dispersion relation is constructed for this system. A particular solution corresponding to the damping of a low-amplitude sound wave is found. For an infinitely short velocity relaxation time, the effective viscosity of the gas-dust medium is derived, which is determined by the viscosity of the gas and the mass fraction of particles in the mixture.

The Lie algebra of symmetries of Burgers–Hopf system is found. Invariant submodels with respect to the basis operators of the symmetry algebra are derived. These submodels are explicitly integrated, except for one that defines stationary motion. For this submodel, a code has been developed for the numerical generation of particular solutions of the system. It is shown that the invariant solution determined by this submodel, in the asymptotic case of infinitely short velocity relaxation time also makes it possible to obtain the effective viscosity of the gas-dust mixture. Moreover, this effective viscosity coincides with the viscosity value determined from the dispersion relation.