Sub-shock formation in Grad 10-moment equations for a binary gas mixture

Shock wave structure of multi-temperature Grad 10-moment equations for a binary gas mixture

“…mean velocity of the whole mixture) while the species temperatures are still well separate. This model is formally obtained starting from a kinetic description, by replacing the distribution functions with suitable Grad approximations [6,7] and assuming that species velocities share the same value.…”

“…We aim at testing such model on the classical problem of shock wave structure. The subject is topical and several interesting results on the occurrence of a smooth solution or on the presence of discontinuities (subshocks) have been proposed [13][14][15][16][17], especially for multi-velocity and multi-temperature Euler-like systems [3,18,19] and Grad 10-moment equations [6,7]. Since the 1950s [20] it has been well known that, even for a single gas, shock wave formation is not accurately described by Navier-Stokes equations, and higher-order closures, as Burnett [21] or Grad equations [22], are desirable.…”

“…The mathematical structure of Grad equations implies the occurrence of non-physical subshocks, namely of discontinuous shock wave solutions, for high Mach numbers, and suitable regularizations have been proposed to prevent these effects [23]. For a gas mixture the situation is much more involved, due to the possible presence of more critical values of the system parameters, corresponding to discontinuities in the shock wave profiles [6,7]. Here, we want to discuss the effects of a single velocity formulation on the shock profiles, by investigating strong and weak solutions and the presence of singular barriers [24] separating the limiting equilibrium states, and compare them with results obtained for multivelocity models [6,7,18].…”

“…For a gas mixture the situation is much more involved, due to the possible presence of more critical values of the system parameters, corresponding to discontinuities in the shock wave profiles [6,7]. Here, we want to discuss the effects of a single velocity formulation on the shock profiles, by investigating strong and weak solutions and the presence of singular barriers [24] separating the limiting equilibrium states, and compare them with results obtained for multivelocity models [6,7,18]. The analysis is performed for varying Mach number.…”

“…In particular, in this paper we consider a macroscopic closure which takes into account only moments of the distribution functions up to second order [29,30]; the resulting 10-moment equations, useful to model the flow in or around micro-electro-mechanical systems [31], can be seen as a proper subsystem of the 13-moment Grad closure, under the assumption of vanishing heat fluxes. Assuming cylindrical symmetry around the x-axis for the distribution functions and Maxwell molecules interaction potential, the following one-dimensional set of equations for a binary gas mixture is obtained [6]:…”