A drift model N of a velocity mixture and the evolution of a discontinuous solution of a two-velocity drift model

At present, to describe the two-velocity flow of a multiphase mixture, either a two-fluid model is used; or a drift model using simplified momentum equations that do not take into account inertial forces. The corresponding system of equations for a two-fluid model with the same phase pressure without special, postulated, stabilizing terms is non-hyperbolic. This can lead to difficulties in finding a solution. Accounting for the difference in phase pressure in this model can lead to instability of solutions, which can also complicate the search for a solution. The drift model, which is the subject of this work, does not have this shortcoming. The case of N phase velocities (N > 2), considered in this article, has not been studied so far, although it has a large area of practical applications, for example, a three-velocity flow of a dispersed-film flow in the energy and chemical industries, and others. Hyperbolicity and characteristics are investigated. The characteristic equation of the system, which has N-1 degree, is analyzed. It is proved that there is one eigenvalue between neighboring speeds. If the k phases have the same speed, then the k-1 eigenvalue is equal to that same speed. So, if there are no coinciding phase velocities, or the number of phases with the same speed does not exceed 2, then all eigenvalues are real and different, which means that the system is hyperbolic. If the number of phases with the same speed exceeds 2, then all eigenvalues are also real, but there are multiples among them. An additional study carried out for this case showed that the system is also hyperbolic. An analytical solution of the discontinuity decay problem for a three-velocity flow (N=3) is found. It is assumed that the speed difference between the first and second phases is much greater than the speed difference between the second and third phases. Without this assumption, the solution can be found numerically. For the case of two phase velocities (N = 2), an analytical solution is found that describes the transition from a continuous distribution to a jump. This solution describes the flow for a given dependence of the slip rate on the regime parameter.