The Investigation On Some Problems Of The Mixed Fluid Mechanics

Mechanics of fluid mixing is a subject which studies the movement law and application of fluid. From a theoretical point of view, mechanics of fluid mixing makes use of the mass conservation law and the momentum conserva-tion law and researches this motion through an accurate mathematical model which is commonly referred to as the fluid dynamics equations. This paper mainly studies the mixed fluid models which include gas-liquid two-phase flow model, fluid-particle interaction model and the mixture of two liquids model derived from the thermodynamic framework.The main results are as follows:We study the blow-up criterion for the strong solution to the three dimensional viscous liquid-gas two-phase flow model in terms of the gradient of the velocity with two types of boundary conditions which are the Dirich-let boundary condition and the Navier-slip boundary condition, respectively. The main results improve the situation of two dimensions and remove the constrains ?< 7? on the viscous coefficient.We establish the convergence rates of the solution for a fluid-particle interaction model in the so-called bubbling regime in the whole space R3 and obtain the existence and uniqueness of global strong solutions when the initial value is near the stationary solution and the potential force are small in some Sobolev norms. By combining the IP-Lq estimates for the linearized equations and an elaborate energy method, the convergence rates are obtained in various norms.We study the global well-posedness for the 2D mixture of two liquids model. We obtain simplified model by assuming the densities of two fluids are constant. Firstly, by using Fridrich's method, we have proved the existence and the uniqueness of the local solution. Then we obtain the global well-posedness by classical method.