The evolution of a cloud of particles in a compressible fluid can be modeled with a Vlasov–Fokker–Planck equation for the distribution function of the particles coupled with Navier–Stokes or Euler equations for the density and velocity of the fluid. Formal calculations have established the convergence of solution to the mesoscopic model to solutions to the macroscopic Navier–Stokes or Euler model coupled with a Smoluchowski equation as the ratio of the settling time for the microscopic velocity fluctuation of the particles to the characteristic macroscopic time scale goes to zero. This paper provides a rigorous asymptotic analysis for a homogeneous mesoscopic fluid–particle interaction model for particles dispersed in a compressible fluid is provided for the bubbling regime. A relative entropy inequality for a mixed hyperbolic/parabolic system of equations is employed.

可以用Vlasov-Fokker-Planck方程对颗粒的分布函数建模，并结合Navier-Stokes或Euler方程对流体的密度和速度进行建模，以模拟可压缩流体中颗粒云的演化。形式上的计算已经建立了介观模型的解与宏观Navier–Stokes或Euler模型以及Smoluchowski方程的解的收敛性，随着粒子微观速度波动的稳定时间与特征宏观时标之比的增加归零。本文提供了一种均匀的介观流体-颗粒相互作用模型的严格渐近分析，该模型为冒泡状态提供了分散在可压缩流体中的颗粒。