Asymptotic analysis for a homogeneous bubbling regime Vlasov-Fokker-Planck/Navier-Stokes system. (English) Zbl 1446.35207
Summary: 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.
MSC:
| 35Q83 | Vlasov equations |
| 35Q84 | Fokker-Planck equations |
| 35Q30 | Navier-Stokes equations |
| 35Q31 | Euler equations |
| 35B40 | Asymptotic behavior of solutions to PDEs |
| 76N06 | Compressible Navier-Stokes equations |
| 76T10 | Liquid-gas two-phase flows, bubbly flows |
Software:
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