In this report, I want to introduce the physics background of the inhomogeneous incompressible Navier-Stokes equations. As is we all known, for homogeneous incompressible Navier-Stokes equations(the usual incompressible Navier-Stokes equations) the viscosity coefficient of the dissipative term is independent of the density. If the viscosity coefficient of the dissipative term is density-dependent, the equations are called the inhomogeneous incompressible Navier-Stokes equations.

However, the inhomogeneous equations are not obtained by technically imposing the condition of density-dependent for the viscosity. There is strong physical meaning which is the asymptotic behavior of a micro-macro complicated fluid model. So I will introduce how derive the inhomogeneous incompressible Navier-Stokes equations from the micro-macro complicated fluid model.