Mathematical study of multi-phase flow under shear through order parameter formulation.

*(English)*Zbl 0937.35123The autor studies a mathematical model for multiphase incompressible fluids in two and three dimensions. The approach chosen uses the order parameter formulation. The model investigates the evolution of a two-phases fluid moving in a channel under shear. The order parameter gives the concentration of the phases in the channel; it takes values between \(-1\) and \(1\), the two extremal values representing the pure phases of the fluid.

The fluid is governed by the incompressible Navier-Stokes equations, while the order parameter obeys the Cahn-Hilliard equation \[ {{\partial \varphi}\over{\partial t}}- \text{div} (B(\varphi)\nabla\mu)=0 \]

The coupling between the two equations is given in Navier-Stokes by the viscosity depending on the order parameter and by a forcing term which is a capillary force acting essentially on the interface between the two phases, while in Cahn-Hilliard there is a transport term. The chemical potential \(\mu\) which appears in the Cahn-Hilliard equation has the form \[ \mu=-\alpha\triangle\varphi+F'(\varphi) \] and the problem is studied with a function \(F\) which has the structure of a double well, an assumption with physical relevance.

The equations are studied in the domain \((-L,L)^{d-1}\times(-1,1)\) with periodic boundary conditions. The author shows the existence of global weak solutions and of unique strong solutions, which are global only in dimension two. Then he studies a degenerate case, in which the function \(F\) is allowed to be singular. In this case the existence of solutions is proved in a weaker sense.

At last an interesting asymptotic stability result is proved. It was already known that in dimension one the stationary solution is stable (in a suitable sense). The author proves the stability of the stationary solution in dimensions two and three in the sense that given initial conditions close to the stationary solution, there exists a unique global strong solution which remains close to the stationary solution.

The fluid is governed by the incompressible Navier-Stokes equations, while the order parameter obeys the Cahn-Hilliard equation \[ {{\partial \varphi}\over{\partial t}}- \text{div} (B(\varphi)\nabla\mu)=0 \]

The coupling between the two equations is given in Navier-Stokes by the viscosity depending on the order parameter and by a forcing term which is a capillary force acting essentially on the interface between the two phases, while in Cahn-Hilliard there is a transport term. The chemical potential \(\mu\) which appears in the Cahn-Hilliard equation has the form \[ \mu=-\alpha\triangle\varphi+F'(\varphi) \] and the problem is studied with a function \(F\) which has the structure of a double well, an assumption with physical relevance.

The equations are studied in the domain \((-L,L)^{d-1}\times(-1,1)\) with periodic boundary conditions. The author shows the existence of global weak solutions and of unique strong solutions, which are global only in dimension two. Then he studies a degenerate case, in which the function \(F\) is allowed to be singular. In this case the existence of solutions is proved in a weaker sense.

At last an interesting asymptotic stability result is proved. It was already known that in dimension one the stationary solution is stable (in a suitable sense). The author proves the stability of the stationary solution in dimensions two and three in the sense that given initial conditions close to the stationary solution, there exists a unique global strong solution which remains close to the stationary solution.

Reviewer: Marco Romito (Firenze)

##### MSC:

35Q30 | Navier-Stokes equations |

76T10 | Liquid-gas two-phase flows, bubbly flows |

35D05 | Existence of generalized solutions of PDE (MSC2000) |