We consider a diffuse interface model for the phase separation of an incompressible and isothermal non-Newtonian binary fluid mixture in three dimensions. The averaged velocity u is governed by a Navier-Stokes system with a shear dependent viscosity controlled by a power p > 2. This system is nonlinearly coupled through the Korteweg force with a convective nonlocal Cahn-Hilliard equation for the order parameter phi, that is, the (relative) concentration difference of the two components. The resulting equations are endowed with the no-slip boundary condition for is and the no-flux boundary condition for the chemical potential mu. The latter variable is the functional derivative of a nonlocal and nonconvex Ginzburg-Landau type functional which accounts for the presence of two phases. We first prove the existence of a weak solution in the case p >= 11/5. Then we extend some previous results on time regularity and uniqueness if p > 11/5.
Nonlocal Cahn–Hilliard–Navier–Stokes systems with shear dependent viscosity / S. Frigeri, M. Grasselli, D. Prazak. - In: JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS. - ISSN 0022-247X. - 459:2(2018), pp. 753-777. [10.1016/j.jmaa.2017.10.078]
Nonlocal Cahn–Hilliard–Navier–Stokes systems with shear dependent viscosity
S. Frigeri;
2018
Abstract
We consider a diffuse interface model for the phase separation of an incompressible and isothermal non-Newtonian binary fluid mixture in three dimensions. The averaged velocity u is governed by a Navier-Stokes system with a shear dependent viscosity controlled by a power p > 2. This system is nonlinearly coupled through the Korteweg force with a convective nonlocal Cahn-Hilliard equation for the order parameter phi, that is, the (relative) concentration difference of the two components. The resulting equations are endowed with the no-slip boundary condition for is and the no-flux boundary condition for the chemical potential mu. The latter variable is the functional derivative of a nonlocal and nonconvex Ginzburg-Landau type functional which accounts for the presence of two phases. We first prove the existence of a weak solution in the case p >= 11/5. Then we extend some previous results on time regularity and uniqueness if p > 11/5.File | Dimensione | Formato | |
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