Allen-Cahn-Navier-Stokes-Voigt systems with moving contact lines. (English) Zbl 07766055
Summary: We consider a diffuse interface model for an incompressible binary fluid flow. The model consists of the Navier-Stokes-Voigt equations coupled with the mass-conserving Allen-Cahn equation with Flory-Huggins potential. The resulting system is subject to generalized Navier boundary conditions for the (volume averaged) fluid velocity \(\mathbf{u}\) and to a dynamic contact line boundary condition for the order parameter \(\phi\). These boundary conditions account for the moving contact line phenomenon. We establish the existence of a global weak solution which satisfies an energy inequality. A similar result is proven for the Allen-Cahn-Navier-Stokes system. In order to obtain some higher-order regularity (w.r.t. time) we propose the Voigt approximation: in this way we are able to prove the validity of the energy identity and of the strict separation property. Thanks to this property, we can show the uniqueness of quasi-strong solutions, even in dimension three. Regularization in finite time of weak solutions is also shown.
MSC:
| 76T06 | Liquid-liquid two component flows |
| 76D05 | Navier-Stokes equations for incompressible viscous fluids |
| 35Q35 | PDEs in connection with fluid mechanics |
Keywords:
Flory-Huggins potential; diffuse interface model; generalized Navier boundary condition; solution existence; energy identity; strict separation property; uniqueness; finite-time regularizationReferences:
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