We consider an unsteady non-isothermal fluid flow subjected to non-homogeneous Dirichlet conditions on a part of the boundary and Tresca’s friction law on the other part. For this problem an existence result has been proved recently in Boukrouche et al. (2014) but uniqueness has been left as an open question. Starting from the approximation of the problem based on a regularization of the free boundary condition due to friction combined with a special penalty method, we establish some sharp a priori estimates leading to better regularity properties for the velocity field and to the uniqueness of the solutions. Finally we study the regularity of the pressure field and of the stress tensor.

Non-isothermal Navier–Stokes system with mixed boundary conditions and friction law: Uniqueness and regularity properties

Existence for non-isothermal fluid flows with Tresca's friction and Cattaneo's heat law

Time fractional Cattaneo-Christov anomalous diffusion in comb frame with finite length of fingers

Comb model for the anomalous diffusion with dual-phase-lag constitutive relation

Highlights

• Fractional Cattaneo-Christov flux model is used in analyzing anomalous diffusion in comb frame subject to finite length of fingers.

• Solutions of formulated governing equation which contains Dirac delta function and mixed partial derivatives are obtained numerically.

• The correctness of numerical method is verified by introducing a source item to construct an exact solution.

• The particles distributions and the distribution for total number of particles on x axis are sketched graphically and the dynamic characteristics with involved parameters effect are analyzed.

Abstract

Fractional Cattaneo-Christov flux model is used in analyzing the anomalous diffusion in comb frame subject to finite length of fingers. Formulated governing equation contains Dirac delta function and mixed partial derivatives. Solutions are obtained by numerical discretization method where the time fractional derivative of order α in (0, 1] is approximated by L1-scheme. The correctness of numerical method is verified by introducing a source item to construct an exact solution. Results show that, when there exists relaxation, the particles distribution displays a parabolic feature with the spatial evolution at small times while a hyperbolic feature appears with a larger relaxation parameter at large times, the particles distribution and the total number of particles on x axis are oscillating with the temporal evolution. Moreover, the effects of involved parameters on dynamic characteristics are graphically analyzed and discussed in detail.

Keywords

 Anomalous diffusion; Cattaneo-Christov flux; Comb frame; Fractional derivative

Motivated by extrusion problems, we consider a non-stationary incompress-ible 3D fluid flow with a non-constant (temperature dependent) viscosity, subjected to mixed boundary conditions with a given time dependent velocity on a part of the boundary and Tresca's friction law on the other part. We construct a sequence of approximate solutions by using a regularization of the free boundary condition due to friction combined with a particular penalty method, reminiscent of the " incompressibility limit " of compressible fluids, allowing to get better insights into the links between the fluid velocity and pressure fields. Then we pass to the limit with compactness arguments to obtain a solution to our original problem.

Imane Boussetouan

Papers

Non-isothermal Navier–Stokes system with mixed boundary conditions and friction law: Uniqueness and regularity properties

Existence for non-isothermal fluid flows with Tresca's friction and Cattaneo's heat law

Unsteady 3D-Navier–Stokes system with Tresca’s friction law

Unsteady 3D-Navier-Stokes System with Tresca's Friction Law