Study of Stokes dynamical system in a thin domain with Fourier and Tresca boundary conditions

Abstract

Asymptotic analysis of an incompressible Stokes fluid in a dynamic regime in a three-dimensional thin domain [Formula: see text] with mixed boundary conditions and Tresca friction law is studied in this paper. The problem statement and variational formulation of the problem are reformulated in a fixed domain. In which case, the estimates on velocity and pressure are proved. These estimates will be useful in order to give a specific Reynolds equation associated with variational inequalities and prove the uniqueness.

Unsteady flow of Bingham fluid in a thin layer with mixed boundary conditions

“…Finally, passing to the limit in δ as in [12], and using the semi-continuous inferior of the function u → T 0 ǎ(u, u)dt and v → T 0 J ε (v) dt for L 2 (0, T ; K ε div ) with the weak topology, to obtain (2.9)-(2.10). The proof of uniqueness is analogous to [8], and this concludes the proof of theorem 3.1.…”

“…J ε is convex and continuous but non differentiable in K ε . Following [5,8], the variational inequality of the problem (2.1)-(2.8) is given by…”

“…Therefore, the Reynolds equation is obtained in a manner similar to the stationary case as in [1], and from [2] the Tresca boundary condition can be recovered. Indeed, the case α = 0 corresponds to the Stokes flow, and has been studied in [8].…”

“…The departure point is the laws of conservation, which includes here the effect of the acceleration-dependent inertia forces. A friction law of Tresca and the Fourier boundary condition are assumed on the boundary, so fall into the scope of the work of [8]. Then we will compare our results to stationary problem in [1,2,4,5].…”

“…In section 3, we give the main results on existence results by the regularization methods. In section 4, we introduce a scaling as in [5,8], we give some needed estimates on the velocity and pressure, also the convergence results. In sections 5 we present the limit problem and we give the mechanical interpretation of the results.…”