On the Asymptotic Behavior of an Interface Problem in a Thin Domain
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RESEARCH ARTICLE
On the Asymptotic Behavior of an Interface Problem in a Thin Domain H. Benseridi1 • Y. Letoufa2 • M. Dilmi1
Received: 6 February 2017 / Revised: 23 January 2019 / Accepted: 5 February 2019 Ó The National Academy of Sciences, India 2019
Abstract Considering a mathematical model for the bilateral, frictionless contact between two Bingham fluids, establish a variational formulation for the problem and prove estimates on the velocity and pressure which are independent of the small parameter. The passage to the limit on e permits us to obtain the existence and uniqueness of the velocity. A specific Reynolds equation associated with variational inequalities is obtained. Keywords Asymptotic behavior Bingham fluid Non-Newtonian fluid Reynolds equation Transmission conditions Tresca law Mathematics Subject Classification 35R35 76F10 78M35
1 Introduction In general, the fluids are three-dimensional, non-stationary, rotational, turbulent, and the domains occupied by the fluid have a complicated geometry. Dilmi et al. [1] worked on the homogeneous case existence and uniqueness results for & H. Benseridi [email protected] Y. Letoufa [email protected] M. Dilmi [email protected] 1
Applied Mathematics Laboratory, Setif 1 University, 19000 Se´tif, Algeria
2
Applied Mathematics Laboratory, El Oued University, El Oued, Algeria
the weak solution, to the stationary equations for Bingham fluid in a three-dimensional bounded domain with Fourier and Tresca boundary condition. The asymptotic analysis of an incompressible Newtonian problem, when one dimension of the fluid domain tends to zero, has been studied [2]. The asymptotic stability of weak solutions for the incompressible non-Newtonian fluid motion in R2 has been investigated [3]. Other similar problems can be found in monographs such as [4, 5], and the literature quoted there. More recently, Benseridi et al. [6] worked on the asymptotic analysis of a nonlinear elasticity in a three-dimensional thin domain with nonlinear friction of Tresca type. The asymptotic behavior of a dynamical problem of nonisothermal elasticity with nonlinear friction of Tresca type has also been studied [7]. Works on the singular behavior of solutions for elasticity system in a nonhomogeneous polygon or a polyhedron are limited only to the results of existence, uniqueness, and regularity of the weak solution under several assumptions, [8–11] and the references cited therein. The goal of this paper is to study the asymptotic behavior of a boundary value problem in a three-dimensional thin domain Xe with nonlinear friction of Tresca type. The novelty here consists in the fact that we study the contact between two Bingham fluids Xe ¼ Xe1 [ Xe2 ; and we assume that on the common part of the boundary there is no separation between the bodies during the process, and thus the contact is bilateral. The use of the small change of x3 variable, z ¼ , transforms the initial problem posed in the e e domain X into a new problem posed on a fixed domain X ¼ X1 [ X2 independen
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