Asymptotic Convergence of a Generalized Non-Newtonian Fluid with Tresca Boundary Conditions
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Wuhan Institute Physics and Mathematics, Chinese Academy of Sciences, 2020
http://actams.wipm.ac.cn
ASYMPTOTIC CONVERGENCE OF A GENERALIZED NON-NEWTONIAN FLUID WITH TRESCA BOUNDARY CONDITIONS∗ Adelkader SAADALLAH
Hamid BENSERIDI Mourad DILMI
Applied Mathematics Laboratory, Department of Mathematics, Faculty of Sciences, University of Ferhat Abbas- S´etif 1, 19000, Algeria E-mail : [email protected]; m [email protected]; [email protected] Abstract The goal of this article is to study the asymptotic analysis of an incompressible Herschel-Bulkley fluid in a thin domain with Tresca boundary conditions. The yield stress and the constant viscosity are assumed to vary with respect to the thin layer parameter ε. Firstly, the problem statement and variational formulation are formulated. We then obtained the existence and the uniqueness result of a weak solution and the estimates for the velocity field and the pressure independently of the parameter ε. Finally, we give a specific Reynolds equation associated with variational inequalities and prove the uniqueness. Key words
Asymptotic approach; Herschel-Bulkley fluid; Reynolds equation; Tresca law
2010 MR Subject Classification
1
35R35; 76F10; 78M35
Introduction
The subject of this work is the study of the asymptotic analysis of a generalized nonNewtonian fluid in a three dimensional thin domain Ωε . The contact boundary conditions considered here are the mixed and the friction law is the Tresca type. Herschel-Bulkley fluid is a generalized model of a non-Newtonian fluid which is introduced for the first time in 1926. In this work, we consider a thin film Ωε of R3 during a time interval [0, T ]. The equations that govern the stationary flow are as follows: − div (σ ε ) = f ε ε ε σij =σ ˜ij − pε δij ,
in Ωε ,
D(uε ) ε r−2 + µ |D(uε )| D(uε ) if D(uε ) 6= 0, in Ω , |D(uε )| |˜ σ ε | ≤ αε if D(uε ) = 0,
σ ˜ ε = αε
where αε ≥ 0 is the yield stress, µ > 0 is the constant viscosity, uε is the velocity field, δij is T the Kronecker symbol, 1 < r ≤ 2, and D(uε ) = 21 (∇uε + (∇uε ) ). ∗ Received November 26, 2018. The first author is supported by MESRS of Algeria (CNEPRU Project No. C00L03UN190120150002).
No.3
A. Saadallah et al: ANALYSIS OF A GENERALIZED FLUID
701
In many problems, which study the asymptotic behavior for a problem of continuum mechanics in a thin domain, we transform the problem into an equivalent problem on a domain Ω independent of the parameter ε. Many research papers were written for similar problems with the same friction boundary conditions, however, these papers were restricted to the particular case of the considered problem. Let us mention some of the works done in [2, 3, 6]. In these articles, the authors consider a problem describing the motion of an incompressible Newtonian and non-Newtonian fluid in a three-dimensional thin domain flow with a non-linear Tresca interface condition. The asymptotic analysis of a Bingham fluid in a three dimensional bounded domain with Fourier and Tresca boundary conditions ar
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