Modeling Of Electrorheological Fluids Research Articles

Keller-Osserman a priori estimates for doubly nonlinear anisotropic parabolic equations with absorption term

We are concerned with divergence type quasilinear parabolic equation with measurable coefficients and lower order terms model of which is a doubly nonlinear anisotropic parabolic equations with absorption term. This class of equations has numerous applications which appear in modeling of electrorheological fluids, image precessing, theory of elasticity, theory of non-Newtonian fluids with viscosity depending on the temperature. But the qualitative theory doesn't construct for these anisotropic equations. So, naturally, that during the last decade there has been growing substantial development in the qualitative theory of second order anisotropic elliptic and parabolic equations. The main purpose is to obtain the pointwise upper estimates in terms of distance to the boundary for nonnegative solutions of such equations. This type of estimates originate from the work of J. B. Keller, R. Osserman, who obtained a simple upper bound for any solution, in any number of variables for Laplace equation. These estimates play a crucial role in the theory of existence or nonexistence of so called large solutions of such equations, in the problems of removable singularities for solutions to elliptic and parabolic equations. Up to our knowledge all the known estimates for large solutions to elliptic and parabolic equations are related with equations for which some comparison properties hold. We refer to I.I. Skrypnik, A.E. Shishkov, M. Marcus , L. Veron, V.D. Radulescu for an account of these results and references therein. Such equations have been the object of very few works because in general such properties do not hold. The main ones concern equations only in the precise choice of absorption term \(f(u)=u^q\). Among the people who published significative results in this direction are I.I. Skrypnik, J. Vetois, F.C. Cirstea, J. Garcia-Melian, J.D. Rossi, J.C. Sabina de Lis. The main result of the paper is a priori estimates of Keller-Osserman type for nonnegative solutions of a doubly nonlinear anisotropic parabolic equations with absorption term that have been proven despite of the lack of comparison principle. To obtain these estimates we exploit the method of energy estimations and De Giorgy iteration techniques.

ON SUPERLINEAR p(x)-LAPLACIAN-LIKE PROBLEM WITHOUT AMBROSETTI AND RABINOWITZ CONDITION

Abstract. This paper deals with the superlinear elliptic problem with-out Ambrosetti and Rabinowitz type growth condition of the form:−div(1 + |∇u| p(x) √ 1+|∇u| 2p(x) )|∇u| p(x)−2 ∇u= λf(x,u), a.e. in Ω,u = 0, on ∂Ω,where Ω ⊂ R N is a bounded domain with smooth boundary ∂Ω, λ > 0 isa parameter. The purpose of this paper is to obtain the existence resultsof nontrivial solutions for every parameter λ. Firstly, by using the moun-tain pass theorem a nontrivial solution is constructed for almost everyparameter λ > 0. Then we consider the continuation of the solutions.Our results are a generalization of that of Manuela Rodrigues. 1. IntroductionDuring the last fifteen years, variational problems and partial differentialequations with various types of nonstandard growth conditions have becomeincreasingly popular. This is partly due to their frequent appearance in ap-plications such as the modeling of electrorheological fluids [1, 12] and imageprocessing [2], but these problems are very interesting from a purely mathe-matical point of view as well.In this paper, we consider the following nonlinear eigenvalue problems forp(x)-Laplacian-like operators originated from a capillary phenomena of thefollowing form:(P)−div(1+

THE STRESS-STRAIN RELATIONSHIP AND THE DYNAMIC YIELD STRESS IN ELECTRORHEOLOGICAL FLUID UNDER A MODIFIED CONDUCTION MODEL

By using a modified conduction model and the dipole approximation model respectively, we simulate the stress-strain curve and evaluate the dynamic yield stress through an ideal microstructure model of electrorheological fluid. The static yield strain is larger in our modified conduction model than in the dipole approximation model. Besides, the dynamic yield stress and static yield stress nearly linearly vary as volume fraction in the dipole model, while in our modified model they both exhibit a maximum at about volume fraction ϕ=0.45. Interpretation about these results is associated with the conduction effect and the inter-chain interactions.