Georges Duvaut

Transfert de chaleur dans un fluide de Bingham dont la viscosité dépend de la température

The paper studies the flow of a so-called Bingham fluid, taking into account the variation of viscosity with temperature. The problem is modelled using the technique of variational inequalities of evolution. One proves the existence of a solution in case of a 2-dimensional flow. Questions of uniqueness, and of existence and uniqueness for 3-dimensional flows, are open. The proof relies on fixed-point theory, previous results of the authors for Bingham flows when viscosity does not depend on the temperature and a regularity theorem of Grisvard for linear parabolic equations.

Rigid Visco-Plastic Bingham Fluid

A rigid visco-plastic fluid is a continuous medium which obeys the general conservation laws stated in Section I.1, and in addition obeys special constituent laws.

Introduction to Plasticity

The term “plastic material” covers materials with various types of behavior, such as “elasto-visco-plastic”, “rigid-perfectly plastic”, “plastic with work hardening”, etc.

Maxwell’s Equations. Antenna Problems

The phenomena of electromagnetism can lead to variational inequalities, either with Maxwell’s equations in a polarizable medium or, when the phenomena are coupled, with the mechanics of Bingham fluids yielding the magnetodynamics of Bingham fluids. This second possibility is treated in Duvaut-Lions [7]. The present chapter is devoted to unilateral problems connected with Maxwell’s equations. Since we cannot refer to a unified exposition of the classical phenomena of electromagnetism, we will set up equations for the phenomena in Section 2 and develop a mathematical theory for the classical solutions in Sections 4–6. This theory is indispensable for the subsequent treatment of Maxwell’s equations in a polarizable medium. Section 3 is a presentation of the physical problems under consideration.

Problems of Heat Control

Orientation. In this chapter, it is assumed that the reader is familiar with Sections 1–5 of Chapter I.

Problems of Semi-Permeable Media and of Temperature Control

In this review, we do not intend to develop the complete theory of continuous media. For such an exposition, we refer the reader to the works of P. Germain [1, 2], G. Mandel [1], W. Noll and C. Truesdell [1], Sedov [1].

Unilateral Phenomena in the Theory of Flat Plates

In this chapter, we will examine unilateral phenomena and phenomena with friction for flat elastic plates.

Classical Problems and Problems with Friction in Elasticity and Visco-Elasticity

In this chapter, we take up problems of elasticity and visco-elasticity. We recall (of course, this will be stated more precisely in what follows) that visco-elasticity differs from elasticity by the fact that the state of stress at present depends, in the visco-elastic case, on all the deformations undergone in the past (and does not depend on them in the elastic case).