On solutions of non-linear equations arising in Rivlin-Ericksen fluids

Abstract

Multicomponent flows whether occurring in nature such us debris flows, avalanches, and mud slides, or in industrial applica.tions, such as fluidized beds, solids transport and many other chemical and agricultural processes, present a formidable challenge to engineers and scientists. To model and study the flow and behavior of slIch complex fluids, one can lise either statistical theories or continuum theories, in addition to the phenomenological/experimental approaches. Due to various properties of real fluids there are many models. The simplest model is NavierStokes model which is used for fluids of low molecular weight. However, it is wen known that materials with complex structures such as solutions and melts of polymers, plastic and synthetic fibers, certain oils and greases, soap and detergents, certain pharmaceutical and biological Ouids fall into the category of non-Newtonian fluids. During the last several years, generalization of Navier-Stokes model to highly non-linear constitutive laws have been proposed because of their interest in applications to industry and technology. In order to explain several non-standard features, such as normal stress effects, rod climbing, shear thinning and shear thickening, RivlinEricksen Buids [1! of differential type are introduced. These fluids are rather complex from the point of view of partial differential equation theory. Nevertheless, several authors in fluid mechanics are now engaged with the equations of motion on non-Newtonian fluids of second and third grade. In particular, some authors are interested in studying a-grade fluids as selfconsistent models and not as approximating models. Therefore, in studying dynamics they ask that all the flows meet the Clausius-Duhem inequality and that the specific Helmholtz free energy of the fluid is II. minimum at equilibrium [2J. On the other hand, it is under the same hypothesis that the Navier-Stokes model is studied. That is, it is always assumed that some real fluids exist such that Navier-Stokes or n-grade fluids are exact models, and Dot truncations of viscoelastic flu ids. Moreover (as Doted in refs. [3,5]), different assumptions could heavily affect the rest state stability. Under these thermodytlamically hypothesis, several results concerning, existence and stability have been obtained....