# Chapter 5 Mathematical Issues Concerning The Navier-Stokes Equations and Some Of Its Generalizations Academic Article

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### abstract

• This chapter primarily deals with internal, isothermal, unsteady flows of a class of incompressible fluids with both constant and shear or pressure dependent viscosity that includes the Navier-Stokes fluid as a special subclass. We begin with a description of fluids within the framework of a continuum. We then discuss We begin with a description of fluids within the framework of a continuum. We then discuss various ways in which the response of a fluid can depart from that of a Navier-Stokes fluid. Next, we introduce a general thermodynamic framework that has been successful in describing the disparate response of continua that includes those of inelasticity, solid-to-solid transformation, viscoelasticity, granular materials, blood and asphalt rheology, etc. Here, it leads to a novel derivation of the constitutive equation for the Cauchy stress for fluids with constant, or shear and/or pressure, or density dependent viscosity within a full thermomechanical setting. One advantage of this approach consists in a transparent treatment of the constraint of incompressibility. We then concentrate on the mathematical analysis of three-dimensional unsteady flows of fluids with shear dependent viscosity that includes the Navier-Stokes model and Ladyzhenskaya's model as special cases. We are interested in the issues connected with mathematical self-consistency of the models, i.e., we are interested in knowing whether (1) flows exist for reasonable, but arbitrary initial data and all instants of time, (2) flows are uniquely determined, (3) the velocity is bounded and (4) the long-time behavior of all possible flows can be captured by a finite-dimensional, small (compact) set attracting all flow trajectories exponentially. For simplicity, we eliminate the choice of boundary conditions and their influence on the flows by assuming that all functions are spatially periodic with zero mean value over a periodic cell. All these results can however be extended to internal flows wherein the tangential component of the velocity satisfies Navier's slip at the boundary. Most of the results also hold for the no-slip boundary condition. While the mathematical consistency understood in the above sense for the Navier-Stokes model in three dimensions has not been established as yet, we will show that Ladyzhenskaya's model and some of its generalization enjoy all above characteristics for a certain range of parameters. We also discuss briefly further results related to generalizations of the Navier-Stokes equations. © 2006 Elsevier B.V. All rights reserved.

### author list (cited authors)

• Málek, J., & Rajagopal, K. R.

• 71

• January 2005