Viscous Incompressible Fluids

Abstract

In this chapter we study the motion of a viscous incompressible fluid. In such a fluid the stress tensor is equal to the difference between the viscosity tensor and the unit tensor times pressure. When the viscosity tensor is zero this model reduces to the model of ideal incompressible fluid. The viscosity tensor is expressed in terms of the strain-rate tensor. We only consider isotropic fluids. Then, using the analogy with the linear elasticity we immediately obtain the expression for the viscous part of the stress tensor in isotropic viscous fluids. In general, this relation is completely defined by two constants, the dynamic viscosity and second viscosity. However, in the case of incompressible fluid it only contains the dynamic viscosity. Using the expression for the viscous stress tensor we derive the famous Navier–Stokes equation describing the motion of a viscous incompressible fluid. Written in dimensionless variables this equation contains only one dimensionless parameter, which if the Reynolds number. We obtain the solutions to the Navier–Stokes equation describing the Couette and Poiseuille flows. We consider Stoke’s approximation valid when the Reynolds number is small. We also consider the theory of boundary layers valid for large Reynolds numbers.