The Navier-Stokes equations

The equivalent in fluid mechanics is the model of the Newtonian fluid for which it is assumed that

Stress oc Rate of strain

(2.86)

However, there is a major difference in status between the two models. At best Hooke’s law is a reasonable approximation for describing small deformations of some solids, particularly structural steel. Whereas the Newtonian fluid is a very accurate model for the behaviour of almost all homogeneous fluids, in particular water and air. It does not give good results for pseudofluids formed from suspensions of particles in homogeneous fluids, e. g. blood, toothpaste, slurries. Various Non­Newtonian fluid models are required to describe such fluids, which are often called non-Newtonian fluids. Non-Newtonian fluids are of little interest in aerodynamics and will be considered no further here.

For two-dimensional flows, the constitutive law (2.86) can be written

(2.87)

where ( ‘ ) denotes time derivatives. The factor 2 is merely used for convenience so as to cancel out the factor 1/2 in the expression (2.72a) for the rate of shear strain. Equation (2.87) is sufficient in the case of an incompressible fluid. For a compressible fluid, however, we should also allow for the possibility of direct stress being gener­ated by rate of change of volume or dilation. Thus we need to add the following to the right-hand side of (2.87)

(2.88)

p and Л are called the first and second coefficients of viscosity. More frequently p is just termed the dynamic viscosity in contrast to the kinematic viscosity v = р/р. If it is required that the actual pressure p — {uxx + ayy) + azz in a viscous fluid be identical to the thermodynamic pressure p, then it is easy to show that

ЗЛ + 2p = 0 or

This is often called Stokes hypothesis. In effect, it assumes that the bulk viscosity, pi, linking the average viscous direct stress to the rate of volumetric strain is zero, i. e.

(2.89)

This is still a rather controversial question. Bulk viscosity is of no importance in the great majority of engineering applications, but can be important for describing the propagation of sound waves in liquids and sometimes in gases also. Here, for the most part, we will assume incompressible flow, so that