Four Linear Continuum Theories - Continuum Mechanics of Anisotropic Materials - page 153

6.4 The Theory of Viscous Fluids

An overview of the theory of viscous fluids, without temperature effects, can be
obtained by considering it as a system of seventeen equations in seventeen scalar
unknowns. The seventeen scalar unknowns are the six components of the stress
tensor T , the fluid pressure p , the fluid density

, the six components of the rate-of-
deformation tensor D , and the three components of the velocity vector v . The
parameters of a viscous fluid problem are the viscosity coefficients

r

l and m (3 l þ 2 m

> 0) and the action-at-a-distance force d , which are assumed to be known.

The system of seventeen equations consists of a constitutive equation relating the
density

0, 2 m >

( p ) (and, in thermal-viscous problems, to the
temperature), the six equations of the Newtonian law of viscosity,

r to the pressure p , r ¼ r

T ¼ p 1 þ lð tr D Þ 1 þ 2 m D ; ð 5 : 11N Þ repeated

the six rate-of-deformation-velocity relations,

T

D ¼ð 1 = 2 Þððr vÞ þrvÞ; ð 2 : 32 Þ repeated

the one equation of the conservation of mass,

r þ rðr v Þ¼ 0 ; ð 3 : 5 Þ repeated

and the three stress equations of motion,

T T

r _ v ¼r T þ r d ; T ¼ : (6.36)

This form of the stress equations of motion differs from (3.37) and ( 6.18 ) only in
notation: the acceleration is here represented by u , respectively, a

result that follows from (2.20) and (2.24). The system of seventeen equations in
seventeen scalar unknowns may be reduced to a system of four equations in four
scalar unknowns, the pressure p and the three components of the velocity v ,by
accomplishing the following algebraic steps: (1) substitute the rate-of-deformation-
velocity (2.32) into the stress-strain relations (5.11N), then (2) substitute the
resulting expression relating the stress to the first derivatives of the velocity into
the three stress equations of motion ( 6.36 ). The result is a system of three equations
in three scalar unknowns, the three components of the velocity vector v :

v rather than _ € x or €

2 v

r _ v ¼r p þðl þ mÞrðr v Þþmr þ r d : (6.37)

This is the Navier-Stokes equation for viscous fluid flow. Substituting the
barotropic relation ( p ) into the conservation of mass (3.5) yields the fourth

equation in the set of four equations for the four unknowns, p and the components of v ,

r ¼ r