Four Linear Continuum Theories - Continuum Mechanics of Anisotropic Materials - page 155

c ) 2 . This function is to be
extremized subject to the constraint that ( x, y, z ) be a point on the surface of the unit
sphere, x 2

a ) 2 b ) 2 ( x, y, z ) and ( a, b, c ), f ( x, y, z ) ¼ ( x þ ( y þ ( z

y 2 z 2 a ) 2 b ) 2 c ) 2 ( x 2 þ þ ¼ 1, thus q ( x, y, z ) ¼ ( x þ ( y þ ( z þ L þ

y 2 z 2

1). Setting the derivatives of q ( x, y, z ) with respect to x , y , and z equal
to zero it follows that x

þ

¼ a /( L þ

1), y

¼ b /( L þ

1), z

¼ c /( L þ

1). From the

constraint condition x 2

y 2 z 2

( a 2

b 2 c 2 ). þ þ ¼ L þ ¼ √ þ þ

1, it follows that 1

In the special case when ( a, b, c )

¼ L þ ¼

2, and ( x, y, z )

¼

(2, 0, 0), ( 1) (1, 0, 0)

or ( x, y, z ) 1, 0, 0); thus the point on the unit sphere closest to (2, 0, 0) is (1, 0, 0)

and the point on the unit sphere furthest from (2, 0, 0) is ( 1, 0, 0).
With this background the problem is to employ Lagrange's method to mini-
mize the dissipation due to the rate of volume change, as opposed to the dissipa-
tion due to the shearing motion, in a viscous fluid. From equation (5.23N) the
stress power or dissipation is given by T : D

¼ (

3)(tr D ) 2

¼ p (tr D) + (3 l þ 2 m þ

tr(dev D) 2 . The constraint condition is that tr D

2 0. The problem is to show

that when this constraint is imposed, the pressure becomes a constant Lagrange
multiplier.

m ¼

)/3)(tr D ) 2 tr(dev D ) 2 Solution : In this case q (tr D ) ¼ p (tr D ) þ ((3 l þ 2 m þ 2 m þ

tr(dev D ) 2 / L (tr D ); thus from q / (tr D ) ¼ ( L p ) þ (2(3 l þ 2 m )/3)(tr D ) ¼ 0,

∂ ∂ ∂

p .
In a paper of 1883 Sir Osborne Reynolds showed that the transition between
laminar flow governed by the Newtonian law of viscosity, and the form of the
Navier-Stokes equations considered here, and the chaotic flow called turbulence
depended upon a dimensionless number that is now called the Reynolds number.
The Reynolds number R is equal to

(tr D ) ¼ 0, and the constraint condition tr D ¼ 0, it follows that L ¼

∂

are the density and
viscosity of the fluid, respectively, and V and d are a representative velocity and a
representative length of the problem under consideration, respectively. Only lami-
nar flows of viscous fluids are considered in this topic, hence there is always a
certain value of a Reynolds number for which the solution no longer describes the
physical situation accurately.
Example 6.4.2 Couette Flow
Consider an incompressible viscous fluid of viscosity m in the domain between two
infinite flat solid plates at x 2 ¼

r Vd / m where r and m

h /2. Action-at-a-distance forces are not present

and the plate at x 2 ¼ h /2 is moving at constant velocity V in the positive x 1

direction. There is no pressure gradient. Determine the velocity distribution and
the stress that must be applied to the top plate to maintain its motion.

Solution : Assume that the only nonzero velocity component is in the x 1 direction
and that it depends only upon the x 2 coordinate, v 1 ¼ v 1 ( x 2 ). This velocity field

automatically satisfies the incompressibility condition, r v ¼ 0, and the reduced

Navier-Stokes equations are

@n 1
@ x 2 ¼

0 and r p ¼ 0 :