CONTINUUM MECHANICS for ENGINEERS

For a fluid in motion the shear stresses are not usually zero, and in this
case we write
σ = − pδ
+ τ
ij ij ij
(7.1-5)
where τ ij is called the viscous stress tensor, which is a function of the motion
and vanishes when the fluid is at rest. In this equation, the pressure p is
called the thermodynamic pressure and is given by the same functional relationship
with respect to θ and ρ as that for the static pressure p o in the
equilibrium state, that is, by
Fp, ( ρθ , )= 0
(7.1-6)
Note from Eq 7.1-5 that, for a fluid in motion, p is not equal to the mean
normal stress, but instead is given by
1
p =− σii −τii
3
( )
(7.1-7)
so that, for a fluid at rest (τ ij = 0), p equates to p o.
In developing constitutive equations for viscous fluids, we first remind
ourselves that this viscous stress tensor must vanish for fluids at rest, and
following the usual practice, we assume that τ ij is a function of the rate of
deformation tensor D ij. Expressing this symbolically, we write
τ ij = fij(
D)
(7.1-8)
If the functional relationship in this equation is nonlinear, the fluid is called
a Stokesian fluid. When f ij defines τ ij as a linear function of D ij, the fluid is
known as a Newtonian fluid, and we represent it by the equation
τ ij = KijpqDpq (7.1-9)
in which the coefficients K ijpq reflect the viscous properties of the fluid.
As may be verified experimentally, all fluids are isotropic. Therefore, K ijpq
in Eq 7.1-9 is an isotropic tensor; this, along with the symmetry properties
of D ij and τ ij, allow us to reduce the 81 coefficients K ijpq to 2. We conclude
that, for a homogeneous, isotropic Newtonian fluid, the constitutive equation
is
* *
σ =− pδ + λδ D + 2µ
D
ij ij ij kk ij
(7.1-10)