CONTINUUM MECHANICS for ENGINEERS

where λ* and µ* are viscosity coefficients which denote the viscous properties
of the fluid. From this equation we see that the mean normal stress for a
Newtonian fluid is
(7.1-11)
* 1 * * where κ = (3λ
+ 2µ
) is known as the coefficient of bulk viscosity. The
condition
3
or, equivalently,
(7.1-12a)
(7.1-12b)
is known as Stokes condition, and we see from Eq 7.1-11 that this condition
assures us that, for a Newtonian fluid at rest, the mean normal stress equals
the (negative) pressure p.
If we introduce the deviator tensors
for stress and
( ) =− +
1 1 * *
σ =− p+ 3λ + 2µ
D p κ*
D
3 3
ii ii ii
* 1 * *
κ = (3λ
+ 2µ ) = 0
3
* *
λ =− µ
2
3
1
Sij = σ − δ σ
3
ij ij kk
βij ij δ D
1
= −
3 D
ij kk
for rate of deformation into Eq 7.1-10, we obtain
1
1
Sij + δσ ij = − pδ
kk ij + δij λ + 2µ kk + 2µβij
3
3
D
*
(3 * *)
(7.1-13a)
(7.1-13b)
which may be conveniently split into the pair of constitutive equations
Sij = 2µ βij
*
( )
*
σ =−3 p−κ D
ii ii
(7.1-14)
(7.1-15a)
(7.1-15b)