Therefore take the average of the diagonal terms of \(\mathbf{T}\) and define a mean pressure (as opposed to thermodynamic pressure \( p \)) as
\[
\bar{p} \equiv -\tfrac{1}{3} T_{ii}
\]
Substitution into \(\boxed{p = -\tfrac{1}{3} T_{ii} + \left( \tfrac{2}{3} \mu + \lambda \right) \nabla \cdot \mathbf{u}}\) gives
\[
p - \bar{p} = \left( \tfrac{2}{3} \mu + \lambda \right) \nabla \cdot \mathbf{u}
\]
For a completely incompressible fluid we can only define a mechanical or mean pressure, because there is no equation of state to determine a thermodynamic pressure
(the absolute pressure in an incompressible fluid is indeterminate, and only its gradients can be determined from the equations of motion).
The λ-term in the constitutive equation \(\boxed{T_{ij} = -p \delta_{ij} + 2\mu S_{ij} + \lambda S_{mm} \delta_{ij}}\) drops out when \( S_{mm} = \nabla \cdot \mathbf{u} = 0 \), and no consideration of \(\boxed{p - \bar{p} = \left( \tfrac{2}{3} \mu + \lambda \right) \nabla \cdot \mathbf{u}}\) is necessary.
So, for incompressible fluids,
the constitutive equation takes the simple form
\[
T_{ij} = -p\delta_{ij} + 2\mu S_{ij} \quad \text{(incompressible)}
\]
where \( p \) can only be interpreted as the mean pressure experienced by a fluid particle.For a compressible fluid, on the other hand, a thermodynamic pressure can be defined, and it seems that \( p \) and \( \bar{p} \) can be different.
In fact, \(\boxed{p - \bar{p} = \left( \tfrac{2}{3} \mu + \lambda \right) \nabla \cdot \mathbf{u}}\) relates this difference to the rate of expansion through the proportionality constant \( \mu_v = \lambda + \frac{2}{3}\mu \),
which is called the coefficient of bulk viscosity.
It has an appreciable effect on sound absorption and shock-wave structure. It is generally found to be nonzero in polyatomic gases because of relaxation effects associated with molecular translation and rotation.
However, the Stokes assumption
\[
\lambda + \frac{2}{3}\mu = 0
\]
is found to be accurate in many situations because either the fluid's \( \mu_v \) or the flow’s dilatation rate is small
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