At thermodynamic equilibrium, the negative-one-third of the trace of the Cauchy stress tensor is often identified with the thermodynamic pressure,
which depends only on equilibrium state variables like temperature and density (equation of state).  In general, the trace of the stress tensor is the sum of thermodynamic pressure contribution and another contribution which is proportional to the divergence of the velocity field. This coefficient of proportionality is called volume viscosity. Common symbols for volume viscosity are  and .
Volume viscosity appears in the classic Navier-Stokes equation if it is written for compressible fluid, as described in most books on general hydrodynamics and acoustics.
where  is the shear viscosity coefficient and  is the volume viscosity coefficient. The parameters  and  were originally called the first and bulk viscosity coefficients, respectively. The operator  is the material derivative. By introducing the tensors (matrices) ,  and  (where e is a scalar called dilation, and  is the identity tensor), which describes crude shear flow (i.e. the strain rate tensor), pure shear flow (i.e. the deviatoric part of the strain rate tensor, i.e. the shear rate tensor) and compression flow (i.e. the isotropic dilation tensor), respectively,
the classic Navier-Stokes equation gets a lucid form.
Note that the term in the momentum equation that contains the volume viscosity disappears for an incompressible flow because there is no divergence of the flow, and so also no flow dilation e to which is proportional:
So the incompressible Navier-Stokes equation can be simply written:
In fact, note that for the incompressible flow the strain rate is purely deviatoric since there is no dilation (e=0).
In other words, for an incompressible flow the isotropic stress component is simply the pressure:
and the deviatoric (shear) stress is simply twice the product between the shear viscosity and the strain rate (Newton's constitutive law):
Therefore, in the incompressible flow the volume viscosity plays no role in the fluid dynamics.
However, in a compressible flow there are cases where , which are explained below. In general, moreover,  is not just a property of the fluid in the classic thermodynamic sense, but also depends on the process, for example the compression/expansion rate. The same goes for shear viscosity. For a Newtonian fluid the shear viscosity is a pure fluid property, but for a non-Newtonian fluid it is not a pure fluid property due to its dependence on the velocity gradient. Neither shear nor volume viscosity are equilibrium parameters or properties, but transport properties. The velocity gradient and/or compression rate are therefore independent variables together with pressure, temperature, and other state variables.