Volume viscosity

Volume viscosity (also called second viscosity or bulk viscosity) becomes important only for such effects where fluid compressibility is essential. Examples would include shock waves and sound propagation. It appears in the Stokes' law (sound attenuation) that describes propagation of sound in Newtonian liquid.

Derivation and Use

The negative-one-third of the trace of the Cauchy stress tensor is often identified with the thermodynamic pressure,

$-{1\over3}T_a^a = p,$

which only depends upon the equilibrium state potentials 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 constant of proportionality is called the volume viscosity.

Volume viscosity appears in the Navier-Stokes equation if it is written for compressible fluid, as described in the most books on general hydrodynamics ^{[1], [2]} and acoustics ^[3], .^[4]

$\rho \left(\frac{\partial \mathbf{v}}{\partial t} + \mathbf{v} \cdot \nabla \mathbf{v}\right) = -\nabla p + \mu \nabla^2 \mathbf{v} + \mathbf{f} +(\frac{1}{3}\mu + \zeta) \nabla (\nabla \cdot \mathbf{v})$

where $\zeta$ is the volume viscosity coefficient. Authors who use the alternative term bulk viscosity for the same parameter include ^[5], .^[6] This additional term disappears for incompressible fluid, when the divergence of the flow equals 0.

This viscosity parameter is additional to the usual dynamic viscosity μ.

Measurement

The volume viscosity of many fluids is inaccurately known, despite its fundamental role for fluid dynamics at high frequencies. The only values for the volume viscosity of simple Newtonian liquids known to us come from the old Litovitz and Davis review, see References. They report the volume viscosity of water at 15 °C is 3.09 centipoise.

Modern acoustic rheometers are able to measure this parameter.

More recent studies have determined the bulk viscosity for a variety of fluids^[6][7]). In the latter study, a number of common fluids were found to have bulk viscosities which were hundreds to thousands of times larger than their shear viscosities. The details of the data used and estimation techniques are provided in Cramer (2012). As discussed by Cramer (2012), fluids having large bulk viscosities include those used as working fluids in power systems having non-fossil fuel heat sources, wind tunnel testing, and pharmaceutical processing.

References

1. Happel, J. and Brenner , H. "Low Reynolds number hydrodynamics", Prentice-Hall, (1965)
2. Landau, L.D. and Lifshitz, E.M. "Fluid mechanics", Pergamon Press,(1959)
3. Litovitz, T.A. and Davis, C.M. In "Physical Acoustics", Ed. W.P.Mason, vol. 2, chapter 5, Academic Press, NY, (1964)
4. Dukhin, A.S. and Goetz, P.J. "Ultrasound for characterizing colloids", Elsevier, (2002)
5. Morse, P.M. and Ingard, K.U. "Theoretical Acoustics", Princeton University Press(1986)
6. Graves, R.E. and Argrow, B.M. "Bulk viscosity:Past to Present", Journal of Thermophysics and Heat Transfer,13, 3, 337-342 (1999)
7. Cramer, M.S. "Numerical estimates for the bulk viscosity of ideal gases.", Phys. Fluids,24,(066102) (2012))