# Volume viscosity

(Redirected from Bulk viscosity)

Volume viscosity (also called second viscosity or bulk viscosity or dilatational viscosity) becomes important only for such effects where fluid compressibility is essential. Dilatational viscosity is a measure of the viscous forces and viscous forces depend on the rate of compression or dilatation. Dilatational viscosity is zero for Monatomic gases at low density. For an incompressible liquid the dilatational viscosity is neglected. The dilatational viscosity is important in describing sound absorption in polyatomic gases and in describing the fluid dynamics of liquids containing gas bubbles.

## Derivation and use

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

${\displaystyle -{1 \over 3}T_{a}^{a}=p,}$

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]

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

If ${\displaystyle \mu }$ and ${\displaystyle \zeta }$ are assumed constant,

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

where ${\displaystyle \zeta }$ is the volume viscosity coefficient. The alternative term bulk viscosity for the same parameter is also used [5], .[6] This additional term disappears for incompressible fluid, when the divergence of the flow equals 0.

The Navier-Stokes equation above includes the dynamic viscosity μ, also known as the shear viscosity.

There are cases where ${\displaystyle \zeta >>\mu }$, which are explained below. And also it should be noted that ${\displaystyle \zeta }$ is not just a property of the fluid, but also depends on the process, for example the compression/expansion rate, whereas the dynamic viscosity is a property of the fluid.

### Landau's explanation

Landau quotes that In compression or expansion, as in any rapid change of state, the fluid ceases to be in thermodynamic equilibrium, and internal processes are set up in it which tend to restore this equilibrium. These processes are usually so rapid(i.e. their relaxation time is so short) that the restoration of equilibrium follows the change in volume almost immediately unless, of course, the rate of change of volume is very large. Further he adds that It may happen, nevertheless, that the relaxation times of the processes of restoration of equilibrium are long, i.e. they take place comparatively slowly. He continues with an example and says at the end Hence, if the relaxation time of these processes is long, a considerable dissipation of energy occurs when the fluid is compressed or expanded, and, since this dissipation must be determined by the second viscosity, we reach the conclusion that ${\displaystyle \zeta }$ is large. This rich explanation can be found in Landau.[7]

## 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][8]). 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. ^ a b Graves, R.E. and Argrow, B.M. "Bulk viscosity:Past to Present", Journal of Thermophysics and Heat Transfer,13, 3, 337-342 (1999)
7. ^ Landau, L. D., and E. M. Lifshitz. "Fluid Mechanics Pergamon." New York 61 (1959).
8. ^ Cramer, M.S. "Numerical estimates for the bulk viscosity of ideal gases.", Phys. Fluids,24, 066102 (2012)
• A. D. McNaught; A. Wilkinson (1997). Compendium of Chemical Terminology (PDF). 2nd Edition. Online version created by created by M. Nic, J. Jirat, B. Kosata; updates compiled by A. Jenkins. Blackwell Scientific Publications. doi:10.1351/goldbook. ISBN 0-9678550-9-8. Retrieved 16 November 2016.
• R. Byron Bird. Transport Phenomenon. 2nd Edition. p. 19.