# Volume viscosity

Volume viscosity (also called bulk viscosity, or dilatational viscosity) is a material property relevant for characterizing fluid flow. Common symbols are ${\displaystyle \zeta ,\mu ',\mu _{\mathrm {b} },\kappa }$ or ${\displaystyle \xi }$. It has dimensions (mass / (length × time)), and the corresponding SI unit is the pascal-second (Pa·s).

Like other material properties (e.g. density, shear viscosity, and thermal conductivity) the value of volume viscosity is specific to each fluid and depends additionally on the fluid state, particularly its temperature and pressure. Physically, volume viscosity represents the irreversible resistance, over and above the reversible resistance caused by isentropic bulk modulus, to a compression or expansion of a fluid.[1] At the molecular level, it stems from the finite time required for energy injected in the system to be distributed among the rotational and vibrational degrees of freedom of molecular motion.[2]

Knowledge of the volume viscosity is important for understanding a variety of fluid phenomena, including sound attenuation in polyatomic gases (e.g. Stokes's law), propagation of shock waves, and dynamics of liquids containing gas bubbles. In many fluid dynamics problems, however, its effect can be neglected. For instance, it is 0 in a monatomic gas at low density, whereas in an incompressible flow the volume viscosity is superfluous since it does not appear in the equation of motion.[3]

Volume viscosity was introduced in 1879 by Sir Horace Lamb in his famous work Hydrodynamics.[4] Although relatively obscure in the scientific literature at large, volume viscosity is discussed in depth in many important works on fluid mechanics,[1][5][6] fluid acoustics,[7][8][9][2] theory of liquids,[10][11] and rheology.[12]

## Derivation and use

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

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

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 ${\displaystyle \zeta }$  and ${\displaystyle \mu _{v}}$ .

Volume viscosity appears in the classic Navier-Stokes equation if it is written for compressible fluid, as described in most books on general hydrodynamics[5][1] and acoustics.[8][9]

${\displaystyle \rho {\frac {D\mathbf {v} }{Dt}}=-\nabla P+\nabla \cdot \left[\mu \left(\nabla \mathbf {v} +\left(\nabla \mathbf {v} \right)^{T}-{\frac {2}{3}}(\nabla \cdot \mathbf {v} )\mathbf {I} \right)\right]+\nabla \cdot [\zeta (\nabla \cdot \mathbf {v} )\mathbf {I} ]+\rho \mathbf {g} }$

where ${\displaystyle \mu }$  is the shear viscosity coefficient and ${\displaystyle \zeta }$  is the volume viscosity coefficient. The parameters ${\displaystyle \mu }$  and ${\displaystyle \zeta }$  were originally called the first and bulk viscosity coefficients, respectively. The operator ${\displaystyle D\mathbf {v} /Dt}$  is the material derivative. By introducing the tensors (matrices) ${\displaystyle \mathbf {S} }$ , ${\displaystyle \mathbf {S} _{0}}$  and ${\displaystyle \mathbf {C} }$ , which describes crude shear flow, pure shear flow and compression flow, respectively,

${\displaystyle \mathbf {S} ={\frac {1}{2}}\left(\nabla \mathbf {v} +\left(\nabla \mathbf {v} \right)^{T}\right)}$
${\displaystyle \mathbf {C} ={\frac {1}{3}}\left(\nabla \!\cdot \!\mathbf {v} \right)\mathbf {I} }$
${\displaystyle \mathbf {S} _{0}=\mathbf {S} -\mathbf {C} }$

the classic Navier-Stokes equation gets a lucid form.

${\displaystyle \rho {\frac {D\mathbf {v} }{Dt}}=-\nabla P+\nabla \cdot \left[2\mu \mathbf {S} _{0}\right]+\nabla \cdot \left[3\zeta \mathbf {C} \right]+\rho \mathbf {g} }$

Note that the term in the momentum equation that contains the volume viscosity disappears for an incompressible fluid because the divergence of the flow equals 0.

There are cases where ${\displaystyle \zeta \gg \mu }$ , which are explained below. In general, moreover, ${\displaystyle \zeta }$  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.

## Landau's explanation

According to Landau,[1]

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.

It may happen, nevertheless, that the relaxation times of the processes of restoration of equilibrium are long, i.e. they take place comparatively slowly.

After an example, he concludes (with ${\displaystyle \zeta }$  used to represent volume viscosity):

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.

## Measurement

A brief review of the techniques available for measuring the volume viscosity of liquids can be found in Dukhin & Goetz[9] and Sharma (2019).[13] One such method is by using an acoustic rheometer.

Below are values of the volume viscosity for several Newtonian liquids at 25 °C (reported in cP):[14]

methanol - 0.8
ethanol - 1.4
propanol - 2.7
pentanol - 2.8
acetone - 1.4
toluene - 7.6
cyclohexanone - 7.0
hexane - 2.4


Recent studies have determined the volume viscosity for a variety of gases, including carbon dioxide, methane, and nitrous oxide. These were found to have volume viscosities which were hundreds to thousands of times larger than their shear viscosities.[13] Fluids having large volume viscosities include those used as working fluids in power systems having non-fossil fuel heat sources, wind tunnel testing, and pharmaceutical processing.

## Modeling

There are many publications dedicated to numerical modeling of volume viscosity. A detailed review of these studies can be found in Sharma (2019)[13] and Cramer.[15] 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.

## References

1. ^ a b c d Landau, L.D. and Lifshitz, E.M. "Fluid mechanics", Pergamon Press, New York (1959)
2. ^ a b Temkin, S., "Elements of Acoustics", John Wiley and Sons, NY (1981)
3. ^ Bird, R. Byron; Stewart, Warren E.; Lightfoot, Edwin N. (2007), Transport Phenomena (2nd ed.), John Wiley & Sons, Inc., p. 19, ISBN 978-0-470-11539-8
4. ^ Lamb, H., "Hydrodynamics", Sixth Edition,Dover Publications, NY (1932)
5. ^ a b Happel, J. and Brenner , H. "Low Reynolds number hydrodynamics", Prentice-Hall, (1965)
6. ^ Potter, M.C., Wiggert, D.C. "Mechaniscs of Fluids", Prentics Hall, NJ (1997)
7. ^ Morse, P.M. and Ingard, K.U. "Theoretical Acoustics", Princeton University Press(1968)
8. ^ a b Litovitz, T.A. and Davis, C.M. In "Physical Acoustics", Ed. W.P.Mason, vol. 2, chapter 5, Academic Press, NY, (1964)
9. ^ a b c Dukhin, A. S. and Goetz, P. J. Characterization of liquids, nano- and micro- particulates and porous bodies using Ultrasound, Elsevier, 2017 ISBN 978-0-444-63908-0
10. ^ Kirkwood, J.G., Buff, F.P., Green, M.S., "The statistical mechanical theory of transport processes. 3. The coefficients of shear and bulk viscosity in liquids", J. Chemical Physics, 17, 10, 988-994, (1949)
11. ^ Enskog, D. "Kungliga Svenska Vetenskapsakademiens Handlingar", 63, 4, (1922)
12. ^ Graves, R.E. and Argrow, B.M. "Bulk viscosity: Past to Present", Journal of Thermophysics and Heat Transfer,13, 3, 337–342 (1999)
13. ^ a b c Sharma, B and Kumar, R "Estimation of bulk viscosity of dilute gases using a nonequilibrium molecular dynamics approach.", Physical Review E,100, 013309 (2019)
14. ^ Dukhin, Andrei S.; Goetz, Philip J. (2009). "Bulk viscosity and compressibility measurement using acoustic spectroscopy". The Journal of Chemical Physics. 130 (12): 124519. doi:10.1063/1.3095471. ISSN 0021-9606. PMID 19334863.
15. ^ Cramer, M.S. "Numerical estimates for the bulk viscosity of ideal gases.", Phys. Fluids,24, 066102 (2012)