Volume viscosity

Volume viscosity (also called second coefficient of viscosity or dilatational viscosity or bulk viscosity) becomes important only for such effects where fluid compressibility is essential. Volume viscosity is mainly related to the vibrational energy of the molecules.[1] It is zero for monatomic gases at low density, but can be large for fluids with larger molecules. The volume viscosity is important in describing sound attenuation (as in e.g. Stokes's law), and the absorption of sound energy into the fluid depends on the sound frequency i.e. the rate of fluid compression and expansion. Volume viscosity is also important in describing the fluid dynamics of liquids containing gas bubbles. For an incompressible liquid the volume viscosity is superfluous, and does not appear in the equation of motion.

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 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 equations if it is written for compressible fluid, as described in the most books on general hydrodynamics[2][3] and acoustics.[4][5]

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

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 second viscosity coefficients, respectively. The operator ${\displaystyle D\mathbf {v} /Dt}$ is the material derivative. By introducing the tensors ${\displaystyle \mathbf {S} }$, ${\displaystyle \mathbf {S} _{0}}$ and ${\displaystyle \mathbf {C} }$, which describe 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. And also it should be noted that ${\displaystyle \zeta }$ 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 etc. in their constitutive equations which are the equivalent to the equation of state for equilibrium properties.

Landau's explanation

According to Landau[3], 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.

He later adds: 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: 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.

Bulk viscosity

The denotation bulk viscosity is sometimes used for the parameter ${\displaystyle \zeta }$, especially in older articles and textbooks,[6][7] but this is not recommended.

There exists a momentum density equation that is a variant of the Navier–Stokes equation

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

where ${\displaystyle \mu }$ is the shear viscosity coefficient and ${\displaystyle \beta }$ is a second viscosity coefficient. Notice that ${\displaystyle \mathbf {S} }$ is not traceless, meaning that it will also model outflow/inflow to the control volume (or grid cell in a 3D model) due to pure expansion/compression. It is quite common in theoretical work to add zero to an equation. If we add and subtract ${\displaystyle 2\mu \mathbf {C} }$ the equation becomes

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

By comparing the variant of the Navier–Stokes equation displayed just above, and the classic Navier–Stokes equation displayed near the top of this article, it follows that

${\displaystyle \beta =\zeta -{\frac {2}{3}}\mu \quad {\text{or}}\quad \zeta =\beta +{\frac {2}{3}}\mu }$

The tensors ${\displaystyle \mathbf {S} _{0}}$ and ${\displaystyle \mathbf {C} }$ describes pure shear flow and pure compression, while the tensor ${\displaystyle \mathbf {S} }$ describes a crude (or unpure) shear flow. The viscosity parameter ${\displaystyle \beta }$ therefore compensates for the excessive/insufficient compression flow caused by using the tensor ${\displaystyle \mathbf {S} }$ instead of ${\displaystyle \mathbf {S} _{0}}$.

When the denotation bulk viscosity is used, it is often given the symbol ${\displaystyle \mu _{b}}$ and defined (as a seemingly composite parameter) by the equation

${\displaystyle \mu _{b}=\beta +{\frac {2}{3}}\mu .}$

Next we return to the classic Navier–Stokes equation and assume that ${\displaystyle \mu }$ and ${\displaystyle \zeta }$ are constants. This gives

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

From Jacobian matrix, tensor derivative, tensor index notation, divergence and vector calculus identities we get the relations

${\displaystyle \nabla \cdot \left(\nabla \!\mathbf {v} \right)=\nabla \left(\nabla \!\cdot \!\mathbf {v} \right)\quad {\text{and}}\quad \nabla \cdot \left[\left(\nabla \!\mathbf {v} \right)^{T}\right]=\nabla ^{2}\mathbf {v} }$
${\displaystyle \nabla ^{2}\mathbf {v} =\nabla \left(\nabla \!\cdot \!\mathbf {v} \right)-\nabla \times \left(\nabla \times \mathbf {v} \right)}$.

The classic Navier–Stokes equation can then be written in two familiar ways:

${\displaystyle \rho {\frac {D\mathbf {v} }{Dt}}=-\nabla P+\mu \nabla ^{2}\mathbf {v} +({\frac {1}{3}}\mu +\zeta )\nabla (\nabla \cdot \mathbf {v} )+\rho \mathbf {g} }$
${\displaystyle \rho {\frac {D\mathbf {v} }{Dt}}=-\nabla P-\mu \nabla \times \left(\nabla \times \mathbf {v} \right)+({\frac {4}{3}}\mu +\zeta )\nabla (\nabla \cdot \mathbf {v} )+\rho \mathbf {g} }$

The equivalent alternative variants of Navier–Stokes equations are:

${\displaystyle \rho {\frac {D\mathbf {v} }{Dt}}=-\nabla P+\mu \nabla ^{2}\mathbf {v} +(\mu +\beta )\nabla (\nabla \cdot \mathbf {v} )+\rho \mathbf {g} }$
${\displaystyle \rho {\frac {D\mathbf {v} }{Dt}}=-\nabla P-\mu \nabla \times \left(\nabla \times \mathbf {v} \right)+(2\mu +\beta )\nabla (\nabla \cdot \mathbf {v} )+\rho \mathbf {g} }$

These last four equations are valid for compressible fluids that is subject to so small variations in pressure, temperature and velocity gradients that both shear and volume viscosity can be treated as constants.

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 classic Litovitz and Davis review[4]. 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.[1] 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).[1] 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. ^ a b c Cramer, M.S. "Numerical estimates for the bulk viscosity of ideal gases.", Phys. Fluids,24, 066102 (2012)
2. ^ Happel, J. and Brenner , H. "Low Reynolds number hydrodynamics", Prentice-Hall, (1965)
3. ^ a b Landau, L.D. and Lifshitz, E.M. "Fluid mechanics", Pergamon Press, New York (1959)
4. ^ a b Litovitz, T. A. and Davis, C. M. In "Physical Acoustics", Ed. W.P.Mason, vol. 2, chapter 5, Academic Press, NY, (1964)
5. ^ Dukhin, A.S. and Goetz, P.J. "Ultrasound for characterizing colloids", Elsevier, (2002)
6. ^ Morse, P.M. and Ingard, K.U. "Theoretical Acoustics", Princeton University Press(1968)
7. ^ Graves, R.E. and Argrow, B.M. "Bulk viscosity: Past to Present", Journal of Thermophysics and Heat Transfer,13, 3, 337–342 (1999)
• IUPAC, Compendium of Chemical Terminology, 2nd ed. (the "Gold Book") (1997). Online corrected version:  (2006–) "volume viscosity (or dilatational viscosity)". doi:10.1351/goldbook.V06650
• R. Byron Bird. Transport Phenomenon. 2nd Edition. p. 19.