Volume viscosity (also called bulk viscosity, or dilatational viscosity) is a material property relevant for characterizing fluid flow. Common symbols are
or
. It has dimensions (mass / (length × time)), and the corresponding
SI unit is the
pascal
Pascal, Pascal's or PASCAL may refer to:
People and fictional characters
* Pascal (given name), including a list of people with the name
* Pascal (surname), including a list of people and fictional characters with the name
** Blaise Pascal, Fren ...
-second (Pa·s).
Like other material properties (e.g.
density
Density (volumetric mass density or specific mass) is the substance's mass per unit of volume. The symbol most often used for density is ''ρ'' (the lower case Greek letter rho), although the Latin letter ''D'' can also be used. Mathematical ...
,
shear viscosity
The viscosity of a fluid is a measure of its resistance to deformation at a given rate. For liquids, it corresponds to the informal concept of "thickness": for example, syrup has a higher viscosity than water.
Viscosity quantifies the inter ...
, and
thermal conductivity
The thermal conductivity of a material is a measure of its ability to conduct heat. It is commonly denoted by k, \lambda, or \kappa.
Heat transfer occurs at a lower rate in materials of low thermal conductivity than in materials of high thermal ...
) the value of volume viscosity is specific to each fluid and depends additionally on the fluid state, particularly its
temperature
Temperature is a physical quantity that expresses quantitatively the perceptions of hotness and coldness. Temperature is measured with a thermometer.
Thermometers are calibrated in various temperature scales that historically have relied o ...
and
pressure
Pressure (symbol: ''p'' or ''P'') is the force applied perpendicular to the surface of an object per unit area over which that force is distributed. Gauge pressure (also spelled ''gage'' pressure)The preferred spelling varies by country and e ...
. Physically, volume viscosity represents the irreversible resistance, over and above the reversible resistance caused by
isentropic
In thermodynamics, an isentropic process is an idealized thermodynamic process that is both adiabatic and reversible. The work transfers of the system are frictionless, and there is no net transfer of heat or matter. Such an idealized process ...
bulk modulus
The bulk modulus (K or B) of a substance is a measure of how resistant to compression the substance is. It is defined as the ratio of the infinitesimal pressure increase to the resulting ''relative'' decrease of the volume.
Other moduli describe ...
, to a compression or expansion of a fluid.
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.
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
In 1851, George Gabriel Stokes derived an expression, now known as Stokes' law, for the frictional force – also called drag force – exerted on spherical objects with very small Reynolds numbers in a viscous fluid. Stokes' law is derived by s ...
), propagation of
shock waves
In physics, a shock wave (also spelled shockwave), or shock, is a type of propagating disturbance that moves faster than the local speed of sound in the medium. Like an ordinary wave, a shock wave carries energy and can propagate through a med ...
, 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
In physics and chemistry, "monatomic" is a combination of the words "mono" and "atomic", and means "single atom". It is usually applied to gases: a monatomic gas is a gas in which atoms are not bound to each other. Examples at standard conditions ...
at low density, whereas in an
incompressible flow
In fluid mechanics or more generally continuum mechanics, incompressible flow ( isochoric flow) refers to a flow in which the material density is constant within a fluid parcel—an infinitesimal volume that moves with the flow velocity. An e ...
the volume viscosity is superfluous since it does not appear in the equation of motion.
Volume viscosity was introduced in 1879 by
Sir Horace Lamb in his famous work ''Hydrodynamics''.
[Lamb, H., "Hydrodynamics", Sixth Edition,''Dover Publications'', NY (1932)] Although relatively obscure in the scientific literature at large, volume viscosity is discussed in depth in many important works on fluid mechanics,
[Potter, M.C., Wiggert, D.C. "Mechaniscs of Fluids", ''Prentics Hall'', NJ (1997)] fluid acoustics,
[Morse, P.M. and Ingard, K.U. "Theoretical Acoustics", ''Princeton University Press''(1968)][Temkin, S., "Elements of Acoustics", ''John Wiley and Sons'', NY (1981)] theory of liquids,
[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)][Enskog, D. "Kungliga Svenska Vetenskapsakademiens Handlingar", 63, 4, (1922)] and rheology.
[Graves, R.E. and Argrow, B.M. "Bulk viscosity: Past to Present", ''Journal of Thermophysics and Heat Transfer'',13, 3, 337–342 (1999)]
Derivation and use
At thermodynamic equilibrium, the negative-one-third of the
trace
Trace may refer to:
Arts and entertainment Music
* Trace (Son Volt album), ''Trace'' (Son Volt album), 1995
* Trace (Died Pretty album), ''Trace'' (Died Pretty album), 1993
* Trace (band), a Dutch progressive rock band
* The Trace (album), ''The ...
of the
Cauchy stress tensor
In continuum mechanics, the Cauchy stress tensor \boldsymbol\sigma, true stress tensor, or simply called the stress tensor is a second order tensor named after Augustin-Louis Cauchy. The tensor consists of nine components \sigma_ that complete ...
is often identified with the thermodynamic
pressure
Pressure (symbol: ''p'' or ''P'') is the force applied perpendicular to the surface of an object per unit area over which that force is distributed. Gauge pressure (also spelled ''gage'' pressure)The preferred spelling varies by country and e ...
,
:
which depends only on equilibrium state variables like temperature and density (
equation of state
In physics, chemistry, and thermodynamics, an equation of state is a thermodynamic equation relating state variables, which describe the state of matter under a given set of physical conditions, such as pressure, volume, temperature, or internal ...
). In general, the trace of the stress tensor is the sum of thermodynamic pressure contribution and another contribution which is proportional to the
divergence
In vector calculus, divergence is a vector operator that operates on a vector field, producing a scalar field giving the quantity of the vector field's source at each point. More technically, the divergence represents the volume density of the ...
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
Compressible flow (or gas dynamics) is the branch of fluid mechanics that deals with flows having significant changes in fluid density. While all flows are compressible, flows are usually treated as being incompressible when the Mach number (the ra ...
, as described in most books on general hydrodynamics
[Happel, J. and Brenner , H. "Low Reynolds number hydrodynamics", ''Prentice-Hall'', (1965)][Landau, L.D. and Lifshitz, E.M. "Fluid mechanics", ''Pergamon Press'', New York (1959)] and acoustics.
[Litovitz, T.A. and Davis, C.M. In "Physical Acoustics", Ed. W.P.Mason, vol. 2, chapter 5, ''Academic Press'', NY, (1964)][Dukhin, A. S. and Goetz, P. J. ''Characterization of liquids, nano- and micro- particulates and porous bodies using Ultrasound'', Elsevier, 2017
]
:
where
is the
shear viscosity
The viscosity of a fluid is a measure of its resistance to deformation at a given rate. For liquids, it corresponds to the informal concept of "thickness": for example, syrup has a higher viscosity than water.
Viscosity quantifies the inter ...
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
, which describes crude shear flow, pure shear flow and compression flow, 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 fluid'' because the
divergence
In vector calculus, divergence is a vector operator that operates on a vector field, producing a scalar field giving the quantity of the vector field's source at each point. More technically, the divergence represents the volume density of the ...
of the flow equals 0.
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
A Newtonian fluid is a fluid in which the viscous stresses arising from its flow are at every point linearly correlated to the local strain rate — the rate of change of its deformation over time. Stresses are proportional to the rate of chang ...
the shear viscosity is a pure fluid property, but for a
non-Newtonian fluid
A non-Newtonian fluid is a fluid that does not follow Newton's law of viscosity, i.e., constant viscosity independent of stress. In non-Newtonian fluids, viscosity can change when under force to either more liquid or more solid. Ketchup, for exa ...
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 variable
A state variable is one of the set of variables that are used to describe the mathematical "state" of a dynamical system. Intuitively, the state of a system describes enough about the system to determine its future behaviour in the absence of a ...
s.
Landau's explanation
According to
Landau
Landau ( pfl, Landach), officially Landau in der Pfalz, is an autonomous (''kreisfrei'') town surrounded by the Südliche Weinstraße ("Southern Wine Route") district of southern Rhineland-Palatinate, Germany. It is a university town (since 1990 ...
,
He later adds:
After an example, he concludes (with
used to represent volume viscosity):
Measurement
A brief review of the techniques available for measuring the volume viscosity of liquids can be found in Dukhin & Goetz
and Sharma (2019).
One such method is by using an
acoustic rheometer
An acoustic rheometer employs a piezo-electric crystal that can easily launch a successive wave of extensions and contractions into the fluid. It applies an oscillating extensional stress to the system. System response can be interpreted in terms ...
.
Below are values of the volume viscosity for several Newtonian liquids at 25 °C (reported in
cP):
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
Carbon dioxide (chemical formula ) is a chemical compound made up of molecules that each have one carbon atom covalently double bonded to two oxygen atoms. It is found in the gas state at room temperature. In the air, carbon dioxide is transpar ...
,
methane
Methane ( , ) is a chemical compound with the chemical formula (one carbon atom bonded to four hydrogen atoms). It is a group-14 hydride, the simplest alkane, and the main constituent of natural gas. The relative abundance of methane on Eart ...
, and
nitrous oxide
Nitrous oxide (dinitrogen oxide or dinitrogen monoxide), commonly known as laughing gas, nitrous, or nos, is a chemical compound, an oxide of nitrogen with the formula . At room temperature, it is a colourless non-flammable gas, and has a ...
. These were found to have volume viscosities which were hundreds to thousands of times larger than their shear viscosities.
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)
[Sharma, B and Kumar, R "Estimation of bulk viscosity of dilute gases using a nonequilibrium molecular dynamics approach.", ''Physical Review E'',100, 013309 (2019)] and Cramer.
[Cramer, M.S. "Numerical estimates for the bulk viscosity of ideal gases.", ''Phys. Fluids'',24, 066102 (2012)] 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
*
*
{{refend
Colloidal chemistry
Fluid dynamics
Viscosity