Bulk viscosity

The bulk viscosity , the viscosity coefficient or the second viscosity (symbol or , dimension M · L ^-1 · t ^-1 , unit Pa · s ) denote the viscosity of fluids in volume changes. With a finite change in volume in a fluid with a uniform temperature distribution, the volume viscosity is responsible for the energy dissipation . ${\displaystyle \zeta ,\mu ',\mu _{\mathrm {b} },\eta _{\mathrm {V} },\nu ^{0}}$ ${\displaystyle \xi }$

In practice, in the case of monatomic gases and pressures that are not too high, the Stokes hypothesis can be assumed, which requires. Even if incompressibility is assumed , the bulk viscosity can be neglected. Even a volume viscosity other than zero does not lead to very noticeable effects under normal conditions . ${\displaystyle \zeta =0}$

However, the volume viscosity has a significant influence in liquids with gas bubbles , in shock waves and in the propagation of sound . Some fluids, especially carbon dioxide , have bulk viscosities that are over a thousand times greater than their shear viscosities , which in such gases has an impact on the hydrodynamic boundary layer in supersonic flows . The bulk viscosity of dilute polyatomic gases plays for entry into planetary atmospheres a role.

definition

In the case of pure compression or expansion of gases, the bulk viscosity occurs as the cause of normal stress acting in all directions

${\displaystyle \sigma ^{V}=-\zeta \,{\frac {\dot {\rho }}{\rho }}=\zeta \,\nabla \cdot {\vec {v}}=\zeta \,\operatorname {div} {\vec {v}}}$,

which acts in addition to the mechanical pressure : ${\displaystyle {\bar {p}}}$

${\displaystyle p(\rho ,T)={\bar {p}}+\sigma ^{V}}$.

It is

• ${\displaystyle \zeta }$ the bulk viscosity
• ${\displaystyle \rho }$the density
• the supreme point the substantial time derivative
• ${\displaystyle {\vec {v}}}$the velocity field of the flow
• "·" The (formal) scalar product with the nabla operator , that the divergence forms ${\displaystyle \nabla }$ ${\displaystyle \operatorname {div} }$
• ${\displaystyle p(\rho ,T)}$the thermodynamic pressure depending on the density and temperature . ${\displaystyle T}$

The divergence of the speed is a measure of the rate of change in volume of an (infinitesimal) small volume element . Because of the constant mass of the volume element, this establishes the mass balance ${\displaystyle {\dot {\mathrm {d} V}}=\operatorname {div} ({\vec {v}})\,\mathrm {d} V}$ ${\displaystyle \mathrm {d} V}$ ${\displaystyle \rho \mathrm {d} V}$

${\displaystyle {\begin{aligned}{\dot {\rho }}+\rho \,&\operatorname {div} {\vec {v}}=0\\\Leftrightarrow \quad &\operatorname {div} {\vec {v}}=-{\frac {\dot {\rho }}{\rho }},\end{aligned}}}$

which was used in the above definition equation.

The relationship applies in a Newtonian fluid

${\displaystyle \zeta ={\frac {2}{3}}\mu +\lambda }$

With

• the shear viscosity ${\displaystyle \mu }$
• the first Lamé constant ${\displaystyle \lambda }$

In the case of incompressibility , the divergence of the speed disappears, so that in this case no volume viscosity can occur.

Newtonian fluids

The motion of a linearly viscous, isotropic Newtonian fluid obeys the Navier-Stokes equations

${\displaystyle \rho \,{\dot {\vec {v}}}=-\nabla p(\rho ,T)+\mu \,\Delta {\vec {v}}+\left(\zeta +{\frac {1}{3}}\mu \right)\nabla (\nabla \cdot {\vec {v}})+{\vec {f}}.}$

Here is

• ${\displaystyle \Delta }$the Laplace operator
• ${\displaystyle {\vec {f}}}$(Vector) stands for the density of a volume force , for example gravitation or the Coriolis force , based on the unit volume.

The Navier-Stokes equations can be derived from the following material model of classical material theory :

${\displaystyle {\begin{alignedat}{2}{\boldsymbol {\sigma }}&=-p(\rho ,T)\mathbf {1} +2\mu \mathbf {d} &&+\lambda \operatorname {Sp} (\mathbf {d} )\mathbf {1} \\&=-p(\rho ,T)\mathbf {1} +2\mu \mathbf {d} ^{\rm {D}}&&+\zeta \operatorname {Sp} (\mathbf {d} )\mathbf {1} \end{alignedat}}}$

Marked here

• ${\displaystyle {\boldsymbol {\sigma }}}$the stress tensor
• ${\displaystyle \mathbf {1} }$the unit tensor
• ${\displaystyle \mathbf {d} }$the symmetrical part of the (spatial) velocity gradient , i.e. the tensor of the distortion velocities

${\displaystyle \mathbf {d} :={\frac {1}{2}}{\begin{pmatrix}2{\frac {\partial v_{x}}{\partial x}}&{\frac {\partial v_{x}}{\partial y}}+{\frac {\partial v_{y}}{\partial x}}&{\frac {\partial v_{x}}{\partial z}}+{\frac {\partial v_{z}}{\partial x}}\\&2{\frac {\partial v_{y}}{\partial y}}&{\frac {\partial v_{y}}{\partial z}}+{\frac {\partial v_{z}}{\partial y}}\\{\text{sym.}}&&2{\frac {\partial v_{z}}{\partial z}}\end{pmatrix}}}$

In this tensor, the speed components of the fluid elements are in -, - or - direction of a Cartesian coordinate system . ${\displaystyle v_{x,y,z}}$ ${\displaystyle x}$ ${\displaystyle y}$ ${\displaystyle z}$

• ${\displaystyle \operatorname {Sp} }$the trace ; the trace of the distortion speed tensor is the divergence of the speed:

${\displaystyle \operatorname {Sp} (\mathbf {d} )={\tfrac {\partial v_{x}}{\partial x}}+{\tfrac {\partial v_{y}}{\partial y}}+{\tfrac {\partial v_{z}}{\partial z}}=\operatorname {div} {\vec {v}}}$

• the superscript the deviator . ${\displaystyle {\rm {D}}}$

From the above model equations it can be deduced that . ${\displaystyle \zeta =\lambda +{\tfrac {2}{3}}\mu }$

The normal stress occurring in the definition is part of the mechanical pressure , which is the negative third of the trace of the stress tensor: ${\displaystyle \sigma ^{V}=-\zeta {\tfrac {\dot {\rho }}{\rho }}}$ ${\displaystyle {\bar {p}}}$

${\displaystyle {\begin{aligned}{\bar {p}}:&=-{\frac {1}{3}}\operatorname {Sp} ({\boldsymbol {\sigma }})\\&=-{\frac {1}{3}}\left[-3p(\rho ,T)+3\zeta \operatorname {Sp} (\mathbf {d} )\right]\\&=\qquad \quad \;p(\rho ,T)-\zeta \operatorname {div} {\vec {v}}\\&=\qquad \quad \;p(\rho ,T)+\zeta {\frac {\dot {\rho }}{\rho }}\end{aligned}}}$

Here it was used that the trace of the unit tensor is equal to the space dimension and that the deviator is, by definition, trace-free.

The fluid dynamic boundary layer is important in flows of viscous fluids. In the boundary layer theory that deals with it, normal stresses are neglected compared to shear stresses , which is why the bulk viscosity in the boundary layer is not needed. However, more recent studies show a significant influence of a high volume viscosity on the boundary layer in supersonic flows.

Pure expansion

In the case of a pure expansion away from the origin, the velocity field has the form with a proportionality factor . The velocity gradient is then because of the symmetrical gradient and ${\displaystyle {\vec {v}}({\vec {x}})=c{\vec {x}}}$ ${\displaystyle c}$ ${\displaystyle \operatorname {grad} {\vec {x}}=\mathbf {1} }$

${\displaystyle \operatorname {grad} {\vec {v}}=c\operatorname {grad} {\vec {x}}=c\mathbf {1} =\mathbf {d} }$

equal to the strain rate tensor.

The stress tensor is calculated with ${\displaystyle \operatorname {Sp} (\mathbf {d} )=3c}$

${\displaystyle {\begin{aligned}{\boldsymbol {\sigma }}&=-p(\rho ,T)\mathbf {1} +2\mu \mathbf {d} ^{\rm {D}}+\zeta \operatorname {Sp} (\mathbf {d} )\mathbf {1} \\&=[-p(\rho ,T)\qquad \qquad +3\,\zeta \,c]\mathbf {1} \end{aligned}}}$

The specific voltage power at the distortion speeds is defined as

${\displaystyle l_{i}:={\tfrac {1}{\rho }}{\boldsymbol {\sigma }}:\mathbf {d} }$.

The colon ":" forms the Frobenius scalar product of two tensors and means ${\displaystyle \mathbf {A} }$ ${\displaystyle \mathbf {B} }$

${\displaystyle \mathbf {A} :\mathbf {B} :=\mathrm {Sp} (\mathbf {A} ^{T}\cdot \mathbf {B} )}$

where the superscript means the transposition . ${\displaystyle T}$

In the case of pure expansion, the specific voltage power is therefore calculated as:

${\displaystyle {\begin{aligned}l_{i}&={\frac {1}{\rho }}[-p(\rho ,T)+3\,\zeta \,c]\mathbf {1} :c\mathbf {1} \\&=-{\frac {3c}{\rho }}p+{\frac {9c^{2}}{\rho }}\zeta \end{aligned}}}$

The first part of the power, proportional to the pressure, is reversible , the second is irreversible and is dissipated .

Stokes' hypothesis

Stokes' hypothesis states:

"In the case of a uniform motion of dilatation the pressure at any instance depends only on the actual density and temperature at that instant, and not on the rate at which the former changes with time ( English in the case of a uniform expansion motion depends on the pressure at any point in time depends only on the current density and temperature at that point in time and not on the rate at which the former changes over time. ) "

- George Gabriel Stokes (1845)

This hypothesis was formulated in a similar way by Barré de Saint-Venant as early as 1843, two years before Stokes .

From the relationship valid for Newtonian fluids

${\displaystyle {\bar {p}}=p(\rho ,T)+\zeta {\frac {\dot {\rho }}{\rho }}}$

and the above hypothesis follows immediately

${\displaystyle \Rightarrow \zeta =0}$

With the facts summarized above, the following equivalent statements about Newtonian fluids result from the hypothesis:

1. In the case of a uniform expansion movement , the pressure at any point in time depends only on the current density and temperature at that point in time. ${\displaystyle \left({\dot {\rho }}={\frac {\mathrm {d} \rho }{\mathrm {d} t}}=\mathrm {konst.} <0\right)}$
2. A pure change in volume is reversible.
3. The thermodynamic pressure and the mechanical pressure are the same.

The measurement of the bulk viscosity is so difficult that at the beginning of the 21st century it was not possible to experimentally test the validity of this postulate for monatomic gases.

Conclusions from the kinetic gas theory

The Chapman-Enskog development of Boltzmann equations of the kinetic theory lead to the Navier-Stokes equations with vanishing bulk viscosity so . This development is based on a distribution function that depends only on the speed of the particles, i.e. neglects their rotational angular momentum . This is a good assumption in monatomic gases at low to medium pressure. ${\displaystyle \zeta =0}$

In the case of polyatomic gases, on the other hand, the rotational angular momentum must not be neglected, because with them energy can be transferred between the translational movement and the molecular movements, i.e. H. the rotational and vibratory movements, which leads to a positive volume viscosity.

To characterize the state of polyatomic gases in non-equilibrium , a generalized distribution function ( density operator ) is to be used, which not only depends on the speed of the particles, but also on their angular momentum. Accordingly, the Boltzmann equation must also be replaced by a generalized kinetic equation (the Waldmann-Snider equation). A temperature relaxation equation can be derived from it, which leads to the following expression for the bulk viscosity:

${\displaystyle \zeta ={\frac {2}{3}}{\frac {n\,k\,{c_{V}}^{\text{intra}}}{c_{V}\,\omega }}T}$

In it is

• ${\displaystyle n}$the particle density
• ${\displaystyle k}$the Boltzmann constant
• ${\displaystyle c_{V}}$the specific heat capacity
• ${\displaystyle c_{V}^{intra}}$ the specific heat capacity, which only results from the molecular movements,
• ${\displaystyle T}$ the temperature
• ${\displaystyle \omega }$the shock frequency .

Because the impact frequency is proportional to the particle density, the bulk viscosity is independent of the particle density and thus of the pressure of the gas.

For monatomic gases is due again . ${\displaystyle c_{V}^{intra}=0}$ ${\displaystyle \zeta =0}$

For a given capacity , the lower the impact frequency , the greater the bulk viscosity , i.e. H. the rarer an exchange between translational and intramolecular energy can take place during collisions. Therefore, the ratio is greater for hydrogen than for nitrogen . ${\displaystyle c_{V}^{intra}}$ ${\displaystyle \omega }$ ${\displaystyle \zeta /\mu }$

Measurement

Acoustic spectrometers are suitable for measuring the volume viscosity of a fluid , since, according to the classical theory of Gustav Robert Kirchhoff, the absorption coefficient of the amplitude per unit length depends on the volume viscosity for plane sound waves in media with not too high a viscosity :

${\displaystyle {\frac {k^{2}}{\omega ^{2}}}={\frac {1}{2}}\left({\frac {4}{3}}\mu +\zeta \right){\frac {1}{c_{0}^{3}\,\rho }}+{\frac {1}{2}}{\frac {\lambda (\kappa -1)}{c_{p}}}{\frac {1}{c_{0}^{3}\,\rho }}}$

With

• the speed of sound for ${\displaystyle c_{0}}$
• very small circular frequencies , ${\displaystyle \omega }$
• the specific heat capacities and at constant pressure or volume ${\displaystyle c_{p}}$ ${\displaystyle c_{v}}$
• ${\displaystyle \kappa =c_{p}/c_{v}}$.

The main contribution to absorption is always made by the first term for liquids , and usually for gases. ${\displaystyle \kappa \approx 1}$

The volume viscosity of a large number of fluids could be determined using other methods:

• Brillouin spectroscopy of relaxing liquids
• Measuring the thickness of a shock wave
• Measurement of the Taylor-Couette flow of a compressible medium.

Individual evidence

1. ^ ^{A b} Hermann Schlichting, Klaus Gersten: boundary layer theory . Springer, Berlin 1997, ISBN 978-3-662-07555-5 , pp. 69 ff . ( limited preview in Google Book search).
2. ↑ ^{a b} Josef Meixner: Principles of Thermodynamics and Statistics . In: S. Flügge (Ed.): Handbuch der Physik . tape III / 2 . Springer, 1959, ISBN 978-3-642-45912-2 , pp. 477 ff . ( limited preview in Google Book search).
3. ↑ MS Cramer: Numerical estimates for the bulk viscosity of ideal gases . In: Physics of Fluids . tape June 24 , 2012, doi : 10.1063 / 1.4729611 (English).
4. ↑ ^{a b} George Emanuel: Effect of bulk viscosity on a hypersonic boundary layer . In: Physics of Fluids . tape 4 , 1992, doi : 10.1063 / 1.858322 (English).
5. ↑ G. Emanuel: Bulk viscosity of a dilute polyatomic gas . In: Physics of Fluids . tape 2 , 1990, doi : 10.1063 / 1.857813 (English).
6. ^ Fritz Kurt Kneubühl: Repetitorium der Physik . Vieweg + Teubner Verlag, Stuttgart 1994, ISBN 978-3-322-84886-4 ( limited preview in the Google book search).
7. ^ Franco M. Capaldi: Continuum Mechanics: Constitutive Modeling of Structural and Biological Materials . Cambridge University Press, 2012, ISBN 978-1-107-01181-6 , pp. 157 ( limited preview in Google Book search).
8. ↑ Mohamed Gad-el-Hak: Stokes' Hypothesis for a newtonian, isotropic fluid. (PDF) March 1, 1995, accessed April 2, 2017 (English). or Mohamed Gad-el-Hak: Questions in Fluid Mechanics . Stokes' Hypothesis for a Newtonian, isotropic fluid. In: American Society of Mechanical Engineers (Ed.): Journal of Fluids Engineering . tape 
   
    117 , no. 1 , March 1, 1995, p. 3–5 ( fluidsengineering.asmedigitalcollection.asme.org [accessed April 5, 2017]).
9. ^ GG Stokes: On the Theories of Internal Friction of Fluids in Motion . In: Transactions of the Cambridge Philosophical Society . tape 8 , 1845, p. 294 f . ( archive.org - Stokes denotes the volume viscosity with κ).
10. ^ ^{A b} R. E. Graves, BM Argrow: Bulk viscosity: Past to Present . In: Journal of Thermophysics and Heat Transfer . tape 13 , no. 3 , 1999, p. 337-342 , doi : 10.2514 / 2.6443 .
11. ↑ Heinz Schade, Ewald Kunz, Frank Kameier, Christian Oliver Paschereit: Fluid Mechanics . Walter de Gruyter, Berlin 2013, ISBN 978-3-11-029223-7 , p. 197 ( limited preview in Google Book search).
12. ^ Sydney Chapman, TG Cowling: The Mathematical Theory of Non-uniform Gases. An Account of the Kinetic Theory of Viscosity, Thermal Conduction and Diffusion in Gases . Cambridge University Press, 1970, ISBN 978-0-521-40844-8 .
13. ↑ ^{a b} Bergmann, Schaefer: Textbook of Experimental Physics. Gases nanosystems liquids . Ed .: Thomas Dorfmüller, Karl Kleinermanns. 2nd Edition. tape 5 . Walter de Gruyter, Berlin 2006, ISBN 978-3-11-017484-7 , p. 45 ff . ( limited preview in Google Book search).
14. ↑ KU Kramm: Measurements on fluids with an acoustic spectrometer . September 25, 2009, doi : 10.1524 / teme.70.11.530.20272 .
15. ↑ Gitis, Mihail: Process and device for quality monitoring of technical one-component and multi-component liquids by means of ultrasound on-line measurements of their viscosity, density, compressibility and volume viscosity . March 19, 2009 ( patent-de.com [accessed April 15, 2017]).
16. ↑ Th. Dorfmüller, G. Fytas, W. Mersch: Brillouin spectroscopy of relaxing liquids. Part I . 1976, doi : 10.1002 / bbpc.19760800503 .
17. ↑ George Emanuel: Linear dependence of the bulk viscosity on shock wave thickness . In: Physics of Fluids . tape May 24 , 1994, doi : 10.1063 / 1.868102 (English).
18. ↑ H. Gonzalez, G. Emanuel: Effect of bulk viscosity on Couette flow . In: Physics of Fluids . tape 5 , 1993, doi : 10.1063 / 1.858612 (English).