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Old   September 23, 2005, 15:07
Default Information on 'second viscosity'
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Can anyone provide references to information on 'the second viscosity'?

Obvious Stokes (gas) & incompressible deletion are understood.

Thanks, diaw...
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Old   September 26, 2005, 12:04
Default Re: Information on 'second viscosity'
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Here are a few thoughts. The Chapman-Enskog procedure for obtaining approximate solutions to the Boltzmann Eqn. is usually truncated at the first term. This is generally referred to as the Navier-Stokes approximation. If you keep the second term, you get the Burnett equations. The stress tensor is the sum of the Euler, Navier-Stokes, Burnett, super-Burnett , etc. We truncate at the NS level. The Euler term is the regular pressure. The NS contribution includes a relaxation pressure and terms proportional to the divergence and deformation of the velocity. The relaxation pressure occurs when there are two distinct relaxation time scales, say rotational and chemical. As to the terms related to velocity:

lambda*Dilatation*IdentityTensor + 2*mu*RateOfDeformation

where lambda is the second viscosity but

RateOfDeformation = RateOfDeformationDeviator - (1/3)*Dilatation*IdentityTensor

Hence, the portion of the viscous stress tensor we care about is

(lambda - 2/3mu)*Dilatation*IdentityTensor + 2*mu*RateOfDeformationDeviator.

In kinetic theory, one usually calls the term

(lambda - 2/3mu) = kappa

We have now separated the material stretching into a constant volume component and what remains. If you want kinetic theory estimates for kappa, look for some papers by Geoffrey Maitland around 1982. He is also the lead author of a book entitled "Intermolecular Forces." If you want to learn about the relaxation pressure, look into the work of Ekaterina Nagnibeda.

Keep in mind that the Chapman-Enskog procedure demands that the stress tensor be composed of gradients of the hydrodynamic variables only. It is not assured that this particular construct is fully capable of representing complicated relaxation phenomena. There's another good book out there by Vladimir Zhdanov entitled "transport processes ..."

Good luck.

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Old   September 26, 2005, 14:22
Default Re: Information on 'second viscosity'
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Wow, thank you so much for answering in so much detail - it is very much appreciated... Some additional areas to work through.

My reason for asking about 'Second Viscosity', is that when a compressible development of the momentum equation is performed, we may not just 'lose' lambda (as for incompressible assumption) & it appears in the final equations. At that point, some level of calculation has to be made regarding its value.


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