### ASA 127th Meeting M.I.T. 1994 June 6-10

## 4pPA6. New constitutive equation for the volume viscosity of fluids.

**Allan J. Zuckerwar
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*NASA Langley Res. Ctr., MS 238, Hampton, VA 23681
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**Robert L. Ash
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*Old Dominion Univ., Norfolk, VA 23508
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The traditional constitutive equation for the volume viscosity of fluids,
written for one-dimensional flow as (sigma)[sub x][sup
']=((lambda)+2(mu))(cursive beta)u/(cursive beta)x, has no theoretical
foundation and no established relationship to known relaxation processes in
fluids. [for notation see H. Schlichting, Boundary Layer Theory (McGraw-Hill
Classic Textbook Reissue Series, New York, 1987)]. When applied to periodic
flow, i.e., sound propagation, this equation yields expressions for the
dispersion and absorption of sound, which indeed have the expected forms for a
single relaxation process; but it inherently leads to the absurd conclusion
that the relaxation strength must equal unity (among other deficiencies). It is
shown here that a constitutive equation of the form (sigma)[sub x][sup
']=(eta)[sub v]((cursive beta)u/(cursive beta)x)+(eta)[sub p](Dp/Dt) yields
expressions which conform to the acoustical single relaxation process, whereby
(eta)[sub v]=-(rho)[sub 0]c[sub 0][sup 2](tau)[sub ps], (eta)[sub p]=-(tau)[sub
vs], and (tau)[sub ps] and (tau)[sub vs] are the isentropic relaxation times at
constant pressure and constant volume, respectively. An application to a simple
problem in steady compressible flow is presented.