New constitutive equation for the volume viscosity of fluids

The traditional constitutive equation for the volume viscosity of fluids, written for one‐dimensional flow as σ^′_x=(λ+2μ)∂u/∂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 σ^′_x=η_v(∂u/∂x)+η_p(Dp/Dt) yields expressions which conform to the acoustical single relaxation process, whereby η_v=−ρ₀c²₀τ_ps, η_p=−τ_vs, and τ_ps and τ_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.