Volume viscosity in fluids with multiple dissipative processes

Abstract

The variational principle of Hamilton is applied to derive the volume viscosity coefficients of a reacting fluid with multiple dissipative processes. The procedure, as in the case of a single dissipative process, yields two dissipative terms in the Navier-Stokes equation: The first is the traditional volume viscosity term, proportional to the dilatational component of the velocity; the second term is proportional to the material time derivative of the pressure gradient. Each dissipative process is assumed to be independent of the others. In a fluid comprising a single constituent with multiple relaxation processes, the relaxation times of the multiple processes are additive in the respective volume viscosity terms. If the fluid comprises several relaxing constituents ͑each with a single relaxation process͒, the relaxation times are again additive but weighted by the mole fractions of the fluid constituents. A generalized equation of state is derived, for which two special cases are considered: The case of "low-entropy production," where entropy variation is neglected, and that of "high entropy production," where the progress variables of the internal molecular processes are neglected. Applications include acoustical wave propagation, Stokes flow around a sphere, and the structure and thickness of a normal shock. Finally, it is shown that the analysis presented here resolves several misconceptions concerning the volume viscosity of fluids.

A non-conventional discontinuous Lagrangian for viscous flow

“…On the other hand, the equations of motion resulting from the variation of Hamilton's principle differ from the Navier–Stokes equations and, therefore, their solutions reveal notable quantitative differences to those of the latter. Similar experiences are reported by Zuckerwar & Ash [14], who made an analogous suggestion for a Lagrangian considering volume viscosity in particular.…”

“…Similar assumptions are made in Zuckerwar & Ash [14]. At first glance, there seems to be a realistic chance to interpret the additional terms and degrees of freedom in the above evolution equations (2.7), (2.9)–(2.11) as an extension of the classical theory towards non-equilibrium thermodynamics.…”

“…Reconsidering the aforementioned hypothesis of Anthony [15] as well as Zuckerwar & Ash [14] to identify the additional forces occurring in the equations of motion as contributions due to a deviation from the thermodynamic local equilibrium, one would expect a limit case leading to the classical dynamics, as already discussed at the end of §(a). Indeed, according to (4.3) the extra forces are scaled down when increasing the parameter ω 0 and the physical dimension of ω 0 is a reciprocal of time, suggesting its interpretation as a relaxation rate.…”

The influence of pressure relaxation on the structure of an axial vortex

“…2 The addition of this constraint results in two volume-dissipative terms in the Navier-Stokes equation: first, the traditional "volume viscosity" term, proportional to the rate of dilatation, and second, a "pressure relaxation" term, proportional to the material time rate of change of the pressure gradient. 2,3 The appearance of two such terms is consistent with the requirement that a relaxation process be characterized by two independent constitutive coefficients. 4 Accordingly, the vector form of the resulting Navier-Stokes equation could be written as…”

Comment on “Roles of bulk viscosity on Rayleigh-Taylor instability: Non-equilibrium thermodynamics due to spatio-temporal pressure fronts” [Phys. Fluids 28, 094102 (2016)]

“…3,4 We believe the more recent work represents a rigorous approach for segregating bulk viscous effects from non-equilibrium thermodynamic effects in unsteady fluid flow processes. The empirical relations which Singupta et al have attributed to Ash et al are not altered directly by this newer theory, but the possibility of non-equilibrium pressure influences on otherwise incompressible rotational flows should be considered as well.…”