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The variational principle of Hamilton is applied to develop an analytical formulation to describe the volume viscosity in fluids. The procedure described here differs from those used in the past in that a dissipative process is represented by the chemical affinity and progress variable ͑sometimes called "order parameter"͒ of a reacting species. These state variables appear in the variational integral in two places: first, in the expression for the internal energy, and second, in a subsidiary condition accounting for the conservation of the reacting species. As a result of the variational procedure, two dissipative terms appear in the Navier-Stokes equation. The first is the traditional volume viscosity term, proportional to the dilatational component of velocity; the second term is proportional to the material time derivative of the pressure gradient. Values of the respective volume viscosity coefficients are determined by applying the resulting volume-viscous Navier-Stokes equation to the case of acoustical propagation and then comparing expressions for the dispersion and absorption of sound. The formulation includes the special case of equilibration of the translational degrees of freedom. As examples, values are tabulated for dry and humid air, argon, and sea water.

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.

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