Navier-Stokes revisited

A revision of Newton's law of viscosity appearing in the role of the deviatoric stress tensor in the Navier-Stokes equation is proposed for the case of compressible fluids, both gaseous and liquid. Explicitly, it is hypothesized that the velocity v appearing in the velocity gradient term ∇v in Newton's rheological law be changed from the fluid's mass-based velocity vm, the latter being the velocity appearing in the continuity equation, to the fluid's volume velocity vv, the latter being a stand-in for the fluid's volume current (volume flux density nv). A similar vm→vv alteration is proposed for the velocity v appearing in the no-slip tangential velocity boundary condition at solid surfaces. These proposed revisions are based upon both experiment and theory, including re-interpretation of the following three items: (i) experimental “near-continuum” thermophoretic and other low Reynolds number phoretic data for the movement of suspended particles in fluids under the influence of mass density gradients ∇ρ, caused either by temperature gradients in single-component fluids undergoing heat transfer or by species concentration gradients in inhomogeneous two-component mixtures undergoing mass transfer; (ii) the hierarchical re-ordering of the Burnett terms appearing in the Chapman-Enskog gas-kinetic theory perturbation expansion of the viscous stress tensor from one of being based upon small Knudsen numbers to one of being based upon small Mach numbers; (iii) Maxwell's (1879) ubiquitous vm-based “thermal creep” or “thermal stress” slip boundary condition used in nonisothermal gas-kinetic theory models, recast in the form of a vv-based no-slip condition. The vv vs. vm dichotomy in the case of compressible fluids is shown to lead to a fundamental distinction between the fluid's tracer velocity as recorded by monitoring the spatio-temporal trajectory of a small non-Brownian particle deliberately introduced into the fluid, and the fluid's “optical” or “colorimetric” velocity as monitored, for example, by the introduction of a dye into the fluid or by some photochromic- or fluorescence-based scheme in circumstances where the individual fluid molecules are themselves responsive to being probed by light. Explicitly, it is argued that the fluid's tracer velocity, representing a strictly continuum nonmolecular notion, is vv, whereas its colorimetric velocity, which measures the mean velocity of the molecules of which the fluid is composed, is vm.