Is the tracer velocity of a fluid continuum equal to its mass velocity?
Owing to its size independence in the so-called near-continuum vanishingly small Knudsen number regime (Kn<<1) , thermophoretic particle motion occurring in an otherwise quiescent gas under the influence of a temperature gradient is here interpreted as representing the motion of a tracer, namely, an effectively point-size test particle monitoring the local velocity of the undisturbed, particle-free, compressible gas continuum through space. "Compressibility" refers here not to the usual effect of pressure on the gas's mass density rho but rather to the effect thereon of temperature. Our unorthodox continuum interpretation of thermophoresis differs from the usual one, which regards the existence of thermophoretic forces in gases as a strictly noncontinuum phenomenon, involving thermal stress-induced Maxwell slip ("thermal creep") of the gas's mass velocity vm at the surface of the particle, with vm denoting the velocity appearing in the continuity equation expressing the law of conservation of mass. Explicitly, instead of regarding the thermally animated particle as moving through the gas, we regard the particle (in its hypothesized role as a tracer of the undisturbed, particle-free, fluid motion) as moving with the gas, through space; that is, the particle is viewed as simply being entrained in the flowing gas, which, as a result of an externally applied temperature gradient, was already in motion prior to the tracer's introduction into the fluid--albeit not mass motion (which is, in fact, identically zero) but rather volume motion. This tracer-particle interpretation of experimental thermophoretic particle velocity measurements raises fundamental issues in regard to the universally accepted Newtonian rheological law constitutively specifying the viscous or deviatoric stress T as being proportional to the (symmetrized, traceless) fluid velocity gradient inverted Deltav , with v identified as being the fluid's mass velocity vm . Rather, it is argued in the case of compressible fluids, including liquids, that v should, instead, be chosen as the fluid's volume flux density or current density nv , the latter being formally equivalent to the fluid's volume velocity vv , which differs from vm except in the case of incompressible fluids. Apart from this strictly constitutive issue in regard to T , it is further argued that the fluid's tracer or Lagrangian velocity vl:=(deltax/deltat) (x0) along the fluid's spatiotemporal trajectory x=x (x0,t) is equal to vv, rather than to vm. This too is contrary to the heretofore unquestioned supposition that the conceptually distinct fluid velocities vl and vm are not only equal but are, in fact, synonymous. To the extent that vl not equal vm in the nonisothermal fluid case, an optical dye- or photochromic-type experiment (each of the latter two experiments presumably serving to measure vm) will record a different velocity than would a comparable tracer particle velocity measurement, one that measures vl .
Brenner
Howard
H
Department of Chemical Engineering, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139-4307, USA. hbrenner@mit.edu