Rheology of Two-Phase Systems

Though much has been written on the dynamics of one, or a few, par­ ticles moving in a viscous fluid (11, 15, 43, 46) , the pragmatic interest in such studies lies more in the realm of polyparticle systems (38, 81) of the type commonly encountered in suspension rheology, sedimentation, flow in porous media, fluidized beds, and the like. Zuber's (86) analysis provides insight into the diversity of engineering applications likely to be encompassed by a satisfactory fundamental theory of such two-phase flows. Under moder­ ately well-defined circumstances these systems may be regarded from a gross view as heterogeneous continua, characterizable by a small set of hetero­ continuum (macrocontinuum) fields. Despite the longevity of this view, dating at least from the work of Darcy 1856 on flow through porous media and of Einstein 1906, 1911 on suspension viscosity, little has been done to formulate a rigorous general theory of the behavior of such systems. That no satisfactory fundamental basis exists is evidenced, for example, by con­ flicting claims as to the legitimacy of various derivations of Darcy's law from first principles, 1 by the innumerable relations (71 , 72) proposed for the higher-order concentration terms in Einstein's (35) viscosity formula, and by the multiplicity of theoretically-derived equations advanced to describe even the first-order concentration corrections to Stokes's law arising in the sedimentation of dilute systems (37) . That divergent views of these elementary problems should exist is symptomatic of a lack of perspective sufficiently general to embrace all such phenomena. The present status of the subject may be likened to that of nonequilibrium molecular statistical mechanics prior to the advent of the work of Kirkwood (53) . What is sorely needed is a comparable statistical mechanics of fluid-particle systems. This promising point of view is currently being pursued by at least several investigators. Clearly, however, an ambi­ tious program of this magnitude must be supplemented by less fundamental, but more tractable, schemes. It is with one such ad hoc scheme that the