Mixture pressure and stress in disperse two-phase flow

The definition and interpretation of average pressure in an incompressible disperse two-phase flow are ambiguous and have been the object of debate in the literature. For example, the physical meaning of definitions involving an internal ‘pressure’ inside rigid particles is unclear. The appearance of the particle internal stresses in averaged equations of the two-fluid types is similarly puzzling as, provided the particles are sufficiently rigid, the precise numerical value of such stresses would not be expected to affect the flow. This paper deals with these matters using a new approach. A proper definition of mixture pressure follows quite naturally by identifying the isotropic component of the mixture stress that — just like the usual pressure in incompressible single-phase flow — is covariant under the gauge transformation p→p+ψ, where ψ can be thought of as the potential of body forces. This transformation includes as special cases the more usual gauge transformation p→p+Πt, with Πt an arbitrary function of time, and p→p−ρg·x, by which gravitational effects are removed from the single-phase equations. The mixture pressure that is identified on the basis of this argument contains the pressure averaged over the surface of the particles, as in some earlier proposals, but also other terms. Explicit examples are given for the case of dilute potential and Stokes flows of spheres. It is also shown that it is possible to completely eliminate the disperse-phase stress field from the averaged equations provided the particle motion is only expressed in terms of the center-of-mass and angular velocity. Finally, the implications for the closure of the averaged equations that derive from the concept of covariance under the general gauge transformation are discussed.