(5bp) Applications of Statistical Mechanics to Studies of Two-Phase Flows

The first system is a suspension of spherical monodisperse bubbles in high-Reynolds number flow. It is an equilibrium system of bubbles treated as dipoles in potential flow. A virtual mass matrix of the system of bubbles is introduced, which depends on the instantaneous positions of the bubbles, and is used to calculate the energy of the bubbly flow as a quadratic form of the bubbles' velocities. The energy is shown to be the system's Hamiltonian and is used to construct a canonical ensemble partition function, which explicitly includes the total impulse of the suspension along with its energy. The Hamiltonian is decomposed into an effective potential due to the bubbles' collective motion and a kinetic term due to the random motion about the mean. An effective bubble temperature?a measure of the relative importance of the bubbles' relative to collective motion?is derived with the help of the impulse-dependent partition function. Two effective potentials are shown to operate: one due to the mean motion of the bubbles, dominates at low bubble temperatures, where it leads to their grouping in flat clusters normal to the direction of the collective motion, while the other, temperature-invariant, is due to the bubbles' position-dependent virtual mass and results in their mutual repulsion. Numerical evidence is presented for the existence of the effective potentials, the condensed and dispersed phases, and a phase transition.

The second system is a non-equilibrium sheared suspension of Brownian spheres in a Newtonian fluid. Here, we employ approaches of liquid state theory to gain useful new insights into the many-particle structure of the suspension. Specifically, triplet microstructure of sheared concentrated suspensions of Brownian monodisperse spherical particles is studied by sampling realizations of a three-dimensional unit cell subject to periodic boundary conditions obtained in accelerated Stokesian Dynamics simulations. Triplets are regarded as a bridge between particle pairs and many-particle clusters thought responsible for shear-thickening. Triplet correlation data for weakly sheared near-equilibrium systems display an excluded volume effect of accumulated correlation for equilateral contacting triplets. As the Peclet number increases, there is a change in the preferred contacting isosceles triplet configuration, away from the ?closed? triplet where the particles lie at the vertices of an equilateral triangle and toward the fully extended rod-like linear arrangement termed the ?open? triplet. This transition is most pronounced for triplets lying in the plane of shear, where the open triplets' angular orientation with respect to the flow is very similar to that of a contacting pair. The correlation of suspension rheology to observed structure signals onset of larger clusters. An investigation of the predictive ability of Kirkwood's superposition approximation (KSA) provides valuable insights into the relationship between the pair and triplet probability distributions and helps achieve a better and more detailed understanding of the interplay of the pair and triplet dynamics. The KSA is seen more successfully to predict the shape of isosceles contacting triplet non-equilibrium distributions in the plane of shear than for similar configurations in equilibrium hard-sphere systems; in the sheared case, the discrepancies in magnitudes of distribution peaks are attributable to two interaction effects when pair average tra jectories and locations of particles change in response to real, or ?hard,? and probabilistically favored (?soft?) neighboring excluded volumes and, in the case of open triplets, due to changes in the correlation of the farthest separated pair caused by the fixed presence of the particle in the middle.