Daniel D. Joseph

Discrete particle simulations are by now well established as an effective tool to study the mechanics of complex gas‐solid flows in gas‐fluidized beds. In this study, a state‐of‐the‐art discrete particle model is used to explore the role of particle‐particle interactions in bubbling gas‐fluidized beds of Geldart A particles. We find that the particle‐particle interactions, including inelastic particle‐particle collision; inter‐particle friction and slightly cohesive forces, only have a negligible effect on the hydrodynamics of bubbling gas‐fluidized beds of Geldart A particles. This is due to the fact that only a very small fraction of the energy input is dissipated during the non‐ideal particle‐particle interaction. We finally show that the selected drag correlation model significantly affects the bed hydrodynamics.

When small particles (e.g., flour, pollen, etc.) come in contact with a liquid surface, they immediately disperse. The dispersion can occur so quickly that it appears explosive, especially for small particles on the surface of mobile liquids like water. This explosive dispersion is the consequence of capillary force pulling particles into the interface causing them to accelerate to a relatively large velocity. The maximum velocity increases with decreasing particle size; for nanometer-sized particles (e.g., viruses and proteins), the velocity on an air-water interface can be as large as approximately 47 m/s. We also show that particles oscillate at a relatively high frequency about their floating equilibrium before coming to stop under viscous drag. The observed dispersion is a result of strong repulsive hydrodynamic forces that arise because of these oscillations.

The rupture of a 3D stationary free liquid film under the competing effects of surface tension and van der Waals forces is studied as a linearized stability problem in a purely irrotational analysis utilizing the dissipation method. The results of the foregoing analysis are compared with a 2D long-wave approximation that has given rise to an extensive literature on the rupture problem. The irrotational and long-wave approximations are here compared with the exact 2D solution. The exact solution and the two approximate theories give the same results for infinitely long waves. The problem considered depends on two dimensionless parameters, the Hamaker number and the Ohnesorge number. The Hamaker number is a dimensionless number defined as a measure of the ratio of van der Waals forces to surface tension. The exact solution and the two approximate solutions differ by < 1% when the Hamaker number is small for all values of the Ohnesorge number. When the Ohhnesorge number is close to one, as in the case of water films separated by distance 100 Å, the long-wave approximation overestimates and the potential flow approximation underestimates the exact solution by similar small amounts. The high accuracy of the dissipation method shows that the effects of vorticity are small for small to moderate Hamaker numbers.

The theory of stress-induced cavitation is applied here to the problem of cavitation of a viscous liquid in the streaming flow past a stationary sphere. This theory is a revision of the pressure theory which states that a flowing liquid will cavitate when and where the pressure drops below a cavitation threshold, or breaking strength, of the liquid. In the theory of stress-induced cavitation the liquid will cavitate when and where the maximum tensile stress exceeds the breaking strength of the liquid. For example, liquids at atmospheric pressure which cannot withstand tension will cavitate when and where additive tensile stresses due to motion exceed one atmosphere. A cavity will open in the direction of the maximum tensile stress, which is 45° from the plane of shearing in pure shear of a Newtonian fluid. This maximum tension criterion is applied here to analyse the onset of cavitation for the irrotational motion of a viscous fluid, the special case imposed by the limit of very low Reynolds numbers and the fluid flow obtained from the numerical solution of the Navier–Stokes equations. The analysis leads to a dimensionless expression for the maximum tensile stress as a function of position which depends on the cavitation and Reynolds numbers. The main conclusion is that at a fixed cavitation number the extent of the region of flow at risk to cavitation increases as the Reynolds number decreases. This prediction that more viscous liquids at a fixed cavitation number are at greater risk of cavitation seems not to be addressed, affirmed nor denied, in the cavitation literature known to us.

In this work, I present the form of the Navier–Stokes equations implied by the Helmholtz decomposition in which the relation of the irrotational and rotational velocity fields is made explicit. The idea of self-equilibration of irrotational viscous stresses is introduced. The decomposition is constructed by first selecting the irrotational flow compatible with the flow boundaries and other prescribed conditions. The rotational component of velocity is then the difference between the solution of the Navier–Stokes equations and the selected irrotational flow. To satisfy the boundary conditions, the irrotational field is required, and it depends on the viscosity. Five unknown fields are determined by the decomposed form of the Navier–Stokes equations for an incompressible fluid: the rotational component of velocity, the pressure, and the harmonic potential. These five fields may be readily identified in analytic solutions available in the literature. It is clear from these exact solutions that potential flow of a viscous fluid is required to satisfy prescribed conditions, like the no-slip condition at the boundary of a solid or continuity conditions across a two-fluid boundary. It can be said that equations governing the Helmholtz decomposition describe the modification of irrotational flow due to vorticity, but the analysis shows the two fields are coupled and cannot be completely determined independently.