The nonlinear stage starts when the amplitude of the unstable flow feature becomes significant. This chapter first studies the nonlinear growth of the interface amplitude and its associated terminal velocity with potential flow models, both for RM and RT. Next, one describes several models intended to predict the evolution of the bubble and spike heights, and the corresponding velocities, for the nonlinear stage. The success and limitations of each model are assessed with comparison to experiments and numerical simulations. The sensitivities to viscosity, density ratio and Mach number are discussed.

This chapter considers the drag force for velocity gradients in the surrounding fluid, particle Mach number and Knudsen number, temperature gradients in the surrounding fluid, particle spin and fluid vorticity, flow turbulence and particle roughness, shape for a solid particle, surface contamination and internal recirculation for a spherical fluid particle, and deformation and drag for a fluid particle. This includes theory, experimental results, and numerical prediction of the drag coefficient for point-force models.

This chapter introduces influence of density change on a flow, i.e., the compressible flow theory. Strictly speaking, any gas flow is both viscous and compressible. In tradition the influence of viscosity and compressibility are dealt with separately to make things easy. In this book, the chapter 6 deals with viscosity, and the chapter 7 deals with compressibility. Sound speed and Mach number are introduced in the beginning, then the equations for steady isentropic flow are derived with statics and total parameters introduced. Some gas dynamic functions are derived that use coefficient of velocity in replace of Mach number. Propagation mode of pressure waves are discussed next, and expansion and compression waves are introduced. Shock wave, as a strong compression wave, is discussed in depth. In the end, transonic and supersonic flow in a variable cross-section pipe is discussed, especially the characteristics of the flow in a Laval nozzle.

This chapter briefly introduces the application of similarity theory and dimensional analysis in fluid mechanics. The concept of flow similarity is discussed at first to give the readers a brief idea what similarity means. Then some important dimensionless number is listed and discussed, which includes Reynolds number, Mach number, Strouhal number, Froude number, Euler number and Weber number. Next, the governing equations are transformed to a dimensionless form to shows how the dimensionless numbers act in the equations. In the end, some flow examples are provided to show the role of the dimensionless numbers.