Collisions among particles, droplets, and bubbles and their growth through coagulation is vital in the understanding of many multiphase problems. Similarly, particles, droplets, and bubbles can also breakup into smaller fragments and daughter droplets and bubbles. For example, it is now well established that collisions and coagulation of droplets play a central role in the formation of precipitation-size raindrops in a cloud (Mason, 1969; Yau and Rogers, 1979; Sundaram and Collins, 1997; Shaw, 2003; Grabowski and Wang, 2013).

In this chapter we will discuss some of the numerical methodologies that are appropriate for particle-resolved simulations of multiphase flows. Our focus will be on PR-DNS, where all the flow scales of fluid motion are resolved along with the surface of the particles. PR-DNS simulations, however, come at a computational cost. The range of multiphase flow problems that can be simulated in a particle-resolved manner is limited. This limitation does not arise from the mathematical formulation. As discussed in Section 2.4, the mathematical formulation of PR-DNS is the easiest among all approaches to dispersed multiphase flows.

We now have all the background information needed to explore the various computational approaches that are available for solving the wide range of multiphase flows we encounter. In fact, you may feel like you are at the cereal aisle in a grocery store wondering which one cereal among the shelf-full to pick. Fortunately, the process of picking the correct computational approach for a particular multiphase flow problem can be simplified through a rational analysis of the strengths and weaknesses of the different approaches and their suitability to the multiphase flow problem at hand.

From Chapter 4 to Chapter 10 we have studied extensively the interaction of an ambient flow with (i) an isolated particle, (ii) an isolated particle in the presence of a nearby wall, (iii) a pair of particles, and (iv) a large collection of particles. These investigations were at the microscale and we paid great attention to solving for the complete details of the flow around the particles. These studies can be classified as “particle-resolved” or “fully resolved,” as they included all the relevant physics. As a result, these studies have yielded reliable results on the hydrodynamic force, torque, and heat transfer on the particles under varying flow conditions.

In this chapter, we will consider particle–particle interactions. Here we distinguish two kinds of interactions. The first is direct interaction between particles in the form of collisions. When two particles collide, the time history of force exchange between them is controlled by the solid mechanics of elastic and plastic deformation between the colliding particles. In the context of multiphase flow computations, such collisions are simplified and treated using either a hard-sphere or a soft-sphere collision model, which will be discussed in this chapter. As a special case we will also consider the problem of particle–wall collisions.

In Chapter 4, we started with a rigorous derivation of force on a spherical particle in the limit of zero Reynolds number in a time-dependent uniform ambient flow, which led to the BBO equation. We then extended the analysis to spatially varying flows in the Stokes limit and obtained the MRG equation. At finite Reynolds number, due to the introduction of fluid inertia, we saw how difficult a complete solution of the hydrodynamic force on a particle can become. In this chapter, we plan to boldly venture into the difficult topic of interaction between a particle and a turbulent flow.

The Euler–Lagrange (EL) approach is also often referred to as the point-particle approach, since the particles are taken to be point masses, as far as their interactions with the surrounding continuous phase are concerned. In the particle-resolved approach, the presence of the particles was fed back to the surrounding continuous phase through the no-slip, no-penetration, isothermal or adiabatic, and other boundary conditions. These boundary conditions, without additional closure assumptions, directly controlled the mass, momentum, and energy exchanges between the particles and the surrounding fluid. Furthermore, these exchanges, which are in the form of tractional force, heat, and mass transfer, are properly distributed around the surfaces of the particles, and they accurately account for the presence of boundary layers, wakes, and other microscale features around the particles.

Multiphase flow is a branch of fluid mechanics that has grown rapidly over the past few decades. The term phase in “multiphase” refers to the solid, liquid, or gaseous state of matter. Thus, a multiphase flow is one that involves more than one phase. Multiphase flow can be a gas–solid flow, as in the case of a sand storm or pneumatic transport of powder.

Until quite recently, discussions on “polycrystals” have been rather concentrated on or confined to how to realistically evaluate the averaged (macroscopic) stress-strain response, focusing on, e.g., relaxed constraint even with FEM simulations. This chapter discusses new perspectives related to Scale C and the attendant theory and modeling for polycrystalline materials including nanocrystals based on the field theory (they mostly are the latest achievements). Emphasis here is placed on the collective effects brought about by a large number of composing grains on the meso- and macroscopic deformation behavior of polycrystals, in the context of hierarchy of polycrystalline plasticity. For this purpose, a series of systematically designed finite element simulations have been conducted.