The Euler–Euler (EE) approach derives its name from the fact that both the continuous and the dispersed phases are solved in the Eulerian frame of reference. For the fluid phase, the Eulerian frame is the natural choice and was pursued both in the particle-resolved (PR) and the Euler–Lagrange (EL) approaches. Particles are, however, inherently Lagrangian, and an Eulerian representation is possible only when the individual nature of the particles is erased. This requires that the particle-related Lagrangian quantities be suitably averaged, so that Eulerian fields of these quantities can be defined. The averaging process will allow particle volume fraction, particle velocity, and particle temperature fields to be defined as functions of space and time.

In this chapter, we investigate the problem of heat transfer from an isolated rigid sphere subjected to a cross flow of different temperature. This thermal problem is analogous to the flow problem considered earlier, and the interest here is to establish an expression for heat transfer in terms of the undisturbed ambient flow, which must now be characterized both in terms of relative velocity and temperature difference. We will start our investigation with rigorous analytical results in the Stokes and the small Péclet number regime.

The Eulerian representation of the continuous phase is natural, where quantities such as fluid velocity u(x,t) represent the average velocity of all the fluid molecules within a suitably chosen volume for continuum description. In the previous section, we considered filtering of these flow quantities over a suitably chosen length scale that is much larger than the size of the individual particles.

In this chapter we will consider in detail the interaction of an isolated rigid particle with the surrounding continuous-phase flow. In the low Reynolds number limit, the problem can be solved analytically. At finite Reynolds number, one must resort to numerical simulations. Nevertheless, in both cases, by simultaneously solving the Navier–Stokes equations for the fluid, equations of rigid-body motion for the particle, and coupling them with no-slip and no-penetration boundary conditions, we can obtain complete details of the flow around the particle.

From the range of topics and the depth of physics that were discussed in the previous chapters, it is quite clear that multiphase flow is a challenging subject even at the level of an individual particle. But clearly we need to move forward and begin to consider more complex multiphase-flow physics. Toward this goal, we will progress beyond an isolated particle in an unbounded medium in two different ways. First, in this chapter we will consider the problem of an isolated particle in an ambient flow, but in the presence of a nearby wall.

In this chapter our attention will primarily be restricted to the dispersed phase. Clearly the continuous phase is also important, but in this chapter we will discuss the state or evolution of the continuous phase only as needed in the context of characterizing the state of the dispersed phase. Consider the case of a turbulent multiphase flow with a random distribution of monosized spherical particles (or droplets or bubbles) within it. Imagine taking pictures of the particle distribution in an experiment (i.e., in one realization) without recording the details of the flow surrounding the particles.

We have completed our discussion of the drag force, where the term “drag” has been used to represent the force on a particle that is in the direction of ambient flow as seen in a frame of reference attached to the particle (i.e., drag is the force component along the direction of relative velocity). But there are many situations where the force on the particle is not only directed along the ambient flow, but also has a component that is perpendicular to the direction of ambient flow. In this case, the particle not only experiences a “drag” force, but also is subjected to a “lift” force.

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.