BBOT equations

This chapter deals with the movement of a small solid particle in a fluid flow. We start by presenting the equations governing particle movements, which we refer to as the Basset, Boussinesq, Oseen, and Tchen (BBOT) equations, to name a few key contributors to this modeling. Rather than deriving the equations, we endeavor to identify and discuss the physical meaning of the different terms acceleration, added mass, Basset term, etc. 

This approach is embodied by applying the BBOT equations to describe the behavior of a particle in three configurations of particular significance by their applications ... 

The BBOT equations allow the determination of the characteristic time with which the dynamics of a solid particle placed in a fluid flow adapts to its environment. This chapter is quite theoretical, although we have presented few derivations. This enables the reader to understand the hypotheses used in Chapters 15 (behavior of particles within gravity field) and 17 (centrifugation). 

In section 16.5, we discuss the lift force applied on a particle in a unidirectional flow. This force is not taken into account by the BBOT equations. Finally, we conclude with the application of the results presented in this chapter to laminar flows, and then to turbulent flows (section 16.7). 

In the following we resume the notations used by Maxey and Riley (1983) and write the Basset, Boussinesq, Oseen, and Tchen (BBOT) equations in a form that is quasi-identical to that of Maxey and Riley. This formulation is interesting insofar as it highlights the relative movement of the particle with respect to the fluid. The three components of the particle s relative velocity with respect to the fluid are obtained by solving the following differential equations ... 

The BBOT equations lead to the classic results on the sedimentation of a small particle within the gravity field when the surrounding fluid is at rest. They describe the acceleration of the particle which is dropped with zero velocity, and determine the characteristic time needed to reach Stokes fall velocity. Since Ui = 0, eqnations... 

The velocity component with index 1 has been chosen to be oriented along the direction of gravity and [16.8] is the BBOT equation in that directioa In the other two directions, the equations reduce to the same equation without the right-hand side. This results in Wi(t) = 0 for / = 2 and i = 3, since fF,(t = 0) = 0. The particle s movement occurs in the direction of gravity. 

The transient stage during which the particle accelerates is described by the BBOT equations. Neglecting the Basset term in a first step, diffeiential equation [16.8] simplifies into the form ... 

We now analyze the movement of a particle in a steady unidirectional sheared flow obtained by solving the BBOT equations. Gravity is not taken into account. The fluid flow occurs in the direction with index 1, and the velocity gradient is along the direction with index 3, which translates into ... 

Figure 16.2. Movement of a fluid particle and balance offorces in a unidirectional sheared flow, modeled by the BBOT equations. The trajectory of the particle, in dashed lines, becomes parallel to the direction of the fluidflow...

The example discussed in the previous section using the BBOT equations shows that the trajectory of a solid particle tends to align with the direction of the fluid flow. The fluid flow does not exert a lift force on the particle, even if it possesses a velocity gradient in the direction perpendicular to that of the flow. This result contradicts some observations of fluid mechanics. The lack of a lift force, for example, does not allow fluid particles to migrate in the direction perpendicular to the direction of the flow, or the fluid to pick up a particle lying on a wall. The smdy of the lift force has given rise to a considerable amount of research, of which we summarize a few important results. The reader will find in the book by Michaehdes" ... 

To conclude our inventory of a few application cases of the BBOT equations, we now consider a rotating flow about Oz axis, omitting gravity. In the cylindrical coordinate system (Figure 16.5), the only non-zero velocity component is the azimuthal component. We consider a velocity field for the flow in the form ... 

The rotating flow is the combination of a solid-body rotation flow and a vortex flow. As will be seen in Chapter 17, this case corresponds to many configurations for which centrifugal separation is implemented. Theoretically speaking, such flows verify the properly AS = 0. This properly greatly simplifies the BBOT equations. It also makes it easier to identify the terms and mechanisms responsible for centrifugal separation. 

As the BBOT equations [16.5] are written in the Cartesian coordinate system (0,xi,X2,X3), the first step is to transcribe the velocity field into the Cartesian coordinate system ... 

The BBOT equation for the radial component is obtained by multiplying respectively [16.42a] and [16.42b] by xj/r and X2 r, then summing the two equations. Lengthy but not especially difficult calculations yield the following equation, in which the relative velocities in the Cartesian coordinate system If j and W2 are eliminated to bring forth the relative velocities If, and cylindrical coordinate system ... 

In a rotating flow, the BBOT equations therefore describe a centrifugation movement of a small particle, the characteristics of which ate driven by rather simple mechanisms. The duration of the transient regime before equilibrium between the centrifugal forcing term and the friction is very short. The centrifugation process is described approximately by ... 

The examples of application of the BBOT equations discussed in this chapter allow some useful guidance to be drawn in order to elucidate and justify the hypotheses formulated in the other chapters of this part. By keeping in mind that these equations are obtained for small particles, the following points should be... 

The BBOT equations have enabled the introduction of the added mass effect, which intervenes during the acceleration and deceleration phases of the particle in the fluid. In a transient regime, the added mass is essential for a bubble in a liquid pp Pf 1), because the bubble has to transmit momentum to the liquid in order to be able to accelerate itself. 

Centrifugal separation of solid particles in a fluid, but also of non-miscible droplets in another liquid or of gas bubbles in a liquid, is a frequently employed process. Its principle has already been described in the previous chapter on the basis of the BBOT equations. 

The principle of centrifugation of a particle within a rotating flow has already been treated in section 16.6 of Chapter 16 using the BBOT equations. Here, we formulate differently the behavior of a particle in a rotational flow, using a physical approach that sets out the main forces governing the dynamics. 

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