Final expressions for the mesoscale acceleration models

In order to complete our discussion on momentum transfer, we must consider the final forms of the mesoscale acceleration models in the presence of all the fluid-particle forces. When the virtual-mass force is included, the mesoscale acceleration models must be derived starting from the force balance on a single particle  

On the right-hand sides of these force balances, we include the body and fluid-particle forces discussed in the previous sections. By moving the time derivatives from the virtual-mass force to the right-hand side, we can combine the balance of forces in matrix form  

From Eqs. (5.93) and (5.94), it is very interesting to note that while the fluid-particle forces are the same as in the absence of the virtual-mass force, the body forces for the particle and the fluid are now coupled whenever Cym + 0. However, body forces that are linearly proportional to mass such as gravity (i.e. Fp = PpVpg and Ff = pfVfg) will remain uncoupled. In the very dilute limit, we have Vp Vf and thus Eq. (5.93) reduces to the classical particle model in which the virtual-mass force affects only the particle mass. The same is true for the fluid seen by the particle in the limit where pf pp. 

The principal fluid-particle forces discussed in this chapter lead to the following expression for the fluid-particle force  

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