Definition of the NDF for fluid-particle systems

The definition of the NDF derived above was carried out in the limit where the particles are surrounded by a vacuum. In the more general case where they are surrounded by a fluid, each particle will be accelerated by fluid stresses acting on its surface. The exact description of the resulting surface forces is very complicated and would involve the complete 

For the present discussion, we do not need to know the exact form of the mesoscale model for the fluid velocity and characteristic properties (e.g. composition and enthalpy). Instead, we will simply assume that and obey  

In order to understand the range of physics contained in the fluid-particle NDF, it is instructive to consider two limiting cases. 

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