Remarks on the Kinetic Theory of Granular Flows

The main advantage associated with the kinetic theory approach for dense suspensions is the appearance of two extra pressure terms in addition to the interstitial fluid phase pressure, one kinetic pressure tensor accounting for the transport phenomena due to the translational particle movement and one collisional pressure tensor accounting for the transport phenomena due to particle collisions. 

In particular, referring to the introduction of the external forces as presented in sect 4.1.3 there are still no complete consensus in the literature regarding the treatment of the interfacial coupling terms like the steady drag-, added mass- and lift forces. In one view it is considered convenient to split the net force exerted by the interstitial fluid on the particle into two different contributions One virtual force applied by an undisturbed flow on a imaginary fluid particle which coincides with the solid particle in volume and shape, and a second contribution that represents the forces due to the perturbations in the flow. These flow disturbances are created by the presence of the particles. The phrase undisturbed flow thus refers to the flow that would be observed if the particle was not present. Neglecting the effects of the perturbations in the flow, the net force exerted on a particle (4.57) might be approximated by  

For undisturbed turbulent flows the local instantaneous velocity of the continuous phase have been decomposed in various ways, not necessarily in accordance with the familiar Reynolds - and Favre averaging procedures. 

To explain the basic problem we use the Reynolds decomposition and averaging procedure, as an example. Introducing the peculiar velocity for the dispersed phase (4.103) can be re-arranged as  

The interpretations of the velocity variables in the first and second terms on the RHS represent a severe problem due to a general lack of knowledge about the physics they represent. Besides, when the F quantity in the Maxwell equation is substituted by this relation and Maxwellian averaged, the resulting covariances consist of velocity quantities which are averaged in an inconsistent manner (Maxwellian- versus time-, volume-, or ensemble averaging). For this reason other somewhat ad hoc decomposition procedures have been defined intending to achieve more consistent closures for these terms. 

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