Transport Equation in Terms of Peculiar Velocity

In this section an alternative derivation of the governing equations for granular flow is examined. In this alternative method the peculiar velocity C, instead of the microscopic particle velocity c, is used as the independent variable in the particle property and distribution functions. The transformation of these functions and the governing equation follows standard mathematical procedures for changing the reference frame. The translational motion of an individual particle may be specified either by its microscopic velocity c relative to a fixed or Galilean frame of reference, or by its velocity relative to a frame of reference moving with the local velocity of the granular material Yd- 

Experience has shown that one might benefit substantially from deriving the governing transport equations after having transformed the property and distribution functions so that they are dependent on the peculiar velocity variable (2.59) instead of the microscopic particle velocity c. This means that the 

To transform the mathematical operators when changing the reference frame, let / = /(r, c,t), fc = /(r, C,f) and /(r, c,t) = /c(r, C,f). It is noted that by performing this frame transformation the interpretations of the mathematical operators d/dt and V in the Boltzmann equation will change as well, because C, not c, is to be kept constant while performing the differentiation. Accordingly, due to the implicit dependence of / upon t and r through the dependence of C upon Vd r,t) as expressed by (2.59), the chain rule theorem (e.g., [17], p 105 [11], pp. 48-49 [22]) has to be invoked to re-formulate the operators in an appropriate manner. 

The chain rule provides a relation between the partial derivative of / with respect to the individual particle velocity c and the partial derivative of fc with respect to the peculiar velocity C. To understand the forthcoming transformation it might be informative to specify explicitly the meaning of the partial derivatives. 

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