The Smoluchowski equation for a system in macroscopic flow

3 The Smoluchowski equation for a system in macroscopic flow Now our aim is to calculate the stress for given history of macroscopic velocity gradient. First we consider the form of the Smoluchowski equation for a given macroscopic velocity field v(r, t) = K(t) t. To do this we have to know the microscopic velocity field v(r, t). This is obtained from the conditions (i) v(r, t) is a solution of eqn (3.104), and (ii) the average of v(r, t) is the macroscopic field, i.e., 

Though at first sight it may seem difficult to find the velocity field v(r, t), the answer is simple  

For this flow, the first condition is obviously satisfied. That the second condition is satisfied is seen by the symmetry argument in the homoge neous flow, the average 

This equation describes the change in the distribution function of a system under macroscopic velocity gradient. 

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