Calculation for weak velocity gradient

Let us now calculate the stress tensor under a weak velocity gradient. If we consider the case 17 = 0, then eqn (8.15) becomes 

To calculate S p we again multiply u up - 8 13 by eqn (8.125) and integrate over . The equation is then rewritten by integration by parts as 

To calculate the first order in k, the coefficients of can be replaced by their equilibrium values. Since at equilibrium 

Note that [ ((u)] has a finite value as which arises from the rigid 

In Section 8.S and 8.6, we considered the case that the perturbation due to the external field is small. This allowed us to replace the unknown fourth moment u UpU u,) by its equilibrium value. If the external field is large, this approximation becomes invalid. 

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