Expansion in powers of the shear rate

To study the shear-rate dependence of the stresses at small y analytically, we first expand I and C in power series of y as 

The numerical coefficient C20 appears from the angular integral, and is calculated in Appendix 9.A. The explicit forms of f ( (f = 1 4) are also presented in Appendix 9.A. 

Since the even-order terms are zero due to symmetry, the shear viscosity is written as 

Rg- 9l9 Thickening-thinning diagram of the shear viscosity (sohd line) and the first normal stress 

The first term agrees with the zero shear viscosity r)o (9.73). The numerical coefficient 

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