Generalized Reynolds number for flow in pipes

For Newtonian flow in a pipe, the Reynolds number is defined by 

In the case of non-Newtonian flow, it is necessary to use an appropriate apparent viscosity. Although the apparent viscosity (ia is defined by equation 1.71 in the same way as for a Newtonian fluid, it no longer has the same fundamental significance and other, equally valid, definitions of apparent viscosities may be made. In flow in a pipe, where the shear stress varies with radial location, the value of fxa varies. As pointed out in Example 3.1, it is the conditions near the pipe wall that are most important. The value of /j.a evaluated at the wall is given by 

Another definition is based, not on the true shear rate at the wall, but on the flow characteristic. This quantity, which may be called the apparent viscosity for pipe flow, is given by 

Representing the fluid s laminar flow behaviour in terms of K and n  

The pipe flow apparent viscosity, defined by equation 3.31, is given by 

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