Generalised approach for laminar flow of time-independent fluids

4 Generalised approach for laminar flow of time-independent fluids 

Approach used in section 3.2 for power-law and Bingham plastic model fluids can be extended to other fluid models. Even if the relationship between shear stress and shear rate is not known exactly, it is possible to use the following approach to the problem. It depends upon the fact that the shear stress distribution over the pipe cross-section is not a function of the fluid rheology and is given simply by equation (3.2), which can be re-written in terms of the wall shear stress, i.e. 

For the no-slip boimdary condition at the wall, the first term on the right hand side is identically zero and therefore  

Now changing the variable of integration from r to using equation (3.18)  

The velocity gradient (or the shear rate) term (—dV /dr) has been replaced by a function of the corresponding shear stress via equation (1.10). The form of the function will therefore depend on the viscosity model chosen to describe the rheology of the fluid. Equation (3.21) can be used in two ways  

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