Isothermal Flow in Non-Uniform Channels

In many practical situations involving the flow of polymer melts through dies and along channels, the cross-sections are tapered. In these circumstances, tensile stresses will be set up in the fluid and their effects superimposed on the effects due to shear stresses as analysed above. Cogswell has analysed this problem for the flow of a power law fluid along coni-cylindrical and wedge channels. The flow in these sections is influenced by three factors  

Each of these will contribute to the behaviour of the fluid although since each results from a different deformation mode, one effect may dominate depending on the geometry of the situation. This will be seen more clearly later when speciflc design problems are tackled. 

Consider the channel section shown in Fig. 5.8(a) from which the element shown in Fig. 5.8(b) may be taken. 

Assuming that the different modes of deformation are separable then considering equilibrium of forces in regard to the shear stress only, gives 

In order to integrate the right hand side of this equation it is necessary to get an equation for a in terms of r. This may be obtained as follows. 

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