Flow Through an Annular Die

To solve this problem we must obtain relationships between the wall shear stress, Tw, and the pressure drop, AP, as well as the volumetric flow rate, Q. This is done by carrying out a momentum or force balance on a differential element of fluid to obtain a differential equation for the stress distribution. A constitutive equation is then substituted into the stress equation to obtain a differential equation for the velocity field. This is then integrated, and the velocity field is found when the appropriate boundary conditions are specified. We make the following assumptions  

Trz is the force per unit area acting in the z direction on a surface at r by a layer of fluid at lesser r. It is customary to take the force per unit area acting at r to be positive while 

FIGURE 2.8 Cylindrical shell of fluid over which the momentum or force balance is performed. 

But this first term is just the derivative, and this gives the following differential equation  

At some distance, fiR, the velocity field must pass through a maximum and (which is proportional to dv fdr) must be zero. Utilizing this information Cl is replaced by —(Pq P l) PRf/2L, which leads to the following equation in place of Eq.2.18  

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