Governing Equations for Axial Velocity

The first step in developing the numerical method is to find a formal solution to Equation 8.40. Observe that Equation 8.40 is variable separable. 

This equation can be integrated twice. Note that dP/dz is a constant when integrating with respect to r. The constants of integration are found using the boundary conditions. 

Equation 8.41 allows the shape of the velocity profile to be calculated (e.g., substitute fi = constant and see what happens), but the magnitude of the velocity depends on the yet unknown value for dP/dz. As is often the case in hydrodynamic calculations, pressure drops are determined through the use of a continuity equation. Here, the continuity equation takes the form of a constant mass flow rate down the tube  

This is the local pressure gradient. It is assumed to vary slowly in the z direction. The pressure at position z is 

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