Start-Up Flow in a Circular Tube - Solution by Separation of Variables

START-UP FLOW IN A CIRCULAR TUBE - SOLUTION BY SEPARATION OF VARIABLES 

In Section B, we considered several time-independent unidirectional flows. It is evident that the resulting steady velocity profiles can be considered as limiting solutions for large times following the imposition of either a steady boundary motion or a steady pressure gradient. An interesting question that we can study in detail for unidirectional flows is how these steady-state velocity profiles evolve in time if the fluid is initially motionless at the instant of application of the boundary motion or pressure gradient. 

In this section, we consider one example of this type of problem, namely, the start-up from rest of the flow in a circular tube when a nonzero pressure gradient is suddenly imposed at some instant (which we shall denote as i = 0) and then held at the same constant value thereafter (t 0). In Section G, we consider the start-up of simple shear flow between two infinite plane boundaries. Other relatively simple examples of start-up flows are left to the reader to solve (for example, see Problem 19 at the end of this chapter). 

In the present problem of the start-up of pressure-gradient-driven flow in a circular tube, we assume that the tube is very long and focus our attention on only the central region away from the ends so that the motion can be approximated as unidirectional, albeit time dependent. Thus we assume, for t 0, that 

It is clear that a much more complicated analysis would be required for considering the entry and exit regions near the ends of the tube, because the axial velocity component uz in that region will be dependent on the distance from the ends of the tube in violation of the basic unidirectional flow assumptions (see Section A). 

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D. Start-Up Flow in a Circular Tube - Solution by Separation of Variables... 

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