Transient Flow in a Duct

Consider a long circular duct in which an incompressible, constant-property fluid is initially at rest. Suddenly a constant pressure gradient is imposed. The axial momentum equation that describes the transient response of the velocity profile for this situation is 

The Hagen-Poiseuille solution at steady state provides a specific relationship among the factors in the nondimensional group of constants involving the pressure gradient. From Eq. 4.59, 

The parabolic partial differential equation can be solved by separation of variables, although the solution shown in Fig. 4.9 is found by a finite-difference technique. Starting from rest (i.e., zero velocity everywhere), the expected steady-state parabolic velocity profile is reached in a dimensionless time of t 1. 

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