Analytical solutions for the incompressible laminar pipe flow rapidly subjected to the arbitrary change in the flow rate

Abstract

A one-dimensional mathematical model is developed for an unsteady incompressible laminar flow in a circular pipe subjected to a rapid change in the flow rate from an initial flow with flow rate, Qi, to a final flow with flow rate, Qf, in a step-like fashion at an arbitrary time, tc. The change in the flow rate may either be an increment, Qf > Qi, or a decrement, Qf < λ a < 1 and 0 < λ d < 1 with 0 < t c < t s i and t c ≥ t s i, where tsi is the time at which the initial flow reaches the temporally developed state. Exact solutions for canonical pipe flow problems reported in the literature are deduced as subsets of the derived generalized solutions. The parametric study reveals the effects of varying λa or λd and tc on the quantities of practical importance, viz., τs and C f R e, where τs is the time required for the final flow to reach the temporally developed state.