Exact solutions for some fully developed laminar pipe flows undergoing arbitrary unsteadiness

New representations of exact solutions are presented for time-unsteady, fully developed laminar pipe flows of constant-property Newtonian fluids. They extend the well-known steady Hagen-Poiseuille solutions to flows that are initially steady or stationary and subsequently undergo transients with arbitrary time unsteadiness. In these unsteady-flow solutions, each Hagen-Poiseuille result is reexpressed as the momentary steady-flow relation for one variable as a function of the other, together with an additive functional of the other’s time derivative. The functional expressions are convolution integrals over the entire history of the transient and represent the memory of the pipe-flow Navier-Stokes equations of earlier values of transient quantities.