Analytical solution for the abrupt cessation of pulsatile flow through a circular pipe

We develop a mathematical model to characterize the abrupt cessation of one-dimensional, incompressible, laminar pulsatile flow at an arbitrary time tc and describe the resulting transient flow in a circular pipe. The initial pulsatile flow velocity is expressed as Ui=Umi+Uoi sin(ωt), where Umi and Uoi are the mean and oscillatory components, ω is the angular frequency, and t is the time. Analytical solutions, derived through Laplace transforms, provide expressions for velocity, velocity gradient, pressure gradient, wall shear stress, and skin-friction factor as functions of radius r and t. By using Umi, Uoi, tc, and the Womersley number α as parameters, we demonstrate that our generalized solution encompasses known results for steady and non-periodic unsteady flows as subsets. We also derive a novel special-case solution describing the decaying flow following the abrupt cessation of oscillatory flow. Using these solutions, we characterize decaying flow behavior in steady, oscillatory, and pulsatile regimes. The parametric study concerns cessation from a periodically developed pulsatile flow. Key findings show that the maximum skin-friction factor depends strongly on λom=UoiUmi (ranging between 1−100), α (ranging between 1−100), and ωtc (ranging between 0°−360°): its absolute peak decreases with increasing α, and the cessation phase at which it is largest shifts from 90° toward 141° as α increases, independently of λom. Under the fixed-mean-velocity comparison adopted here, the peak is largest for pulsatile flow, but since the solution is linear in the mean and oscillatory amplitudes, this reflects the superposition of a non-zero oscillation on a fixed mean rather than an intrinsic ranking of the flow classes. For applications requiring higher wall friction forces, cessation should occur at a phase angle between 90° and 141°, whereas for lower friction forces, cessation should occur at a phase angle outside this range.