Time-dependent Axial Fluid Flow in an Annulus: A Semi-Analytical Approach

Abstract

This study presents a semi-analytical solution for the start-up pressure-driven axial flow of an incompressible Newtonian fluid between two stationary concentric cylinders. The governing unsteady momentum equation is transformed using the Laplace technique, yielding closed-form expressions for the velocity and wall-shear distributions in terms of modified Bessel functions. The inverse transforms are evaluated by a Riemann-sum approximation, while exact steady-state solutions are derived for validation. For 𝜆 = 0.2, the velocities at 𝑅 = 0.4, 0.6, and 0.8 reach 94.28%, 94.39%, and 95.06% of their steady values at 𝑇 = 0.2, respectively. By 𝑇 = 0.4, the maximum deviation from the steady solution decreases to 0.41%, and at 𝑇 = 10^4 agreement is obtained to four decimal places. Increasing the radius ratio from 0.4 to 0.8 reduces the steady inner-wall shear from 0.3730 to 0.1042, a 72.06% decrease, while the outer-wall shear magnitude decreases from 0.2708 to 0.0967, corresponding to a 64.29% reduction. The main contribution is a compact, root-free benchmark formulation that combines the transient axial velocity, signed wall-shear histories and exact steady-state limits within a single computational framework.