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 \( \lambda = 0.2 \), 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.