Exact solution to Navier-Stokes equation for developed radial flow between parallel disks

Laminar radial flow between two parallel disks is a fundamental nonlinear fluid mechanics problem described by the Navier-Stokes (NS) equation, but is unsolved because (1) an exact solution is not found even with extensive references, and (2) it is unclear why radial flow remains laminar at high Reynolds numbers. This paper first presents exact velocity distribution solutions for developed radial inflows and outflows, proving that both flows are described by brief Jacobi elliptic sine-squared functions but with different characteristics. For inflow, a stable velocity distribution forms; for outflow, the velocity distribution may have an inflection point inducing flow instability or separation. Both velocity distributions become the classic parabolic law at low Reynolds numbers, but uniform (similar to turbulent velocity distributions) at high Reynolds numbers. Furthermore, both pressure and boundary shear stress follow an inverse-square law, but the friction factor is invariant. These results are instructive for studying nonuniform open-channel flow for which nonlinear inertia is of importance.