Inhomogeneous Ekman flow

This study presents a new exact solution describing the non-uniform distribution of velocity and pressure fields in an isothermal steady shear flow of a viscous incompressible fluid. The obtained exact solutions remain valid when kinematic viscosity is replaced with turbulent viscosity in the Navier-Stokes equations. We demonstrate that within the class of functions linear in some coordinates, any joint non-uniform solution for the velocity field must have a specific structure characterized by constant spatial accelerations. Specifically, either only two particular accelerations vanish, or all four spatial accelerations equal zero (corresponding to the homogeneous velocity field in the Ekman solution). No other joint solutions exist within this specified class. We analyze in detail the case with two non-zero spatial accelerations and provide the complete exact solution. To elucidate the fundamental properties of this solution, we investigate the corresponding boundary value problem and present comprehensive illustrative material.