Fluid Driven by Tangential Velocity and Shear Stress: Mathematical Analysis, Numerical Experiment, and Implication to Surface Flow

This paper investigates behaviors of flows driven by tangential velocity and shear stress on their boundaries such as solid walls and water surfaces. In a steady flow between two parallel plates with one of them in motion, analytic solutions are the same when a velocity and a shear stress boundary condition are applied on the moving plate. For an unsteady, impulsively started flow, however, analysis shows that solutions for velocity profiles as well as energy transferring and dissipation are different under the two boundary conditions. In an air-water flow, if either a velocity or a stress condition is imposed at the air-water interface, the problem becomes ill-posed because it has multiple solutions. Only when both of the conditions are specified, it will have a unique solution. Numerical simulations for cavity flows are made to confirm the theoretical results; a tangential velocity and a shear stress boundary condition introduce distinct flows if one considers an unsteady flow, whereas the two conditions lead to a same solution if one simulates a steady flow. The results in this paper imply that discretion is needed on selection of boundary conditions to approximate forcing on fluid boundaries such as wind effects on surfaces of coastal ocean waters.