Flow Behaviors under Surface Shear and Velocity Conditions

Flows with free surfaces are commonly encountered in the nature and engineering applications. We investigate behaviors of flows generated by velocity and shear stress boundary conditions at water surfaces. First, analysis is made for a steady flow between two horizontal infinite plates, and it indicates that the solution for the flow driven by velocity condition at the top plate is exactly same as that by shearing condition. Then, an analytical solution is derived for an impulsively started flow initiated by a constant shear stress on a surface using Laplace transform, and it is shown that the flow is distinct from that caused by suddenly applying a velocity on it. At last, numerical solutions of cavity flows are made and they indeed confirm the conclusions drawn by the analyses; surface velocity and stress conditions lead to same results if one only considers steady state flows, whereas they produce flows with distinct velocity profiles if one simulates them as unsteady flows. These conclusions suggest that the conditions at water surfaces should be selected with discretion in studying flows with surfaces such as those in rivers and oceans.