Torques on Solid Boundaries Caused by Inviscid Laminar Flows

摘要

Certain homogeneous laminar fluid flows are examined for their time rates of change in orbital angular momentum in order to try to learn about the torques exerted by the fluid on nearby solid boundaries. A time rate of change in the angular momentum of the fluid must be caused by a torque induced in the fluid by the boundary as the fluid moves past it. Conversely, by an extension of Newton 's third law to rotary motion the fluid causes an equal but opposite torque on the boundary about the same axis. Friction is assumed negligible and the moving fluid is taken to be a two-dimensional shear flow without any vortices. First we consider the flow past a thin circular arc bump on an otherwise flat wall. Computing the angular momentum per unit volume about the center of the circular arc gives a time rate of increase in the angular momentum at the front half of the arc (encountered first by the flow), followed by a time rate of decrease at the back half. As a consequence of action equals reaction, the torque per unit volume imparted by the fluid to the front half of the arc tries to rotate it counterclockwise, while the torque on the back half tries to rotate it clockwise. There is no net torque on the arc. In magnitude the torques per unit volume on the arc are proportional to the fluid density and to the square of the mean speed of the fluid, and they increase with increasing thickness for constant horizontal length of the arc. Second, an example of flow past an asymmetrical bump is discussed in which it is argued that there is a net torque on the bump whose magnitude and direction are independent of the mean direction of flow. The sense of the net torque is to rotate the side of the bump with the lesser surface curvature toward the side with the greater surface curvature. Finally, the solitary wave is predicted to cause torques on a completely flat bottom as it passes by. Forces induced on solid boundaries that are directed parallel to the mean flow are mentioned.