The Force and Torque Acting on an Open Irrotational Surface Due to Viscous Stresses

Summary The force experienced by any closed surface due to viscous stresses vanishes when the flow is irrotational and incompressible over that surface (J. E. Sader and D. I. Pullin, J. Fluid Mech. 987 (2024), A19). The behaviour of the flow away from the closed surface can be arbitrary and is immaterial. The same holds for the torque in three-dimensional flows. For two-dimensional plane flows, however, this contribution to the torque is non-zero in the presence of circulation. We provide the generalisation of these results for open surfaces/contours over which the flow is irrotational and compressible. In so doing, we uncover the origin of the above-mentioned non-zero torque in two-dimensional plane flows on closed contours. We also show that the force due to viscous stresses acting on any open surface/contour, over which the flow is irrotational and incompressible, depends only on the flow velocities at the surface boundaries/contour end-points, that is, it is independent of the surface/contour shape. The reported formulas provide a non-trivial connection between the viscous stress over the surface/contour and its boundary/end-points, which can be used in the numerical analysis of viscous flows.