The forces on a solid body moving through viscous fluid

Abstract

The following theorem will be proved. The drag force on a solid body of any size and shape moving with uniform velocity U through otherwise still fluid of density ρ , and of any viscosity, is ρ U multiplied by the inflow along the wake, the inflow being taken at an infinite distance behind the body. The force in any direction at right angles to the velocity U can be obtained by taking a cylinder, everywhere at a great distance from the solid body, with generators at right angles to the velocity U and to the direction of the force required; integrating, along the cylinder, the circulation around sections by planes perpendicular to the generators; and multiplying the integral by ρ U. In taking the circulation, we must cut the wake behind the solid body at right angles. When the motion of the fluid is not steady, but oscillates within fixed limits, the theorem remains true if average values are taken over a sufficiently long time.