Drag is Due to Flux of Spanwise Vorticity from Solid Surfaces

Wall drag is exactly related to vorticity flux from solid surfaces [1]. Potential flow, with the same in-flow and out-flow and with no-flow through the wall, is used for comparison. Total drag from skin friction and pressure is given instantaneously by flux of spanwise vorticity across potential-flow streamlines, integrated over flow volume. This follows from incompressible Navier-Stokes for any Reynolds number and generally for interior flows through pipes, channels, etc. and for exterior flows past solid bodies. Drag is thus due entirely to flux away from the wall of spanwise vorticity, generated at the body surface by streamwise pressure gradients and body acceleration and then transported through the flow by nonlinear advection, stretching and viscous diffusion. Streamwise vorticity does not contribute directly. Our result is inspired by the Josephson-Anderson relation for drag in inviscid superfluids due to motion of quantized vortices. Drag for classical fluids in the infinite Reynolds limit is likewise explained by Euler solutions with vorticity flux, distinct from d'Alembert's. Drag reduction at any Reynolds number can be achieved only by reducing spanwise vorticity flux