How Does Flow Go Round a Bend?

The steady flow of an incompressible fluid round a bend of uniform width is considered, and it is shown that the speed at a point on an inner streamline exceeds that at the corresponding point on an outer streamline. The normal equation of motion requires the pressure to increase outward, and Bernoulli’s theorem, applied to streamlines of common total head, then requires the speed to decrease outward; together they yield the free vortex distribution V = C/r. A numerical potential-flow solution in the actual bend geometry confirms the result to within three per cent. The argument that the outer particle must move faster because its path is longer is shown to rest on an unsupported assumption of equal transit times. Viscous effects that may reverse the distribution near the bend exit are discussed. Files: the paper (PDF), a particle-tracking animation (GIF)