Resolution of d'Alembert's paradox using slip boundary conditions: The effect of the friction parameter on the drag coefficient

Abstract

d'Alembert's paradox is the contradictory observation that for incompressible and inviscid (potential) fluid flow, there is no drag force experienced by a body moving with constant velocity relative to the fluid. This paradox can be straightforwardly resolved by considering Navier's slip boundary condition. Potential flow around a cylinder then solves the Navier--Stokes equations using friction parameter $\beta=-2\nu$. This negative friction parameter can be interpreted physically as the fluid being accelerated by the cylinder wall. This explains the lack of drag. In this paper, we introduce the Navier slip boundary condition and show that choosing the friction parameter positive resolves d'Alembert's paradox. We then further examine the effect of the friction parameter $\beta$ on the drag coefficient. In particular, we show that for large $\beta$ the drag coefficient corresponds well with experimental values. Moreover, we provide numerical evidence that the Newton continuation method (moving from small to large Reynold's numbers) requires fewer iterations to succeed. Thus the slip boundary condition is advantageous also from a computational perspective.