OVERVIEW

Potential flow arises in a variety of natural contexts and engineering applications; a familiar example is high-Reynolds-number flow past a streamlined body discussed in Chapters 7 and 8. Since the vorticity is confined within thin boundary layers and narrow wakes, the main body of the flow is virtually irrotational and may thus be described in terms of a velocity potential ?, setting u = ? ?. The continuity equation requires ? to be a harmonic function, ? ² ? = 0, and this reduces the computation of the flow to solving Laplace's equation subject to the no-penetration condition over the impermeable boundaries. A related example concerns the flow due to the propagation of gravity waves on a free surface.

Another example of potential flow from a different physical context is provided by the flow of a viscous fluid through the Hele Shaw cell discussed in Section 6.3. Equation (6.3.2) shows that the average velocity of the fluid across the width of the channel U is proportional to the gradient of the modified pressure P; hence, P plays the role of a velocity potential with U = ? ?, where ? = ? h ² P/(12 ?). Conservation of mass requires that the modified pressure be a harmonic function, ? ² P = 0, and this reduces the computation of the flow to solving Laplace's equation subject to Dirichlet, Neumann, or mixed-type boundary conditions, as discussed in...