Potential Flows

In order to make the integration of the partial differential equation of fluid mechanics possible by simple mathematical means, the introduction of irrotationality of the flow field is necessary. The introduction of irrotationality is necessary to yield a replacement for the momentum equations and it is this fact that permits simpler mathematical methods to be applied. In Sect. 5.8.1, a transport equation equivalent to the momentum equation was derived for the vorticity which for viscosity-free flows is reduced to the simple form Dω/Dt = 0. From this equation, two things follow. On the one hand, it becomes evident that irrotational fluids obey automatically a simplified form of the momentum equation. On the other hand, Kelvin’s theorem results immediately, according to which all flows of viscosity-free fluids are irrotational, when at any point in time the irrationality of the flow field was detected. This can be understood graphically by considering that all surface forces acting on a non-viscous fluid element act normal to the surface and as a resultant go through the center of mass of the fluid element. At the same time, the inertia forces also act on the center of mass, so that no resultant momentum comes about which can lead to a rotation. Hence the conclusion is possible that rotating fluid elements cannot receive an additional rotation due to pressure and inertia forces acting on ideal fluids. This is indicated in Fig. 10.1.