ABSTRACT
The potential flow (Chapter 12) is based on the two assumptions that it is (a) irrotational (Section 12.1); (b) incompressible (Section 12.2). To identify the physical conditions under that these approximations hold, it is necessary to consider the fundamental equations of fluid mechanics, viz.: (i) the equation of continuity (Section 14.1), stating the conservation of mass, implies that a flow is incompressible (a) if the mass density is conserved along streamlines, or equivalently the divergence of the velocity is zero (Section 12.1); (ii) the momentum equation shows that the fluid is accelerated in the direction of decreasing pressure, in the inviscid case in the absence of external forces (Section 14.2); (iii) in the absence of viscosity and thermal conduction and other dissipative processes an adiabatic condition (Section 14.3) holds, excluding heat exchanges; (iv) the temperature can be specified by the equation of state (Section 14.3), for example, a perfect gas is a reasonable approximation for air at ambient temperatures. It can be shown that the circulation is conserved along streamlines (a) for an steady irrotational flow (Section 14.4) with two restrictions: (i) homentropic, that is, with constant entropy; (ii) under external forces that are conservative, that is, derive from a potential, like gravity. Also for a homentropic steady irrotational flow the assumption of incompressibility (b) holds if (Section 14.5) the flow velocity is small compared with the sound speed. For an incompressible irrotational (that is, potential) steady flow the difference between the stagnation pressure (at a stagnation point of zero velocity) and the free stream pressure, is the dynamic pressure (Section 14.6) that is equal to the kinetic energy per unit volume; this form of Bernoulli’s equation has many applications, for example, to measure the flow velocity using a Pitot (Venturi) tube in a free stream (Section 14.6) [ducted flow (Section 14.7)], from the pressure at two points. The simplest potential flow is uniform and corresponds to a dipole at infinity; the higher order multipoles at the origin (Sections 12.4-12.9) by inversion with regard to the origin lead to multipoles at infinity, corresponding to corner flows (Section 14.8), for example, a quadrupole (higher order 2n-pole) at infinity corresponds to the flow in a rectangular (acute) corner with angle π/4(π/2n) at the vertex. The extension to a corner with arbitrary angle 0< β ≤ 2π includes the sharp edge for β = 2π or n = 1/2; this allows the consideration of a stream incident on a wedge at an angle-of-attack (Section 14.9). The equations of fluid mechanics are deduced with the aim to particulize to potential flows, while indicating the restrictions involved; the simple properties of potential flows will be modified as the various restrictions are lifted in the sequel, for example, effects of vorticity, compressibility, viscosity, turbulence, and energy and mass transfer, chemical reactions, are introduced.
