Forces and Moments on Bodies

The kinetic energy of a potential flow contained in a domain can be expressed as an integral over the boundary (Section 28.1) and likewise for the electric and magnetic energies. A body in a stream (Section 28.2) may be acted by: (i) two force components, viz. a drag (thrust) force along (opposite) to the motion, and a transverse lift (down force); (ii) a component of the moment orthogonal to the plane, viz. the pitching moment. The drag/thrust (lift/down force) is associated with source/sinks (vortices), and is analogous (Section 28.3) to the electric (magnetic) force on electric charges (currents). The gravity field has differences as regards the energy (Section 28.3) and similarities as regards the force (Section 28.1). A source (sink) in a uniform stream specifies (Section 28.4) a Rankine semiinfinite body (fairing) that is subject to a drag (thrust); a source-and-sink pair aligned with the flow specifies a Rankine finite body with an oval shape that is subject (Section 28.5) to a pitching moment in a flow when at angle-of-attack. Reversing the position of the source and sink can lead to (i/ii) a valley between mountains (i) and a throated nozzle (ii); (iii) a gap between two semiinfinite bodies. In the dipole limit of coincidence of source-and-sink the Rankine oval becomes a cylinder, whose motion in a potential flow entrains an added mass (Section 28.6) equal to the mass of fluid it displaces. The cylinder as a streamline is preserved by adding a vortex, whose circulation produces a lift or down force (Section 28.7), and displaces the stagnation points where the velocity is zero. The cylinder as a streamline is preserved by a nearby source/sink and/or vortex in the presence of two opposite images; the system of three singularities is not at rest (Section 28.8) because of the forces between the images and the monopole. The forces are modified in the case of a dipole (Section 28.9) near a cylinder. The calculation of forces and energies can be made by similar methods for gravity (Chapter 18), electrostatic (Chapter 24), and magnetostatic (Chapter 26) fields, and potential flows (Chapter 28); the latter receive more emphasis in the present section as a continuation of Chapters 12, 14, and 16.