The energy of a body moving in an infinite fluid, with an application to airships

Abstract

Abstract When a rigid body moves without rotation in an infinite fluid it sets up a flow in the fluid the velocity potential of which may be represented by a series of spherical harmonics in the form ϕ = (ax + by + cz)r-3 + r-3s2 + r-4s3 + ... r-m-1sm + ... , (1) where sm is a surface spherical harmonic of degree m and the origin is chosen at some point inside the body. The energy of the flow, T, may be expressed in the form 2T/ρ = Au2 + Bv2 + Cw2 + 2A'vw + 2b'wu + 2C'uv, (2) where u, v, w are the components of velocity of the body and the six constants A, B, C, A', B', C' depend only on the shape of the body. It is proposed to show that T depends only on the shape of the body. It is proposed to show that T depends only on the harmonic terms of the first degree in the expansion (1) (namely, on (ax + by + cz) r-3) and on the volume V of the body, the other terms in (1) making no contribution to T. T may be expressed in the form of the surface integral 2T/ρ = - ∫∫iϕ ∂ϕ/∂nds, where the suffix i indicates that the integration is carried out over the surface of the body.