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

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 -3 s 2 + r -4 s 3 + ... r - m -1 s m + ... , (1) where s m 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 = A u 2 + B v 2 + C w 2 + 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 = - ∫∫ i ϕ ∂ ϕ /∂ n ds , where the suffix i indicates that the integration is carried out over the surface of the body.