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

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 formϕ = (ax +by +cz )r -3 +r -3 s 2 +r -4 s 3 + ...r -m -1s m + ... , (1) wheresm is a surface spherical harmonic of degreem 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 = Au 2 + Bv 2 + Cw 2 + 2A'vw + 2b'wu + 2C'uv , (2) whereu, 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 suffixi indicates that the integration is carried out over the surface of the body.