The force–moments on a deforming body introduced impulsively into an inviscid incompressible fluid with uniform vorticity

This paper is concerned with the derivation of dynamical equations for freely deforming bodies with more than six degrees of freedom which are immersed in an inviscid incompressible fluid. Following Proudmanʹs pioneering work for a sphere our method is applied to a fluid with uniform vorticity but otherwise arbitrary non-uniform strain-rate at the instant after the body has been impulsively introduced into the fluid. The rotational disturbance field is consequently zero thus enabling the generalised force–moments of arbitrary order to be determined from a Laplace problem through the use of Greenʹs theorem and generalised Kirchhoff potentials. An infinite system of equations is obtained each which contains an inertial term, given by the rate of change of the generalised Kelvin Impulse, a generalised lift, a deformation-induced surface momentum flux and a surface kinetic energy. The assumption of an impulsive start places no constraint on the use of our force–moment formulae in irrotational flow but they can only be applied at the starting instant in rotational flow or, when the strain-rate is weak, for early times in the bodyʹs motion. Nonetheless, the start conditions for the rotational case can be created experimentally and be applied to initially free tumbling bodies when they start to deform. This newly identified equation system provides the foundation for new analytical and numerical approaches to the macroscopic modelling of freely deforming bodies and bubbly two-phase flow. In particular, the equations show that the added masses are not sufficient to characterise the bodyʹs geometry and that independent geometric constants are also required, here referred to as the added Kirchhoff energies. Finally, the zero- and first-order force–moment equations are used to derive the force and torque that apply to bodies with six degrees of freedom and their analytic forms are shown to agree with independent results for arbitrarily shaped deforming bodies in both rotational and irrotational flows.