The planar translational motion of a pair of bodies and the hydrodynamic interaction forces acting on them in an inviscid fluid are studied based on Lagrange's equations of motion. In a relative coordinate system moving with the fluid at infinity, the kinetic energy of the fluid is expressed as a function of six added masses due to motions parallel and perpendicular to the line joining the centers of two bodies. The velocity components and the moving trajectories of each body are obtained by integrating the equations of motion in terms of the added masses, which are evaluated in terms of source distributions on the surfaces of two bodies by solving a set of Fredholm integral equations of the second kind. Numerical results for several practical engineering problems involving central and oblique motions between two bodies are presented. it is found that the hydrodynamic interaction force depends on the separation distance between two bodies and on the direction of the flow with respect to the centerline joining the centers of two bodies. The velocity component along the centerline produces a repulsive force, which prevents the collision of two bodies, while the component perpendicular to it produces an attractive force.
Hydrodynamic interactions between two floating bodies, or between a floating body and a fixed body, have a variety of applications in the offshore and polar engineering. Hicks (1880) and Herman (1887) first analyzed the kinetic energy of the fluid, due to the motion of two spheres along the line joining their centers, and obtained analytical solutions of added masses in terms of doublets interior to each body. Their expressions about the strengths and positions of the doublets were alternatively reduced to a set of recurrence formulas, which were suitable for computation, as shown by Lamb (1932).