The hydrodynamic interaction between two bodies with rotational motion through an inviscid and incompressible fluid is investigated theoretically. The dynamical behavior of an elliptic cylinder moving around a fixed circular cylinder is described first based on the dynamical equations of motion in the plane of motion. In a relative coordinate system moving with the stream, the kinetic energy of the fluid is expressed as a function of fifteen generalized added masses due to the planar motion of the two cylinders. By means of the generalized added masses, the planar motion of an elliptic cylinder around a fixed circular cylinder can be computed without considering the flow field. The trajectories of an elliptic cylinder around a fixed circular cylinder in planar motion are obtained and the effects of non-circularity, initial position and initial velocity on the interaction between two cylinders are discussed. Similarly, the planar motion of a prolate spheroid around a sphere is investigated. The numerical results show explicitly that the dynamical behaviors of the moving bodies with rotational motion appear nonlinear. Their moving properties exhibit significant difference from those in the particle dynamics.

A variational formulation and solution of general three-dimensional potential flows gave rise to the construction of a special family of ‘trial functions’. This family is composed by circular-sector vortex rings, here named α-rings, i.e., rings that are positioned on the border of a circular sector with aperture angle α. An explicit formula for the velocity potential describing the α-rings family is here derived. A particular case is the well-known circular vortex-ring. The formula is given in terms of a uniformly valid series involving trigonometric and Hypergeometric functions. Results concerning the complete circular ring are compared to the well-known solution given, in closed form, in terms of Bessel functions, validating the present formula. Convergence is discussed. Graphical examples are shown for various rings of different sector angles. As an elementary application, the steady potential flow around three-dimensional bodies in unbounded fluid is formulated and solved under the variational approach. The variational method is fully validated through the sphere problem and for a family of spheroids. Examples concerning either translatory or rotatory motion around a transversal axis are presented for the spheroid family.

The classical equations used to describe the incompressible flow of an ideal fluid may be obtained on an elegant and compact structure with a single equation using geometric algebra. This article proposes a simple way to explain and introduce this approach to a reader unfamiliar with the subject.