Inviscid and viscous interaction of a line or a circular vortex filament with a solid sphere

Inviscid and viscous fluid flows induced by a line vortex or a circular vortex ring in the presence of a solid sphere of radius $a$ are studied. Euler equations are used for the inviscid fluid flow model while equations of low-Reynolds number flow (Stokes flow) are employed for the viscous fluid model. The velocity potential formulation yields exact solution for the line vortex-sphere non-viscous interaction problem. Our analytic solution reveals extreme values for the radial and axial velocities indicating a strong interaction. An expression for the force on the sphere is calculated in an integral form containing a logarithmic term. The force is higher when the line vortex is closer to the sphere and is weaker when it is placed farther. Analytic solution for the corresponding viscous problem is found using a general solution representation. It is observed that the interaction in the viscous case is weaker. Closed form solutions for the axisymmetric vortex ring-sphere problem are also determined using stream function methods. The radius and location parameters have significant impact on the interaction in all cases. Our results form a basis for the investigation of the motion of vortex lines and rings in the vicinity of a spherical boundary.