A numerical treatment of the dynamic motion of a zero bending rigidity cylinder in a viscous stream

Previous attempts to derive the homogeneous closed form solution to the problem of the dynamic motion of a zero bending rigidity cylinder in a viscous stream have expressed the solution as an infinite series involving Bessel functions of complex argument and order, which are often impractical to evaluate because of their complexity. Moreover, when these solutions are extended to nonhomogeneous situations, a harmonic time dependence is assumed that requires forcing the system by an arbitrary time function using multiple solutions combined in the Fourier sense. This paper presents a general purpose numerical treatment formulated to overcome these difficulties. The numerical approach is based on finite difference schemes applied in conjunction with powerful numerical ordinary differential equation methods. The theory is examined with respect to consistency, stability, and convergence of these numerical procedures. A numerical example is included to demonstrate the validity of the treatment. Although an explicit boundary condition is absent from this study, a derived boundary condition is demonstrated to be adequate.