Unsteady Flow About a Solid Cylinder Falling Through a Viscous Fluid Contained in a Vertical Tube

Abstract

This study is concerned with the unsteady flow field developed in a viscous fluid contained in a vertical tube by a solid cylinder of considerable length falling vertically from rest in the tube. Based on simplifying assumptions, the Navier-Stokes equations are linearized and reduced to a time-dependent one dimensional equation in cylindrical polar coordinates. A solution of the equation preceded by the determination of eigen values from boundary conditions peculiar to the problem of interest has yielded time-dependent velocity profiles across an axisymmetric cross-section of laminar flow in the annular space and other physical quantities related to the flow phenomena about the falling cylinder. Example problems were solved with the use of an electronic digital computer and the numerical results are presented in both tabular and graphical forms. In addition, a simple experiment of the flow system was performed and recorded on high speed movie films. The experimental results have verified reasonably well the theoretical results.