A ‘Boundary Layer’ Method for Obtaining an Approximate Equation for the Velocity–Time History of an Accelerating Sphere

A sphere starting from rest in a stationary fluid accelerates under the action of gravity and eventually reaches a terminal velocity. The descriptive equation for the acceleration of the sphere is a first order, non-linear, ordinary differential equation, which can be solved using a finite difference scheme. In this study, we present a method (obtained from boundary layer methods) of generating a velocity—time equation to obtain an alternative solution. The method is used to solve a specific example and the results are compared to those obtained with the numerical solution. The comparison yields a small error, and so the polynomial approximation method can be used as an alternative technique to describing the motion of an accelerating sphere.

The accelerating sphere problem is usually encountered in a study of flow past immersed objects. Boundary layer methods are usually taught at the graduate level. The advantage of the method presented here is that boundary layer methods can be introduced to undergraduates in a first or second course in fluid mechanics.