Exact Solution for the Stokes Problem of an Infinite Cylinder in a Fluid with Harmonic Boundary Conditions at Infinity

TL;DR

This paper derives an exact, analytical solution for the time-dependent Stokes flow around an infinite cylinder with harmonic boundary conditions at infinity, simplifying a 3D problem to a 2D analysis due to symmetry.

Contribution

It provides a novel exact solution for the Stokes problem with harmonic boundary conditions, including velocity fields and boundary condition satisfaction for an infinite cylinder.

Findings

Exact velocity field derived for the flow

Solution satisfies no-slip boundary condition

Analysis of flow behavior for air at sea level

Abstract

We present an exact solution for the time-dependent Stokes problem of an infinite cylinder of radius r=a in a fluid with harmonic boundary conditions at infinity. This is a 3-dimensional problem but, because of translational invariance along the axis of the cylinder it effectively reduces to a 2-dimensional one. The Stokes problem being a linear reduction of the full Navier-Stokes equations, we show how to satisfy the no-slip boundary condition at the cylinder surface and the harmonic boundary condition at infinity, exhibit the full velocity field for radius r>a, and discuss the nature of the solutions for the specific case of air at sea level.