Sphere oscillating in a rarefied gas

Abstract

Flow generated by an oscillating sphere in a quiescent fluid is a classical problem in fluid mechanics whose solution is used ubiquitously. Miniaturisation of mechanical devices to small scales and their operation at high frequencies in fluid, which is common in modern nanomechanical systems, can preclude the use of the unsteady Stokes equation for continuum flow. Here, we explore the combined effects of gas rarefaction and unsteady motion of a sphere, within the framework of the unsteady linearised Boltzmann–BGK (Bhatnagar–Gross–Krook) equation. This equation is solved using the method of characteristics, and the resulting solution is valid for any oscillation frequency and arbitrary degrees of gas rarefaction. The resulting force provides the non-continuum counterpart to the (continuum) unsteady Stokes drag on a sphere. In contrast to the Stokes solution, where the flow is isothermal, non-continuum effects lead to a temperature jump at the sphere surface and non-isothermal flow. Unsteady effects and heat transport are found to mix strongly, leading to marked differences relative to the steady case. The solution to this canonical flow problem is expected to be of significant practical value in many applications, including the optical trapping of nanoparticles and the design and application of nanoelectromechanical systems. It also provides a benchmark for computational and approximate methods of solution for the Boltzmann equation.