First-order analysis of slip flow at the microscale and nanoscale

Abstract

A convenient approach to derive simple expressions for properties of Stokes flows with low levels of slip is presented. The method is based on a series expansion of a Stokes-flow solution (one satisfying a Navier slip boundary condition) with non-dimensional slip length as the small expansion parameter. Most notably, first-order predictions of surface moments of the traction force (e.g., drag and torque) can be obtained purely from no-slip solutions to the same problem. The analysis is directly applicable to microscale rarefied gas flows in the so-called `slip regime' and relevant to a range of liquid flows at the microscale and nanoscale. A number of application examples are considered, with expressions derived for: the drag and torque on translating and rotating Janus particles and spheroids (prolate and oblate); the efficiency of a micro journal bearing; the speed of a self-propelled particle (a `squirmer'); and the pressure drop required to drive flow through long, straight micro/nano channels. Where appropriate, accurate numerical calculations provide verification of the derived expressions. Certain general results are also obtained. For example, for low-slip Stokes flow: any surface distribution of positive slip length will reduce the drag on any translating particle; any perimetric distribution of positive slip length will reduce the pressure loss through a straight channel flow of arbitrary cross-section; unlike in no-slip flows, the rate of work done by a bounding solid surface on the fluid is not balanced by dissipation in the fluid volume -- there is additional dissipation at the fluid-solid interface.