Variational Approximations for Steady Unidirectional Slip Flows in Microchannels

Abstract

Consider steady flows in a channel, cross section Ω, with the Navier slip boundary condition, and let the volume flowrate be denoted by Q. We present a new simple approximation, a rigorous lower bound on Q, requiring, along with the channel's area and perimeter, the calculation of just the torsional rigidity and two other domain functionals. This avoids the need for solving the partial differential equation repeatedly for differing values of the slip parameter. It also provides the opportunity to give tables for different shapes, requiring, for each shape, just its area and perimeter and the three domain functionals previously mentioned. We expect that for shapes used in practice, the approximation will be good for the entire range of slip parameter. This is illustrated with the case of Ω being rectangular.