Compressible flows through micro-channels with sharp edged turns and bifurcations

Bends and bifurcations connecting straight channels are common features in micro-channel networks. We apply our recently extended Hele-Shaw asymptotic scheme to study the viscous compressible gas flow through a shallow bend (whose depth is small relative to all other characteristic dimensions). By the use of analytic-function theory, we obtain closed-form solutions for arbitrary values of the turn angle and the ratio between the upstream- and downstream- channel widths, respectively. The viscous resistance is thereby evaluated in terms of the geometrical parameters characterizing the bend. The linearity of the problem in terms of an appropriately defined quadratic form of the pressure allows us to briefly consider the role of bifurcations in the viscous resistance of tree-like micro-channel networks.