Friction factor at low Knudsen number for the duct with sine-shaped cross-section

A numerical method is applied to analyse hydrodynamically developed internal forced laminar flow in microgeometries, for a Newtonian fluid in ducts of sinusoidal cross-section. The solution to the momentum equation is obtained through a co-ordinate transformation that maps the sinusoidal cross-section into the interior of a square. The hypotheses of constant physical properties and fully isothermal developed flow are supposed to hold. Slip flow conditions are imposed at the walls, with a first-order boundary condition. The effects of the fluid rarefaction appear in the Knudsen number, which introduces the linear dependence between the slip velocity and the velocity gradient. The 2D velocity distribution is obtained as a function of the Knudsen number, and consequently, the friction factor (in fully developed isothermal flow) is deduced.