Steady slip flow of Newtonian fluids through tangential polygonal microchannels

The concern in this paper is the problem of finding—or, at least, approximating—functions, defined within and on the boundary of a tangential polygon, functions whose Laplacian is $-1$ and which satisfy a homogeneous Robin boundary condition on the boundary. The parameter in the Robin condition is denoted by $\beta $ . The integral of the solution over the interior, denoted by $Q$ , is, in the context of flows in a microchannel, the volume flow rate. A variational estimate of the dependence of $Q$ on $\beta $ and the polygon's geometry is studied. Classes of tangential polygons treated include regular polygons and triangles, especially isosceles: the variational estimate $R(\beta)$ is a rational function which approximates $Q(\beta)$ closely.