On the Flow of a Viscous Fluid past a thin Screen at Small Reynolds Numbers

The slow steady flow of a viscous incompressible fluid past a thin screen with holes or slits is investigated on the basis of the Stokes equations of motion. It is found that the flow conductance σ of the holes or slits (i.e. the ratio of the total flow Q to the pressure drop P ) is given by {aligned} σ{=}Q/P{=}M/8ρμ, {aligned} where ρ is the density of the fluid, µ the viscosity, and M the virtual mass of disks or strips which are congruent with the holes or slits, moving broadside-on in a perfect fluid. The drag D acting on a part of the wall δW of the screen is also given by {aligned} D{=}Pqₚ, {aligned} where q p , is the total flow of the perfect fluid through δW in this movement with unit velocity. As examples, the cases of a single elliptic hole, a single slit, two parallel equal slits, and a series of parallel equal and equidistant slits are considered. Especially, in the last case, the total drag for an intervening strip is shown to be 4πµ U /log |cos ( ε π/2)| where U is the velocity of the viscous fluid at infinity, and ε the relative aperture of the screen (0< ε <1).