The hydrodynamic problem of laminar viscous flow in a filtration channel with an elliptic cross section is solved analytically. Similar already published solutions of cross-flow filtration of pure solvent in tubes and slits are generalized. The solution predicts a qualitative change in the flow pattern along the channel depending on the inlet pressure : at low inlet pressures the pressure can drop to zero along the channel; at higher values of the inlet pressure the pressure gradient (and not the pressure itself) can drop to zero, corresponding to a vanishing longitudinal flow at finite values of the transmembrane pressure. An analytical formula relates the "fimbrillating" inlet pressure, separating the two regions, to the eccentricity of the elliptic section. The theoretical predictions are illustrated for typical microfiltration conditions.