In this paper, analytical developments of the biharmonic equation representing two-dimensional Stokes flow are performed with elliptical coordinates. It is found that the streamfunction is expressed with series expansions based on Gegenbauer polynomials of first and second kinds with order one and for n natural integer number. Application to an elliptical fiber enclosed in an elliptical boundary with uniform flow is made. It is found that a particular solution responsible of the drag must be added to the general solution. Following this, the flow through rectangular arrays of elliptical fibers is studied, and the permeability of this medium is determined as a function of porosity.