text: Let ,  be the directions parallel to the flat plates, and  the perpendicular direction, with  being the gap between the plates (at ) and  be the relevant characteristic length scale in the -directions. Under the limits mentioned above, the incompressible Navier–Stokes equations, in the first approximation becomes
  where  is the viscosity. These equations are similar to boundary layer equations, except that there are no non-linear terms. In the first approximation, we then have, after the non-slip boundary conditions at , 
  The equation for  is obtained from the continuity equation. Integrating the continuity equation from across the channel and imposing no-penetration boundary conditions at the walls, we have
  which leads to the Laplace Equation:
  This equation is supplemented by appropriate boundary conditions. For exmaple, no-penetration boundary conditions on the side walls become: , where  is a unit vector perpendicular to the side wall (note that on the side walls, non-slip boundary conditions cannot be imposed). The boundary cal also be regions exposed to constant pressure in which a Dirichlet boundary conditions are appropirtae. Similayly, periodic boundary conditions can also be used. It can also be noted that the vertical velocity component in the first approximation is 
  While the velocity magnitude  varies in the  direction, the velocity-vector direction  is independent of  direction, that is to say, streamline patterns at each level are similar. Eliminating pressure in the above equation, one obtains
  where  is the vorticity in the  direction. The streamline patterns thus correspond to potential flow (irrotational flow). Unlike potential flow, here the circulation  around any closed contour , whether it encloses a solid object or not, is zero,
  where the last integral is set to zero because  is a single-valued function and the integration is done over a closed contour. If
  then the depth-averaged velcity vector , satisfies Darcy's law,