P. Gondret, N. Rakotomalala, M. Rabaud, D. Salin and P. Watzky, "Viscous parallel flow in finite aspect ratio Hele-Shaw cell," Physics of Fluids 9, 1841-1843 (1997).
Abstract : The parallel flow of one or two fluids of contrasted viscosities through a rectangular channel of large aspect ratio is studied. The usual result for an infinite aspect ratio is that the velocity profile is parabolic throughout the gap and flat in the other direction. For a finite aspect ratio a deviation from this usual profile is found in boundary layers along the edges of the channel or close to the interface. The extension of these boundary layers is of the order of the small dimension of the channel. In the two-fluid case we find however that the velocity profile at the interface is strictly a parabola. The velocity profiles obtained by a 3D lattice BGK simulation are successfully compared to the analytical results in the one and two-fluid cases.