S 4

Consider an under-the-sea pipeline for intercontinental Natural oil flow network. In varying temperatures, the viscosity of different zones of oil extraction lead to different viscosities and densities. Such two streams of viscous oil layers are flowing in the z direction in a horizontal thin slit of inter-planar distance of 2b and assume that length L and unit width under the influence of a horizontal pressure gradient [(po - PL)/L]. The fluid flow rates are adjusted so that the slit is half filled with fluid I (the denser phase) and half filled with fluid II (the less dense phase). The fluids are flowing slow which dictates the interface remaining exactly planar throughout L. Velocity distribution- Less dense, less viscous fluid X Interface - More dense, ul more viscous b fluid Prove that the velocity profile for the fluid II (the upper half plane) is given by (Po - PL)62 2 ull + 2 u "L ut ull + u"