To reduce pumping costs in the transmission of viscous crude oil in long (~ 10 miles) pipes, a thin layer of a low viscosity lubricant such as water is used between the pipe wall and crude oil. This is shown schematically below. The lower viscosity of the water reduces pumping costs but also reduces the volumetric flow rate of oil at a given pipe size and pressure drop. Obtain expressions for the velocity field and volumetric flow rate of the oil in the lubricated flow case. Then compare the flow rate when < 1, QK, (lubricated case) to the flow rate in the unlubricated case k = 1, Q, to find the optimum value of K. Let a = "> 1 and assume the pressure is the same in the oil and water phases. R KR r WATER OIL W To reduce pumping costs in the transmission of viscous crude oil in long (~ 10 miles) pipes, a thin layer of a low viscosity lubricant such as water is used between the pipe wall and crude oil. This is shown schematically below. The lower viscosity of the water reduces pumping costs but also reduces the volumetric flow rate of oil at a given pipe size and pressure drop. Obtain expressions for the velocity field and volumetric flow rate of the oil in the lubricated flow case. Then compare the flow rate when < 1, QK, (lubricated case) to the flow rate in the unlubricated case k = 1, Q, to find the optimum value of K. Let a = "> 1 and assume the pressure is the same in the oil and water phases. R KR r WATER OIL W