Two-phase immiscible flowM. shows an apparatus for measuring the pressure drop of two immiscible liquids as they flow horizontally between two parallel plates that extend indefinitely normal to the...



Two-phase immiscible flow—M. shows an apparatus for measuring the pressure drop of two immiscible liquids as they flow horizontally between two parallel plates that extend indefinitely normal to the plane of the diagram. The liquids, A and B, have viscosities μA and μB, densities ρA and ρB, and volumetric flow rates QA and QB (per unit depth normal to the plane of the figure), respectively. Gravity may be considered unimportant, so that the pressure is essentially only a function of the horizontal distance, x. (a) What type of flow is involved? (b) Considering layer A, start from the differential equations of mass and momentum, and, clearly stating your assumptions, simplify the situation so that you obtain a differential equation that relates the horizontal velocity vxA to the vertical distance y. (c) Integrate this differential equation so that you obtain vxA in terms of y and any or all of d, dp/dx, μA, ρA, and (assuming the pressure gradient is uniform) two arbitrary constants of integration, say, c1A and c2A. Assume that a similar relationship holds for vxB. (d) Clearly state the four boundary and interfacial conditions, and hence derive expressions for the four constants, thus giving the velocity profiles in the two layers. (e) Sketch the velocity profiles and the shear-stress distribution for τyx between the upper and lower plates. (f) Until now, we have assumed that the interface level y = d is known. In reality, however, it will depend on the relative flow rates QA and QB. Show clearly how this dependency could be obtained, but do not actually carry the analysis through to completion.



Dec 31, 2021
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