Numerical analysis of stratified laminar flow of two immiscible Newtonian liquids in a circular pipe

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Abstract
Complete numerical solutions were determined in dimensionless form, of the velocity profiles for cocurrent laminar stratified flow of two immiscible Newtonian liquids in a circular pipe, for viscosity ratios of 1, 10, 100 and 1000, at eight positions of the horizontal liquid‐liquid interface, which was incorporated into the relaxation procedure by a twodimensional finite difference method described by D. N. de G. Allen. The velocity profiles were used to compute the theoretical hold‐up ratios, pressure gradient reduction (or volumetric flow rate enhancement) factors and power reduction factors for the various viscosity ratios and interface positions; the maximum reduction factors for each viscosity ratio; and the input volume fractions of the two liquids corresponding to each of the above.The theoretical results for laminar flow were compared with experimental data in the literature, in particular those of Russell, Hodgson and Govier for the horizontal flow of a mineral oil and water (viscosity ratio = 20.1) in a circular pipe. Good agreement was obtained for both hold‐up and pressure drop data when both water and oil were in laminar flow. As the water entered the transitional and turbulent regions, however, a deviation between the experimental and the computed results developed, the deviation increasing in the anticipated direction as the Reynolds number of the water increased.

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