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The flow of a long bubble in an otherwise liquid-filled tube is a hydrodynamics problem with interesting practical applications in enhanced oil recovery, the coating of monolithic structures, and more recently the design and operation of three-phase monolithic reactors. For intermediate to large capillary numbers the ‘‘bubble’’ is axisymmetric and the theoretical approach of Bretherton [J. Fluid Mech. 10, 166 (1961)] can be applied if the velocity profile of the fluid flowing between the bubble and the square tube wall is known. The velocity profile can be computed using an infinite series expansion of harmonic polynomials. A film evolution equation—which includes all the terms that are important throughout the entire bubble profile—is integrated using a parametric representation as a function of arclength and free-surface angle. The flow is analyzed and complete bubble profiles are generated. The results are shown to be in good agreement with experiment.

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