A Variational Approach to the Hydrodynamics of Immiscible Viscous Flow in Vertical Pipes

Abstract

In this study, a dedicated variational framework is proposed to model the steady-state transport of a non-mixing mass within a perfectly circular pipe. By applying consistent mathematical analysis, analytical solutions are obtained for key physical parameters of such systems, including the velocity distribution across the pipe radius for both vertical and horizontal orientations, the volumetric flow rate (derived in the form of the Hagen-Poiseuille relation), and the shear stress along the pipe wall. These results are particularly relevant to applications such as gas-lift oil well production, water-cut and sand-producing wells, and various two-phase or multiphase flow regimes. In deep hydrocarbon wells, the temperature typically increases by approximately 1 °C for every 10 m of depth; for wells extending several kilometers, this effect makes temperature a significant factor influencing flow behavior. As fluids rise from the wellbore to the surface, cooling occurs in a non-uniform manner, causing even a homogeneous fluid at the outlet to behave like a non-mixing system due to temperature-dependent variations in viscosity and density. Consequently, the flow can be treated as that of a non-mixing viscous medium. These considerations, along with associated operational implications, form the basis of the analysis presented in this article.