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Summary Fundamental understanding of the flow inside progressing-cavity pumps (PCPs) represents an important step in the optimization of the efficiency of these pumps, which are largely used in artificial-lift processes in the petroleum industry. The computation of the flow inside a PCP is extremely complex because of the transient character of the flow, the moving boundaries, and the difference in length scale of the channel height between the stator and rotor. This complexity makes the use of computational fluid dynamics (CFD) as an engineering tool almost impossible. This work presents an asymptotic model to describe the single-phase flow inside PCPs using lubrication theory. The model was developed for Newtonian fluid, and lubrication theory was used to reduce the 3D Navier-Stokes equations in cylindrical coordinates to a 2D Poisson's equation for the pressure field at each timestep, which is solved numerically by a second-order finite-difference method. The predictions are close to the experimental data and the results obtained by solving the complete 3D, transient Navier-Stokes equations with moving boundaries, available in the literature. Although the accuracy is similar to the complete 3D model, the computing time of the presented model is orders of magnitude smaller. The model was used to study the effect of geometry, fluid properties, and operating parameters in the pump-performance curves and can be used in the design of new pumping processes.

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This article use the lubrication theory to simplify the Navier-Stokes equations for the flow between two cylinders with a small thickness of oil film. An effective parameterization of the geometry, proposed in Mota (2020) [6], is used to analyse traditional hydrodynamic bearings and two other configurations of them: elliptical and worn bearings. The simplifications in the model result in a elliptical partial differential equation, that are solved by the centered finite difference method which solution is the pressure field between the cylinders. The results agreed with the literature and different geometries are analysed. This modeling, as well as all its simulations were developed to integrate Ross-Rotordynamics, a library in Python for rotodynamic analysis, available on the GitHub platform.