Deriving thin-film averaged equations using computer algebra

TL;DR

This work addresses deriving thin-film reduced-order models for two-phase core-annular flows using the weighted residual integral boundary layer (WRIBL) method. It leverages open-source computer algebra (SymPy) to perform the intricate $O(\epsilon^{2})$-level derivation, including second-order terms, interfacial conditions, and cylindrical geometry, yielding explicit expressions for the leading velocity fields, weight functions, and WRIBL coefficients. The authors provide a transparent, step-by-step computational workflow, validate the derived model against known results, and illustrate its utility through Rayleigh-Plateau-Driven dynamics in two core-annular configurations, backed by open-source code and a Jupyter notebook. This approach lowers barriers to constructing and validating reduced-order models for multiscale thin-film flows and enables classroom-friendly, reproducible, and easily extensible analyses with substantial computational savings relative to full DNS.

Abstract

We demonstrate the use of computer algebra for facilitating the derivation of thin film reduced-order models. We focus on the weighted residual integral boundary layer (WRIBL) method, which has proven to be a very effective technique for developing reduced-order models by averaging the Navier-Stokes equations over the thin-gap direction. In particular, we use SymPy (the symbolic computing library in Python) to derive the core-annular WRIBL model of Dietze and Ruyer-Quil (J. Fluid Mech. 762, 60, 2015); the derivation is especially involved due to the inclusion of second-order terms, the presence of two hydrodynamically active phases, the enforcement of interfacial boundary conditions, and the cylindrical geometry. We show, using excerpts of code, how each step of the derivation can be broken down into substeps that are amenable to symbolic computation. To illustrate the application of the derived model, we solve it numerically using scientific computing libraries in Python, and briefly explore the dynamics of the Rayleigh-Plateau instability. The use of open-source computer algebra, in the manner described here, greatly eases the derivation of averaged models, thereby facilitating their use for the study of multiscale flows, as well as for computationally-efficient prediction and optimization.

Figures (3)

• Figure 1: Schematic of an axisymmetric core-annular flow in a cylindrical coordinate system.
• Figure 2: Rayleigh-Plateau instability of a annular viscous liquid film linning a tube; the fluid properties are chosen to mimic mucus and air in lung airways (case 1 of Table \ref{['prop']}). (a) Evolution of the minimum interface position $d_{min}$. The blue triangles correspond to the results of Dietze2015. (b) Space-time kymograph showing contours of the film thickness, $1-d(z,t)$. Panels (c) and (d) present the streamlines in the two phases, in a stationary reference frame, for two time instances [see the labels in panel (a)]. A Jupyter notebook that simulates the WRIBL model and generates this figure is available at https://cocalc.com/share/public_paths/93b59c92b2c7b5b66f21df18b052b5941dff1cd5 .
• Figure 3: Gravity driven flow of air (core) and water (annulus) down a vertical tube (case 2 in Table \ref{['prop']}). (a) Evolution of the minimum interface position $d_{min}$. (b) Space-time kymograph showing contours of the film thickness, $1-d(z,t)$. Panels (c) and (d) present the streamlines in the two phases in a stationary frame [panel (c)] and in a frame that moves with the long-time asymptotic speed of the water hump [panel (d)]. The corresponding time instant is indicated by the red marker in panel (a).