Motivated by stability problems arising in the context of chemical enhanced oil recovery, we perform linear stability analysis of Hele-Shaw and porous media flows in radial geometry involving an arbitrary number of immiscible fluids. Key stability results obtained and their relevance to the stabilization of fingering instability are discussed. Some of the key results, among many others, are (i) absolute upper bounds on the growth rate in terms of the problem data; (ii) validation of these upper bound results against exact computation for the case of three-layer flows; (iii) stability enhancing injection policies; (iv) asymptotic limits that reduce these radial flow results to similar results for rectilinear flows; and (v) the stabilizing effect of curvature of the interfaces. Multi-layer radial flows have been found to have the following additional distinguishing features in comparison to rectilinear flows: (i) very long waves, some of which can be physically meaningful, are stable; and (ii) eigenvalues can be complex for some waves depending on the problem data, implying that the dispersion curves for one or more waves can contact each other. Similar to the rectilinear case, these results can be useful in providing insight into the interfacial instability transfer mechanism as the problem data are varied. Moreover, these can be useful in devising smart injection policies as well as controlling the complexity of the long-term dynamics when drops of various immiscible fluids intersperse among each other. As an application of the upper bound results, we provide stabilization criteria and design an almost stable multi-layer system by adding many layers of fluid with small positive jumps in viscosity in the direction of the basic flow.

Lagrangian particle dispersion is studied using the one-dimensional turbulence (ODT) model in homogeneous decaying turbulence configurations. The ODT model has been widely and successfully applied to a number of reacting and nonreacting flow configurations, but only limited application has been made to multiphase flows. Here, we present a version of the particle implementation and interaction with the stochastic and instantaneous ODT eddy events. The model is characterized by comparison to experimental data of particle dispersion for a range of intrinsic particle time scales and body forces. Particle dispersion, velocity, and integral time scale results are presented. The particle implementation introduces a single model parameter β p , and sensitivity to this parameter and behavior of the model are discussed. Good agreement is found with experimental data and the ODT model is able to capture the particle inertial and trajectory crossing effects. These results serve as a validation case of the multiphase implementations of ODT for extensions to other flow configurations.

We develop a deep autoencoder architecture that can be used to find a coordinate transformation which turns a non-linear partial differential equation (PDE) into a linear PDE. Our architecture is motivated by the linearising transformations provided by the Cole–Hopf transform for Burgers’ equation and the inverse scattering transform for completely integrable PDEs. By leveraging a residual network architecture, a near-identity transformation can be exploited to encode intrinsic coordinates in which the dynamics are linear. The resulting dynamics are given by a Koopman operator matrix K. The decoder allows us to transform back to the original coordinates as well. Multiple time step prediction can be performed by repeated multiplication by the matrix K in the intrinsic coordinates. We demonstrate our method on a number of examples, including the heat equation and Burgers’ equation, as well as the substantially more challenging Kuramoto–Sivashinsky equation, showing that our method provides a robust architecture for discovering linearising transforms for non-linear PDEs.

High-throughput sequencing measurements of the vaginal microbiome have yielded intriguing potential relationships between the vaginal microbiome and preterm birth (PTB; live birth prior to 37 weeks of gestation). However, results across studies have been inconsistent. Here we perform an integrated analysis of previously published datasets from 12 cohorts of pregnant women whose vaginal microbiomes were measured by 16S rRNA gene sequencing. Of 1926 women included in our analysis, 568 went on to deliver prematurely. Substantial variation between these datasets existed in their definition of preterm birth, characteristics of the study populations, and sequencing methodology. Nevertheless, a small group of taxa comprised a vast majority of the measured microbiome in all cohorts. We trained machine learning (ML) models to predict PTB from the composition of the vaginal microbiome, finding low to modest predictive accuracy (0.28-0.79). Predictive accuracy was typically lower when ML models trained in one dataset predicted PTB in another dataset. Earlier preterm birth (<32 weeks, <34 weeks) was more predictable from the vaginal microbiome than late preterm birth (34 - 37 weeks), both within and across datasets. Integrated differential abundance analysis revealed a highly significant negative association between L. crispatus and PTB that was consistent across almost all studies. The presence of the majority (18 out of 25) of genera was associated with a higher risk of PTB, with L. iners, Prevotella, and Gardnerella showing particularly consistent and significant associations. Some example discrepancies between studies could be attributed to specific methodological differences, but not most study-to-study variations in the relationship between the vaginal microbiome and preterm birth. We believe future studies of the vaginal microbiome and PTB will benefit from a focus on earlier preterm births, and improved reporting of specific patient metadata shown to influence the vaginal microbiome and/or birth outcomes.

Boundary value problems (BVPs) play a central role in the mathematical analysis of constrained physical systems subjected to external forces. Consequently, BVPs frequently emerge in nearly every engineering discipline and span problem domains including fluid mechanics, electromagnetics, quantum mechanics, and elasticity. The fundamental solution, or Green's function, is a leading method for solving linear BVPs that enables facile computation of new solutions to systems under any external forcing. However, fundamental Green's function solutions for nonlinear BVPs are not feasible since linear superposition no longer holds. In this work, we propose a flexible deep learning approach to solve nonlinear BVPs using a dual-autoencoder architecture. The autoencoders discover an invertible coordinate transform that linearizes the nonlinear BVP and identifies both a linear operator L and Green's function G which can be used to solve new nonlinear BVPs. We find that the method succeeds on a variety of nonlinear systems including nonlinear Helmholtz and Sturm-Liouville problems, nonlinear elasticity, and a 2D nonlinear Poisson equation. The method merges the strengths of the universal approximation capabilities of deep learning with the physics knowledge of Green's functions to yield a flexible tool for identifying fundamental solutions to a variety of nonlinear systems.