Understanding the generation of mechanical stress in drying, particle-laden films is important for a wide range of industrial processes. One way to study these stresses is through the cantilever experiment, whereby a thin film is deposited onto the surface of a thin plate that is clamped at one end to a wall. The stresses that are generated in the film during drying are transmitted to the plate and drive bending. Mathematical modelling enables the film stress to be inferred from measurements of the plate deflection. The aim of this paper is to present simplified models of the cantilever experiment that have been derived from the time-dependent equations of continuum mechanics using asymptotic methods. The film is described using nonlinear poroelasticity and the plate using nonlinear elasticity. In contrast to Stoney-like formulae, the simplified models account for films with non-uniform thickness and stress. The film model reduces to a single differential equation that can be solved independently of the plate equations. The plate model reduces to an extended form of the Föppl-von Kármán (FvK) equations that accounts for gradients in the longitudinal traction acting on the plate surface. Consistent boundary conditions for the FvK equations are derived by resolving the Saint-Venant boundary layers at the free edges of the plate. The asymptotically reduced models are in excellent agreement with finite element solutions of the full governing equations. As the Péclet number increases, the time evolution of the plate deflection changes from $t$ to $t^{1/2}$ , in agreement with experiments.

The mixing of immiscible oil and water by a pitched blade turbine in a cylindrical vessel is studied numerically. Three-dimensional simulations combined with a hybrid front-tracking/level-set method are employed to capture the complex flow and interfacial dynamics. A large eddy simulation approach, with a Lilly–Smagorinsky model, is employed to simulate the turbulent two-phase dynamics at large Reynolds numbers $Re=1802{-}18\ 026$. The numerical predictions are validated against previous experimental work involving single-drop breakup in a stirred vessel. For small $Re$, the interface is deformed but does not reach the impeller hub, assuming instead the shape of a Newton's Bucket. As the rotating speed increases, the deforming interface attaches to the impeller hub which leads to the formation of long ligaments that subsequently break up into small droplets. For the largest $Re$ studied, the system dynamics becomes extremely complex wherein the creation of ligaments, their breakup and the coalescence of drops occur simultaneously. The simulation outcomes are presented in terms of spatio-temporal evolution of the interface shape and vortical structures. The results of a drop size analysis in terms of the evolution of the number of drops, and their size distribution, is also presented as a parametric function of $Re$.

We consider the mixing dynamics of an air–liquid system driven by the rotation of a pitched blade turbine (PBT) inside an open, cylindrical tank. To examine the flow and interfacial dynamics, we use a highly parallelised implementation of a hybrid front-tracking/level-set method that employs a domain-decomposition parallelisation strategy. Our numerical technique is designed to capture faithfully complex interfacial deformation, and changes of topology, including interface rupture and dispersed phase coalescence. As shown via transient, a three-dimensional (3-D) LES (large eddy simulation) using a Smagorinsky–Lilly turbulence model, the impeller induces the formation of primary vortices that arise in many idealised rotating flows as well as several secondary vortical structures resembling Kelvin–Helmholtz, vortex breakdown, blade tip vortices and end-wall corner vortices. As the rotation rate increases, a transition to ‘aeration’ is observed when the interface reaches the rotating blades leading to the entrainment of air bubbles into the viscous fluid and the creation of a bubbly, rotating, free surface flow. The mechanisms underlying the aeration transition are probed as are the routes leading to it, which are shown to exhibit a strong dependence on flow history.

Reduced-order models (ROMs) are computationally inexpensive simplifications of high-fidelity complex ones. Such models can be found in computational fluid dynamics where they can be used to predict the characteristics of multiphase flows. In previous work, we presented a ROM analysis framework that coupled compression techniques, such as autoencoders, with Gaussian process regression in the latent space. This pairing has significant advantages over the standard encoding–decoding routine, such as the ability to interpolate or extrapolate in the initial conditions’ space, which can provide predictions even when simulation data are not available. In this work, we focus on this major advantage and show its effectiveness by performing the pipeline on three multiphase flow applications. We also extend the methodology by using deep Gaussian processes as the interpolation algorithm and compare the performance of our two variations, as well as another variation from the literature that uses long short-term memory networks, for the interpolation.

Multicomponent polymer systems are of interest in organic photovoltaic and drug delivery applications, among others where diverse morphologies influence performance. An improved understanding of morphology classification, driven by composition-informed prediction tools, will aid polymer engineering practice. We use a modified Cahn–Hilliard model to simulate polymer precipitation. Such physics-based models require high-performance computations that prevent rapid prototyping and iteration in engineering settings. To reduce the required computational costs, we apply machine learning (ML) techniques for clustering and consequent prediction of the simulated polymer-blend images in conjunction with simulations. Integrating ML and simulations in such a manner reduces the number of simulations needed to map out the morphology of polymer blends as a function of input parameters and also generates a data set which can be used by others to this end. We explore dimensionality reduction, via principal component analysis and autoencoder techniques, and analyze the resulting morphology clusters. Supervised ML using Gaussian process classification was subsequently used to predict morphology clusters according to species molar fraction and interaction parameter inputs. Manual pattern clustering yielded the best results, but ML techniques were able to predict the morphology of polymer blends with ≥90% accuracy.

We study the effect of insoluble surfactants on the wave dynamics of vertically falling liquid films. We use three-dimensional numerical simulations and employ a hybrid interface-tracking/level-set method, taking into account Marangoni stresses induced by gradients of interfacial surfactant concentration. Our numerical predictions for the evolution of the surfactant-free, three-dimensional wave topology are validated against the experimental work of Park & Nosoko (AIChE J., vol. 49, 2003, pp. 2715–2727). The addition of surfactants is found to influence significantly the development of horseshoe-shaped waves. At low Marangoni numbers, we show that the wave fronts exhibit spanwise oscillations before eventually acquiring a quasi-two-dimensional shape. In addition, the presence of Marangoni stresses is found to suppress the peaks of the travelling waves and preceding capillary wave structures. At high Marangoni numbers, a near-complete rigidification of the interface is observed.

In chemical process engineering, surrogate models of complex systems are often necessary for tasks of domain exploration, sensitivity analysis of the design parameters, and optimization. A suite of computational fluid dynamics (CFD) simulations geared toward chemical process equipment modeling has been developed and validated with experimental results from the literature. Various regression-based active learning strategies are explored with these CFD simulators in-the-loop under the constraints of a limited function evaluation budget. Specifically, five different sampling strategies and five regression techniques are compared, considering a set of four test cases of industrial significance and varying complexity. Gaussian process regression was observed to have a consistently good performance for these applications. The present quantitative study outlines the pros and cons of the different available techniques and highlights the best practices for their adoption. The test cases and tools are available with an open-source license to ensure reproducibility and engage the wider research community in contributing to both the CFD models and developing and benchmarking new improved algorithms tailored to this field.

The dynamics of a gas bubble in a square channel with a linearly increasing temperature at the walls in the vertical direction is investigated via three-dimensional numerical simulations. The channel contains a so-called ‘self-rewetting’ fluid whose surface tension exhibits a parabolic dependence on temperature with a well-defined minimum. The main objectives of the present study are to investigate the effect of Marangoni stresses on bubble rise in a self-rewetting fluid using a consistent model fully accounting for the tangential surface tension forces, and to highlight the effects of three-dimensionality on the bubble rise dynamics. In the case of isothermal and non-isothermal systems filled with a ‘linear’ fluid, the bubble moves in the upward direction in an almost vertical path. In contrast, strikingly different behaviours are observed when the channel is filled with a self-rewetting fluid. In this case, as the bubble crosses the location of minimum surface tension, the buoyancy-induced upward motion of the bubble is retarded by a thermocapillary-driven flow acting in the opposite direction, which in some situations, when thermocapillarity outweighs buoyancy, results in the migration of the bubble in the downward direction. In the later stages of this downward motion, as the bubble reaches the position of arrest, its vertical motion decelerates and the bubble encounters regions of horizontal temperature gradients, which ultimately lead to the bubble migration towards one of the channel walls. These phenomena are observed at sufficiently small Bond numbers (high surface tension). For stronger self-rewetting behaviour, the bubble undergoes spiralling motion. The mechanisms underlying these three-dimensional effects are elucidated by considering how the surface tension dependence on temperature affects the thermocapillary stresses in the flow. The effects of other dimensionless numbers, such as Reynolds and Froude numbers, are also investigated.

The mechanisms driving the surfactant-enhanced spreading of droplets on the surface of solid substrates, and particularly those underlying the superspreading behaviour sometimes observed, are investigated theoretically. Lubrication theory for the droplet motion, together with advection–diffusion equations and chemical kinetic fluxes for the surfactant transport, leads to coupled evolution equations for the drop thickness, interfacial concentrations of surfactant monomers and bulk concentrations of monomers and micellar, or other, aggregates. The surfactant can be adsorbed on the substrate either directly from the bulk monomer concentrations or from the liquid–air interface through the contact line. An important feature of the spreading model developed here is the surfactant behaviour at the contact line; this is modelled using a constitutive relation, which is dependent on the local surfactant concentration. The evolution equations are solved numerically, using the finite-element method, and we present a thorough parametric analysis for cases of both insoluble and soluble surfactants with concentrations that can, in the latter case, exceed the critical micelle, or aggregate, concentration. The results show that basal adsorption of the surfactant plays a crucial role in the spreading process; the continuous removal of the surfactant that lies upon the liquid–air interface, due to the adsorption at the solid surface, is capable of inducing high Marangoni stresses, close to the droplet edge, driving very fast spreading. The droplet radius grows at a rate proportional to ta with a = 1 or even higher, which is close to the reported experimental values for superspreading. The spreading rates follow a non-monotonic variation with the initial surfactant concentration also in accordance with experimental observations. An accompanying feature is the formation of a rim at the leading edge of the droplet. In some cases, the drop spreads to form a ‘pancake’ or creates a ‘secondary’ front separated from the main droplet.

We describe measurements of the surface slope and reconstruction of the interface shape during the spreading of an oleic acid film on the surface of a thin aqueous glycerol mixture. This experimental system closely mimics the behaviour of an insoluble surfactant film driven to spread on a thin viscous layer under the action of a tangential (Marangoni) surface stress. Refracted image Moiré topography is used to monitor the evolution of the surface slope over macroscopic distances, from which the time variant interface shape and advancing speed of the surfactant film are inferred. The interfacial profile exhibits a strong surface depression ahead of the surfactant source capped by an elevated rim at the surfactant leading edge. The surface slope and shape as well as the propagation characteristics of the advancing rim can be compared directly with theoretical predictions. The agreement is quite strong when the model allows for a small level of pre-existing surface contamination of the initial liquid layer. Comparison between theoretical and experimental profiles reveals the importance of the initial shear stress in determining the evolution in the film thickness and surfactant distribution. This initial stress appears to thin the underlying liquid support so drastically that the surfactant droplet behaves as a finite and not an infinite source, even though there is always an excess of surfactant present at the origin.

The effect of gravity modulation on long-wavelength disturbances at the free surface of a surfactant-covered thin liquid layer is analysed. The surfactants are assumed to be insoluble so that variations in their concentration along the free surface produce Marangoni flows in the underlying liquid. Lubrication theory is applied to obtain nonlinear partial differential equations that describe the behaviour of the free surface height and surfactant concentration, and the stability of these equations to small-amplitude disturbances is examined by applying Floquet theory. It is found that long-wavelength disturbances are destabilized by gravity modulation when surfactants are present, whereas such disturbances are stable when surfactants are absent. Results from additional calculations indicate that the instability becomes more difficult to excite as the Marangoni forces, body forces, capillary forces and surfactant diffusivity increase, and becomes easier to excite as the van der Waals forces increase.

Within lubrication theory, we investigate the hydrodynamic stability of a thin surfactant coated liquid film spreading strictly by Marangoni stresses. These stresses are generated along the air-liquid interface because of local variations in surfactant concentration. The evolution equations governing the unperturbed film thickness and surface surfactant concentration admit simple self-similar solutions for rectilinear geometry and global conservation of insoluble surfactant. A linear stability analysis of these self-similar flows within a quasi steady-state approximation (QSSA) yields an eigenvalue problem for a single third-order nonlinear differential equation. The analysis indicates that a thin film driven purely by Marangoni stresses is linearly stable to small perturbations of all wavenumbers. The insights gained from this calculation suggest a flow mechanism that can potentially destabilize the spreading process.