We investigate sharp-interface cohesive fracture models formulated as energy minimization problems. We argue that models with arbitrary cohesive interfaces are incompatible with linear bulk elasticity, in the sense that they cannot feature solutions in the form of a regular crack with a simple tip. To this end, we provide analytical and numerical solutions for a model problem consisting of a single straight crack under mode-III loading, where we show that the stress magnitude exceeds the cohesive yield threshold in a finite region around the crack tip. Our findings are consistent with the unavailability of existence results for such models, related to the lack of lower semicontinuity of the associated variational problem. In the mathematical literature, lower semicontinuity and existence of solutions is recovered by introducing a relaxed functional combining the cohesive surface energy on the crack set with a bulk behavior comparable to perfect plasticity, where the bulk strength is determined by the maximal allowable traction of the cohesive law. The relaxed energy provides a homogenised macroscopic model of the possible microscopic structuring of a dense distribution of cracks with vanishing displacement jumps. We report numerical simulations in antiplane shear that illustrate that the relaxed model admits an equilibrium solution in the form of straight cracks that capture both crack nucleation and propagation. Cracks emerging from pre-existing flaws and notches exhibit a smooth transition from classical crack tip plasticity solutions near the notch to a propagating cohesive crack accompanied by an elongated zone around the tip where the nonlinear bulk behavior is active and the stress is constant. We discuss how these observations can inform the development of mathematically consistent coupled models with a minimal number of constitutive parameters, highlighting the inconsistencies observed when arbitrarily combining models with different surface and bulk strengths.

The mesoscopic-level formation mechanism of carbon nanotube growth, particularly the transition from a graphene patch to a tube, remains unclear. Recent studies have focused on finding a complete description of this phenomenon to understand its origin and to have total control over the formation of carbon nanotubes. In this study, a thermodynamic model is proposed that follows the sharp interface model approach, in which all energy contributions to the total energy of the nanotube are identified as bulk or surface terms. This study proposes an additive model to calculate the total energy of these structures and identify the dominant energy contributions that determine whether the adsorbed carbon atoms form a growing cap on the catalyst nanoparticle or initiate a new carbon nanotube perpendicular to the surface. By comparing the total energy of these two mechanisms, the research aims to elucidate the energetic driving force behind each pathway for optimizing future nanotube growth processes. We also discuss the generality of the model introduced in this article, showing how it can incorporate temperature and disorder effects, thereby enabling the description of both vapor-solid-solid and vapor-liquid-solid mechanisms of carbon nanotube formation at relevant operating conditions. Finally, we outline a strategy for a genuine multiscale framework that combines the microscopic, atomistic picture of carbon nanotube formation with a meso/macroscopic nucleation and growth representation based on a sharp-interface model.

We study the dynamic interaction of two viscous gravity currents in a confined porous layer using laboratory experiments in a vertically placed bead-packed Hele-Shaw cell. By varying the injection rate, along with the density and viscosity of the injecting and ambient fluids, these experiments cover three exact and eight approximate regimes of gravity current interaction, as identified based on the one-dimensional sharp-interface model. By superimposing the theoretically predicted profile shapes and time-dependent frontal locations, a verification is obtained in the different asymptotic regimes of viscous current interaction. Overall, fairly good agreement has been observed between the time-dependent numerical solutions and laboratory measurements on the profile shapes, particularly in the bulk region, where the aspect ratio of the interface shape is fairly large. Such an observation indicates the applicability of the sharp-interface model of viscous current interaction, including the very interesting dynamics of overriding and coflowing. However, the self-similar solutions in some of the exact regimes fail to make reasonable predictions in these experiments, presumably due to the influence of unfinished time transition. We have also observed some degree of disagreement in the frontal regions, which is likely due to the influence of fluid mixing, two-dimensional flow, local heterogeneity and the development of hydrodynamic instabilities for the viscously unstable experiments. The theoretical predictions of the model problem, along with the laboratory experimental observations, offer useful insights into the potential application of, e.g. the technology of co-flooding CO $_2$ and water in oil fields for the co-profits of geological CO $_2$ sequestration and enhanced oil recovery.

Sharp interface modeling and simulations of two-phase ferrofluid flows

• Development of a thermodynamically consistent phase field model • Chemical potential calculations considering the non-ideality of saline solutions using the Pitzer Model • Simulation of interface dynamics and dendritic ice growth under various cooling conditions • Validation against the Stefan problem demonstrates higher interface velocities and increased solute trapping in dendritic regimes • Improved predictive capability for solidification phenomena beyond conventional models This article explores the freeze crystallization process in water-sodium chloride ( H 2 O − N a C l ) solutions, occurring between 273.15 K and the eutectic point at 252.05 K. During this process, two phases are formed: a solid ice phase and a liquid phase of concentrated solution. To simulate this phenomenon, a thermodynamically consistent phase field model with diffuse interface is developed. Key innovations include a unified energy formulation, the use of the Pitzer model to simplify chemical potential calculations, and a novel pseudo-component approach to ensure consistency with the H 2 O − N a C l phase diagram. The model is validated against the sharp interface model as conventional method in Chemical engineering and provides critical insights into the destabilization of interfaces caused by solute expulsion, leading to dendritic growth under high undercooling. These findings highlight the phase field model’s capability to accurately capture complex interface dynamics and offer valuable guidance for optimizing freeze crystallization processes.

On generalizing the induced surface charge method to heterogeneous Poisson-Boltzmann models for electrostatic free energy calculation.

Mass transfer of gaseous components from rising bubbles to the ambient liquid depends not only on the chemical potential difference of the transfer component but also on the interfacial free energy and composition. The latter is strongly affected by surface active agents that are present in many applications. Surfactants lead to local changes in the interfacial tension, which influence the mass transfer rates in two different ways. On the one hand, inhomogeneous interfacial tension leads to Marangoni stress, which can strongly change the local hydrodynamics, thus impacting the local mass transfer. On the other hand, the coverage by surfactant molecules results in a mass transfer resistance. This hindrance effect is not included in current continuum physical models. The present work provides the experimental validation of a recently introduced extended sharp-interface model for two-phase flows with mass transfer that also accounts for the mass transfer hindrance due to adsorbed surfactant. The crucial feature is to account for area-specific concentrations not only of adsorbed constituents but also of transfer species, and to model mass transfer as a series of two bi-directional sorption-type bulk-interface exchange processes. The resulting model is shown to quantitatively describe experimental measurements on mass transfer reduction for the dissolution of CO 2 bubbles in different surfactant solutions. • Mass transfer model including local resistance effect due to surfactant coverage. • Mass transfer is a series of two bi-directional bulk-surface exchange processes. • Interfacial chemical potentials cross-couples surfactants and transfer species. • Mass transfer resistance depends mainly on surface tension reduction. • Experimental validation of mass transfer model for dissolving CO 2 bubbles.

The shape of liquid polystyrene (PS) droplets obtained via the dewetting of nanometer thin PS films from soft viscoelastic polydimethylsiloxane (PDMS) substrates are investigated. For a range of droplet sizes and substrate elasticities we measure the profiles of all the interfaces by combining lift-off techniques with atomic force microscopy and compare them to the predictions of fully time-dependent sharp-interface models for the PS/PDMS system, that are derived through energy minimization methods and allow to follow the dewetting dynamics towards their equilibrium states. Our analysis shows that there is a thin layer of uncrosslinked PDMS molecules that cloaks the PS droplets. By incorporating the effect of cloaking into the surface energies of our theoretical model, the experimental droplet and substrate profiles are shown to be in excellent quantitative agreement for all considered droplet sizes and substrate elasticities. Interestingly, our comparisons also establish small but systematic discrepancies between the experimental results and the theoretical predictions in the vicinity of the three-phase contact line. These discrepancies tend to increase for softer substrates and smaller droplets. Our analysis shows that global variations in system parameters, such as surface tension and elastic shear modulus, cannot account for these differences but instead point to a locally larger elastocapillary length, whose possible origins we investigate in detail.

We investigate a novel Marangoni-induced instability that arises exclusively in diffuse fluid interfaces, that is absent in classical sharp-interface models. Using a validated phase-field Navier–Stokes–Allen–Cahn framework, we linearise the governing equations to analyse the onset and development of interfacial instability driven by solute-induced surface tension gradients. A critical interfacial thickness scaling inversely with the Marangoni number, $\delta _{\textit{cr}} \sim \textit{Ma}^{-1}$ , emerges from the balance between advective and diffusive transport. Unlike sharp-interface scenarios where matched viscosity and diffusivity stabilise the interface, finite thickness induces asymmetric solute distributions and tangential velocity shifts that destabilise the system. We identify universal power-law scalings of velocity and concentration offsets with a modified Marangoni number $\textit{Ma}_\delta$ , independent of capillary number and interfacial mobility. A critical crossover at $ \textit{Ma}_\delta \approx 590$ distinguishes diffusion-dominated stabilisation from advection-driven destabilisation. These findings highlight the importance of diffuse-interface effects in multiphase flows, with implications for miscible fluids, soft matter, and microfluidics where interfacial thickness and coupled transport phenomena are non-negligible.

Co-existence of planar and non-planar traveling waves in a sharp interface model

As popular approximations to sharp-interface models, the Cahn–Hilliard type phase-field models are usually used to simulate interface dynamics with volume conservation. However, the convergence rate of the volume enclosed by the interface to its sharp-interface limit is usually at first order of the interface thickness in the classical Cahn–Hilliard model with constant or degenerate mobilities. In this work, we propose a variational framework for developing new Cahn–Hilliard dynamics with enhanced volume conservation by introducing a more general conserved quantity. In particular, based on Onsager’s variational principle (OVP) and a modified conservation law, we develop an anisotropic Cahn–Hilliard (ACH) equation with improved conservation (ACH-IC) for approximating anisotropic surface diffusion. The ACH-IC model employs a new conserved quantity that approximates step functions more effectively, and yields second-order volume conservation while preserving energy dissipation for the classical anisotropic surface energy. The second-order volume conservation as well as the convergence to the sharp-interface surface diffusion dynamics is derived through comprehensive asymptotic analysis. Numerical evidence not only reveals the underlying physics of the proposed model in comparison with the classical one, but also demonstrates its exceptional performance in simulating anisotropic surface diffusion dynamics.

Abstract In this study, we propose a parametric finite element method for a degenerate multi-phase Stefan problem with triple junctions. This model describes the energy-driven motion of a surface cluster whose distributional solution was studied by Garcke and Sturzenhecker. We approximate the weak formulation of this sharp interface model by an unfitted finite element method that uses parametric elements for the representation of the moving interfaces. We establish existence and uniqueness of the discrete solution and prove unconditional stability of the proposed scheme. Moreover, a modification of the original scheme leads to a structure-preserving variant, in that it conserves the discrete analogue of a quantity that is preserved by the classical solution. Some numerical results demonstrate the applicability of our introduced schemes.

Dynamic wetting poses a well-known challenge in classical sharp-interface formulation as the no-slip wall condition leads to a contact line singularity that is typically regularized with a Navier boundary condition, often requiring empirical fitting for the slip length. On the other hand, this paradox does not appear in phase-field models as the contact line moves through diffusive mass transport. In this work, we present a toy model that accounts for mass diffusion at the contact line within a sharp-interface framework. This model is based on a theoretical relation derived from the Cahn-Hilliard equations, which links the total diffusive mass transport to the curvature at the wall. From an estimate of the chemical potential on a curved interface, we obtain an expression for the width of the highly curved region δ and the apparent angle. In the sharp-interface model, we then introduce a fictitious boundary, displaced by a distance δ into the domain, where a Navier boundary condition is applied along a dynamic apparent contact angle that accounts for the local interface bending. The robustness of the model is assessed by comparing the toy model results with standard phase-field ones on two cases: the steady state profiles of a liquid bridge between two plates moving in opposite directions and the transient behaviors of a drop spreading on a solid with a prescribed equilibrium angle. This offers a practical and efficient alternative to solve contact line problems at lower cost in a sharp-interface framework with input parameters from phase-field models.

Abstract In the sharp-interface modeling of three phase contact lines special care has to be taken with respect to boundary and interface conditions, to avoid singularities. In this work, one such singularity, arising from an incompatibility in a standard model for evaporation/condensation at the contact line, is investigated numerically. By means of employing an extended Discontinuous Galerkin method, established in preliminary works, we study the severity of the singularity in the model. In a first step, the experimental order of convergence is determined in absence of evaporation/condensation and thus without the incompatibility under investigation. The extent of the incompatibility is then measured by computing the convergence orders again, repeating the same simulation with evaporation/condensation present. As one possible model alteration to dispose of the incompatibility causing the singularity and to assess the properties of the numerical method we propose the introduction of a slip condition on the fluid–fluid interface. This part of the study is complemented by an examination of the experimental order of convergence for the test case using the model without and with slip on the interface.

We present a theoretical framework for porous media gravity currents propagating over rigid curvilinear surfaces. By reducing the flow dynamics to low-dimensional models applicable on surfaces where curvature effects are negligible, we demonstrate that, for finite-volume releases, the flow behaviour in both two-dimensional and axisymmetric configurations is primarily governed by the ratio of the released viscous fluid volume to the characteristic volume of the curvilinear surface. Our theoretical predictions are validated using computational fluid dynamics simulations based on a sharp-interface model for macroscopic flow in porous media. In the context of carbon dioxide geological sequestration, our findings suggest that wavy cap rock geometries can enhance trapping capacity compared with traditional flat-surface assumptions, highlighting the importance of incorporating realistic topographic features into subsurface flow models.

We are concerned with the sharp interface limit for the Beris–Edwards system in a bounded domain Ω⊂R3 in this paper. The system can be described as the incompressible Navier–Stokes equations coupled with an evolution equation for the Q-tensor. We prove that the solutions to the Beris–Edwards system converge to the corresponding solutions of a sharp interface model under well-prepared initial data, as the thickness of the diffuse interfacial zone tends to zero. Moreover, we give not only the spatial decay estimates of the velocity vector field in the H1 sense but also the error estimates of the phase field. The analysis relies on the relative entropy method and elaborated energy estimates.

Vertical equilibrium (VE) simulation has re-emerged recently as an efficient approach for simulating geological CO2 storage in aquifers. The approach is well-established and traditionally applied for simulating storage in homogeneous formations. In this study, we assess its application to large-scale heterogeneous aquifers. We focus on sharp-interface models due to their utility during the screening phase of storage aquifer assessment. The study demonstrates that neglecting geological heterogeneity in the vertically-averaged relative permeability and pseudo-capillary pressure may lead to significant errors in predicted CO2 plume extents. These errors result from unjustifiably assuming homogeneous-acting formations to highly heterogeneous formations. Their magnitudes depend on the heterogeneity level and the simulation timescale. We leverage existing simulators to run VE models and discuss their shortcomings. Our investigation highlights VE simulation as a promising tool for dedicated future development in the industry, particularly through automation of the proposed screening workflows for probabilistic prediction of CO2 plume migration. We also propose treating VE models as metamodels (models of vertically-discretized models with infinite vertical resolution), particularly in contexts where geological data are limited. The study concludes with reporting a successful implementation of a sharp-interface VE simulation in a real, heterogeneous formation. While our conclusions are drawn specifically for geological CO2 storage, they are applicable to other fluids in similar settings.

Abstract To enhance heat dissipation with boiling flows, simulations of bubble merging need to be examined from a fundamental perspective with a focus on the mechanisms near the interface. The current study develops a model for 3D multiphase boiling flows using the volume-of-fluid (VOF) interface tracking method by customizing ansys-fluent. The software is customized to incorporate sharp interface modeling and localized adaptive mesh refinement (AMR) for improved interface tracking. The simulation focuses on the heat transfer and fluid transport mechanisms during bubble merging in water at atmospheric conditions. The developed approach can capture 3D bubble growth and merging dynamics for both two and three bubbles cases at 5 K wall superheat. Detailed visualization and quantification of the heat transfer mechanisms near the interface are explored for the three bubble merger case. The influence region, quantified by the wall shear stress, is 3.1 times the bubble diameter at departure. Peaks in the local heat transfer coefficient (HTC) due to trapped liquid when bubbles are merging were detected. An average heat transfer coefficient of 13,150 W/m2 K was observed near departure. Total computational time required to achieve bubble departure is quantified; the simulation with adaptive mesh refinement of two bubbles required 86 h, and three bubbles required 103 h on a 64-core machine.

Abstract Optimal-order convergence in the $H^{1}$ norm is proved for an arbitrary Lagrangian–Eulerian (ALE) interface tracking finite element method (FEM) for the sharp interface model of two-phase Navier–Stokes flow without surface tension, using high-order curved evolving mesh. In this method, the interfacial mesh points move with the fluid’s velocity to track the sharp interface between two phases of the fluid, and the interior mesh points move according to a harmonic extension of the interface velocity. The error of the semidiscrete ALE interface tracking FEM is shown to be $O(h^{k})$ in the $L^\infty (0, T; H^{1}(\varOmega ))$ norm for the Taylor–Hood finite elements of degree $k \geqslant 2$. This high-order convergence is achieved by utilizing the piecewise smoothness of the solution on each subdomain occupied by one phase of the fluid, relying on a low global regularity on the entire moving domain. Numerical experiments illustrate and complement the theoretical results.

We present and analyze a variational front-tracking method for a sharp-interface model of multiphase flow. The fluid interfaces between different phases are represented by curve networks in two space dimensions (2d) or surface clusters in three space dimensions (3d) with triple junctions where three interfaces meet, and boundary points/lines where an interface meets a fixed planar boundary. The model is described by the incompressible Navier–Stokes equations in the bulk domains, with classical interface conditions on the fluid interfaces, and appropriate boundary conditions at the triple junctions and boundary points/lines. We propose a weak formulation for the model, which combines a parametric formulation for the evolving interfaces and an Eulerian formulation for the bulk equations. We employ an unfitted discretization of the coupled formulation to obtain a fully discrete finite element method, where the existence and uniqueness of solutions can be shown under weak assumptions. The constructed method admits an unconditional stability result in terms of the discrete energy. Furthermore, we adapt the introduced method so that an exact volume preservation for each phase can be achieved for the discrete solutions. Numerical examples for three-phase flow and four-phase flow are presented to show the robustness and accuracy of the introduced methods.