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Experiments conducted by NASA measured the pressure drop due to gas-liquid flow through a packed-bed reactor under microgravity conditions. From these experiments, we develop correlations for the gas-liquid $f_{gl}$ interphase drag in a two-fluid model (TFM). We use an Ergun-type closure for liquid-solid drag. Then, under a 1D flow assumption, $f_{gl}$ is the only unknown in the TFM. Using a data-driven approach, we determine $f_{gl}$ and correlate it (via composite fits) with the liquid and gas Reynolds numbers, $Re_{l}$ and $Re_{g}$, respectively, and the Suratman number $Su_{l}$. To validate the proposed $f_{gl}(Re_{l},Re_{g},Su_{l})$ closure, we perform two-dimensional transient simulations at microgravity conditions using ANSYS Fluent and employing an Euler-Euler formulation. We find good agreement between the simulations based on the proposed $f_{gl}$ closure and the experimental data., Comment: 25 pages, 5 figures, preprint mode; v2: minor updates/enhancements

We employ the Lorentz reciprocal theorem to derive a closed-form expression for the pressure drop reduction due to the coupling between shear-thinning fluid flow and a weakly deformable channel wall. The methodology is applied in parallel to fluids for which the generalized Newtonian viscosity depends on either the shear rate or the shear stress magnitude. The only limitation on the viscosity model is that it allows for a closed-form solution for the axial velocity profile in a straight and rigid channel. The power-law and Ellis viscosity models are considered as featured examples and to enable comparisons to previous works. The analytical expression obtained via the reciprocal theorem shows that the pressure drop reduction under the power-law model can also be obtained from the pressure drop reduction under the Ellis model in the regime of large Carreau number (small Ellis number)., Comment: 10 pages, 3 figures

We show how bright, fully coherent, hard x-ray beams can be generated through nonlinear upconversion of femtosecond laser light. By using longer-wavelength mid-infrared driving lasers of moderate peak intensity, full phase matching of the high harmonic generation process can extend, in theory, into the hard x-ray region of the spectrum. We identify the dominant phase matching mechanism for long wavelength driving lasers, and verify our predictions experimentally by demonstrating phase-matched up-conversion into the soft x-ray region of the spectrum around 330 eV using an extended, high-pressure, gas medium that is weakly ionized by the laser. Scaling of the overall conversion efficiency is surprisingly favorable as the wavelength of the driving laser is increased, making useful, fully coherent, multi-keV x-ray sources feasible. Finally, we show that the rapidly decreasing microscopic single-atom yield at longer driving wavelengths is compensated macroscopically by an increasing optimal pressure for phase matching and a rapidly decreasing reabsorption of the generated light at higher photon energies., Comment: 43 pages, 8 figures, 1 table

We present spatial coherence measurements of extreme-ultraviolet light generated using the process of high-harmonic upconversion of a femtosecond laser. Using a phase-matched hollow-fiber geometry, the generated beam is found to exhibit essentially full spatial coherence. The coherence of this laser-like EUV source is demonstrated by recording Gabor holograms of small objects. This work demonstrates the capability to do EUV holography using a tabletop experimental setup. Such an EUV source, with low divergence and high spatial coherence, can be used for experiments such as high-precision metrology, inspection of optical components for EUV lithography (1), and for microscopy and holography (2) with nanometer resolution. Furthermore, the short time duration of the EUV radiation (a few femtoseconds) will enable EUV microscopy and holography to be performed with ultrahigh time resolution., Comment: 16 pages, 4 figures

A linear sixth-order partial differential equation (PDE) of ``parabolic'' type describes the dynamics of thin liquid films beneath surfaces with elastic bending resistance when deflections from the equilibrium film height are small. On a finite domain, the associated sixth-order Sturm--Liouville eigenvalue value problem is self-adjoint for the boundary conditions corresponding to a thin film in a closed trough, and the eigenfunctions form a complete orthonormal set. Using these eigenfunctions, we derive the Green's function for the governing sixth-order PDE on a finite interval and compare it to the known infinite-line solution. Further, we propose a Galerkin spectral method based on the constructed sixth-order eigenfunctions and their derivative expansions. The system of ordinary differential equations for the time-dependent expansion coefficients is solved by standard numerical methods. The numerical approach is applied to versions of the governing PDE with a second-order derivative (in addition to the sixth-order one), which arises from gravity acting on the film. In the absence of gravity, we demonstrate the self-similar intermediate asymptotics of initially localized disturbances on the film surface, at least until the disturbances ``feel'' the finite boundaries, and show that the derived Green's function is the global attractor for such solutions. In the presence of gravity, we use the proposed spectral numerical method to demonstrate that self-similar behavior persists, albeit for shortened intervals of time, even for large values of the gravity-to-bending ratio., Comment: 20 pages, 6 figures

We provide an experimental framework to measure the flow rate--pressure drop relation for Newtonian and shear-thinning fluids in two common deformable configurations: (\textit{i}) a rectangular channel and (\textit{ii}) an axisymmetric tube. Using the Carreau model to describe the shear-dependent viscosity, we identify the key dimensionless rheological number, $Cu$, which characterizes shear thinning, and we show that our experiments lie within the power-law regime of shear rates. To rationalize the experimental data, we derive the flow rate-pressure drop relation taking into account the two-way-coupled fluid-structure interaction between the flow and its compliant confining boundaries. We thus identify the second key dimensionless number, $\alpha$, which characterizes the compliance of the conduit. We then compare the theoretical flow rate-pressure drop relation to our experimental measurements, finding excellent agreement between the two. We further contrast our results for shear-thinning and Newtonian fluids to highlight the influence of $Cu$ on the flow rate-pressure drop relation. Finally, we delineate four distinct physical regimes of flow and deformation by mapping our experimental flow rate-pressure drop data for Newtonian and shear-thinning fluids into a $Cu-\alpha$ plane., Comment: 10 pages, 4 figures; v2 accepted for publication in Physical Review Fluids

We demonstrate the use of the Lorentz reciprocal theorem in obtaining corrections to the flow rate for oscillatory flow in channels. We start from the unsteady Stokes equations and derive the suitable reciprocity relations for both rigid and deformable channels, assuming all quantities can be expressed as time-harmonic phasors. For the case of a rigid channel, the auxiliary problem is a steady flow solution, and the reciprocal theorem allows us to calculate the first-order correction in the Womersley number to the steady flow rate. For the case of a deformable channel, we account for the fluid--structure interaction using domain perturbation under the assumption of a small compliance number. Using the oscillatory flow solution in a rigid channel as the auxiliary problem, we obtain the first-order correction in the compliance number to the rigid channel flow rate. This correction, validated against numerical simulations, provides the leading-order effect of the interplay between the oscillations of the fluid flow and the compliance of the channel for arbitrary values of the Womersley number., Comment: 13 pages, 4 figures

Gas-liquid flows through packed bed reactors (PBRs) are challenging to predict due to the tortuous flow paths that fluid interfaces must traverse. Experiments at the International Space Station showed that bubble and pulse flows are predominately observed under microgravity conditions, while the trickle and spray flows observed under terrestrial conditions are not present in microgravity. To understand the physics behind the former experiments, we simulate bubble flow through a PBR for different packing-particle-diameter-based Weber numbers and under different gravity conditions. We demonstrate different pore-scale mechanisms, such as capillary entrapment, buoyancy entrapment, and inertia-induced bubble displacement. Then, we perform a quantitative analysis by introducing new dynamic scales, dependent upon the evolving gas-liquid interfacial area, to understand the dynamic trade-offs between the inertia, capillary, and buoyancy forces on a bubble passing through a PBR. This analysis leads us to define new dimensionless Weber-like numbers that delineate bubble entrapment from bubble displacement., Comment: 12 pages, 9 figures; v2: minor updates, to appear in the AIChE Journal

The quantum entanglement entropy of the electrons in one-dimensional hydrogen molecule is quantified locally using an appropriate partitioning of the two-dimensional configuration space. Both the global and the local entanglement entropy exhibit a monotonic increase when increasing the inter-nuclear distance, while the local entropy remains peaked at the middle between the nuclei with its width decreasing. Our findings show that at the inter-nuclear distance where stable hydrogen molecule is formed, the quantum entropy shows no peculiarity thus indicating that the entropy and the energy measures display different sensitivity with respect to the interaction between the two identical electrons involved. One possible explanation is that the calculation of the quantum entropy does not account explicitly for the distance between the nuclei, which contrasts to the total energy calculation where the energy minimum depends decisively on that distance. The numerically exact and the time-dependent quantum Monte Carlo calculations show close results.

We develop a method for modeling and simulating a class of two-phase flows consisting of two immiscible incompressible dielectric fluids and their interactions with imposed external electric fields in two and three dimensions. We first present a thermodynamically-consistent and reduction-consistent phase field model for two-phase dielectric fluids. The model honors the conservation laws and thermodynamic principles, and has the property that, if only one fluid component is present in the system, the two-phase formulation will exactly reduce to that of the corresponding single-phase system. In particular, this model accommodates an equilibrium solution that is compatible with the zero-velocity requirement based on physics. This property provides a simpler method for simulating the equilibrium state of two-phase dielectric systems. We further present an efficient numerical algorithm, together with a spectral-element (for two dimensions) or a hybrid Fourier-spectral/spectral-element (for three dimensions) discretization in space, for simulating this class of problems. This algorithm computes different dynamic variables successively in an un-coupled fashion, and involves only coefficient matrices that are time-independent in the resultant linear algebraic systems upon discretization, even when the physical properties (e.g. permittivity, density, viscosity) of the two dielectric fluids are different. This property is crucial and enables us to employ fast Fourier transforms for three-dimensional problems. Ample numerical simulations of two-phase dielectric flows under imposed voltage are presented to demonstrate the performance of the method herein and to compare the simulation results with theoretical models and experimental data., Comment: 38 pages, 19 figures, 2 tables

Thermal greases, often used as thermal interface materials, are complex paste-like mixtures composed of a base polymer in which dense metallic (or ceramic) filler particles are dispersed to improve the heat transfer properties of the material. They have complex rheological properties that impact the performance of the thermal interface material over its lifetime. We perform rheological experiments on thermal greases and observe both stress relaxation and stress buildup regimes. This time-dependent rheological behavior of such complex fluid-like materials is not captured by steady shear-thinning models often used to describe these materials. We find that thixo-elasto-visco-plastic (TEVP) and nonlinear-elasto-visco-plastic (NEVP) constitutive models characterize the observed stress relaxation and buildup regimes respectively. Specifically, we use the models within a data-driven approach based on physics-informed neural networks (PINNs). PINNs are used to solve the inverse problem of determining the rheological model parameters from the dynamic response in experiments. This training data is generated by startup flow experiments at different (constant) shear rates using a shear rheometer. We validate the ``learned'' models by comparing their predicted shear stress evolution to experiments under shear rates not used in the training datasets. We further validate the learned TEVP model by solving a forward problem numerically to determine the shear stress evolution for an input step-strain profile. Meanwhile, the NEVP model is further validated by comparison to a steady Herschel--Bulkley fit of the material's flow curve., Comment: 13 pages, 10 figures; v2 included new validation appendices; to appear in J. Rheol

Discrete dynamical models of walking droplets ("walkers") have allowed swift numerical experiments revealing heretofore unobserved quantum statistics and related behaviors in a classical hydrodynamic system. We present evidence that one such model of walking droplets exhibits the empirically elusive $n$-bounce resonances that are traditionally seen in the scattering of solitary waves governed by covariant nonlinear field theories with polynomial self-interaction. A numerical investigation of the chosen model of interacting walking droplets reveals a fractal structure of resonances in the velocity in--velocity out diagram, much like the usual maps constructed for collisions of solitary waves. We suggest avenues for further theoretical analysis of walker collisions, which may connect this discrete model to the field-theoretic setting, as well as directions towards new experimental realizations $n$-bounce resonances., Comment: 13 pages, 5 figures. v2: minor updates, accepted for publication in a Special Issue of Mechanics Research Communications. This paper is dedicated to the memory of our friend, mentor, and colleague, Denis Blackmore. Sk\r{a}l

We develop a theory of fluid--structure interaction (FSI) between an oscillatory Newtonian fluid flow and a compliant conduit. We consider the canonical geometries of a 2D channel with a deformable top wall and an axisymmetric deformable tube. Focusing on the hydrodynamics, we employ a linear relationship between wall displacement and hydrodynamic pressure, which has been shown to be suitable for a leading-order-in-slenderness theory. The slenderness assumption also allows the use of lubrication theory, and the flow rate is related to the pressure gradient (and the tube/wall deformation) via the classical solutions for oscillatory flow in a channel and in a tube (attributed to Womersley). Then, by two-way coupling the oscillatory flow and the wall deformation via the continuity equation, a one-dimensional nonlinear partial differential equation (PDE) governing the instantaneous pressure distribution along the conduit is obtained, without \textit{a priori} assumptions on the magnitude of the oscillation frequency (\textit{i.e.}, at arbitrary Womersley number). We find that the cycle-averaged pressure (for harmonic pressure-controlled conditions) deviates from the expected steady pressure distribution, suggesting the presence of a streaming flow. An analytical perturbative solution for a weakly deformable conduit is obtained to rationalize how FSI induces such streaming. In the case of a compliant tube, the results obtained from the proposed reduced-order PDE and its perturbative solutions are validated against three-dimensional, two-way-coupled direct numerical simulations. We find good agreement between theory and simulations for a range of dimensionless parameters characterizing the oscillatory flow and the FSI, demonstrating the validity of the proposed theory of oscillatory flows in compliant conduits at arbitrary Womersley number., Comment: 18 pages, 4 figures; v2 makes some revisions to the presentation, introduces a Strouhal number, which is then fixed, and compares the model against new 3D simulations; v3 corrects minor typos, to appear in Phys. Rev. Fluids

A ferrofluid droplet confined in a Hele-Shaw cell can be deformed into a stably spinning ``gear,'' using crossed magnetic fields. Previously, fully nonlinear simulation revealed that the spinning gear emerges as a stable traveling wave along the droplet's interface bifurcates from the trivial (equilibrium) shape. In this work, a center manifold reduction is applied to show the geometrical equivalence between a two-harmonic-mode coupled system of ordinary differential equations arising from a weakly nonlinear analysis of the interface shape and a Hopf bifurcation. The rotating complex amplitude of the fundamental mode saturates to a limit circle as the periodic traveling wave solution is obtained. An amplitude equation is derived from a multiple-time-scale expansion as a reduced model of the dynamics. Then, inspired by the well-known delay behavior of time-dependent Hopf bifurcations, we design a slowly time-varying magnetic field such that the timing and emergence of the interfacial traveling wave can be controlled. The proposed theory allows us to determine the time-dependent saturated state resulting from the dynamic bifurcation and delayed onset of instability. The amplitude equation also reveals hysteresis-like behavior upon time reversal of the magnetic field. The state obtained upon time reversal differs from the state obtained during the initial (forward-time) period, yet it can still be predicted by the proposed reduced-order theory., Comment: 16 pages, 11 figures; v2: minor updates; v3: clarifications & new appendices

Cerebral aneurysm progression is a result of a complex interplay of the biomechanical and clinical risk factors that drive aneurysmal growth and rupture. Subjects with multiple aneurysms are unique cases wherein clinical risk factors are expected to affect each aneurysm equally, thus allowing for disentangling the effect of biomechanical factors on aneurysmal growth. Towards this end, we performed a comparative computational fluid--structure interaction analysis of aneurysmal biomechanics in image-based models of stable and growing aneurysms in the same subjects, using the cardiovascular simulation platform SimVascular. We observed that areas exposed to low shear and the median peak systolic arterial wall displacement were higher by factors of 2 or more and 1.5, respectively, in growing aneurysms as compared to stable aneurysms. Furthermore, we defined a novel metric, the oscillatory stress index (OStI), that indicates locations of oscillating arterial wall stresses. We observed that growing aneurysms were characterized by regions of combined low wall shear and high OStI, which we hypothesize to be associated with regions of collagen degradation and remodeling. Such regions were either absent or below 5% of the surface area in stable aneurysms. Our results lay the groundwork for future studies in larger cohorts of subjects, to evaluate the statistical significance of these biomechanical parameters in cerebral aneurysm growth., Comment: 21 pages (8 figures) + 6 pages of supplementary appendices (4 figures); v2: fix typos

We analyze the steady non-Newtonian fluid-structure interaction between the flow of an Oldroyd-B fluid and a deformable channel. Specifically, we provide a theoretical framework for calculating the leading-order effect of the fluid's viscoelasticity on the flow rate-pressure drop relation and on the deformation of the channel's elastic wall. We first identify the characteristic scales and dimensionless parameters governing the fluid-structure interaction in slender and shallow channels. Applying the lubrication approximation for the flow and employing a perturbation expansion in powers of the Deborah number $De$, we derive a closed-form expression for the pressure as a function of the non-uniform shape of the channel in the weakly viscoelastic limit up to $\mathrm{O}(De)$. Coupling the hydrodynamic pressure to the elastic deformation, we provide the leading-order effect of the interplay between the viscoelasticity of the fluid and the compliance of the channel on the pressure and deformation fields, as well as on the flow rate-pressure drop relation. For the flow-rate-controlled regime and in the weakly viscoelastic limit, we show analytically that both the compliance of the deforming top wall and the viscoelasticity of the fluid decrease the pressure drop. Furthermore, we reveal a trade-off between the influence of compliance of the channel and the fluid's viscoelasticity on the deformation. While the channel's compliance increases the deformation, the fluid's viscoelasticity decreases it., Comment: 5 figures

Using numerically exact solution of the time-dependent Schroedinger equation together with time-dependent quantum Monte Carlo (TDQMC) calculations we compare the effects of spatial nonlocality versus nonlocal causality for the ground state and for real-time evolution of two entangled electrons in parabolic potential in one spatial dimension. It was found that the spatial entanglement quantified by the linear quantum entropy is predicted with good accuracy using the spatial nonlocality, parameterized naturally within the TDQMC approach. At the same time, the nonlocal causality predicted by the exact solution leads to only small oscillations in the quantum trajectories which belong to the idler electron as the driven electron is subjected to a strong high frequency electric field, without interaction between the electrons.

We investigate the steady-state fluid--structure interaction between a Newtonian fluid flow and a deformable microtube in two novel geometric configurations arising in recent microfluidics experiments. The first configuration is a cylindrical fluidic channel surrounded by an annulus of soft material with a rigid outer wall, while the second one is a cylindrical fluidic channel extruded from a soft rectangular slab of material. In each configuration, we derive a mathematical theory for the nonlinear flow rate--pressure drop relation by coupling lubrication theory for the flow with linear elasticity for the inner tube wall's deformation. Using the flow conduit's axial slenderness and its axisymmetry, we obtain an analytical expression for the radial displacement in each configuration from a plane-strain configuration. The predicted displacement field, and the resulting closed-form flow rate--pressure drop relation, are each validated against three-dimensional direct numerical simulations via SimVascular's two-way-coupled fluid--structure interaction solver, svFSI, showing good agreement. We also show that weak flow inertia can be easily incorporated in the derivation, further improving the agreement between theory and simulations for larger imposed flow rates., Comment: 8 pages, 4 figures; v2: minor revisions; to appear in Brian Straughan 75th Birthday Celebratory Issue of Mechanics Research Communications

Viscous flows through configurations manufactured from soft materials apply both pressure and shear stress at the solid-liquid interface, leading to deformation of the cross-section, which affects the flow rate-pressure drop relation. Conventionally, calculating this flow rate-pressure drop relation requires solving the complete elastohydrodynamic problem, which couples the fluid flow and elastic deformation. In this work, we use the reciprocal theorems for Stokes flow and linear elasticity to derive a closed-form expression for the flow rate-pressure drop relation in deformable channels, bypassing the detailed calculation of the solution to the fluid-structure-interaction problem. For small deformations (under a domain perturbation scheme), our theory provides the leading-order effect, of the interplay between the fluid stresses and the compliance of the channel, on the flow rate-pressure drop relation. Our approach uses solely the fluid flow solution and the elastic deformation due to the fluid stress distribution in an undeformed channel, eliminating the need to solve the coupled elastohydrodynamic problem. Unlike previous theoretical studies that neglected the presence of lateral sidewalls and considered shallow geometries of effectively infinite width, our approach allows to determine the influence of confining sidewalls on the flow rate-pressure drop relation. For the flow-rate-controlled situation and the plate-bending theory for the elastic deformation, we show a trade-off between the effect of compliance of the deforming top wall and the drag due to sidewalls on the pressure drop. While increased compliance decreases the pressure drop, the effect of the sidewalls increases it. Our theoretical framework may provide insight into existing experimental data and pave the way for the design of novel optimized soft microfluidic configurations of different cross-sectional shapes., Comment: 9 pages, 2 figures

Experiments have shown that flow in compliant microchannels can become unstable at a much lower Reynolds number than the corresponding flow in a rigid conduit. Therefore, it has been suggested that the wall's elastic compliance can be exploited towards new modalities of microscale mixing. While previous studies mainly focused on the local instability induced by the fluid--structure interactions (FSIs) in the system, we derive a one-dimensional (1D) model to study the FSI's effect on the global instability. The proposed 1D FSI model is tailored to long, shallow rectangular microchannels with a deformable top wall, similar to the experiments. Going beyond the usual lubrication flows analyzed in these geometries, we include finite fluid inertia and couple the reduced flow equations to a novel reduced 1D wall deformation equation. Although a quantitative comparison to previous experiments is difficult, the behaviors of the proposed model show qualitative agreement with the experimental observations, and capture several key effects. Specifically, we find the critical conditions under which the inflated base state of the 1D FSI model is linearly unstable to infinitesimal perturbations. The critical Reynolds numbers predicted are in agreement with experimental observations. The unstable modes are highly oscillatory, with frequencies close to the natural frequency of the wall, suggesting that the observed instabilities are resonance phenomena. Furthermore, during the start-up from an undeformed initial state, self-sustained oscillations can be triggered by FSI. Our modeling framework can be applied to other microfluidic systems with similar geometric scale separation under different operating conditions., Comment: 34 pages, 15 figures + supplementary materials attached; v2: minor revisions

We harness the physics-informed neural network (PINN) approach to extend the utility of phenomenological models for particle migration in shear flow. Specifically, we propose to constrain the neural network training via a model for the physics of shear-induced particle migration in suspensions. Then, we train the PINN against experimental data from the literature, showing that this approach provides both better fidelity to the experiments, and a novel understanding of the relative roles of the hypothesized migration fluxes. We first verify the PINN approach for solving the inverse problem of radial particle migration in a non-Brownian suspension in an annular Couette flow. In this classical case, the PINN yields the same value (as reported in the literature) for the ratio of the two parameters of the empirical model. Next, we apply the PINN approach to analyze experiments on particle migration in both non-Brownian and Brownian suspensions in Poiseuille slot flow, for which a definitive calibration of the phenomenological migration model has been lacking. Using the PINN approach, we identify the unknown/empirical parameters in the physical model through the inverse solver capability of PINNs. Specifically, the values are significantly different from those for the Couette cell, highlighting an inconsistency in the literature that uses the latter value for Poiseuille flow. Importantly, the PINN results also show that the inferred values of the empirical model's parameters vary with the shear P\'eclet number and the particle bulk volume fraction of the suspension, instead of being constant as assumed in some previous literature., Comment: 19 pages, 9 figures; v2: minor updates & provided data accessibility statement; v3: accepted for publication in International Journal of Multiphase Flow

Soft hydraulics, which addresses the interaction between an internal flow and a compliant conduit, is a central problem in microfluidics. We analyze Newtonian fluid flow in a rectangular duct with a soft top wall at steady state. The resulting fluid--structure interaction (FSI) is formulated for both vanishing and finite flow inertia. At the leading-order in the small aspect ratio, the lubrication approximation implies that the pressure only varies in the streamwise direction. Meanwhile, the compliant wall's slenderness makes the fluid--solid interface behave like a Winkler foundation, with the displacement fully determined by the local pressure. Coupling flow and deformation and averaging across the cross-section leads to a one-dimensional reduced model. In the case of vanishing flow inertia, an effective deformed channel height is defined rigorously to eliminate the spanwise dependence of the deformation. It is shown that a previously-used averaged height concept is an acceptable approximation. From the one-dimensional model, a friction factor and the corresponding Poiseuille number are derived. Unlike the rigid duct case, the Poiseuille number for a compliant duct is not constant but varies in the streamwise direction. Compliance can increase the Poiseuille number by a factor of up to four. The model for finite flow inertia is obtained by assuming a parabolic vertical variation of the streamwise velocity. To satisfy the displacement constraints along the edges of the channel, weak tension is introduced in the streamwise direction to regularize the Winkler-foundation-like model. Matched asymptotic solutions of the regularized model are derived., Comment: 17 pages, 6 figures; accepted for publication in Physics of Fluids as a contribution to the special issue "Tribute to Frank M. White on his 88th Anniversary"l v2: corrects a minor typos

In 1995, C. I. Christov and M. G. Velarde introduced the concept of a dissipative soliton in a long-wave thin-film equation [Physica D 86, 323--347]. In the 25 years since, the subject has blossomed to include many related phenomena. The focus of this short note is to survey the conceptual influence of the concept of a "production-dissipation (input-output) energy balance" that they identified. Our recent results on nonlinear periodic waves as dissipative solitons (in a model equation for a ferrofluid interface in a parallel-flow rectangular geometry subject to an inhomogeneous magnetic field) have shown that the classical concept also applies to nonlocalized (specifically, spatially periodic) nonlinear coherent structures. Thus, we revisit the so-called KdV-KSV equation studied by C. I. Christov and M. G. Velarde to demonstrate that it also possesses spatially periodic dissipative soliton solutions. These coherent structures arise when the linearly unstable flat film state evolves to sufficiently large amplitude. The linear instability is then arrested when the nonlinearity saturates, leading to permanent traveling waves. Although the two model equations considered in this short note feature the same prototypical linear long-wave instability mechanism, along with similar linear dispersion, their nonlinearities are fundamentally different. These nonlinear terms set the shape and eventual dynamics of the nonlinear periodic waves. Intriguingly, the nonintegrable equations discussed in this note also exhibit multiperiodic nonlinear wave solutions, akin to the polycnoidal waves discussed by J. P. Boyd in the context of the completely integrable KdV equation., Comment: 8 pages, 3 figures, REVTeX, submitted to the proceedings of the Thirteenth Conference of the Euro-American Consortium for Promoting the Application of Mathematics in Technical and Natural Sciences

The time dependent quantum Monte Carlo method for fermions is introduced and applied for calculation of entanglement of electrons in one-dimensional quantum dots with several spin-polarized and spin-compensated electron configurations. The rich statistics of wave functions provided by the method allows one to build reduced density matrices for each electron and to quantify the spatial entanglement using measures such as quantum entropy by treating the electrons as identical or distinguishable particles. Our results indicate that the spatial entanglement in parallel-spin configurations is rather small and it is determined mostly by the quantum nonlocality introduced by the ground state. By contrast, in the spin-compensated case the outermost opposite-spin electrons interact like bosons which prevails their entanglement, while the inner shell electrons remain largely at their Hartree-Fock geometry. Our findings are in a close correspondence with the numerically exact results, wherever such comparison is possible.

Microfluidic devices manufactured from soft polymeric materials have emerged as a paradigm for cheap, disposable and easy-to-prototype fluidic platforms for integrating chemical and biological assays and analyses. The interplay between the flow forces and the inherently compliant conduits of such microfluidic devices requires careful consideration. While mechanical compliance was initially a side-effect of the manufacturing process and materials used, compliance has now become a paradigm, enabling new approaches to microrheological measurements, new modalities of micromixing, and improved sieving of micro- and nano-particles, to name a few applications. This topical review provides an introduction to the physics of these systems. Specifically, the goal of this review is to summarize the recent progress towards a mechanistic understanding of the interaction between non-Newtonian (complex) fluid flows and their deformable confining boundaries. In this context, key experimental results and relevant applications are also explored, hand-in-hand with the fundamental principles for their physics-based modeling. The key topics covered include shear-dependent viscosity of non-Newtonian fluids, hydrodynamic pressure gradients during flow, the elastic response (deformation and bulging) of soft conduits due to flow within, the effect of cross-sectional conduit geometry on the resulting fluid--structure interaction, and key dimensionless groups describing the coupled physics. Open problems and future directions in this nascent field of soft hydraulics, at the intersection of non-Newtonian fluid mechanics, soft matter physics, and microfluidics, are noted., Comment: 32 pages, 14 figures, 267 references; invited topical review to appear in J. Phys.: Condens. Matter; v2: minor updates/correct typos

We study the dynamics of a ferrofluid thin film confined in a Hele-Shaw cell, and subjected to a tilted nonuniform magnetic field. It is shown that the interface between the ferrofluid and an inviscid outer fluid (air) supports traveling waves, governed by a novel modified Kuramoto--Sivashinsky-type equation derived under the long-wave approximation. The balance between energy production and dissipation in this long-wave equations allows for the existence of dissipative solitons. These permanent traveling waves' propagation velocity and profile shape are shown to be tunable via the external magnetic field. A multiple-scale analysis is performed to obtain the correction to the linear prediction of the propagation velocity, and to reveal how the nonlinearity arrests the linear instability. The traveling periodic interfacial waves discovered are identified as fixed points in an energy phase plane. It is shown that transitions between states (wave profiles) occur. These transitions are explained via the spectral stability of the traveling waves. Interestingly, multiperiodic waves, which are a non-integrable analog of the double cnoidal wave, are also found to propagate under the model long-wave equation. These multiperiodic solutions are investigated numerically, and they are found to be long-lived transients, but ultimately abruptly transition to one of the stable periodic states identified., Comment: 27 pages, 10 figures, 2 videos; v2: added new appendix E and section 5.1.2 on multiple-scale analysis of velocity correction; v3: minor revisions, accepted for publication in Proceedings of the Royal Society A

We develop a two-fluid model (TFM) for heat transfer in dense non-Brownian suspensions. Specifically, we propose closure relations for the inter-phase heat transfer coefficient and the thermal diffusivity of the particle phase based on calibration against experimental data. The model is then employed to simulate non-isothermal flow in an annular Couette cell. We find that, when the shear rate is controlled by the rotation of the inner cylinder, both the shear and thermal gradients are responsible for particle migration. Within the TFM framework, we identify the origin and functional form of a "thermo-rheological" migration force that rationalizes our observations. Furthermore, we apply our model to flow in eccentric Couette cells. Our simulations reveal that the system's heat transfer coefficient is affected by both the classic shear-induced migration of particles and the newly identified thermo-rheological migration effect., Comment: 20 pages, 11 figures; v2: minor revisions, accepted for publication in International Journal of Heat and Mass Transfer

In the sense of distributions, the derivative of the Heaviside unit step function $H(t)$ is a generalized Dirac-$\delta$ distribution. If the velocity $V(t)$ of a flat plate is impulsive, as $V(t)=H(t)$ (i.e., it is suddenly set into motion with unit velocity at $t=0^+$), then its acceleration is $V'(t)=\delta(t)$. The Dirac-$\delta$ distribution has no point values. However, when the Dirac-$\delta$ is the forcing term of an ODE (in $t$), it contributes to the solution. The recently published paper [Chaos Solitons Fractals 117 (2018) 68] incorrectly treats the Dirac-$\delta$ function as being identically 0. This Comment analyzes the source of this error, and provides guidance on how to correct it (based on the established literature). The mathematical error identified is in addition to some issues about rheological models with fractional derivatives, which are also noted. That is to say, whether or not an "Atangana--Baleanu fractional derivative" is used in [Chaos Solitons Fractals 117 (2018) 68], the solution to Stokes' first problem provided therein is not correct., Comment: 2 pages, peer-reviewed Letter to the Editor of Chaos, Solitons & Fractals

We study the spreading and leveling of a gravity current in a Hele-Shaw cell with flow-wise width variations as an analog for flow {in fractures and horizontally heterogeneous aquifers}. Using phase-plane analysis, we obtain second-kind self-similar solutions to describe the evolution of the gravity current's shape during both the spreading (pre-closure) and leveling (post-closure) regimes. The self-similar theory is compared to numerical simulations of the partial differential equation governing the evolution of the current's shape (under the lubrication approximation) and to table-top experiments. Specifically, simulations of the governing partial differential equation from lubrication theory allow us to compute a pre-factor, which is \textit{a priori} arbitrary in the second-kind self-similar transformation, by estimating the time required for the current to enter the self-similar regime. With this pre-factor calculated, we show that theory, simulations and experiments agree well near the propagating front. In the leveling regime, the current's memory resets, and another self-similar behavior emerges after an adjustment time, which we estimate from simulations. Once again, with the pre-factor calculated, both simulations and experiments are shown to obey the predicted self-similar scalings. For both the pre- and post-closure regimes, we provide detailed asymptotic (analytical) characterization of the universal current profiles that arise as self-similarity of the second kind., Comment: 33 pages, 7 figures, REVTeX 4-2; v2,v3: minor revision; accepted for publication in Physical Review Fluids

We employ a generalization of Einstein's random walk paradigm for diffusion to derive a class of multidimensional degenerate nonlinear parabolic equations in non-divergence form. Specifically, in these equations, the diffusion coefficient can depend on both the dependent variable and its gradient, and it vanishes when either one of the latter does. It is known that solutions of such degenerate equations can exhibit finite speed of propagation (so-called localization property of solutions). We give a proof of this property using a De Giorgi--Ladyzhenskaya iteration procedure for non-divergence-from equations. A mapping theorem is then established to a divergence-form version of the governing equation for the case of one spatial dimension. Numerical results via a finite-difference scheme are used to illustrate the main mathematical results for this special case. For completeness, we also provide an explicit construction of the one-dimensional self-similar solution with finite speed of propagation function, in the sense of Kompaneets--Zel'dovich--Barenblatt. We thus show how the finite speed of propagation quantitatively depends on the model's parameters., Comment: 17 pages, 3 figures; to appear in Math. Meth. Appl. Sci

The shape of a microchannel during flow through it is instrumental to understanding the physics that govern various phenomena ranging from rheological measurements of fluids to separation of particles and cells. Two commonly used approaches for obtaining a desired channel shape (for a given application) are (i) fabricating the microchannel in the requisite shape and (ii) actuating the microchannel walls during flow to obtain the requisite shape. However, these approaches are not always viable. We propose an alternative, passive approach to {\it a priori} tune the elastohydrodynamics in a microsystem, towards achieving a pre-determined (but not pre-fabricated) flow geometry when the microchannel is subjected to flow. That is to say, we use the interaction between a soft solid layer, the viscous flow beneath it and the shaped rigid wall above it, to tune the fluid domain's shape. Specifically, we study a parallel-wall microchannel whose top wall is a slender soft coating of arbitrary thickness attached to a rigid platform. We derive a nonlinear differential equation for the soft coating's fluid--solid interface, which we use to infer how to achieve specific conduit shapes during flow. Using this theory, we demonstrate the tuning of four categories of microchannel geometries, which establishes, via a proof-of-concept, the viability of our modeling framework. We also explore slip length patterning on the rigid bottom wall of the microchannel, a common technique in microfluidics, as an addition `handle' for microchannel shape control. However, we show that this effect is much weaker in practice., Comment: 20 pages, 10 figures; v2: minor update to parameters and figures; v3: reorganization + regenerated figures for clarity; accepted for publication in Phys. Rev. E; v4: corrects minor typos

Using advanced stochastic methods (time-dependent quantum Monte Carlo, TDQMC) we explore the ground state of 1D and 2D artificial atoms with up to six bosons in harmonic trap where these interact by long-range and short-range Coulomb-like potentials (bosonic quantum dots). It is shown that the optimized value of the key variational parameter in TDQMC named nonlocal correlation length is close to the standard deviation of the Monte Carlo sample for one boson and it is slightly dependent on the range of the interaction potential. Also it is almost independent on the number of bosons for the 2D system thus confirming that the spatial quantum non-locality experienced by each particle is close to the spatial uncertainty exhibited by the rest of the particles. The intimate connection between spatial non-locality and quantum correlations is clearly evidenced.

Two dimensional free surface flows in Hele-Shaw configurations are a fertile ground for exploring nonlinear physics. Since Saffman and Taylor's work on linear instability of fluid--fluid interfaces, significant effort has been expended to determining the physics and forcing that set the linear growth rate. However, linear stability does not always imply nonlinear stability. We demonstrate how the combination of a radial and an azimuthal external magnetic field can manipulate the interfacial shape of a linearly unstable ferrofluid droplet in a Hele-Shaw configuration. We show that weakly nonlinear theory can be used to tune the initial unstable growth. Then, nonlinearity arrests the instability, and leads to a permanent deformed droplet shape. Specifically, we show that the deformed droplet can be set into motion with a predictable rotation speed, demonstrating nonlinear traveling waves on the fluid-fluid interface. The most linearly unstable wavenumber and the combined strength of the applied external magnetic fields determine the traveling wave shape, which can be asymmetric., Comment: 9 pages, 7 figures, REVTeX; v2: combines main text and supplementary material, a few minor changes, accepted for publication in Phys. Rev. E

We analyze the linear stability of the base state of the problem of coupled flow and deformation in a long and shallow rectangular soft hydraulic conduit with a thick top wall. Specifically, the steady base state is computed at low but finite Reynolds number. Then, we show that with the upstream flux fixed and the outlet pressure set to gauge, the flow is linearly stable to infinitesimal flow-wise perturbations. Multiple oscillatory but stable eigenmodes are computed in a range of the reduced Reynolds number, $\hat{Re}$, and the so-called fluid--structure interaction (FSI) parameter, $\lambda$, indicating the stiffness of this FSI system. These results provide a framework to address, in future work, the individual effects of various aspects of two-way FSI coupling on instability and flow transition in soft hydraulic conduits., Comment: 15 pages, 5 figures, REVTeX, submitted to the proceedings of the Twelfth Conference of the Euro-American Consortium for Promoting the Application of Mathematics in Technical and Natural Sciences

Motivated by problems arising in the pneumatic actuation of controllers for micro-electromechanical systems (MEMS), labs-on-a-chip or biomimetic soft robots, and the study of microrheology of both gases and soft solids, we analyze the transient fluid--structure interaction (FSIs) between a viscoelastic tube conveying compressible flow at low Reynolds number. We express the density of the fluid as a linear function of the pressure, and we use the lubrication approximation to further simplify the fluid dynamics problem. On the other hand, the structural mechanics is governed by a modified Donnell shell theory accounting for Kelvin--Voigt-type linearly viscoelastic mechanical response. The fluid and structural mechanics problems are coupled through the tube's radial deformation and the hydrodynamic pressure. For small compressibility numbers and weak coupling, the equations are solved analytically via a perturbation expansion. Three illustrative problems are analyzed. First, we obtain exact (but implicit) solutions for the pressure for steady flow conditions. Second, we solve the transient problem of impulsive pressurization of the tube's inlet. Third, we analyze the transient response to an oscillatory inlet pressure. We show that an oscillatory inlet pressure leads to acoustic streaming in the tube, attributed to the nonlinear pressure gradient induced by the interplay of FSI and compressibility. Furthermore, we demonstrate an enhancement in the volumetric flow rate due to FSI coupling. The hydrodynamic pressure oscillations are shown to exhibit a low-pass frequency response (when averaging over the period of oscillations), while the frequency response of the tube deformation is similar to that of a band-pass filter., Comment: 30 pages, 11 figures; v2 revision removes Knudsen number effects and clarifies a few points; accepted for publication in Physics of Fluids; v3: update to REVTeX 4.2

Periodically-driven flows are known to generate non-zero, time-averaged fluxes of heat or solute species, due to the interactions of out-of-phase velocity and temperature/concentration fields, respectively. Herein, we investigate such transport (a form of the well-known Taylor--Aris dispersion) in the gap between two parallel plates, one of which oscillates vertically, generating a time-periodic squeeze flow of either a newtonian or Maxwellian fluid. Using the method of multiple time-scale homogenization, the mass/heat balance equation describing transport in this flow is reduced to a one-dimensional advection--diffusion--reaction equation. This result indicates three effective mechanisms in the mass/heat transfer in the system: an effective diffusion that spreads mass/heat along the concentration/temperature gradient, an effective advective flux, and an effective reaction that releases or absorbs mass/heat - in the time-averaged frame. Our results demonstrate that there exist resonant modes under which the velocity peaks when the dimensionless plate oscillation frequency (embodied by the Womersley number, the ratio of the transient inertia to viscous forces) approaches specific values. As a result, transport in this flow is significantly influenced by the dimensionless frequency. On the one hand, the effective, time-averaged dispersion coefficient is always larger than the molecular diffusivity, and is sharply enhanced near resonance. The interaction between fluid elasticity and the oscillatory forcing enhances the efficiency of transport in the system. On the other hand, the identified effective advection and reaction mechanisms may transport mass/heat from regions of high concentration/temperature to those of low concentration/temperature, or vice versa, depending on the value of dimensionless frequency., Comment: 29 pages, 8 figures, REVTeX; v2: minor updates, accepted for publication in Physical Review Fluids

We study collisions of coherent structures in higher-order field-theoretic models, such as the $\phi^8$, $\phi^{10}$ and $\phi^{12}$ ones. The main distinguishing feature, of the example models considered herein, is that the collision arises due to the long-range interacting algebraic tails of these solitary waves. We extend the approach to suitably initialize the relevant kinks, in the additional presence of finite initial speed, in order to minimize the dispersive wave radiation potentially created by their slow spatial decay. We find that, when suitably initialized, these models still feature the multi-bounce resonance windows earlier found in models in which the kinks bear exponential tails, such as the $\phi^4$ and $\phi^6$ field theories among others. Also present is the self-similar structure of the associated windows with three- and more-bounce windows at the edges of two- and lower-bounce ones. Moreover, phenomenological, but highly accurate (and predictive) scaling relations are derived for the dependence of the time between consecutive collisions and, e.g., the difference in kinetic energy between the incoming one and the critical one for one-bounces. Such scalings are traced extensively over two-bounce collision windows throughout the three models, hinting at the possibility of an analytical theory in this direction.

We study the long-time asymptotics of prototypical non-linear diffusion equations. Specifically, we consider the case of a non-degenerate diffusivity function that is a (non-negative) polynomial of the dependent variable of the problem. We motivate these types of equations using Einstein's random walk paradigm, leading to a partial differential equation in non-divergence form. On the other hand, using conservation principles leads to a partial differential equation in divergence form. A transformation is derived to handle both cases. Then, a maximum principle (on both an unbounded and a bounded domain) is proved, in order to obtain bounds above and below for the time-evolution of the solutions to the non-linear diffusion problem. Specifically, these bounds are based on the fundamental solution of the linear problem (the so-called Aronson's Green function). Having thus sandwiched the long-time asymptotics of solutions to the non-linear problems between two fundamental solutions of the linear problem, we prove that, unlike the case of degenerate diffusion, a non-degenerate diffusion equation's solution converges onto the linear diffusion solution at long times. Select numerical examples support the mathematical theorems and illustrate the convergence process. Our results have implications on how to interpret asymptotic scalings of potentially anomalous diffusion processes (such as in the flow of particulate materials) that have been discussed in the applied physics literature., Comment: 25 pages, 3 figures, REVTeX 4.1; v2: minor updates/improvements

Characterizing the elastic properties of soft materials through bulge testing relies on accurate measurement of deformation, which is experimentally challenging. To avoid measuring deformation, we propose a hydrodynamic bulge test for characterizing the material properties of thick, pre-stressed elastic sheets via their fluid--structure interaction with a steady viscous fluid flow. Specifically, the hydrodynamic bulge test relies on a pressure drop measurement across a rectangular microchannel with a deformable top wall. We develop a mathematical model using first-order shear-deformation theory of plates with stretching, and the lubrication approximation for Newtonian fluid flow. Specifically, a relationship is derived between the imposed flow rate and the total pressure drop. Then, this relationship is inverted numerically to yield estimates of the Young's modulus (given the Poisson ratio), if the pressure drop is measured (given the steady flow rate). Direct numerical simulations of two-way-coupled fluid--structure interaction are carried out in ANSYS to determine the cross-sectional membrane deformation and the hydrodynamic pressure distribution. Taking the simulations as ``ground truth,'' a hydrodynamic bulge test is performed using the simulation data to ascertain the accuracy and validity of the proposed methodology for estimating material properties. An error propagation analysis is performed via Monte Carlo simulation to characterize the susceptibility of the hydrodynamic bulge test estimates to noise. We find that, while a hydrodynamic bulge test is less accurate in characterizing material properties, it is less susceptible to noise, in the input (measured) variable, than a hydrostatic bulge test., Comment: 14 pages, 8 figures and 4 pages supplementary material with 1 figure; v2: minor corrections, fixing typos; v3: minor fixes

Here, we identify a new mechanism where the early stage of electron ionization in the field of strong femtosecond laser pulse determines the final yield of non-sequential double ionization. By using simple trajectory methods we prove that the powerful short wavelength laser field causes an injection locking between its phase and the phases of the electron trajectories, which is next responsible for the enhanced or suppressed ionization for given intensities in the knee region. It is shown that both ionization and entanglement of the final electron state can be easily controlled by introducing frequency chirp of the laser field. Our methods allow one to quantify the quantum correlations due to the different mechanisms of strong field ionization.

Long, shallow microchannels embedded in thick soft materials are widely used in microfluidic devices for lab-on-a-chip applications. However, the bulging effect caused by fluid--structure interactions between the internal viscous flow and the soft walls has not been completely understood. Previous models either contain a fitting parameter or are specialized to channels with plate-like walls. This work is a theoretical study of the steady-state response of a compliant microchannel with a thick wall. Using lubrication theory for low-Reynolds-number flows and the theory for linearly elastic isotropic solids, we obtain perturbative solutions for the flow and deformation. Specifically, only the channel's top wall deformation is considered, and the ratio between its thickness $t$ and width $w$ is assumed to be $(t/w)^2 \gg 1$. We show that the deformation at each stream-wise cross-section can be considered independently, and that the top wall can be regarded as a simply supported rectangle subject to uniform pressure at its bottom. The stress and displacement fields are found using Fourier series, based on which the channel shape and the hydrodynamic resistance are calculated, yielding a new flow rate--pressure drop relation without fitting parameters. Our results agree favorably with, and thus rationalize, previous experiments., Comment: 23 pages, 10 figures; v2 minor revisions

A number of errors, both mathematical and conceptual, are identified, in a recent article by Farhat \textit{et al.}\ [Phys.\ Rev.\ Appl.\ \textbf{11}, 044089 (2019)] on cloaking of thermal waves in solids, and corrected. The differences between the two thermal flux laws considered in the latter article are also critically discussed, specifically showing that the chosen model does not, in fact, correspond to the Maxwell--Cattaneo hyperbolic (wave) theory of heat transfer., Comment: 4 pages, 2 figures, REVTeX 4-1; comment on arXiv:1812.10063

We discuss the numerical solution of nonlinear parabolic partial differential equations, exhibiting finite speed of propagation, via a strongly implicit finite-difference scheme with formal truncation error $\mathcal{O}\left[(\Delta x)^2 + (\Delta t)^2 \right]$. Our application of interest is the spreading of viscous gravity currents in the study of which these type of differential equations arise. Viscous gravity currents are low Reynolds number (viscous forces dominate inertial forces) flow phenomena in which a dense, viscous fluid displaces a lighter (usually immiscible) fluid. The fluids may be confined by the sidewalls of a channel or propagate in an unconfined two-dimensional (or axisymmetric three-dimensional) geometry. Under the lubrication approximation, the mathematical description of the spreading of these fluids reduces to solving the so-called thin-film equation for the current's shape $h(x,t)$. To solve such nonlinear parabolic equations we propose a finite-difference scheme based on the Crank--Nicolson idea. We implement the scheme for problems involving a single spatial coordinate (i.e., two-dimensional, axisymmetric or spherically-symmetric three-dimensional currents) on an equispaced but staggered grid. We benchmark the scheme against analytical solutions and highlight its strong numerical stability by specifically considering the spreading of non-Newtonian power-law fluids in a variable-width confined channel-like geometry (a "Hele-Shaw cell") subject to a given mass conservation/balance constraint. We show that this constraint can be implemented by re-expressing it as nonlinear flux boundary conditions on the domain's endpoints. Then, we show numerically that the scheme achieves its full second-order accuracy in space and time. We also highlight through numerical simulations how the proposed scheme accurately respects the mass conservation/balance constraint., Comment: 36 pages, 9 figures, Springer book class; v2 includes improvements and corrections; to appear as a contribution in "Applied Wave Mathematics II"

We provide new concepts for understanding transport phenomena in flows of granular materials by using a non-Fickian macroscopic model of axial diffusion of a granular material in a finite cylindrical tumbler. The model accounts for diffusion induced by particle collisions only in a thin surface flowing layer due to localization of shear within the cross-section of the drum. All model parameters are related to measurable quantities in a granular flow. It is shown that the proposed model is a member of the general class of linear constitutive relations with memory. An exact solution for the spreading of a finite-width pulse initial condition under the proposed non-Fickian model is derived and compared to the solution of a Fickian model (i.e., the ``classical'' diffusion equation)., Comment: 8 pages, 3 figures; v3: updates and filled in some details; v4: published in a Mechanics Research Communications Special Issue in Memory of Prof. Denis L. Blackmore

In this Letter, we address the {long-range interaction} between kinks and antikinks, as well as kinks and kinks, in $\varphi^{2n+4}$ field theories for $n>1$. The kink-antikink interaction is generically attractive, while the kink-kink interaction is generically repulsive. We find that the force of interaction decays with the $(\frac{2n}{n-1})$th power of their separation, and we identify the general prefactor for {\it arbitrary} $n$. Importantly, we test the resulting mathematical prediction with detailed numerical simulations of the dynamic field equation, and obtain good agreement between theory and numerics for the cases of $n=2$ ($\varphi^8$ model), $n=3$ ($\varphi^{10}$ model) and $n=4$ ($\varphi^{12}$ model)., Comment: 6 pages, 3 figures, revtex4-1 class; v2 includes minor revisions; v3 corrects typos, accepted for publication in Physical Review Letters

We present a theoretical and numerical study on the (in)stability of the interface between two immiscible liquids, i.e., viscous fingering, in angled Hele-Shaw cells across a range of capillary numbers ($Ca$). We consider two types of angled Hele-Shaw cells: diverging cells with a positive depth gradient and converging cells with a negative depth gradient, and compare those against parallel cells without a depth gradient. A modified linear stability analysis is employed to derive an expression for the growth rate of perturbations on the interface and for the critical capillary number ($Ca_c$) for such tapered Hele-Shaw cells with small gap gradients. Based on this new expression for $Ca_c$, a three-regime theory is formulated to describe the interface (in)stability: (i) in Regime I, the growth rate is always negative, thus the interface is stable; (ii) in Regime II, the growth rate remains zero (parallel cells), changes from negative to positive (converging cells), or from positive to negative (diverging cells), thus the interface (in)stability possibly changes type at some location in the cell; (iii) in Regime III, the growth rate is always positive, thus the interface is unstable. We conduct three-dimensional direct numerical simulations of the full Navier--Stokes equations, using a phase field method to enforce surface tension at the interface, to verify the theory and explore the effect of depth gradient on the interface (in)stability. We demonstrate that the depth gradient has only a slight influence in Regime I, and its effect is most pronounced in Regime III. Finally, we provide a critical discussion of the stability diagram derived from theoretical considerations versus the one obtained from direct numerical simulations., Comment: 25 pages, 7 figures; v2 and v3: minor corrections/revisions; v4: added a figure, minor revisions; v5: merged two figures, minor updates/revisions

Suspension flows are ubiquitous in nature (hemodynamics, subsurface fluid mechanics, etc.) and industrial applications (hydraulic fracturing, CO$_2$ storage, etc.). However, such flows are notoriously difficult to model due to the variety of fluid-particle and particle-particle interactions that can occur. In this work, we focus on non-Brownian shear-dominated suspensions, where kinetic collisions are negligible and frictional effects play a dominant role. Under these circumstances, irreversible phenomena such as particle diffusion and migration arise, requiring anisotropic stress models to describe the suspension rheology. On a continuum level, reduced-order models such as the suspension balance model (SBM) or the diffusive flux model are commonly used to predict particle migration phenomena. We propose a new method based on a two-fluid model (TFM), where both the phases are considered as interpenetrating continua with their own conservation of mass and momentum equations. Without employing the nowadays customary simplifications in applying the SBM, we close the ``full'' TFM instead. Specifically, we show that when an anisotropic stress analogous to that used in the SBM is added to the equilibrium equations for the particle phase, the TFM is able to accurately predict particle migration. Thus, the TFM does not require the assumptions of a steady suspension velocity and a Stokesian (inertialess) fluid, and the TFM can be easily extended to include buoyancy and even kinetic collisional models. We present several benchmark simulations of our TFM implementation in OpenFOAM{\textsuperscript\textregistered}, including in curvilinear coordinates and three-dimensional flow. Good agreement between the TFM solutions and previous experimental and numerical results is found., Comment: 22 pages, 13 figures; v2 corrects some typos; v3 and v4: minor revisions