Delayed Hopf bifurcation and control of a ferrofluid interface via a time-dependent magnetic field.

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 cycle 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 hysteresislike 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.

Comparative Assessment of Biomechanical Parameters in Subjects With Multiple Cerebral Aneurysms Using Fluid-Structure Interaction Simulations.

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. Toward 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), which 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.

Challenges in Modeling Hemodynamics in Cerebral Aneurysms Related to Arteriovenous Malformations.

PURPOSE: The significantly higher incidence of aneurysms in patients with arteriovenous malformations (AVMs) suggests a strong hemodynamic relationship between these lesions. The presence of an AVM alters hemodynamics in proximal vessels by drastically changing the distal resistance, thus affecting intra-aneurysmal flow. This study discusses the challenges associated with patient-specific modeling of aneurysms in the presence of AVMs. METHODS: We explore how the presence of a generic distal AVM affects upstream aneurysms by examining the relationship between distal resistance and aneurysmal wall shear stress using physiologically realistic estimates for the influence of the AVM on hemodynamics. Using image-based computational models of aneurysms and surrounding vasculature, aneurysmal wall-shear stress is calculated for a range of distal resistances corresponding to the presence of AVMs of various sizes and compared with a control case representing the absence of an AVM. RESULTS: In the patient cases considered, the alteration in aneurysmal wall shear stress due to the presence of an AVM is considerable, as much as 19 times the base case wall shear stress. Furthermore, the relationship between aneurysmal wall shear stress and distal resistance is shown to be highly geometry-dependent and nonlinear. In most cases, the range of physiologically realistic possibilities for AVM-related distal resistance are so large that patient-specific flow measurements are necessary for meaningful predictions of wall shear stress. CONCLUSIONS: The presented work offers insight on the impact of distal AVMs on aneurysmal wall shear stress using physiologically realistic computational models. Patient-specific modeling of hemodynamics in aneurysms and associated AVMs has great potential for understanding lesion pathogenesis, surgical planning, and assessing the effect of treatment of one lesion relative to another. However, we show that modeling approaches cannot usually meaningfully quantify the impact of AVMs if based solely on imaging data from CT and X-ray angiography, currently used in clinical practice. Based on recent studies, it appears that 4D flow MRI is one promising approach to obtaining meaningful patient-specific flow boundary conditions that improve modeling fidelity.

Soft hydraulics: from Newtonian to complex fluid flows through compliant conduits.

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.

Cell spinpods are a simple inexpensive suspension culture device to deliver fluid shear stress to renal proximal tubular cells.

Rotating forms of suspension culture allow cells to aggregate into spheroids, prevent the de-differentiating influence of 2D culture, and, perhaps most importantly of all, provide physiologically relevant, in vivo levels of shear stress. Rotating suspension culture technology has not been widely implemented, in large part because the vessels are prohibitively expensive, labor-intensive to use, and are difficult to scale for industrial applications. Our solution addresses each of these challenges in a new vessel called a cell spinpod. These small 3.5 mL capacity vessels are constructed from injection-molded thermoplastic polymer components. They contain self-sealing axial silicone rubber ports, and fluoropolymer, breathable membranes. Here we report the two-fluid modeling of the flow and stresses in cell spinpods. Cell spinpods were used to demonstrate the effect of fluid shear stress on renal cell gene expression and cellular functions, particularly membrane and xenobiotic transporters, mitochondrial function, and myeloma light chain, cisplatin and doxorubicin, toxicity. During exposure to myeloma immunoglobulin light chains, rotation increased release of clinically validated nephrotoxicity cytokine markers in a toxin-specific pattern. Addition of cisplatin or doxorubicin nephrotoxins reversed the enhanced glucose and albumin uptake induced by fluid shear stress in rotating cell spinpod cultures. Cell spinpods are a simple, inexpensive, easily automated culture device that enhances cellular functions for in vitro studies of nephrotoxicity.

Profiling a soft solid layer to passively control the conduit shape in a compliant microchannel during flow.

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 a priori tune the elastohydrodynamics in a microsystem toward achieving a predetermined (but not prefabricated) flow geometry when the microchannel is subjected to flow. That is, 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 additional "handle" for microchannel shape control. However, we show that this effect is much weaker in practice.

Tuning a magnetic field to generate spinning ferrofluid droplets with controllable speed via nonlinear periodic interfacial waves.

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 wave number and the combined strength of the applied external magnetic fields determine the traveling wave shape, which can be asymmetric.

Theory of the flow-induced deformation of shallow compliant microchannels with thick walls.

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≫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 favourably with, and thus rationalize, previous experiments.

Kink-Kink and Kink-Antikink Interactions with Long-Range Tails.

In this Letter, we address the long-range interaction between kinks and antikinks, as well as kinks and kinks, in φ^{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 2n/(n-1)th power of their separation, and we identify the general prefactor for 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 (φ^{8} model), n=3 (φ^{10} model), and n=4 (φ^{12} model).

Cutting and shuffling with diffusion: Evidence for cut-offs in interval exchange maps.

Low-dimensional dynamical systems are fruitful models for mixing in fluid and granular flows. We study a one-dimensional discontinuous dynamical system (termed "cutting and shuffling" of a line segment), and we present a comprehensive computational study of its finite-time mixing properties including the effect of diffusion. To explore a large parameter space, we introduce fit functions for two mixing metrics of choice: the number of cutting interfaces (a standard quantity in dynamical systems theory of interval exchange transformations) and a mixing norm (a more physical measure of mixing). We compute averages of the mixing metrics across different permutations (shuffling protocols), showing that the latter averages are a robust descriptor of mixing for any permutation choice. If the decay of the normalized mixing norm is plotted against the number of map iterations rescaled by the characteristic e-folding time, then universality emerges: mixing norm decay curves across all cutting and shuffling protocols collapse onto a single stretched-exponential profile. Next, we predict this critical number of iterations using the average length of unmixed subsegments of continuous color during cutting and shuffling and a Batchelor-scale-type diffusion argument. This prediction, called a "stopping time" for finite Markov chains, compares well with the e-folding time of the stretched-exponential fit. Finally, we quantify the effect of diffusion on cutting and shuffling through a Péclet number (a dimensionless inverse diffusivity), showing that the system transitions more sharply from an unmixed initial state to a mixed final state as the Péclet number becomes large. Our numerical investigation of cutting and shuffling of a line segment in the presence of diffusion thus presents evidence for the latter phenomenon, known as a "cut-off" for finite Markov chains, in interval exchange maps.

Static response of deformable microchannels: a comparative modelling study.

We present a comparative modelling study of fluid-structure interactions in microchannels. Through a mathematical analysis based on plate theory and the lubrication approximation for low-Reynolds-number flow, we derive models for the flow rate-pressure drop relation for long shallow microchannels with both thin and thick deformable top walls. These relations are tested against full three-dimensional two-way-coupled fluid-structure interaction simulations. Three types of microchannels, representing different elasticity regimes and having been experimentally characterized previously, are chosen as benchmarks for our theory and simulations. Good agreement is found in most cases for the predicted, simulated and measured flow rate-pressure drop relationships. The numerical simulations performed allow us to also carefully examine the deformation profile of the top wall of the microchannel in any cross section, showing good agreement with the theory. Specifically, the prediction that span-wise displacement in a long shallow microchannel decouples from the flow-wise deformation is confirmed, and the predicted scaling of the maximum displacement with the hydrodynamic pressure and the various material and geometric parameters is validated.

Comment on "Locomotion of a microorganism in weakly viscoelastic liquids".

We point out, and show the implications of resolving, an apparent conceptual difficulty in a recent article by De Corato et al. [Phys. Rev. E 92, 053008 (2015)PLEEE81539-375510.1103/PhysRevE.92.053008] on the locomotion of certain microorganisms in a second-grade fluid. The difficulty arises due to the assumption that α_{1}<0, where α_{1} is the first normal stress modulus of the (non-Newtonian) liquid, was chosen for this study. In particular, this choice of sign for α_{1} is inconsistent with thermodynamics, and as such casts considerable doubt on De Corato et al.'s assumption regarding the existence of a steady-state solution of the equations of motion of the fluid.

Introduction: energy and the subsurface.

This theme issue covers topics at the forefront of scientific research on energy and the subsurface, ranging from carbon dioxide (CO2) sequestration to the recovery of unconventional shale oil and gas resources through hydraulic fracturing. As such, the goal of this theme issue is to have an impact on the scientific community, broadly, by providing a self-contained collection of articles contributing to and reviewing the state-of-the-art of the field. This collection of articles could be used, for example, to set the next generation of research directions, while also being useful as a self-study guide for those interested in entering the field. Review articles are included on the topics of hydraulic fracturing as a multiscale problem, numerical modelling of hydraulic fracture propagation, the role of computational sciences in the upstream oil and gas industry and chemohydrodynamic patterns in porous media. Complementing the reviews is a set of original research papers covering growth models for branched hydraulic crack systems, fluid-driven crack propagation in elastic matrices, elastic and inelastic deformation of fluid-saturated rock, reaction front propagation in fracture matrices, the effects of rock mineralogy and pore structure on stress-dependent permeability of shales, topographic viscous fingering and plume dynamics in porous media convection.This article is part of the themed issue 'Energy and the subsurface'.

Corrections to Morse and Ingard's variational-based treatment of weakly-nonlinear acoustics in lossless gases.

Errors in Morse and Ingard's treatment of the topic of weakly-nonlinear acoustics in Sec. 6.2 of their book [Theoretical Acoustics (1968)] are noted and corrected.

Successive phase transitions and kink solutions in ϕ(8), ϕ(10), and ϕ(12) field theories.

We obtain exact solutions for kinks in ϕ(8), ϕ(10), and ϕ(12) field theories with degenerate minima, which can describe a second-order phase transition followed by a first-order one, a succession of two first-order phase transitions and a second-order phase transition followed by two first-order phase transitions, respectively. Such phase transitions are known to occur in ferroelastic and ferroelectric crystals and in meson physics. In particular, we find that the higher-order field theories have kink solutions with algebraically decaying tails and also asymmetric cases with mixed exponential-algebraic tail decay, unlike the lower-order ϕ(4) and ϕ(6) theories. Additionally, we construct distinct kinks with equal energies in all three field theories considered, and we show the coexistence of up to three distinct kinks (for a ϕ(12) potential with six degenerate minima). We also summarize phonon dispersion relations for these systems, showing that the higher-order field theories have specific cases in which only nonlinear phonons are allowed. For the ϕ(10) field theory, which is a quasiexactly solvable model akin to ϕ(6), we are also able to obtain three analytical solutions for the classical free energy as well as the probability distribution function in the thermodynamic limit.

Two-phase fluid displacement and interfacial instabilities under elastic membranes.

We study the elastic version of the Saffman-Taylor problem: the Hele-Shaw displacement of a viscous liquid by a gas underneath an elastic membrane. We derive the dynamics of the propagating gas-liquid interface and of the deforming membrane. Even though the displacement of a viscous liquid by a gas is susceptible to viscous fingering, the presence of the elastic boundary can lead to the suppression of the instability. We demonstrate how the mechanism of suppression is provided by surface tension at the gas-liquid interface owing to the tapered flow geometry underneath the deflected membrane. We also determine the critical conditions for the onset of the fingering instability in the presence of the elastic boundary.

Parameters and scalings for dry and immersed granular flowing layers in rotating tumblers.

The scaling properties of the continuous flowing layer in a quasi-2D circular tumbler half filled with a granular material are studied experimentally in the presence of three different interstitial fluids (air, water, and glycerine). In the dry case, the dimensionless flowing layer thickness δ(0)/d scales with the dimensionless flow rate Q(dry)(*) = Q/(dsqrt[gd]), where Q is the flow rate, d is the particle diameter, and g is the acceleration due to gravity, in agreement with previous studies. However, unlike previous studies, we show that the exponent for the power-law relation between the two depends on the range of Q(dry)(*) Meanwhile, the angle of repose increases linearly with Q(dry)(*). In the immersed case, the interstitial fluid changes the relevant time scales, which can be accommodated by considering the fluid properties. The result is that there are two different expressions for the dimensionless flow rate in the immersed flow; one corresponding to a free fall regime for a large Stokes number, and one corresponding to a viscous regime at small Stokes number. On this basis, a single dimensionless flow rate that incorporates both buoyancy and viscous friction is proposed. The effect of side walls is also investigated. For dry flows and those immersed in water, the thickness of the flowing layer decreases while the slope of the free surface increases as the gap separating the walls becomes smaller. For immersed granular flows with glycerine as the interstitial fluid, however, the ratio of the thickness of the flowing layer to the bead diameter is independent of the distance the between the side walls because viscous effects dominate.

Resolving a paradox of anomalous scalings in the diffusion of granular materials.

Granular materials do not perform Brownian motion, yet diffusion can be observed in such systems when agitation causes inelastic collisions between particles. It has been suggested that axial diffusion of granular matter in a rotating drum might be "anomalous" in the sense that the mean squared displacement of particles follows a power law in time with exponent less than unity. Further numerical and experimental studies have been unable to definitively confirm or disprove this observation. We show two possible resolutions to this apparent paradox without the need to appeal to anomalous diffusion. First, we consider the evolution of arbitrary (non-point-source) initial data towards the self-similar intermediate asymptotics of diffusion by deriving an analytical expression for the instantaneous collapse exponent of the macroscopic concentration profiles. Second, we account for the concentration-dependent diffusivity in bidisperse mixtures, and we give an asymptotic argument for the self-similar behavior of such a diffusion process, for which an exact self-similar analytical solution does not exist. The theoretical arguments are verified through numerical simulations of the governing partial differential equations, showing that concentration-dependent diffusivity leads to two intermediate asymptotic regimes: one with an anomalous scaling that matches the experimental observations for naturally polydisperse granular materials, and another with a "normal" diffusive scaling (consistent with a "normal" random walk) at even longer times.

Chaotic mixing via streamline jumping in quasi-two-dimensional tumbled granular flows.

We study, numerically and analytically, the singular limit of a vanishing flowing layer in tumbled granular flows in quasi-two-dimensional rotating containers. The limiting behavior is found to be identical under the two versions of the kinematic continuum model of such flows, and the transition to the limiting dynamics is analyzed in detail. In particular, we formulate the no-shear-layer dynamical system as a piecewise isometry. It is shown how such a discontinuous map, through the concordant mechanism of streamline jumping, leads to the physical mixing of granular matter. The dependence of the dynamics of Lagrangian particle trajectories on the tumbler fill fraction is also established through Poincaré sections, and, in the special case of a half-full tumbler, chaotic behavior is shown to disappear completely in the singular limit. At other fill levels, stretching in the sense of shear strain is replaced by spreading due to streamline jumping. Finally, we use finite-time Lyapunov exponents to establish the manifold structure and understand "how chaotic" the limiting piecewise isometry is.