Isotropic Case Research Articles

Falling film on an anisotropic porous medium

The stability and dynamics of a falling liquid film over an anisotropic porous medium are studied using a one-domain approach. Our stability analysis shows a significant departure from the effective no-slip boundary condition in the isotropic case. Anisotropy does not affect the threshold of linear instability. However, a non-trivial dual effect of anisotropy on the film stability is observed depending on the permeability of the porous medium. This dual effect results from the net balance of the enhancement of viscous diffusion at the top Brinkman sublayer and the mitigation of viscous damping in the core Darcy sublayer. Three-equation models have been derived from the lubrication theory approximation in terms of the exact mass balance and averaged momentum balances in the porous and liquid layers. In the nonlinear regime, anisotropy has a dual effect by damping capillary waves at large permeabilities and enhancing them at low permeabilities. Anisotropy also affects wave speeds and shapes, modifies travelling-wave branches of solutions, affects the development of a time-periodic wavetrain by inlet forcing and alters the noise-driven dynamics of the flow. These effects result from the mitigation of mass exchange at the liquid–porous interface and the contribution of the cross-stream permeability in the Brinkman top sublayer to the viscous diffusion.

Critical fields of superconductors with magnetic impurities

The upper critical field ${H}_{c2}$, the field ${H}_{c3}$ for nucleation of the surface superconductivity, and the thermodynamic field ${H}_{c}$ are evaluated within the weak-coupling theory for the isotropic s-wave case with arbitrary transport, and pair-breaking scattering. We find that, for the standard geometry of a half-space sample in a magnetic field parallel to the surface, the ratio $\mathcal{R}={H}_{c3}/{H}_{c2}$ is within the window $1.55\ensuremath{\lesssim}\mathcal{R}\ensuremath{\lesssim}2.34$, regardless of temperature or the scattering type. While the nonmagnetic impurities tend to flatten the $\mathcal{R}(T)$ variation, magnetic scattering merely shifts the maximum of $\mathcal{R}(T)$ to lower temperatures. Surprisingly, while reducing the transition temperature, magnetic scattering has a milder impact on $\mathcal{R}$ than nonmagnetic scattering. The surface superconductivity is quite robust; in fact, the ratio $\mathcal{R}\ensuremath{\approx}1.7$ even in the gapless state. We used Eilenberger's energy functional to evaluate the condensation energy ${F}_{c}$ and the thermodynamic critical field ${H}_{c}$ for any temperature and scattering parameters. By comparing ${H}_{c2}$ and ${H}_{c}$, we find that, unlike transport scattering, the pair-breaking pushes materials toward type-I behavior. We find a peculiar behavior of ${F}_{c}$ as a function of the pair-breaking scattering parameter at the low-$T$ transition from gapped to gapless phases, which has recently been associated with the topological transition in the superconducting density of states.

An analytical prediction method for the bifurcation of an asymmetric rotor system partially filled with viscous incompressible fluid

Instability of a rotor partially filled with viscous incompressible fluid will cause the amplitudes of perturbations to increase exponentially. Many models of an isotropic rotor partially filled with fluid have been proposed to investigate its stability. However, the bifurcation of an anisotropic rotor partially filled with viscous incompressible fluid is complicated, which has rarely been studied. To investigate this problem, a continuous model is first established for the isotropic case and the hydrodynamic forces are calculated. The D-decomposition method is then used to determine the stable and unstable regions of the isotropic rotor. An analytical prediction method is then proposed in this paper, and the results for stable and unstable regions are the same as those obtained with the D-decomposition method. Then, this novel analytical prediction model is applied to an anisotropic rotor partially filled with viscous incompressible fluid, and the stable and unstable regions are analyzed. One isotropic and two anisotropic conditions are compared to verify the correctness of the proposed analytical method. The results show that the dimensionless damping and stiffness have significant effects on the stability of an anisotropic rotor partially filled with viscous incompressible fluid; in particular, it is found that there exists a single stable region for low values of the dimensionless damping coefficient and stiffness. Furthermore, the bifurcation law of different anisotropic parameters is first explored, which can provide theoretical guidance for the chosen external stiffness and damping coefficients.

On the Steiner property for planar minimizing clusters. The isotropic case

We consider the isoperimetric problem for clusters in the plane with a double density, that is, perimeter and volume depend on two weights. In this paper, we consider the isotropic case, in the parallel paper [V. Franceschi, A. Pratelli and G. Stefani, On the Steiner property for planar minimizing clusters. The anisotropic case, preprint (2020)] the anisotropic case is studied. Here we prove that, in a wide generality, minimal clusters enjoy the “Steiner property”, which means that the boundaries are made by [Formula: see text] regular arcs, meeting in finitely many triple points with the [Formula: see text] property.

Feasibility of Kitaev quantum spin liquids in ultracold polar molecules

Ultracold atoms and molecules trapped in optical lattices are expected to serve as simulators of strongly correlated systems and topological states of matter. A fascinating example is to realize the Kitaev quantum spin liquid by using ultracold polar molecules. However, although experimental implementation of the Kitaev-type interaction was proposed, the stability of the Kitaev quantum spin liquid has not been fully investigated thus far. Here we study a quantum spin model with long-range angle-dependent Kitaev-type interactions proposed for the polar molecules, by the pseudofermion functional renormalization group method. We reveal that the ground state is magnetically ordered in both ferromagnetic and antiferromagnetic models regardless of the spatial anisotropy of the interactions, while the isotropic case is most frustrated and closest to the realization of the Kitaev quantum spin liquid. Furthermore, by introducing a cutoff in the interaction range, we clarify how the Kitaev quantum spin liquid is destroyed by the long-range interactions. The results urge us to reconsider the feasibility of the Kitaev quantum spin liquid in ultracold polar molecules.

Unifying approach to Heisenberg models for spin distributions on curved surfaces of revolution with isothermal (conformal) metrics

In the present article, a unifying approach to description of different Heisenberg models (i.e., XY model, isotropic case, easy‐plane, and easy‐axis regimes) for spin distributions on curved surfaces is discussed. After introduction of isothermal (conformal) metrics on surfaces of revolution, the axially symmetric solution of the Euler–Lagrange equation for the quasi‐one‐dimensional magnetic Hamiltonian has been obtained for all four discussed cases in the form of an incomplete elliptic integral of the third kind. It has been shown that the limiting case of XY model can be achieved from an easy‐plane regime when the parameter of anisotropy of interaction amongst spins is approaching −1, while the isotropic case can be achieved from both easy‐plane and easy‐axis regimes when is approaching 0. After the application of the corresponding boundary conditions, the soliton‐like solutions have been obtained that either cover the whole surface (in a one‐twist or multi‐twist manner) or can be arranged in the form of a periodic lattice structure on the curved surface. The general unifying scheme has been illustrated using the exemplary surfaces of revolution with zero (cylinder), negative (catenoid), and positive (sphere) Gaussian curvatures.

Electric current and magnetization distributions’ self-organization features in a magnetoresistive film nanoelement under the influence of an external magnetic field

The new generalized Laplace’s equation for an electric potential is presented, which in the general case, is nonlinear for media with resistance anisotropy caused by the anisotropic magnetoresistance (AMR) effect and determined by the orientation of the magnetization vector relative to the current density vector at an arbitrary point of the sample. The solution to this equation is obtained for an anisotropic medium with a non-uniform distribution of current density and magnetization in the case of an oblique-shaped plate of nanoscale thickness. In addition, the analytical solution was obtained for the current density distribution in such a medium using conformal mapping, which confirms the numerical solution in the case of an isotropic conductor. An analysis of the results showed that the presence of the AMR effect leads to a significant change in the current density distribution in the sample compared to the isotropic case, which is expressed in the deviation of the current density vector in perpendicular to the magnetization vector direction in the plate. It has been established that the presence of an external magnetic field in the film plane, which leads to the emergence of an inhomogeneous magnetization distribution, entails self-organization of the electric current distribution. The graphs of the plate resistance as a function of applied magnetic field demonstrate asymmetry for the cases of the direct and reverse directions of the external magnetic field, which cannot be obtained within the framework of simplified models of current flow in media with the AMR effect.

Anisotropy profoundly alters stress fields within contractile cells and cell aggregates.

Many biological phenomena such as cell proliferation and death are correlated with stress fields within cells. Stress fields are quantified using computational methods which rely on fundamental assumptions about local mechanical properties. Most existing methods such as Monolayer Stress Microscopy assume isotropic properties, yet experimental observations strongly suggest anisotropy. We first model anisotropy in circular cells analytically using Eshelby's inclusion method. Our solution reveals that uniform anisotropy cannot exist in cells due to the occurrence of substantial stress concentration in the central region. A more realistic non-uniform anisotropy model is then introduced based on experimental observations and implemented numerically which interestingly clears out stress concentration. Stresses within the entire aggregate also drastically change compared to the isotropic case, resulting in better agreement with observed biomarkers. We provide a physics-based mechanism to explain the low alignment of stress fibers in the center of cells, which might explain certain biological phenomena e.g., existence of disrupted rounded cells, and higher apoptosis rate at the center of circular aggregates.

Numerical study on the effect of anisotropy on deformation behavior of aluminum bilayer sheets in single point incremental metal forming

The anisotropy of metals is an important property that has a significant impact on the dimensional accuracy of components in manufacturing. In metal forming processes significant attention has been given to multilayer composite sheets as they may combine the advantages of materials with various mechanical properties. In this study, the effect of anisotropy is investigated in the process of single point incremental forming (SPIF) of aluminum bilayer sheets. The finite element method (FEM) is used to study the effect of layer arrangement and the anisotropy of metal sheets relative to the rolling direction. For the effective description of sheet metal anisotropy, the two-dimensional Yld2000-2D yield function is implemented in Abaqus software using a material subroutine (VUMAT). The calculation and calibration of the coefficients of Yld2000-2D in accordance with the experimental data were performed using a MATLAB code. For comparison, the anisotropic coefficients of Yld2000-2D were replaced by unit values in the same VUMAT to investigate forming behavior in the isotropic case with the von Mises yield function. Anisotropic and isotropic models are compared in a conical geometry by assessing equivalent plastic strain, sheet thickness, and forming tool reaction forces. Our findings show that anisotropy of sheet metal causes less variation in material properties compared to the isotropic case, significantly affecting the stiffness and, subsequently, the final shape dimensional accuracy of the formed component. The results of the study have a practical application in that they can be used to identify strategies for anisotropic bimetal sheets to improve such material forming processes

A simple application of quantum metrology to isotropic and anisotropic spacetimes

In this work, we study quantum metrology protocol in the context of relativistic quantum fields in curved spacetimes. In particular, we restrict our attention to the isotropic and anisotropic models. We investigate how the oscillatory anisotropy affects a quantum metrology protocol. Here, the oscillatory anisotropy is incorporated as a small gravitational disturbance. We calculate the Fisher information for both cosmological parameters [Formula: see text] (expansion volume) and [Formula: see text] (expansion rate). We find that, unlike the isotropic case, the Fisher information of both the parameters [Formula: see text] and [Formula: see text] are oscillating functions of the momentum [Formula: see text]. The oscillatory spectra of Fisher information can theoretically be understood as results of the dependence of the Bogoliubov coefficients on the anisotropic oscillations. In principle, this framework can be used to infer about the underlying spacetime structure and examine the effects of the universe scale factor oscillations.

Effect of electron temperature anisotropy on TEM in reversed-field-pinch plasmas

For the first time in the reversed-field-pinch configuration, trapped electron mode (TEM) with anisotropies of electron temperature and its gradient is studied by solving the gyrokinetic integral eigenmode equation. Detailed numerical analyses indicate that TEM is enhanced by the anisotropy with temperature in the direction perpendicular to the magnetic field that is higher than that in the direction parallel to the magnetic field when the latter is kept constant. However, the enhancement is limited, such that TEM is weakened and even stabilized when the anisotropy is higher than a critical value, due to strong Landau damping. In comparison with the isotropic case, the lower Landau damping with the higher parallel electron temperature makes TEM instability easier to excite, which expands the TEM unstable region in the diagram of density and temperature scale lengths. In addition, it is found that the electron temperature gradient in the perpendicular direction offers a stronger driving force on TEM instability than that in the parallel direction. The overall effects of the temperature gradients of electrons and ions, magnetic shear, safety factor and density gradient on TEM in the presence of the anisotropies are presented in detail.

Vacuum Einstein field equations in smooth metric measure spaces: the isotropic case* *Supported by projects PID2019-105138GB-C21/AEI/10.13039/501100011033 (Spain) and ED431C 2019/10 (Xunta de Galicia, Spain).

On a smooth metric measure spacetime (M, g, e −f dvol g ), we define a weighted Einstein tensor. It is given in terms of the Bakry–Émery Ricci tensor as a tensor which is symmetric, divergence-free, concomitant of the metric and the density function. We consider the associated vacuum weighted Einstein field equations and show that isotropic solutions have nilpotent Ricci operator. Moreover, the underlying manifold is a Brinkmann wave if it is two-step nilpotent and a Kundt spacetime if it is three-step nilpotent. More specific results are obtained in dimension 3, where all isotropic solutions are given in local coordinates as plane waves or Kundt spacetimes.

Villain transformation for XXZ anisotropic ferrimagnetic spin chain

We investigate the spin excitation spectrum and the thermodynamic properties of the XXZ anisotropic ferrimagnetic spin-(1/2, 1) chain under the presence of a magnetic field using the Villain transformation method. The effect of the XXZ anisotropic parameter and external magnetic field on the thermodynamic observables are calculated at finite temperature. The energy spectra exhibit the coexistence of a gapless branch and a gapful branch, both of them are sensitive to the change of XXZ anisotropic parameter. The gapless branch exhibits a V-shape structure, while the gapful branch is reversed and presents an omega-shape for the XY model which is completely different from previous results for the XXZ model. Thermodynamic quantities such as free energy, internal energy, entropy, specific heat and magnetization are present. These findings for the isotropic case are compared with the results calculated by the other established methods and it turns out that Villain transformation well describes this system at low temperature. • The spin excitation spectrum and thermodynamic properties of the XXZ anisotropic ferrimagetic spin-(1/2, 1) chain under the presence of magnetic field are investigated by the Villain transformation method. • Two branches of the excitation energy spectra are obtained, which are sensitive to the size of the XXZ anisotropic parameter. • The gapless branch exhibits a V-shape structure, while the higher branch is reversed and exhibits an omega-shape structure for the XY model. • The dependence of the thermodynamic quantities on the XXZ anisotropic parameter and magnetic field are studied.

Subgrid-scale modelling using deconvolutional artificial neural networks in large eddy simulations of chemically reacting compressible turbulence

Based on the framework of deconvolutional artificial neural network (DANN) proposed by Yuan et al.[DOI:https://doi.org/10.1063/5.0027146], we extend the DANN approach to model subgrid-scale (SGS) terms in large eddy simulation (LES) of chemically reacting compressible turbulent flow with evident heat release. In constructing the DANN, the normalized density-weighted filtered variables in the neighbouring stencils are taken as the inputs while the outputs are unfiltered density-weighted variables. The SGS stress, SGS heat flux, SGS scalar flux and chemical reaction source terms are modelled using those unfiltered variables which are recovered by the DANN framework to close the governing equations. The DANN framework is evaluated by two chemically reacting compressible isotropic turbulent flow cases adopting a simple one-step irreversible chemical reaction mechanism at turbulent Mach numbers 0.4 and 0.8. In the a priori study, the DANN method shows better performance compared to the classical approximate-deconvolution method (ADM) and the velocity-gradient model (VGM). In the a posteriori test, the DANN method outperforms the dynamic-Smagorinsky model (DSM), and the dynamic-mixed model (DMM). In addition, the DANN framework can predict flow variables with high accuracy by using limited training samples which are constructed from any single instantaneous flow data during the reaction process.

Entire Vortex Solutions of Negative Degree for the Anisotropic Ginzburg–Landau System

The anisotropic Ginzburg-Landau system \[ \Delta u+\delta\, \nabla (\mathrm{div}\: u) +\delta\, \mathrm{curl}^*(\mathrm{curl}\: u)=(|u|^2-1) u, \] for $u\colon\mathbb R^2\to\mathbb R^2$ and $\delta\in (-1,1)$, models the formation of vortices in liquid crystals. We prove the existence of entire solutions such that $|u(x)|\to 1$ and $u$ has a prescribed topological degree $d\leq -1$ as $|x|\to\infty$, for small values of the anisotropy parameter $|\delta| < \delta_0(d)$. Unlike the isotropic case $\delta=0$, this cannot be reduced to a one-dimensional radial equation. We obtain these solutions by minimizing the anisotropic Ginzburg-Landau energy in an appropriate class of equivariant maps, with respect to a finite symmetry subgroup.

Passive and active fiber reorientation in anisotropic materials

We present a continuum model to describe the reorientation of an anisotropic material structure, characterized by two fiber families able to modify their orientations following different evolution dynamics. The evolution equations are derived in a thermodynamically consistent way, and passive and active contributions to the reorientation process are identified. It is shown that a weaker extension of a well-known coaxiality result holds. The transversely isotropic and orthotropic cases are then recovered by imposing the proper constraint on the fiber rotation. Applications to biological experiments on cell layers under stretch are discussed, showing a good agreement between the model and the experimental results. Even though we focus on cell layers, our framework remains general and may be employed to describe reorientation in engineering materials.

New constraints on anisotropic expansion from supernovae Type Ia

ABSTRACT We re-examine the contentious question of constraints on anisotropic expansion from Type Ia supernovae (SNIa) in the light of a novel determination of peculiar velocities, which are crucial to test isotropy with SNe out to distances ⪝200h-1 Mpc. We re-analyse the Joint Light-Curve Analysis (JLA) Supernovae (SNe) data, improving on previous treatments of peculiar velocity corrections and their uncertainties (both statistical and systematic) by adopting state-of-the-art flow models constrained independently via the 2M++ galaxy redshift compilation. We also introduce a novel procedure to account for colour-based selection effects, and adjust the redshift of low-z SNe self-consistently in the light of our improved peculiar velocity model. We adopt the Bayesian hierarchical model BAHAMAS to constrain a dipole in the distance modulus in the context of the Lambda cold dark matter (ΛCDM) model and the deceleration parameter in a phenomenological Cosmographic expansion. We do not find any evidence for anisotropic expansion, and place a tight upper bound on the amplitude of a dipole, |Dμ| &amp;lt; 5.93 × 10−4 (95 per cent credible interval) in a ΛCDM setting, and $|D_{q_0}| \lt 6.29 \times 10^{-2}$ in the Cosmographic expansion approach. Using Bayesian model comparison, we obtain posterior odds in excess of 900:1 (640:1) against a constant-in-redshift dipole for ΛCDM (the Cosmographic expansion). In the isotropic case, an accelerating universe is favoured with odds of ∼1100:1 with respect to a decelerating one.

On Some Properties of the First Brocard Triangle in the Isotropic Plane

In this paper we introduce the first Brocard triangle of an allowable triangle in the isotropic plane and derive the coordinates of its vertices in the case of a standard triangle. We prove that the first Brocard triangle is homologous to the given triangle and that these two triangles are parallelogic. We consider the relationships between the first Brocard triangle and the Steiner axis, the Steiner point, and the Kiepert parabola of the triangle. We also investigate some other interesting properties of this triangle and consider relationships between the Euclidean and the isotropic case.

Nanoparticle-Based Magnetorheological Elastomers with Enhanced Mechanical Deflection for Haptic Displays

Haptics allows tactile interactions between humans and digital interfaces. Magnetorheological elastomers (MREs) constitute a promising candidate material for creating the tactile interface of the future─one able to recreate 3D shapes that can be sensed with touch. Furthermore, an MRE formed by using nanoparticles, as opposed to previously used microparticles, is necessary to generate a variety of shapes involving sharp curvatures over small, micrometer-scale horizontal distances to pave the way for haptic displays with microtexture resolution. Here we fabricated both isotropic and anisotropic MREs with different concentrations (2-8 vol % nanoparticles) of soft, low-remanence ferromagnetic nanoparticles. When placed in a magnetic field gradient, isotropic MREs, nonintuitively, show higher deflection than anisotropic MREs, with the former achieving displacement on the order of a millimeter at just 100 mT. This enhanced performance in the isotropic case is explained based on the soft magnetic nature of the nanoparticles. We show that performance improves with magnetic content up to a composition of 6 vol %, where it plateaus. This behavior is attributed to the stiffness of the composite material increasing at a faster rate than the magnetization as the rigid magnetic nanoparticles are added to the elastomeric matrix. Moreover, 6 vol % microparticle-based isotropic and anisotropic MREs were fabricated and compared with the nanoparticle-based MREs. Anisotropic nanoparticle-based films show higher deflection when compared with their microparticle-based counterparts. The latter is only able to match the nanoparticle film deflection at higher applied fields of almost 300 mT. This performance difference between nanoparticle and microparticle-based films is attributed to the increased anisotropic film stiffness resulting from the larger micrometer-size particles. Finally, the optimally designed nanoparticle-based isotropic film was utilized to create a programmable and real-time reconfigurable braille-inspired interface.

Evidence of Magnetic Reconnection with Multiple X Lines and Flux Ropes in Thin Magnetotail Currents Observed by the MMS Spacecraft: Results of Grad–Shafranov Reconstruction

Abstract There is some observational evidence for the existence of multiple X line magnetic reconnection (MR) in various planetary magnetotails but the overall observationally based MR topology in two or three dimensions is still not available. This study reports the first 2D structures of MR with multiple X lines and magnetic islands observed by the Magnetospheric Multiscale (MMS) spacecraft in the Earth’s magnetotail based on the Grad–Shafranov (GS) reconstruction model with temperature anisotropy. The tearing mode geometry is revealed within the spatial domain of 3800 km × 800 km with multiple X lines and magnetic islands on the spatial scale of the sub-ion inertial length or a few times the electron gyroradius. The MR event is seen by all four MMS spacecraft but the magnetic islands are caught only by the MMS3 spacecraft, and exhibit large firehose-type temperature anisotropy. The GS reconstructed maps based on the MMS1, 2, and 4 show a single X line and partial ion-scale magnetic islands with a smaller degree of temperature anisotropy. The reconstruction results remain the same for various energy closures, and the firehose-type anisotropy is found to yield smaller magnetic islands than the isotropic cases, which is opposite to the previous findings for MR events with mirror-type temperature anisotropy.