Elliptic Nature Research Articles

Implicit methods for viscous incompressible flows

Numerical methods and simulation tools for incompressible flows have been advanced largely as a subset of the computational fluid dynamics (CFD) discipline. Especially within the aerospace community, simulation of compressible flows has driven most of the development of computational algorithms and tools. This is due to the high level of accuracy desired for predicting aerodynamic performance of flight vehicles. Conversely, low-speed incompressible flow encountered in a wide range of fluid engineering problems has not typically required the same level of numerical accuracy. This practice of tolerating relatively low-fidelity solutions in engineering applications for incompressible flow has changed. As the design of flow devices becomes more sophisticated, a narrower margin of error is required. Accurate and robust CFD tools have become increasingly important in fluid engineering for incompressible and low-speed flow. Accuracy depends not only on numerical methods but also on flow physics and geometry modeling. For high-accuracy solutions, geometry modeling has to be very inclusive to capture the elliptic nature of incompressible flow resulting in large grid sizes. Therefore, in this article, implicit schemes or efficient time integration schemes for incompressible flow are reviewed from a CFD tool development point of view. Extension of the efficient solution procedures to arbitrary Mach number flows through a unified time-derivative preconditioning approach is also discussed. The unified implicit solution procedure is capable of solving low-speed compressible flows, transonic, as well as supersonic flows accurately and efficiently. Test cases demonstrating Mach-independent convergence are presented.

Relating shear sonic anisotropy directions to stress in deviated wells

We develop a model that relates the polarization direction of a dipole sonic fast shear wave propagating along the borehole (hereafter called FSA) to the relative deviatoric stress tensor for arbitrary well orientations. We first define stress quantities, or “subsidiary principal stress,” relevant for sonic dipole shear characterized sufficiently far away into the formation to be unaffected by borehole stress concentration. Next, we show analytically for wells oriented within principal stress planes and numerically for wells outside principal stress planes that the stress-induced FSA coincides with the maximum normal stress direction orthogonal to the borehole (“maximum subsidiary principal stress”) using a nonlinear elastic model for isotropic unstressed background media and plane wave solutions. This model is independent of stress sensitivity parameters. This result is a consequence of, first, the direct relationship between the stressellipsoid factor and the shear stiffness difference ratio, [Formula: see text], and, second, the almost elliptical nature of stress-induced orthorhombic media. Using published laboratory data relating stiffnesses to applied differential stress for isotropic unstressed limestone and sandstone rocks, the model is confirmed for most well orientations except in the vicinity of the stress nodal point within the maximum and minimum principal stress plane, due to small anellipticity as stress increases. The orientations near the nodal point are also the zones of weakest stress-induced anisotropy zones and are therefore the least likely to be measured in practice as long as a quality indicator depending on both the degree of anellipticity and the strength of the shear anisotropy is below 0.5. Finally, we present a synthetic example of the application of this model to estimate the relative deviatoric stress tensor using two wells with different orientations.

3D3C velocimetry measurements of an electrothermal microvortex using wavefront deformation PTV and a single camera

We study the three-dimensional fluid transport in an electrothermal microvortex (EMV), by using wavefront deformation particle-tracking velocimetry (PTV) developed at Universitat der Bundeswehr Munchen. By using a cylindrical lens in conjunction with a microscope objective lens, systematic wavefront deformations in the particle images are created. The particles are observed by a single camera and appear as ellipses. The elliptical nature of the particle images encodes out-of-plane information regarding the particle’s position. This new technique is ideally suited for measuring transport in the EMV and provides full three-dimensional, time-resolved particle trajectories with Lagrangian velocity and acceleration. Measurements reveal the toroidal nature of the EMV and the experimentally obtained velocities are used to validate a simplistic model, which describes the interaction between the applied laser illumination and the microfluidic device. The model allows one to conduct numerical simulations of the complex fluid transport in the EMV.

Mathematical modeling of plasma drifts over equatorial low latitude regions

This paper presents a mathematical model to simulate ionospheric plasma drifts at equatorial low latitude regions by coupling of E- and F-regions. The governing non-linear differential equations (of elliptic and parabolic nature) are solved numerically through finite-difference schemes and obtained neutral winds and electric fields. The temperature and electron density profiles are generated utilizing MSIS-86 atmospheric model. The continuity equation is employed to obtain night-time E-region density profile using measured ionograms at Trivandrum (India). The computed vertical and zonal plasma drifts are comparable with measured Jacamarca plasma drifts with little variations during noon and evening times. The plasma drifts at Trivandrum (8.5° N, 76.5° E, dip 0.5° N) are compared with those of Jicamarca (12° S, 76.9° W, dip 2° N). Neutral wind simulations of present model agree well with those of horizontal wind model (HWM-93). The post-sunset enhancement and its reversal are also discussed.

The role of ellipticity on the preferential binding site of Ce and La in C78-D3h—A density functional theory study

Endohedral metallofullerenes that encapsulate one or several atoms, or a cluster of atoms have molecular properties making them useful both in technology and in bio-medical applications. Some fullerenes are found to have two metal atoms incarcerated and it has been recently found that two Ce atoms are incorporated into the C(78)-D(3h) (78 : 5) cage. In this study, we report calculations on the structural and electronic properties of Ce(2)@C(78) using density functional theory (DFT). While Ce(2)@C(80)-I(h) (D(3d)) and La(2)@C(80)-I(h) (D(2h)) have different ground state structures, we have found that Ce(2)@C(78) has a D(3h) ground state structure just as La(2)@C(78). The encapsulated Ce atoms bind strongly to the C(78)-D(3h) cage with a binding energy (BE) of 5.925 eV but not as strong as in Ce@C(82)-C(2v) nor in Ce(2)@C(80)-I(h). The elliptical nature of the cage plays a crucial role and accommodates the two Ce atoms at opposite ends of the C(3) axis with a maximized inter atomic distance (4.078 A). This means that the effect of the additional f-electron repulsion in M(2)@C(78) with M = Ce compared to M = La, is less pronounced than in Ce(2)@C(80) compared to La(2)@C(80). We compare the results to the elliptical M(2)@C(72) (#10611) (M = La, Ce), and with a range of additional Ce and La endohedral fullerenes and explain the role ellipticity has in the preferential binding site of Ce and shed light on the formation mechanism of these nanostructures.

Towards physically realizable and hyperbolic moment closures for kinetic theory

The numerical prediction of continuum and non-equilibrium flows by using fully hyperbolic and realizable mathematical descriptions that follow from moment closures of gas kinetic theory is reviewed. A brief review is first given of some of the current capabilities and limitations of moment closures for predicting a range of continuum and non-equilibrium flow. Next, an extended but hyperbolic Gaussian closure for diatomic gases that does not account for heat-transfer effects, as well as a regularized version of this closure that incorporates anisotropic thermal-diffusion effects via the inclusion of higher order terms having an elliptic nature, are both reviewed and applied to a number of canonical flow problems. The numerical results for the Gaussian closures clearly demonstrate the capabilities and potential of moment closures and purely hyperbolic treatments. Following these reviews, a somewhat novel hierarchy of physically realizable and hyperbolic moment closures is considered and described. This alternative hierarchy is based on modifications to the more common maximum-entropy hierarchies, so as to ensure the validity and integrability of the approximate distribution function for all values of the velocity moments that are physically realizable. The predictive capabilities of this new closure hierarchy are then explored by considering numerical solutions of the closures for a one-dimensional kinetic equation with a relaxation-time collision operator and by comparing the closure solutions to discrete numerical solutions of this simplified kinetic equation. The study concludes with a brief summary of the findings and a discussion of the potential of the physically realizable and hyperbolic moment closures for application to fully three-dimensional physics.

Phase relation of oscillations near the planetary period of Saturn's auroral oval and the equatorial magnetospheric magnetic field

Previous analyses of Hubble Space Telescope (HST) images of Saturn's southern auroras obtained during two campaigns, in January 2007 and February 2008, have revealed that the auroral oval oscillates at a period close to the planetary rotation period, with its center describing an elongated ellipse of semimajor axis ∼2° colatitude aligned along the prenoon to premidnight direction. Previous analyses of Cassini magnetic field data from Saturn's near‐equatorial quasi‐dipolar magnetosphere have also established the presence of a rotating pattern of magnetic field perturbations near the planetary period, a phase model which has been derived from data over the interval from mid‐2004 to the end of 2007, whose extrapolation is verified here for use during the February 2008 HST campaign. In this paper we compare the phases of these oscillatory phenomena and show that the southern oval displacement was directed approximately opposite to the rotating equatorial perturbation field during both HST campaign intervals. We also examine the relation of the southern oval oscillations to the periodic power modulations in Saturn kilometric radiation (SKR) and show that the southern oval was displaced sunward at SKR maxima. It is suggested that the oval displacements are related to magnetospheric field line distortions associated with the rotating magnetic field perturbations, this picture also being consistent with recently reported periodic tilting of the equatorial plasma sheet. We note, however, that this picture provides no immediate explanation for the significantly elliptical nature of the observed oval motion.

Viscosity solutions for a model of contact line motion

This paper considers a free boundary problem that describes the motion of contact lines of a liquid droplet on a flat surface. The elliptic nature of the equation for droplet shape and the monotonic dependence of contact line velocity on contact angle allows us to introduce a notion of “viscosity” solutions for this problem. Unlike similar free boundary problems, a comparison principle is only available for a modified short-time approximation because of the constraint that conserves volume. We use this modified problem to construct viscosity solutions to the original problem under a weak geometric restriction on the free boundary shape. We also prove uniqueness provided there is an upper bound on front velocity.

Band-like Stress Fiber Propagation in a Continuum and Implications for Myosin Contractile Stresses

Stress fibers are band-like features that form with sarcomere-like actin and myosin arrangement between cell regions, resisting myosin contractility. We consider three aspects of stress fiber formation: (1) they form by cytoskeletal actin–myosin interaction when myosin contractile forces are resisted, (2) they propagate in a band-like manner, and (3) they maintain a level of stress and material continuity with the cytoskeleton. This suggests that any description of myosin force should capture the band-like propagation of stress fibers within the constraints of a continuum model. Recent studies describe myosin force as increasing proportional to the cytoskeletal resistance in that direction, but do not capture the band-like propagation of myosin stresses in a continuum. While the spreading of myosin stresses in continuum models is commonly attributed to the elliptic nature of continuum equations, we show that it comes from an incomplete description of the myosin force. Qualitative observations of cytoskeletal actin–myosin interaction indicate the interaction to be ‘zipper-like’; myosin contractile forces get transmitted by bending actin filaments in directions away from that of the cytoskeletal resistance. A simple coarse-grained implementation of the lateral myosin forces that arise from the zippering action reproduces band-like stress propagation within a continuum model for the first time. This model also shows actin packing into the stress channel and its propagation along the edge for square and triangular constrained cells; features not captured earlier. Physically, the lateral contractile forces prevent stress spreading by balancing perpendicular shear forces that arise when stress channels through a continuum. Mathematically, these forces render the continuum stress equation hyperbolic. This paper presents a theoretical argument, based on continuum mechanics principles, that it is the zippering actin–myosin action that allows for band-like stress fiber propagation within a coarse-grained cytoskeletal continuum, and that any visualization of the cytoskeletal stress field should account for lateral contractile forces accompanying the much-acknowledged contractile force along a stress fiber.

On Linear Instability of 2D Solitary Water Waves

We consider instability of 2D irrotational solitary water waves. The maxima of energy and the travel speed of solitary waves are known to not be attained at the highest wave that has a 120 angle at the crest. Under the assumption of nonexistence of secondary bifurcation, which had been confirmed numerically, we prove linear instability of solitary waves, which are higher than the wave of maximal energy and lower than the wave of maximal travel speed. It is also shown that there exist a sequence of unstable solitary waves approaching the highest wave, which suggests the instability of the highest wave. These unstable waves are of large amplitude and therefore this type of instability cannot be captured by the approximate models derived under small-amplitude assumptions. For the proof, we introduce a family of nonlocal dispersion operators to relate the linear instability problem with the elliptic nature of solitary wave problem. A continuity argument with a moving kernel formula is used to study these dispersion operators and yield the instability criteria.

The Borealis basin and the origin of the martian crustal dichotomy

The most prominent feature on the surface of Mars is the near-hemispheric dichotomy between the southern highlands and northern lowlands. The root of this dichotomy is a change in crustal thickness along an apparently irregular boundary, which can be traced around the planet, except where it is presumably buried beneath the Tharsis volcanic rise. The isostatic compensation of these distinct provinces and the ancient population of impact craters buried beneath the young lowlands surface suggest that the dichotomy is one of the most ancient features on the planet. However, the origin of this dichotomy has remained uncertain, with little evidence to distinguish between the suggested causes: a giant impact or mantle convection/overturn. Here we use the gravity and topography of Mars to constrain the location of the dichotomy boundary beneath Tharsis, taking advantage of the different modes of compensation for Tharsis and the dichotomy to separate their effects. We find that the dichotomy boundary along its entire path around the planet is accurately fitted by an ellipse measuring approximately 10,600 by 8,500 km, centred at 67 degrees N, 208 degrees E. We suggest that the elliptical nature of the crustal dichotomy is most simply explained by a giant impact, representing the largest such structure thus far identified in the Solar System.

Contrast‐enhanced timing robust acquisition order with a preparation of the longitudinal signal component (CENTRA plus) for 3D contrast‐enhanced abdominal imaging

To investigate a new image acquisition method that enables an accurate hepatic arterial phase definition and the visualization of contrast agent uptake processes in abdominal organs like liver, spleen, and pancreas. A 3D turbo gradient echo method where a fat suppression prepulse is followed by the acquisition of several profiles was combined with an elliptical centric k-space ordering technique and 3D dynamic elliptical centric keyhole. The new k-space ordering method (CENTRA+) was validated experimentally. In an initial clinical evaluation phase the method was employed in five patients to assess the accuracy of the hepatic arterial phase definition and the visualization of the contrast uptake processes in dynamic scanning in abdominal organs like liver, spleen, and pancreas. In total, five patients were evaluated using the new k-space order. Our initial results indicate that the new k-space order allows consistent capture of the hepatic arterial phase. In dynamic scanning the extreme short temporal resolution obtained with 3D elliptical centric keyhole enables contrast enhancement to be followed in organs with fast contrast uptake characteristics. The elliptical centric nature of the new image acquisition method effectively allows capture of the contrast enhancement processes with good fat suppression.

High-precision orbital parameters for the short-period spectroscopic binary HD 159656 – an eccentric orbit confirmed for an old but active system

HD 159656 is a seventh magnitude short-period (P ∼ 10 d) binary system with a solar-type (G2/3V) primary component. The single-lined spectroscopic orbital solution has been recomputed with high precision. This work confirms the orbit's elliptical nature (e = 0.116 16 ± 0.000 20) which had been brought into doubt based on an earlier less precise solution (Barker, Evans & Laing 1967) and the comprehensive study conducted by Lucy & Sweeney (1971). Short-period binary orbits of this eccentricity are uncommon, particularly when the system has the considerable age that we estimate to be τise = 9.3 ± 0.6 Gyr. Thus HD 159656 may be a system that challenges our present theories of circularization time-scales. Alternatively, HD 159656 could be considered to be suitable for future studies aiming to establish whether or not a third star is interfering with the circularization process. Apsidal motion, undetectable based on this and the previous solution, may also be revealed in the future, this possibility being greatly increased by the high precision of our results. Attempts to detect any companions directly have so far been unsuccessful. The age claimed provides indirect support for the proposition that the system is pseudo-synchronized. This in turn gives us reason to claim an estimate for the inclination, i = 32 +14 -12 degrees with a corresponding mass ratio q ∼ 0.3 that implies that the secondary is an M3 dwarf, or an unusually low-mass white dwarf, since M 2 ∼ 0.3 M ⊙ . HD 159656 is a standard star in many photometric systems. However, a small sample of our spectra indicates that the Ca II H&K line emission, that we demonstrate arises principally from the primary, is variable. Furthermore, ROSAT data of an X-ray source probably coincident with HD 159656 provide evidence of flare activity. Such behaviour is more typical of M-dwarf stars and so may be more properly associated with the secondary, if it is of this spectral type. If the binary is tidally locked, this could explain the activity of this system, which implies a chromospheric age an order of magnitude less than the isochrone age. The spectra reveal a very low lithium abundance for the primary component, together with many other spectral characteristics and photometric colours that are typical of solar analogues. Thus the primary star represents an example of the behaviour of solar analogues in this astrophysical context.

Illustrations of the analytic memo as reflexivity for preservice teachers

This study illustrates the use of analytic memos during the action research process as a space to support preservice teachers’ emerging teacher identity and construction of practice through critical reflexivity. The authors reviewed 34 sets of analytic memos written by graduate preservice teachers by asking, ‘How are preservice teachers using the space of the analytic memo during the action research process as critical reflexivity?’. Reflexivity is understood in this work as a method of self‐inquiry, and positions teacher identity as fluid and subject to multiple and often competing discourses. The outcome of this analysis demonstrates how emerging action research questions reflect emerging teacher identity; the power of dialogue within the memos to act as a ‘corrective moment’; and the elliptical nature of action research as preservice teachers confront contradictions and paradoxes in learning to teach. The authors suggest that the analytic memo can act as a vehicle of agency and transformation, a space where preservice teachers may practice reflexivity as they begin to critically construct their practice, create and re‐create their teacher identity, and confront assumptions about teaching and learning.

Development of an electro‐osmotic flow model to study the dynamic behaviour in human meridian

AbstractA numerical study has been performed on the proposed bio‐fluid dynamics model to explore and provide some human meridian characteristics. The proposed meridian model involves tissue fluids, which contain ions and nutrition, in the meridian passage. The tissue fluid under investigation can interact with the blood in the capillary vessel at the acupuncture points a through a complex electro‐osmosis transport process occurring in the meridian path. The investigated physical domain consists of a meridian path having three acupuncture points, namely, gall bladder 37, 38, 39, which are connected with their associated small meridian bodies. Based on the proper physiological coefficients, the simulated mean velocity of the tissue fluid has the result of 3.37 cm/min, and is observed to increase and decrease with the hydraulic pressure of the arteriole with the maximum and minimum values of 4.34 and 2.40 cm/min, respectively. Our simulated results are in good agreement with the experimental findings reported in the literature. In the meridian, both blood and tissue fluid flows exhibited complex electro‐osmosis nonlinear behaviour. The immediate response due to the externally applied acupuncture is analysed to reveal the elliptic meridian nature, which is intrinsic in the body fluid. The interaction between the body fluid and blood is as the master and mother, and the influence of the blood circulation on the meridian system and vice versa are also analysed in detail. Copyright © 2007 John Wiley & Sons, Ltd.

Geometrical Variations of Gain Patterns

This paper studies the effect of the altitude and the elliptical nature of the Earth on projected antenna pattern functions (APFs). These variations can play a significant role in spatial filter algorithms, such as the Backus-Gilbert method. In this paper, a solution for projecting an APF from an arbitrary sensor and platform onto an elliptical Earth is presented. A general time integration of this projection is presented, which is important for dynamically generating the effective antenna pattern functions (EAPFs) in cases where the variations are nonnegligible. It is shown that the elliptical nature of the Earth provides a variation of order 1% in the area of the EAPF. These results suggest that approximating the Earth as a sphere is valid. It is also shown that elliptical orbits should consider the implication of the altitude variation before precomputing EAPFs

On Boundary Condition Capturing for Multiphase Interfaces

This review paper begins with an overview of the boundary condition capturing approach to solving problems with interfaces. Although the authors' original motivation was to extend the ghost fluid method from compressible to incompressible flow, the elliptic nature of incompressible flow quickly quenched the idea that ghost cells could be defined and used in the usual manner. Instead the boundary conditions had to be implicitly captured by the matrix formulation itself, leading to the novel approach. We first review the work on the variable coefficient Poisson equation, noting that the simplicity of the method allowed for an elegant convergence proof. Simplicity and robustness also allowed for a quick extension to three-dimensional two-phase incompressible flows including the effects of viscosity and surface tension, which is discussed subsequently. The method has enjoyed popularity in both computational physics and computer graphics, and we show some comparisons with the traditional delta function approach for the visual simulation of bubbles. Finally, we discuss extensions to problems where the velocity is discontinuous as well, as is the case for premixed flames, and show an example of multiple interacting liquids that includes all of the aforementioned phenomena.

Effects of Elliptical Nature of Rotating Electric Field on Partial Discharge Pattern in a Three-phase Construction

Discharge characteristics in single-phase equipment are a well-known and well-established art, however, those in three-phase equipment are far less understood. Because the electric field in the insulating material of three-phase equipment is not linear, but rotates with power frequency, it can be expected that the discharge characteristics in three-phase equipment will be different from those in single-phase equipment. In this paper, the effects of rotating electric field on partial discharge pattern in a three-phase construction are reported. The simplified model of a three-phase GIS with air inside is used. The electric field in the construction was analyzed, and the discharge characteristics were observed. The electric field vector locus in a three-phase construction is elliptical, although the shape differs locally. The elliptical nature of rotating electric field is expressed as an electric field ratio. The experimental results showed that the phase widths where partial discharge current pulses appeared were different depending on the electric field ratio even for the same applied voltage. The phase width where partial discharge current pulses appeared was also influenced by the applied electric field. These results are analysed from the relationship between the value and the locus of the applied electric field and the critical electric field of air.

Transverse permeability of chopped fibre bundle beds

The out-of-plane permeability of 25 mm-long chopped glass fibre bundle beds was measured. These bundles, used in sheet moulding compound (SMC) composites, have an elliptical cross-section with the major axis perpendicular to the flow direction. An adapted experimental set-up was developed, which prevented flow along the sides of the packed bundle bed. The set-up was tested with needled glass fibre mats, and satisfactory agreement was found with previous results. The transverse permeability of the loose SMC bundle beds was then measured and linked to the bundle sizing type, geometry and volume fraction. Finally, the results could be well described by models taking into account the elliptical nature of the bundles.

EXPONENTIAL CONVERGENCE OF hp-FEM FOR MAXWELL EQUATIONS WITH WEIGHTED REGULARIZATION IN POLYGONAL DOMAINS

The time-harmonic Maxwell equations do not have an elliptic nature by themselves. Their regularization by a divergence term is a standard tool to obtain equivalent elliptic problems. Nodal finite element discretizations of Maxwell's equations obtained from such a regularization converge to wrong solutions in any non-convex polygon. Modification of the regularization term consisting in the introduction of a weight restores the convergence of nodal FEM, providing optimal convergence rates for the h version of finite elements. We prove exponential convergence of hp FEM for the weighted regularization of Maxwell's equations in plane polygonal domains provided the hp-FE spaces satisfy a series of axioms. We verify these axioms for several specific families of hp finite element spaces.