Non-zero Divergence Research Articles

Calculation of divergenceless magnetically induced current density in molecules

A method for the calculation of divergenceless, magnetically induced quantum mechanical current densities in molecules that approximates the exact current is presented. This was achieved by adding to the calculated conventional current density, i.e., a current that typically has a non-zero divergence, a corrective term that is the negative of the irrotational field of its Helmholtz decomposition. The solenoidal field of the decomposition is the divergence-free current density, which is still an approximation of the exact current but which now satisfies the continuity requirement regardless of the quality of the basis set. Based on calculations performed on several simple molecules (LiH, H2O, benzene, and zethrene), adopting different kinds of low-level theoretical approaches, clear improvements are observed in the correspondence of vortices, sources, and sinks for which the conventional current density shows a lack of continuity. A little improvement is also observed for the calculated diagonal components of the magnetizability tensor.

Shear-free inhomogeneous energy density in 4D Einstein-Gauss-Bonnet spherical systems

We explore the inhomogeneity factors for the initially regular relativistic spheres in 4D-Einstein-Gauss-Bonnet (EGB) theory. The corresponding equations of motion are derived once the generic expressions for the kinematical variables are obtained for spherically symmetric self-gravitating system. By using the non-zero divergence of the stress-energy tensor, the independent components of Bianchi identities are constructed. To enable a thorough explanation of the inhomogeneity of the particular shear free matter distribution, we computed two distinct components of evolution equations employing the Weyl tensor. We then investigate the requisite variables for the irregularity by looking at particular scenarios in both the adiabatic and non-adiabatic domains. These instances demonstrate how, in addition to other factors, the Gauss-Bonnet terms contribute to the regularity requirements of the collapsing fluid.

Explicit and implicit large eddy simulations of variable-density low-speed flows

The application of the Non-oscillatory Forward-in-Time (NFT) class of solvers to facilitate conventional explicit Large Eddy Simulations (LES) and implicit LES (ILES) of low-speed turbulent flow encompassing substantial local density variations is explored. Schemes solving two different sets of Low Mach Number (LMN) equations are proposed with the aim of capturing variable-density effects arising from either a non-isothermal distribution in single-species flow or compositional variation occurring in a binary-species mixing under isothermal conditions. Both schemes employ the semi-implicit NFT integrators based on the Multidimensional Positive Definite Advection Transport Algorithm and a non-symmetric Krylov subspace solver for the variable-coefficient Poisson problems arising from different treatments applied to the non-zero velocity divergence term in the two sets of the LMN equations. The developments are successfully validated using three diverse test cases, differentially heated cavity flow, non-isothermal free-jets, and a helium plume. The reported simulations show that both the dynamic Smagorinsky model in LES and ILES subgrid-scale treatments yield similar accuracy in capturing turbulent effects for the considered flows.

Regularity for parabolic equations with singular non-zero divergence vector fields

We establish two-sided Gaussian bounds on the heat kernel of divergence-form parabolic equation with singular time-inhomogeneous vector field satisfying some minimal assumptions.

On the random G equation with nonzero divergence

On the random G equation with nonzero divergence

ALTERNATIVE DIRECT INTERPOLATION BOUNDARY ELEMENT METHOD APPLIED TO ADVECTIVE-DIFFUSIVE PROBLEMS WITH VARIABLE VELOCITY FIELD

The wide range of physical phenomena of industrial interest which can be properly represented by advection-diffusion transport models motivates a constant effort in the development of new numerical methods capable of dealing with strong advective effects such as compressibility ones. The recent direct interpolation technique (DIBEM) proved to be an accurate and reliable tool for the representation of problems with constant velocity field and initial tests were also performed for problems with variable velocity field, where the results are reasonably satisfactory, but not so robust, since the integral relative to the velocity divergence, in general, seems to disturb the performance of the formulation. The current article presents a new formulation of the direct interpolation technique for solving variable velocity problems with non-zero velocity divergence. The accuracy of the new proposal is measured against a known analytical solution and, also, contrasted with the classical formulation of DIBEM and dual reciprocity technique (DRBEM) for the same case. Preliminary results show that the alternative DIBEM formulation proposed promotes a consistent improvement in precision, outperforming the two techniques in cross-comparison.

Background-oriented schlieren technique with vector tomography for measurement of axisymmetric pressure fields of laser-induced underwater shock waves

This study aims to overcome the problems that existing background-oriented schlieren (BOS) techniques based on computed tomography (CT-BOS) face when measuring pressure fields of laser-induced underwater shock waves. To do this, it proposes a novel BOS technique based on vector tomography (VT-BOS) of an axisymmetric target. The remarkable feature of the proposed technique is the reconstruction of an axisymmetric vector field with nonzero divergence, such as the field of a laser-induced underwater shock wave. This approach is based on an approximate relation between the projection of the axisymmetric vector field and the reconstructed vector field. For comparison, the pressure fields of underwater shock waves are measured with VT-BOS, CT-BOS, and a needle hydrophone. It is found that VT-BOS is significantly better than CT-BOS in terms of better convergence, less dependence on the spatial resolution of the acquired images, and lower computational cost. The proposed technique can be applied not only to fluid dynamical fields, but also to other axisymmetric targets in other areas, such as electromagnetics and thermodynamics.Graphical abstract

Background Oriented Schlieren Technique With Reconstruction Of Three-Dimensional Vector Fields For Pressure Measurement Of Laser Induced Underwater Shock Waves

We propose a novel reconstruction of the three-dimensional vector field (VR) of an axisymmetric target to overcome the problems that existing background-oriented schlieren (BOS) techniques based on reconstruction of three-dimensional scalar field (SR-BOS). This approach is based on an approximate relation between the projection of the axisymmetric vector field and the reconstructed vector field. The remarkable feature of the proposed technique is the reconstruction of an axisymmetric vector field with nonzero divergence, such as the field of a laser-induced underwater shock wave. Reconstruction of 3D vector fields, such as conventional vector tomography, is usually used in the field of electromagnetics with vector fields of zero divergence, and its application to fluid phenomena has been di fficult. We have successfully applied this technique to BOS (VR-BOS) and measured the pressure fields of underwater shock waves. Using the experimental data and previous research data, the superiority of VR-BOS was clarified by comparing the measurement results of SR-BOS and VR-BOS, for example, measurement accuracy, in better convergence, less dependence on the spatial resolution of the acquired images, and lower computational cost. In particular, the fact that the measurement accuracy of VR-BOS is almost independent of resolution is expected to expand the maximum pressure values that can be measured by the BOS technique. We also used synthetic data to investigate the effect of noise on VR-BOS. If the synthetic displacement contains noise, the noise is amplified in the vicinity of the axisymmetry axis in the reconstruction distribution calculated by VT. If the distance from the axisymmetry axis is more than 1/4 of the measurement range, the measurement accuracy of the reconstructed distribution can be sufficiently guaranteed. This proposed technique can be applied fluid dynamical fields, as well as other axisymmetric targets with vector reconstruction fields in other areas, such as electromagnetics and material mechanism.

On the Free Surface Motion of Highly Subsonic Heat-Conducting Inviscid Flows

For the free surface problem of highly subsonic heat-conducting inviscid flow in 2-D and 3-D, a priori estimates for geometric quantities of free surfaces, such as the second fundamental form and the injectivity radius of the normal exponential map, and the Sobolev norms of fluid variables, are proved by investigating the coupling of the boundary geometry and the interior solutions. An interesting feature for the free surface problem studied in this paper is the loss of one more derivative than the problem of incompressible Euler equations, for which a geometric approach was introduced by Christodoulou and Lindblad [11]. Due to the loss of the one more derivative and loss of the symmetry of equations which create significant difficulties in closing the estimates in Sobolev spaces of finite regularity, the geometric approach in [11] needs to be substantially developed and extended by exploring the interaction of large variation of temperature, heat-conduction, non-zero divergence of the fluid velocity and the evolution of free surfaces.

Impact of f(R,T) gravity in evolution of charged viscous fluids

In this work, the evolution of spherically symmetric charged anisotropic viscous fluids is discussed in framework of [Formula: see text] gravity. In order to conduct the analysis, modified Einstein Maxwell field equations are constructed. Nonzero divergence of modified energy momentum tensor is taken that implicates dynamical equations. The perturbation scheme is applied to dynamical equations for stability analysis. The stability analysis is carried out in Newtonian and post-Newtonian limits. It is observed that charge, fluid distribution, electromagnetic field, viscosity and mass of the celestial objects greatly affect the collapsing process as well as stability of stars.

Projection methods for the incompressible Navier-Stokes equations

The incompressible Navier-Stokes equations (INSE) are the basic governing equations of fluid dynamics, and their numerical solutions are of great significance. In this review paper, we first recollect some classical projection methods and their relatives in the past 50 years and then fully explain the recent fourth-order projection method called GePUP [Zhang Q <ext-link ext-link-type="uri" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="http://doi.org/10.1007/s10915-015-0122-4">2016 <i>J. Sci. Comput.</i> <b>67</b> 1134</ext-link>]. Based on a generic projection operator and the UPPE formulation of the INSE [Liu J G, Liu J, Pego R L <ext-link ext-link-type="uri" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="http://doi.org/10.1016/j.jcp.2011.08.011">2007 <i>Comm. Pure Appl. Math</i>. <b>60</b> 1443</ext-link>], we derive the GePUP equations, which retain the advantage of UPPE that the velocity divergence is governed by a heat equation and is thus well under control. In comparison with UPPE, the GePUP formulation is advantageous in three aspects: (1) its derivation depends on none of the properties of the Leray-Helmholtz projection; (2) the evolutionary velocity can be divergent, thus it is directly applicable to numerical calculations with nonzero velocity divergence; (3) the Leray-Helmholtz projection does appear on the right-hand sides of the governing equations, thus making it transparent to analyze the accuracy and stability issues raised by numerically approximating the Leray-Helmholtz projection. As the most appealing feature of GePUP, temporal integration and spatial discretization are completely decoupled and can be treated as black boxes, so that the user can choose his favorite methods for the two parts to form his own GePUP method. In particular, high-order accuracy in time can be easily obtained since no internal details of the ODE solver are needed. The flexibility in time makes the GePUP method applicable to both low-Reynolds-number flow and high-Reynolds-number flow. The flexibility in space makes the GePUP method applicable to both rectangular boxes and irregular domains. The numerical results and elementary analysis show that the fourth-order GePUP method may be much more accurate and efficient than classical second-order projection methods by many orders of magnitude.

Accuracy aspects of conventional discretization methods for scalar transport with nonzero divergence velocity field arising from the energy balance equation

AbstractWe are concerned with the numerical solution of a linear transport problem with nonzero divergence velocity field that originates from the spectral energy balance equation describing the evolution of wind waves and swells in coastal seas. The discretization error of the commonly used first‐order upwind finite difference and first‐order vertex‐centered upwind finite volume schemes in one space dimension is analyzed. Smoothness of nondivergent velocity field plays a crucial role in this. No such analysis has been attempted to date for such problems. The two schemes studied differ in the manner in which they treat the scalar flux numerically. The finite difference variant is shock captured, whereas the vertex‐centered finite volume approximation employs an arithmetic mean of the velocity and appears not to be flux conservative. The methods are subsequently extended to two dimensions on triangular meshes. Numerical experiments are provided to verify the convergence analysis. The main finding is that the finite difference scheme displays optimal rates of convergence and offers higher accuracy over the finite volume scheme, regardless the regularity of the velocity field. The latter scheme notably yields convergence rates of 0.5 and 0 in L2‐norm and L∞‐norm, respectively, when the velocity field is not smooth. A test case illustrating wave shoaling and refraction over submerged shoals is also presented and demonstrates the practical importance of flux conservation.

A discontinuity in the electromagnetic field of a uniformly accelerated charge

The electric field of a uniformly accelerated charge shows a plane of discontinuity, where the field extending only on one side of the plane, terminates abruptly on the plane with a finite value. This indicates a non-zero divergence of the electric field in a source-free region, implying a violation of Gauss law. In order to make the field compliant with Maxwell’s equations everywhere, an additional field component, proportional to a δ-function at the plane of discontinuity, is required. Such a ‘δ-field’ might be the electromagnetic field of the charge, moving with a uniform velocity approaching c, the speed of light, prior to the imposition of acceleration at infinity. However, a range of attempts to derive this δ-field for such a case, have not been entirely successful. Some of the claims of the derivation involve elaborate calculations with some not-so-obvious mathematical approximations. Since the result to be derived is already known from the constraint of its compliance with Maxwell’s equations, and the derivation involves the familiar text-book expressions for the field of a uniformly moving charge, one would expect an easy, simple approach, to lead to the correct result. Here, starting from the electromagnetic field of a uniformly accelerated charge in the instantaneous rest frame, in terms of the position and motion of the charge at the retarded time, we derive this δ-field, consistent with Maxwell’s equations, in a fairly simple manner. This is followed by a calculation of the energy in the δ-field, in an analytical manner without making any approximation, where we show that this energy is exactly the one that would be lost by the charge because of the radiation reaction on the charge, proportional to its rate of change of acceleration, that was imposed on it at a distant past.

New Deployable Expandable Electrodes in the Electroporation Treatment in a Pig Model: A Feasibility and Usability Preliminary Study.

The aim of the study is to evaluate the usability aspects of new deployable, expandable, electrode prototypes, in terms of suitability solutions for laparoscopic applications on the liver, endoscopic trans-oral and trans-anal procedures, electroporation segmentation in several steps, mechanical functionality (flexibility, penetrability), visibility of the electrode under instrumental guidance, compatibility of the electrode with laparoscopic/endoscopic accesses, surgical instruments, and procedural room and safety compatibility. The electroporation was performed on an animal model (Sus Scrofa Large White 60 kg) both in laparoscopy and endoscopy, under ultrasound guidance, and in open surgery. Electrodes without divergence, with needles coming out straight, parallel to each other, and electrodes with peripheral needles (four needles), diverging from the electrode shaft axis (electrode with non-zero divergence) have been tested. To cause an evaluable necrosis effect, the number of electrical pulses was increased to induce immediate liver cell death. Histological samples were analyzed by staining with Haematoxylin/Eosin or by immunohistochemical staining to confirm complete necrosis. The prototypes of expandable electrodes, tested in laparoscopy and endoscopy and in open surgery, respectively, are suitable in terms of usability, electroporation segmentation in several steps, mechanical functionality (flexibility, penetrability), visibility under instrumental guidance, compatibility with laparoscopic/endoscopic accesses, surgical instruments and procedural room safety, patient safety (no bleeding and/or perforation), and treatment efficacy (adequate ablated volume). Electroporation treatment using new deployable expandable electrode prototypes is safe and feasible. Moreover, electrode configurations allow for a gradual increase in the ablated area in consecutive steps, as confirmed by histology and immunohistochemistry.

Solar photosphere magnetization

Context.A recent review shows that observations performed with different telescopes, spectral lines, and interpretation methods all agree about a vertical magnetic field gradient in solar active regions on the order of 3 G km−1, when a horizontal magnetic field gradient of only 0.3 G km−1is found. This represents an inexplicable discrepancy with respect to the divB = 0 law.Aims.The objective of this paper is to explain these observations through the lawB = μ0(H+M) in magnetized media.Methods.Magnetization is due to plasma diamagnetism, which results from the spiral motion of free electrons or charges about the magnetic field. Their usual photospheric densities lead to very weak magnetizationM, four orders of magnitude lower thanH. It is then assumed that electrons escape from the solar interior, where their thermal velocity is much higher than the escape velocity, in spite of the effect of protons. They escape from lower layers in a quasi-static spreading, and accumulate in the photosphere. By evaluating the magnetic energy of an elementary atom embedded in the magnetized medium obeying the macroscopic lawB = μ0(H+M), it is shown that the Zeeman Hamiltonian is due to the effect ofH. Thus, what is measured isH.Results.The decrease in density with height is responsible for non-zero divergence ofM, which is compensated for by the divergence ofH, in order to ensure div B = 0. The behavior of the observed quantities is recovered.Conclusions.The problem of the divergence of the observed magnetic field in solar active regions finally reveals evidence of electron accumulation in the solar photosphere. This is not the case of the heavier protons, which remain in lower layers. An electric field would thus be present in the solar interior, but as the total charge remains negligible, no electric field or effect would result outside the star.

A critical comparison of smooth and sharp interface methods for phase transition

In this study, the Local Front Reconstruction Method (LFRM) is extended to allow for the direct numerical simulation of flows with phase transition. The LFRM is a hybrid front tracking method without connectivity, which can easily handle complex topological changes. The expansion due to phase change is incorporated as a non-zero divergence condition at the interface. The energy equation is treated with two different approaches: smooth interface approach and sharp interface approach. The smooth interface approach uses a one fluid formulation to solve the energy equation with an interfacial source term accounting for phase change. This interfacial source term enforces the saturation temperature at the interface. However, in the sharp interface approach, the thermal properties are not volume-averaged near the interface and the saturation temperature is imposed as a boundary condition at the interface. A detailed mathematical formulation and numerical implementation pertaining to both approaches is presented. Both implementations are verified using 1D and 3D test cases and produce a good match with analytical solutions. A comparison of results highlights certain advantages of the sharp interface approach over the smooth interface approach such as better accuracy and convergence rate, reduced fluctuations in the velocity field and a physically bounded temperature field near the interface. Finally, both approaches are validated with a 3D simulation of the rise and growth of a vapor bubble in a superheated liquid under gravity, where a good agreement with experimental data is observed for the bubble growth rate.

Diffuse interface immersed boundary method for low Mach number flows with heat transfer in enclosures

A novel diffuse interface immersed boundary (IB) approach in the finite volume framework is developed for non-Boussinesq flows with heat transfer. These flows are characterized by variable density, large temperature differences, nonzero velocity divergence, and low Mach numbers. The present IB methodology assumes that the solid body immersed in the domain is filled with a “virtual” fluid and constructs a unified momentum equation that is solved everywhere in the domain. The unified momentum equation is obtained as a convex combination of the Navier-Stokes equation and the no-slip boundary condition employing the solid volume fraction. The hydrodynamic pressure (p) that drives the flow is obtained by the solution of a variable density Poisson equation that is constructed by assuming that the velocity field inside the solid always remains solenoidal although the velocity divergence is nonzero in the fluid domain. The unified Poisson equation is also solved everywhere in the domain and has source terms that depend on the solid volume fraction, temperature gradients, and the spatially invariant thermodynamic pressure (P) that vanish in the Boussinesq limit. The thermodynamic pressure in closed domains follows from the principle of global mass conservation and is used to determine the density field everywhere in the domain except inside the solid where the density remains constant. Numerical simulations are carried out for natural and mixed convective flows in enclosures with stationary and moving heated bodies encompassing both Boussinesq and strongly non-Boussinesq flow regimes. The results of these investigations show that the local Nusselt number distribution over the body surface is oscillatory particularly when grid lines are not aligned with the surface of the body. However, the proposed approach can reasonably accurately compute the average heat transfer in both Boussinesq and non-Boussinesq flows. Investigations show that the heat transfer is significantly enhanced in the non-Boussinesq regime as compared to the Boussinesq regime. A comparison of results from the present approach with those obtained using a body-fitted finite volume solver for stationary bodies demonstrates that the proposed IB approach can compute the flow dynamics quite accurately even on Cartesian meshes that do not conform to the geometry. The IB approach presented herein is a generic approach for quasi-incompressible flows and may be applied to other low Mach number flows such as mixing and reacting flows.

A new linear theory of light and matter

A new theory is proposed with similar form and mechanics to the Dirac equation, but where the rest-mass is introduced in a more fundamental way. The new theory proves to encompass and extend classical electromagnetism, and reduce to it exactly in the field only case. A sharpening of the principle of relativity allows fully relativistic field only solutions, which take on some of the properties more usually associated with quantised states. The extended theory has four new solutions which exhibit a non-zero electric field divergence and a double-covering symmetry. These are necessarily charged, essentially fermionic states, two with positive and two with negative charge, differing by an inner spin symmetry. The new configurations are identified with the spin “up” and spin “down” electron and positron.

Vaporization of liquid droplet with large deformation and high mass transfer rate, II: Variable-density, variable-property case

A novel approach for treating vaporization of liquid droplets with large deformation and high mass transfer rate is developed. The formulation is based on decomposed vaporization mechanisms and accommodates the effects of density and property variations in both the gas and liquid phases. In this model, the mass and energy transfer across the interface are realized by a source layer in the gas phase and a sink layer in the liquid phase around the interface. The flow field in each phase is described by a set of unified conservation equations in the low Mach number limit. The equations are then solved by a modified projection method, in which the nonzero velocity divergence is treated as a source term in the pressure Poisson equation. The vaporization model is implemented in a volume-of-fluid (VOF) based code “Gerris” featuring adaptive mesh refinement. The vaporization of a single n-decane droplet in both quiescent and convective air environments of 1000 K at 1 atm is investigated to verify and validate the present model in terms of interface localization and mass conservation. The results demonstrate the accuracy and robustness of the present work. The vaporization of an impulsively started n-decane droplet with large deformation and breakup is also studied. The droplet surface dynamics are captured well, as are the vaporization behaviors.

Topological Monopoles in Quantum Antiferromagnets

While the observation of magnetic monopoles has defied all experimental attempts in high-energy physics and astrophysics, sound theoretical approaches predict that they should exist, and they have indeed been observed as quasiparticle excitations in certain condensed-matter systems. This indicates that, even though they are not ubiquitous contrary to electrons, it is possible to get them as excitations above a background. In this report, we show that phonons or lattice shear strain generate topological monopoles in some low-dimensional quantum antiferromagnets. For the Heisenberg ladder, phonons are found to generate topological monopoles with nonzero density due to quantum spin fluctuations. For the four-leg Heisenberg tube, longitudinal shear stress generates topological monopoles with density proportional to the strain deformation. The present theory is based on mapping the spin degrees of freedom onto spinless fermions using the generalized Jordan–Wigner transformation in dimensions higher than one. The effective magnetic field generated by the motion of the spinless fermions has nonzero divergence when phonons or shear stress are present. A possible material where the present kind of monopoles could be observed is BiCu
 
 
 
 
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