Non-constant Density Research Articles

Fluid-structure interaction: Extended-FEM approach to solidification

The extended finite element method (XFEM) and the level set method (LSM) are applied to simulate the solidification phenomenon and the behavior of the liquid-solid phase transition. The temperature-based energy equation is loosely-coupled with the incompressible Navier-Stokes (INS) equations and solved by XFEM using the Stefan condition to express the energy conservation law for phase change. The INS equations are additionally supplemented with the Boussinesq approximation for the buoyancy force that drives the ensuing melt flow. The temperature, pressure, and fluid velocity are discontinuous at the interface, and the LSM implicitly captures its location. A modified abs-enrichment scheme (where abs stands for the absolute value function) is used for the weakly-discontinuous temperature field, and a sign-enrichment scheme is employed for the strongly-discontinuous pressure field. The penalty method imposes the interface temperature and velocity and allows for fluid-structure interactions. The numerical model is verified with several benchmark tests: 1D solidification, infinite corner solidification, Frank sphere, flow over a cylinder, as well as tin melting with non-constant density. Once the simulation results have been shown to be in good agreement with analytical solutions and results obtained with other methods, the present methodology is applied to a melting ice cylinder at a high Reynolds number.

Periodic pattern formation in the coupled chemotaxis-(Navier–)Stokes system with mixed nonhomogeneous boundary conditions

AbstractWe consider the coupled chemotaxis-fluid model for periodic pattern formation on two- and three-dimensional domains with mixed nonhomogeneous boundary value conditions, and prove the existence of nontrivial time periodic solutions. It is worth noticing that this system admits more than one periodic solution. In fact, it is not difficult to verify that (0, c, 0, 0) is a time periodic solution. Our purpose is to obtain a time periodic solution with nonconstant bacterial density.

Influence of liquid density variation on the bubble and gas dynamics of a single acoustic cavitation bubble

The effects of liquid density variation at the bubble surface on the dynamics of a single acoustic cavitation bubble are numerically studied. The Gilmore model together with a comprehensive hydrochemical model is used. The evaporation and condensation of water vapor are included in the hydrochemical model. The simulation results are compared to those resulting from the widely known Keller-Miksis model, which assumes a constant liquid density at the bubble surface. The numerical results for a single argon bubble in water reveal that the pressure and the temperature inside the bubble in collapse phase significantly increase, when the non-constant liquid density is used. These differences increase by raising the ultrasonic amplitude and by decreasing the bubble ambient radius and ultrasonic frequency. More importantly, at higher ultrasonic frequencies, the models give the same results regarding the cavitation dynamics and much more remarkably on the thermodynamic behavior of the bubble contents. Also, it is revealed that the entered number of water vapor molecules into the bubble in expansion phase through evaporation are less than the simulated one by the diffusion limited model. Notably, in the case of an argon bubble in aqueous solution of H2SO4(85wt%), a better match between the results of two models is observed. In addition, it is shown that considering the liquid bulk viscosity, arising from the rapid liquid density variation at the end of bubble collapse, in the Gilmore model leads to a slight growth in the collapse strength, temperature, and pressure within the bubble.

The second-order stabilized Gauge-Uzawa method for incompressible flows with variable density

The Navier-Stokes equations with variable density are challenging problems in numerical analysis community. We recently built the 2nd order stabilized Gauge-Uzawa method [SGUM] to solve the Navier-Stokes equations with constant density and have estimated theoretically optimal accuracy. Also we proved that SGUM is unconditionally stable. In this paper, we apply SGUM to the Navier-Stokes equations with nonconstant variable density and find out the stability condition of the algorithms. Because the condition is rather strong to apply to real problems, we consider Allen-Cahn scheme to construct unconditionally stable scheme.

Connection and curvature in crystals with non-constant dislocation density

Given a smooth defective solid crystalline structure defined by linearly independent ‘lattice’ vector fields, the Burgers vector construction characterizes some aspect of the ‘defectiveness’ of the crystal by virtue of its interpretation in terms of the closure failure of appropriately defined paths in the material and this construction partly determines the distribution of dislocations in the crystal. In the case that the topology of the body manifold M is trivial (e.g., a smooth crystal defined on an open set in ), it would seem at first glance that there is no corresponding construction that leads to the notion of a distribution of disclinations, that is, defects with some kind of ‘rotational’ closure failure, even though the existence of such discrete defects seems to be accepted in the physical literature. For if one chooses to parallel transport a vector, given at some point P in the crystal, by requiring that the components of the transported vector on the lattice vector fields are constant, there is no change in the vector after parallel transport along any circuit based at P. So the corresponding curvature is zero. However, we show that one can define a certain (generally non-zero) curvature in this context, in a natural way. In fact, we show (subject to some technical assumptions) that given a smooth solid crystalline structure, there is a Lie group acting on the body manifold M that has dimension greater or equal to that of M. When the dislocation density is non-constant in M the group generally has a non-trivial topology, and so there may be an associated curvature. Using standard geometric methods in this context, we show that there is a linear connection invariant with respect to the said Lie group, and give examples of structures where the corresponding torsion and curvature may be non-zero even when the topology of M is trivial. So we show that there is a ‘rotational’ closure failure associated with the group structure – however, we do not claim, as yet, that this leads to the notion of a distribution of disclinations in the material, since we do not provide a physical interpretation of these ideas. We hope to provide a convincing interpretation in future work. The theory of fibre bundles, in particular the theory of homogeneous spaces, is central to the discussion.

Two-Dimensional Defective Crystals with Non-constant Dislocation Density and Unimodular Solvable Group Structure

In Parry and Zyskin (J. Elast. 127:249–268, 2017) we outlined mathematical methods which seemed to be necessary in order to discuss crystal structures with non-constant dislocation density tensor (ddt). This was part of a programme to investigate the geometry of continuously defective crystals and the symmetries of associated discrete structures—one can think of the programme as an attempt to generalize the use of crystallographic groups as material symmetries in non-linear elasticity theory, for perfect crystals, to deal with the case where defects are present.The methods used rely on the following fact: when the ddt is non-constant, (given technical assumptions), there is a Lie group that acts on the set of material points, and the dimension of the group is strictly greater than that of the ambient space in which the crystal resides. So there is a non-trivial isotropy group associated with the group action. We develop ideas, and recap the requisite mathematical apparatus, in the context of Davini’s model of defective crystals, then focus on a particular case where the ddt is such that a solvable three dimensional Lie group acts on a two dimensional crystal state. We construct the corresponding discrete structures too.The paper is an extension of Parry and Zyskin (J. Elast. 127:249–268, 2017), where the analogous group was nilpotent.

Variationally consistent inertia templates for B-spline- and NURBS-based FEM: Inertia scaling and customization

In this contribution, variationally consistent inertia templates for B-spline and NURBS-based finite elements are proposed as a unified concept for two different purposes: Customization of the template allows construction of masses and reciprocal masses with desired properties like higher-order accuracy or improved dispersion behavior;Inertia scaling allows substantial speed-up for explicit dynamics by increased critical time steps.The derivation of the template is based on a three-field parametrized functional as in previous works of the authors’ group, but with modified primary variables, namely displacement, velocity and mass-specific linear momentum. The latter allows for mass-preservation for non-constant density throughout the domain and is therefore an enhancement to the formulation proposed in Tkachuk and Bischoff (2015).With the focus on B-spline and NURBS-based finite elements, the proposed template provides alternatives to the row-sum-lumped mass matrix, which is only 2nd order accurate independent of the polynomial order. Earlier proposed algebraically constructed higher order masses from the literature can be reconstructed in the variational setting described here as special instances. Furthermore, higher-order reciprocal masses can be constructed from the template. They are especially attractive for explicit dynamics as there is no extra expense per time step compared with lumped mass for linear problems. For non-linear problems only small overhead is expected, but this paper focuses on linear problems only and mainly undistorted meshes. Tuning the method towards inertia scaling, a reduction of the maximum eigenfrequency by 25%–40% is obtained in the examples herein, whereas the accuracy is higher than for lumped or consistent mass.

Phase transitions, inhomogeneous horizons and second-order hydrodynamics

We use holography to study the spinodal instability of a four-dimensional, strongly-coupled gauge theory with a first-order thermal phase transition. We place the theory on a cylinder in a set of homogeneous, unstable initial states. The dual gravity configurations are black branes afflicted by a Gregory-Laflamme instability. We numerically evolve Einstein’s equations to follow the instability until the system settles down to a stationary, inhomogeneous black brane. The dual gauge theory states have constant temperature but non-constant energy density. We show that the time evolution of the instability and the final states are accurately described by second-order hydrodynamics. In the static limit, the latter reduces to a single, second-order, non-linear differential equation from which the inhomogeneous final states can be derived.

Critical temperature of metallic hydrogen sulfide at 225-GPa pressure

The Eliashberg theory generalized for electron—phonon systems with a nonconstant density of electron states and with allowance made for the frequency behavior of the electron mass and chemical potential renormalizations is used to study T c in the SH3 phase of hydrogen sulfide under pressure. The phonon contribution to the anomalous electron Green’s function is considered. The pairing within the total width of the electron band and not only in a narrow layer near the Fermi surface is taken into account. The frequency and temperature dependences of the complex mass renormalization ReZ(ω), the density of states N(e) renormalized by the electron—phonon interactions, and the electron—phonon spectral function obtained computationally are used to calculate the anomalous electron Green’s function. A generalized Eliashberg equation with a variable density of electron states has been solved. The frequency dependence of the real and imaginary parts of the order parameter in the SH3 phase has been obtained. The value of T c ≈ 177 K in the SH3 phase of hydrogen sulfide at pressure P = 225 GPa has been determined by solving the system of Eliashberg equations.

Geometrical Structure of Two-Dimensional Crystals with Non-Constant Dislocation Density

We outline mathematical methods which seem to be necessary in order to discuss crystal structures with non-constant dislocation density tensor(ddt) in some generality. It is known that, if the ddt is constant (in space), then material points can be identified with elements of a certain Lie group, with group operation determined in terms of the ddt - the dimension of the Lie group equals that of the ambient space in which the body resides, in that case. When the ddt is non-constant, there is also a relevant Lie group (given technical assumptions), but the dimension of the group is strictly greater than that of the ambient space. The group acts on the set of material points, and there is a non-trivial isotropy group associated with the group action. We introduce and discuss the requisite mathematical apparatus in the context of Davini's model of defective crystals, and focus on a particular case where the ddt is such that a three dimensional Lie group acts on a two dimensional crystal state - this allows us to construct corresponding discrete structures too.

Global existence and decay of solution for the nonisentropic Euler–Maxwell system with a nonconstant background density

In this paper, the Cauchy problem for the nonisentropic Euler–Maxwell system with a nonconstant background density is studied. The global existence of classical solution is constructed in three space dimensions provided the initial perturbation is sufficiently small. The proof is mainly based on classical energy estimate and the techniques of symmetrizer. And the time decay of the solution is also established by combining the decay estimate of the Green’s function with some time-weighted estimate.

Impact of heterogeneous permeability distribution on the groundwater flow systems of a small sedimentary basin

Ground water flow systems of shallow sedimentary basins are studied in general by analyzing the fluid dynamics at the real world example of the Thuringian Basin. The impact of the permeability distribution and density differences on the flow velocity pattern, the salt concentration, and the temperature distribution is quantified by means of transient coupled simulations of fluid flow, heat, and mass transport processes. Simulations are performed with different permeabilities in the sedimentary layering and heterogeneous permeability distributions as well as with a non-constant fluid density. Three characteristic numbers are useful to describe the effects of permeability on the development of flow systems and subsurface transport: the relation of permeability between aquiclude and aquifer, the variance, and the correlation length of the log-normal permeability distribution. Density dependent flow due to temperature or concentration gradients is of minor importance for the distribution of the flow systems, but can lead to increased mixing dissolution of salt. Thermal convection is in general not present. The dominant driver of groundwater flow is the topography in combination with the permeability distribution. The results obtained for the Thuringian Basin give general insights into the dynamics of a small sedimentary basin due to the representative character of the basin structure as well as the transferability of the settings to other small sedimentary basins.

A spatially explicit capture-recapture estimator for single-catch traps.

Single-catch traps are frequently used in live-trapping studies of small mammals. Thus far, a likelihood for single-catch traps has proven elusive and usually the likelihood for multicatch traps is used for spatially explicit capture-recapture (SECR) analyses of such data. Previous work found the multicatch likelihood to provide a robust estimator of average density. We build on a recently developed continuous-time model for SECR to derive a likelihood for single-catch traps. We use this to develop an estimator based on observed capture times and compare its performance by simulation to that of the multicatch estimator for various scenarios with nonconstant density surfaces. While the multicatch estimator is found to be a surprisingly robust estimator of average density, its performance deteriorates with high trap saturation and increasing density gradients. Moreover, it is found to be a poor estimator of the height of the detection function. By contrast, the single-catch estimators of density, distribution, and detection function parameters are found to be unbiased or nearly unbiased in all scenarios considered. This gain comes at the cost of higher variance. If there is no interest in interpreting the detection function parameters themselves, and if density is expected to be fairly constant over the survey region, then the multicatch estimator performs well with single-catch traps. However if accurate estimation of the detection function is of interest, or if density is expected to vary substantially in space, then there is merit in using the single-catch estimator when trap saturation is above about 60%. The estimator's performance is improved if care is taken to place traps so as to span the range of variables that affect animal distribution. As a single-catch likelihood with unknown capture times remains intractable for now, researchers using single-catch traps should aim to incorporate timing devices with their traps.

Stimulated neutrino transformation through turbulence on a changing density profile and application to supernovae

We apply the model of stimulated neutrino transitions to neutrinos traveling through turbulence on a non-constant density profile. We describe a method to predict the location of large amplitude transitions and demonstrate the effectiveness of this method by comparing to numerical calculations using a model supernova (SN) profile. The important wavelength scales of turbulence, both those that stimulate neutrino transformations and those that suppress them, are presented and discussed. We then examine the effects of changing the parameters of the turbulent spectrum, specifically the root-mean-square amplitude and cutoff wavelength, and show how the stimulated transitions model offers an explanation for the increase in both the amplitude and number of transitions with large amplitude turbulence, as well as a suppression or absence of transitions for long cutoff wavelengths. The method can also be used to predict the location of transitions between antineutrino states which, in the normal hierarchy we are using, will not undergo Mikheev-Smirnov-Wolfenstein (MSW) transitions. Finally, the stimulated neutrino transitions method is applied to the turbulence derived found in a 2D supernova simulation and explains the minimal observed effect on neutrino oscillations in the simulation as as being due to excessive long wavelength modes suppressing transitions and the absence of modes that fulfill the parametric resonance condition.

Charge transport and rectification in molecular junctions formed with carbon-based electrodes.

Molecular junctions formed using the scanning-tunneling-microscope-based break-junction technique (STM-BJ) have provided unique insight into charge transport at the nanoscale. In most prior work, the same metal, typically Au, Pt, or Ag, is used for both tip and substrate. For such noble metal electrodes, the density of electronic states is approximately constant within a narrow energy window relevant to charge transport. Here, we form molecular junctions using the STM-BJ technique, with an Au metal tip and a microfabricated graphite substrate, and measure the conductance of a series of graphite/amine-terminated oligophenyl/Au molecular junctions. The remarkable mechanical strength of graphite and the single-crystal properties of our substrates allow measurements over few thousand junctions without any change in the surface properties. We show that conductance decays exponentially with molecular backbone length with a decay constant that is essentially the same as that for measurements with two Au electrodes. More importantly, despite the inherent symmetry of the oligophenylamines, we observe rectification in these junctions. State-of-art ab initio conductance calculations are in good agreement with experiment, and explain the rectification. We show that the highly energy-dependent graphite density of states contributes variations in transmission that, when coupled with an asymmetric voltage drop across the junction, leads to the observed rectification. Together, our measurements and calculations show how functionality may emerge from hybrid molecular-scale devices purposefully designed with different electrodes beyond the so-called "wide band limit," opening up the possibility of assembling molecular junctions with dissimilar electrodes using layered 2D materials.

Probing the mass and anisotropy of the Milky Way gaseous halo: sight-lines toward Mrk 421 and PKS 2155-304

We recently found that the halo of the Milky Way contains a large reservoir of warm-hot gas that accounts for large fraction of the missing baryons from the Galaxy. The average physical properties of this circumgalactic medium (CGM) are determined by combining average absorption and emission measurements along several extragalactic sightlines. However, there is a wide distribution of both, the halo emission measure and the O vii column density, suggesting that the Galactic warm-hot gaseous halo is anisotropic. We present Suzaku observations of fields close to two sightlines along which we have precise O vii absorption measurements with Chandra. The column densities along these two sightlines are similar within errors, but we find that the emission measures are different: 0.0025±0.0006 cm−6 pc near the Mrk 421 direction and 0.0042±0.0008 cm−6 pc close to the PKS 2155-304 sightline. Therefore the densities and pathlengths in the two directions must be different, providing a suggestive evidence that the warm-hot gas in the CGM of the Milky Way is not distributed uniformly. However, the formal errors on derived parameters are too large to make such a claim. In the Mrk 421 direction we derive the density of \(1.6^{+2.6}_{-0.8} \times 10^{-4}~\mbox{cm}^{-3}\) and pathlength of \(334^{+685}_{-274}~\mbox{kpc}\). In the PKS 2155-304 direction we measure the gas density of \(3.6^{+4.5}_{-1.8} \times10^{-4}~\mbox{cm}^{-3}\) and path-length of \(109^{+200}_{-82}~\mbox{kpc}\). Thus the density and pathlength along these sightlines are consistent with each other within errors. The average density and pathlength of the two sightlines are similar to the global averages, so the halo mass is still huge, over 10 billion solar masses. With more such studies, we will be able to better characterize the CGM anisotropy and measure its mass more accurately. We can then compare the observational results with theoretical models and investigate if/how the CGM structure is related to the larger scale environment of the Milky Way.We also show that the Galactic disk makes insignificant contribution to the observed O vii absorption; a similar conclusion was also reached by Henley and Shelton (2013) about the emission measure. We further argue that any density inhomogeneity in the warm-hot gas, be it from clumping, from the disk, or from a non-constant density gradient, would strengthen our result in that the Galactic halo path-length and the mass would become larger than what we estimate here. As such, our results are conservative and robust.

Numerical Simulation of the Gravity Waves Generated by a Moving Obstacle

The article deals with the numerical simulation of the flow pattern around a moving body in a stratified fluid. The flow is assumed to be unsteady, incompressible and stratified. The mathematical model is based on the Boussinesq approximation of the Navier-Stokes equations for viscous incompressible flow wit non-constant density. The resulting set of PDE's is then solved by the AUSM MUSCLE scheme in finite volume approximation. For the time integration the three stage BDF method of the second order is used. Three different obstacle models were tested.

Hybrid aeroacoustic computation of a low Mach number non-isothermal shear layer

Direct numerical simulation of noise generated by low speed flows requires strong numerical constraints related to the different scales in space and time for the dynamics of the flow and the propagation of sound waves. At low Mach numbers, the aeroacoustic hybrid approaches initiated by Hardin and Pope (1994) [5] based on separate calculations for the flow and for the acoustic radiation, are therefore attractive. In this paper, we show that such methods can be used for the general case of non-constant density or temperature. The starting point is an asymptotic expansion of the full Navier–Stokes equations that gives a set of equations that retain the presence of density and temperature inhomogeneities, allowing access to the dynamic quantities without the stability constraints related to acoustic waves. Then starting from the solutions of flow fluctuating quantities, we propose several possible developments of the equations to obtain the acoustic field. They lead to different sets of equations and source terms depending on the level of simplifying assumptions: the Perturbed Low Mach Number Approximation (PLMNA) or the linearized Euler equations (LEE) linearized with respect to the mean flow. An isothermal and a non-isothermal spatially evolving mixing layer are taken as test problems. The solutions of the proposed hybrid methods show a satisfactory behavior compared with the reference solution given by a compressible DNS.

Compressible impurity flow in the TJ-II stellarator

Fully ionized carbon impurity flow is studied in ion-root, neutral beam heated plasmas by means of charge exchange recombination spectroscopy (CXRS) in the TJ-II stellarator. Perpendicular flows are found to be in reasonable agreement with neoclassical calculations of the radial electric field. The parallel flow of the impurity is obtained at two locations of the same flux surface after subtraction of the calculated Pfirsch–Schlüter parallel velocity. For the medium density plasmas studied, , the measured impurity flow is found to be inconsistent with a total incompressible flow, i.e. ∇ · uz ≠ 0, thus implying a non-constant impurity density on those flux surfaces. The experimentally observed velocity deviations are compared with the parallel return flow calculated from a modelled impurity density redistribution driven by ion-impurity friction. Although the calculated return flow substantially modifies the incompressible velocity pattern, modifications at the locations of the CXRS measurements are generally smaller in magnitude and opposite in sign when compared to the experimentally observed deviations. Small inhomogeneities of the electrostatic potential in a surface are also shown to affect the impurity redistribution but do not provide a better understanding of the measurements.

A Current-Density Centric Logical Effort Delay and Power Model for High-Speed CML Gates

This paper presents a logical effort delay and power model for high-speed current-mode logic (CML) circuits. Current density centric and voltage swing dependent logical effort parameters are defined in terms of the characteristic current density for peak transistor cutoff frequency, which remains relatively constant across different technology nodes. Based on this model, constant and non-constant current density biasing schemes in data-paths of CML circuits are investigated and optimized for delay, power, and energy-delay metrics. The proposed model is simple yet sufficiently accurate for technology nodes in the constant-field scaling regime.