Steady Solutions Research Articles

Structure and Decay Properties of Th Isotopes Using E-RMFT Formalism

In the present scenario, the search for the thermally fissile nuclei is crucial and also important not only for the research background of nuclear physics but also for the great social and economic impact on the country. Many theoretical works have been performed to analyze a series of Th and U-isotopes and found that some of these isotopes are stable against α-decays and spontaneous fission. Here, we have chosen the isotopic chain of Th-nuclei for the present analysis using relativistic mean-field formalism. The work also explores a few stable isotopes in this region of the nuclear landscape, which is crucial for understanding the exotic region of the nuclear landscape. The objective of this work is to study the bulk properties such as binding energies, root mean square charge radii, neutron-proton radii, neutron skin-thickness as well as intrinsic properties such as excitation energy and specific heat for the 216-238 Th-isotopic chain. Furthermore, the stability of these isotopes is investigated through their possible decay chain analysis. The relativistic mean-field theory was used to obtain the nuclear bulk properties, namely, binding energies, root-mean-square charge radii, neutron skin-thickness, and excitation energy. The steady solution of the temperature-dependent effective relativistic mean-field equations was obtained self-consistently by taking different inputs of the initial deformations. All the calculations were done for NL3, FSUGarnet and IOPB-I parameter sets for 216-238 Th-isotopes. The decay energy of α (Q α ) and β-decay (Q β ) were calculated from the binding energies and were further used to obtain the corresponding half-lives. We have analyzed the structural and decay properties of 216-238 Th isotopes. The excitation energy and specific heat are also estimated for these considered nuclei by using the temperature-dependent effective relativistic mean-field (E-RMFT) formalism for NL3, FSUGarnet and IOPB-I parameters sets. The calculated results are compared with the available experimental data and found similar observations for all the parameter sets at a given temperature. The excitation energy study signifies the shell melting point where maybe the shape transition occurs. Three phenomenological formulae such as Viola-Seaborg, Royer and modified universal decay law are adopted for the calculation of α-decay half-lives. We found lower values of α-decay half-lives indicating a higher rate of β-decay for the isotopic chain.

A Very Easy High-Order Well-Balanced Reconstruction for Hyperbolic Systems with Source Terms

When adopting high-order finite volume schemes based on MUSCL reconstruction techniques to approximate the weak solutions of hyperbolic systems with source terms, the preservation of the steady states turns out to be very challenging. Indeed, the designed reconstruction must preserve the steady states under consideration in order to get the required well-balancedness property. A priori, to capture such a steady state, one needs to solve some strongly nonlinear equations. Here, we design a very easy correction to high-order finite volume methods. This correction can be applied to any scheme of order greater than or equal to 2, such as a MUSCL-type scheme, and ensures that this scheme exactly preserves the steady solutions. In contrast to usual techniques, our scheme avoids the inversion of the nonlinear function that governs the steady solutions. Moreover, for underdetermined steady solutions, several nonlinear functions must be considered simultaneously. Since the derived correction only considers the evaluation of the governing nonlinear functions, we are able to deal with underdetermined stationary systems. Several numerical experiments illustrate the relevance of the proposed well-balanced correction, as well as its main limitation, namely the fact that it may fail at being both well-balanced and more than second-order accurate for a specific class of initial conditions.

Three-dimensional unsteady axisymmetric vortex solutions to the Bellamy-Knights equation and the distribution of boundary conditions

The Bellamy-Knights equation gives three-dimensional unsteady axisymmetric incompressible vortex solutions that are disconnected from the known exact steady solutions. In this paper, by introducing a new parameter that allows a change of the time scale, the Bellamy-Knights equation is transformed into an integro-differential equation that continuously connects with the equation for steady vortices. The new equation is viewed as describing the one-dimensional motion of a particle under conserved and nonconserved forces and leads to new regular solutions, in which the particle initially undergoes almost sinusoidal oscillations around the minimum of potential followed by a very rapid approach to the extrema of the potential. The regular solutions distribute discretely in the space of boundary conditions and are classified into types that are specified by the pattern of sequence of small and large oscillations of the axial component of velocity. Extending the solutions to singular ones, the boundary conditions for solutions of the same type are shown to form a continuous one-dimensional set, which we call a branch. The set of branches forms dense self-similar patterns with accumulation points, some of which coalesce to a single point. In an arbitrary neighborhood of the accumulation or coalescent point, any two distinct boundary conditions whose distance in the space of boundary conditions is arbitrarily small give mutually distinctive solutions with a finite difference at some region of the spatial distance. This feature of the equations signifies the extreme sensitivity of inflowing vortex solutions to boundary conditions.

Conduit Flow Dynamics During the 1986 Sub-Plinian Eruption at Izu-Oshima Volcano

We investigated conduit flow dynamics during the 1986 sub-Plinian eruption of Izu-Oshima volcano, Japan, using a 1-D steady conduit flow model. Following observations that the magma plumbing system beneath Izu-Oshima, characterized by the feeding of a dyke, generated a fissure-type sub-Plinian eruption, we considered a dyke-like conduit geometry by applying a pseudo-dyke conduit with an ellipsoidal horizontal cross-section. Under appropriate parameter settings constrained by geological, petrological, and geophysical observations, we identified a conduit geometry that enables steady solutions of a conduit flow that generates the sub-Plinian eruption. A dyke-like conduit geometry allows us to widen the range of geometric parameters of the solution. We found that the distribution of magma overpressure in the conduit strongly depends on conduit geometry. When the conduit geometry is composed of deeper and shallower dykes with large and small aspect ratios, respectively, localized overpressurization occurs in the region before magma fragmentation. This overpressurization in the dyke-like conduit may induce a characteristic crustal deformation similar to that caused by a vertical tensile fault. It is crucial to consider the effects of conduit flow with a dyke-like geometry on deformation for precise monitoring of eruption sequences based on geodetic signals in future eruptions at Izu-Oshima volcano.

Shallow katabatic flow on a non-uniformly cooled slope

We examine katabatic flow driven by a non-uniformly cooled slope surface but unaffected by Coriolis acceleration. A general formulation is given, valid for non-uniform surface buoyancy distributions over a down-slope length scale Lgg delta _0, where delta _0=nu /(Nsin alpha )^{1/2} is the slope-normal Prandtl depth, for a kinematic viscosity nu, buoyancy frequency N and slope angle alpha. We demonstrate that the similarity solution of Shapiro and Fedorovich (J Fluid Mech 571:149–175, 2007) can remain quantitatively relevant local to the end of a non-uniformly cooled region. The usefulness of the steady similarity solution is determined by a spatial eigenvalue problem on the L length scale. Broadly speaking, there are also two modes of temporal instability; stationary down-slope aligned vortices and down-slope propagating waves. By considering the limiting inviscid stability problem, we show that the origin of the vortex mode is spatial oscillation of the buoyancy profile normal to the slope. This leads to vortex growth in a region displaced from the slope surface, at a point of buoyancy inflection, just as the propagating modes owe their existence to an inflectional velocity. Non-uniform katabatic flows that detrain fluid to the ambient are shown to further destabilise the vortex mode whereas entraining flows lead to weaker vortex growth rates. Rayleigh waves dominate in general, but the vortex modes become more significant at small slope angles and we quantify their relative growth rates.

Stability and bifurcation of dynamic contact lines in two dimensions

The moving contact line between a fluid, liquid and solid is a ubiquitous phenomenon, and determining the maximum speed at which a liquid can wet/dewet a solid is a practically important problem. Using continuum models, previous studies have shown that the maximum speed of wetting/dewetting can be found by calculating steady solutions of the governing equations and locating the critical capillary number,$Ca_{{crit}}$, above which no steady-state solution can be found. Below$Ca_{{crit}}$, both stable and unstable steady-state solutions exist and if some appropriate measure of these solutions is plotted against$Ca$, a fold bifurcation appears where the stable and unstable branches meet. Interestingly, the significance of this bifurcation structure to the transient dynamics has yet to be explored. This article develops a computational model and uses ideas from dynamical systems theory to show the profound importance of the unstable solutions on the transient behaviour. By perturbing the stable state by the eigenmodes calculated from a linear stability analysis it is shown that the unstable branch is an ‘edge’ state that is responsible for the eventual dynamical outcomes and that the system can become transient when$Ca< Ca_{{crit}}$due to finite-amplitude perturbations. Furthermore, when$Ca>Ca_{{crit}}$, we show that the trajectories in phase space closely follow the unstable branch.

Multiple hysteresis effects and their destabilization mechanism in the enclosed mixed convection enclosure with three representative aspect ratios

Mixed convection in the lid-driven cavity model widely represented real engineering applications, especially electronic cooling and wind across buildings. In past decades, avalanche experiments and numerical calculations have been done on the lid-driven cavity model. Unfortunately, one unusual flow behavior in the cavity mixed convections—multiple steady solutions, i.e., identical boundary conditions and governing parameters whereas different initial conditions or loading perturbations may lead to two or more flow states, and it was hardly investigated. Multiple steady enclosure flow behaviors essentially complicate the convective transport of heat and air, which has been vividly analyzed by streamlines, heatlines and isotherms. In the present work, the flow mechanisms and evolution driven by mixed convections in lid-driven cavities for multiple aspect ratios will be investigated through the numerical methodology of computational fluid dynamics. The effects of dimensionless parameter (Grashof number, Reynolds number, and Richardson number) on multiple flow motions have been disclosed, regarding of hysteresis effect and their destabilization mechanism. Investigations on these complicated flow motions essentially facilitate the optimization of engineering flows similar to lid driven enclosures, whatever mechanical cooling of electronics, or wind across street canyons.

FINITE ELEMENT METHOD FOR NONLINEAR EDDY CURRENT PROBLEMS IN POWER TRANSFORMERS

An efficient finite element method to take account of the nonlinearity of the magnetic materials when analyzing three dimensional eddy current problems is presented in this paper. The problem is formulated in terms of vector and scalar potentials approximated by edge and node based finite element basis functions. The application of Galerkin techniques leads to a large, nonlinear system of ordinary differential equations in the time domain. The excitations are assumed to be time-periodic and the steady state periodic solution is of interest only. This is represented in the frequency domain as a Fourier series for each finite element degree of freedom and a finite number of harmonics is to be determined, i.e. a harmonic balance method is applied. Due to the nonlinearity, all harmonics are coupled to each other, so the size of the equation system is the number of harmonics times the number of degrees of freedom. The harmonics would be decoupled if the problem were linear, therefore, a special nonlinear iteration technique, the fixed-point method is used to linearize the equations by selecting a timeindependent permeability distribution, the so called fixed-point permeability in each nonlinear iteration step. This leads to uncoupled harmonics within these steps resulting in two advantages. One is that each harmonic is obtained by solving a system of algebraic equations with only as many unknowns as there are finite element degrees of freedom. A second benefit is that these systems are independent of each other and can be solved in parallel. The appropriate selection of the fixed point permeability accelerates the convergence of the nonlinear iteration. The method is applied to the analysis of a large power transformer. The solution of the electromagnetic field allows the computation of various losses like eddy current losses in the massive conducting parts (tank, clamping plates, tie bars, etc.) as well as the specific losses in the laminated parts (core, tank shielding, etc.). The effect of the presence of higher harmonics on these losses is investigated.

Positivity-preserving and unconditionally energy stable numerical schemes for MEMS model

In this paper, we propose and analyze the positivity-preserving, unconditionally energy stable, and linear second order fully discrete schemes for Micro-Electromechanical system (MEMS). More precisely, we use the first order backward difference formulation (BDF1) and second order Crank-Nicolson (CN) formulation for the temporal discretization, and the central finite difference method for spatial discretization. A variant of the exponential scalar auxiliary variable (ESAV) approach is involved in our numerical schemes to deal with the singular nonlinear term. The unconditional energy stability of the numerical schemes is rigorously proved, without any restriction for the time step sizes. Furthermore, we derive that the numerical solutions always preserve the positivity property of the MEMS model, that is the distance variable is always between 0 and the steady solution, at a point-wise level. A series of numerical simulations are presented to demonstrate the positivity and energy stability of our numerical schemes.

Trapped waves on interfacial hydraulic falls over bottom obstacles

Work on the interfacial hydraulic falls of a two-layer stratified flow generated by an obstacle at the bottom is extended. Interfacial hydraulic falls for transcritical flows are calculated numerically by solving the full Euler equations. New steady solutions characterized by a train of waves trapped between two successive obstacles with various geometrical properties, of which the wavelength can be well predicted by the linear dispersion relation, are found.

Extraction of density-layered fluid from a porous medium

We consider axisymmetric flow towards a point sink from a stratified fluid in a vertically confined aquifer. We present two approaches to solve the equations of flow for the linear density gradient case. Firstly, a series method results in an eigenfunction expansion in Whittaker functions. The second method is a finite difference method. Comparison of the two methods verifies the finite difference method is accurate, so that more complicated nonlinear, density stratification can be considered. Interesting results for the case where the density stratification changes from linear to almost two-layer are presented, showing that in the nonlinear stratification case, there are certain values of flow rate for which a steady solution does not occur. A spectral method is then implemented to consider cases in which there is a stagnant region beneath a sharp interface between two layers of different, but constant, density. In this situation, flows also exist only for flow rates beneath a critical flux value, consistent with the results for the continuous density stratification.

Improved Body Force Model for Estimating Off-Design Axial Compressor Performance

Based on the COMSOL software, body forces substituted into the Reynolds-averaged Navier–Stokes (RANS) equations as the source terms instead of the actual blade rows were improved to better predict the compressor performance. Improvements in parallel body force modeling were implemented, central to which were the local flow quantities. This ensured accurate and reliable off-design performance prediction. The parallel force magnitude mainly depended on the meridional entropy gradient extracted from three-dimensional (3D) steady single-passage RANS solutions. The COMSOL software could easily and accurately translate the pitchwise-averaged entropy into the grid points of the body force domain. A NASA Rotor 37 was used to quantify the improved body force model to represent the compressor. Compared with the previous model, the improved body force model was more efficient for the numerical calculations, and it agreed well with the experimental data and computational fluid dynamics (CFD) results. The results indicate that the improved body force model could quickly and efficiently capture the flow field through a turbomachinery blade row.

On the steady solutions of the inviscid two-dimensional quasi-geostrophic equation

In this study, we consider the two-dimensional quasi-geostrophic equation with fractional order. We prove the existence of steady solutions for this equation, and examine the stability of steady solutions. Our method is based on using the variational principles on rearrangements of a given function.

A well-balanced discontinuous Galerkin method for the shallow water flows on erodible bottom

In this paper, we investigate the shallow water (SW) flow over the erodible layer using a fully coupled mathematical model in two-dimensional (2D) space. A well-balanced discontinuous Galerkin (DG) scheme is proposed for solving the SW equations with sediment transport and bed evolution. To achieve the well-balanced property of the numerical scheme easily, the nonlinear SW equations are first reformulated into a new form by introducing an auxiliary variable. Then the DG method is used to discretize the model, in which the FORCE flux is used. By choosing an appropriate value of the auxiliary variable, we can prove that the numerical method can accurately maintain the steady solution in still water, so it is indeed an equilibrium preserving scheme. The well-balanced property can be extended to any numerical flux which satisfies the consistency. Moreover, we investigate the impact on the numerical results if the appropriate value of auxiliary variable is slightly modified. The effectiveness of the numerical method is finally verified by numerical experiments.

Global bifurcation of capillary-gravity dark solitary waves on the surface of a conducting fluid under normal electric fields

Summary This article is concerned with capillary-gravity waves travelling on the interface of a dielectric gas and a conducting fluid under the effect of a vertical electric field. A boundary integral equation method is employed to compute fully nonlinear steady travelling wave solutions. The global bifurcation diagram of periodic waves, solitary waves, generalised solitary waves and dark solitary waves is presented and discussed in detail.

Regularity of Boltzmann Equation with Cercignani--Lampis Boundary in Convex Domain

The Boltzmann equation is a fundamental kinetic equation that describes the dynamics of dilute gas. In this paper we study the regularity of both dynamical and steady Boltzmann equation in strictly convex domain with the Cercignani-Lampis (C-L) boundary condition. The C-L boundary condition describes the intermediate reflection law between diffuse reflection and specular reflection via two accommodation coefficients. We construct local weighted $C^1$ dynamical solution using repeated interaction through the characteristic. When we assume small fluctuation to the wall temperature and accommodation coefficients, we construct weighted $C^1$ steady solution.

Improved Prediction of Aerodynamic Loss Propagation as Entropy Rise in Wind Turbines Using Multifidelity Analysis

Several physics-based enhancements are embedded in a low-fidelity general unducted rotor design analysis tool developed, py_BEM, including the local Reynolds number effect, rotational corrections to airfoil polar, stall delay model, high induction factor correction, polar at large angle of attack, exergetic efficiency calculation and momentum-based loss. A wind turbine rotor is analyzed in high fidelity designed from py_BEM using a 3D blade generator. It is a design derived from the NREL Phase VI rotor. Three design variations are analyzed using steady 3D CFD solutions to demonstrate the effect of geometry on aerodynamics. S809 and NACA 2420 airfoil properties are used for calculating the aerodynamic loading. Momentum, vorticity and energy transport are explained in depth and connected to entropy production as a measure of performance loss. KE dissipation downstream of the rotor is shown to be a significant contributor of entropy rise. Wake analysis demonstrates mixing with the free stream flow, which begins after 3 diameters downstream of the rotor and extends to about 25 diameters until the decay is very small. Vorticity dynamics is investigated using a boundary vorticity flux technique to demonstrate the relationship between streamwise vorticity and lift generated in boundary layers. Drag components are accounted as well. It is demonstrated using rothalpy that shaft power is not only torque multiplied by rotational velocity but a viscous power loss term must also be included. A multifidelity analysis of wind turbine aerodynamics is demonstrated by capturing flow physics at several levels.

Higher order compact finite difference schemes for steady and transient solutions of space–time neutron diffusion model

One of the most difficult tasks in the numerical simulation of nuclear reactors is to find solution of a system arising from spatial discretization of multidimensional multigroup neutron diffusion equations by finite difference method. The big disadvantage is that it requires smaller mesh spacing. Using smaller mesh spacing produces reasonable results but it consumes huge computational effort. In this work, we overcome this flaw by constructing higher order compact finite difference schemes to approximate the neutron leakage terms. The technique is based on using concentrated points around a central point instead of minimizing the mesh spacing taking into account the location of the diffusion coefficients. Beside the accuracy of the proposed schemes, the dimension of the coefficient matrix and therefore the complexity of the system still the same. Five point stencils with equally distance h in each direction are used to produce more than one of standard central difference formulas to be combined producing the fourth order finite difference scheme with order of accuracy O(h4). Also, seven point stencils are used to get the sixth order scheme with accuracy O(h6). The proposed fourth and sixth order schemes are highlighted through two theories. The maximum error norms and the convergence rates of the proposed schemes are discussed. For the time discretization, multi-step differential transform method MDTM is employed to solve the neutron diffusion model. To illustrate the applicability and accuracy of the proposed schemes, results are tested on some multidimensional homogenous and heterogeneous benchmark problems. Our static normalized assembly power densities that are obtained by dividing the whole core into just few dozens or hundreds of fuel assemblies are compared with the Improved SIDK, FLUENT and BRBFCM methods that divide the core into many thousands of fuel assemblies. It is proved that the obtained results are very close to those methods.

Temporal instability of curved viscous fibers with a radial electric field

Abstract One-dimensional equations are derived for a rotating viscous slender liquid jet in a radial electric field using asymptotic methods. The trajectory of the curved Newtonian liquid jets is found by solving the nonlinear one-dimensional equations. The temporal instability of the steady solutions is analysed. It was found that the electric force enhances the growth rate and increases its corresponding maximum wavenumber.

Uniform Structural Stability of Hagen–Poiseuille Flows in a Pipe

In this paper, we prove the uniform nonlinear structural stability of Hagen–Poiseuille flows with arbitrary fluxes in the axisymmetric case in an infinitely long pipe. This uniform nonlinear structural stability is the first step to study Liouville type property for steady solutions of Navier–Stokes system in a pipe, which may play an important role in proving the existence of solutions with arbitrary flux to steady Navier–Stokes system in a nozzle with Poiseuille flows as far field asymptotic states (Leray’s problem). A key step is the a priori estimate for the associated linearized problem for Navier–Stokes system around Hagen–Poiseuille flows. The linear structural stability is established as a consequence of elaborate analysis on the governing equation for the partial Fourier transform of the stream function. The uniform estimates are obtained based on the analysis for the solutions with different fluxes and frequencies. One of the most involved cases is to analyze the solutions with large flux and intermediate frequency, where the boundary layer analysis plays a crucial role.