Continuity Equation Research Articles

Comparative study on volume conservation among various SPH models for flows of different levels of violence

The weakly-compressible smoothed particle hydrodynamics (WCSPH) models offer distinct advantages in simulating free-surface flow. Within the rapidly evolving WCSPH field, Classical WCSPH model, based on the artificial viscous term, and its derived models, which additionally introduce numerical diffusive terms into the continuity equations - including the δ-SPH, δ-SPHC, δ-SPHC+, δ-LES-SPH, and δ-SPHF models have gained prominence recently. The numerical diffusive term is found to alter the particle distributions in the free-surface region, leading to a change in the mean water level that reflects the volume conservation, which might result in inaccuracies in some simulations. This study evaluates the volume conservations of the aforementioned WCSPH models for flows at quasi-hydrostatic states evolved from varying levels of flow violence under prolonged simulations, through a series of numerical tests involving hydrostatic, standing-wave, and dam-break cases. For the hydrostatic and standing-wave cases, the performance of different types of WCSPH models remains consistent overall. The Classical WCSPH and δ-LES-SPH models exhibit minimal changes in volume, with almost no decrease in the mean water level occurring. The δ-SPH, δ-SPHC, δ-SPHC + models demonstrate decreases in the mean water level converging to around 0.2 times the SPH particle diameter (Δx), indicating favorable volume conservation. Conversely, the δ-SPHF model consistently exhibits a decrease in the mean water level exceeding 1 Δx at t = 2000s in a linear fashion, resulting in a noticeable reduction in volume. Under the dam-break case, most of the mean water levels simulated by different models experience a small increase, which is close to the decrease observed under the hydrostatic and standing-wave cases with corresponding models. Only the δ-SPHF model continues to show a continuous decreasing tendency. In summary, considering volume conservation, the δ-LES-SPH model demonstrates the best performance (excluding the Classical WCSPH model as it cannot simulate violent flow), followed by the δ-SPHC, δ-SPHC+, δ-SPH and δ-SPHF models.

Quantum statistical mechanics from a Bohmian perspective

We develop a general formulation of quantum statistical mechanics in terms of probability currents that satisfy continuity equations in the multi-particle position space, for closed and open systems with a fixed number of particles. The continuity equation for any closed or open system suggests a natural Bohmian interpretation in terms of microscopic particle trajectories, that make the same measurable predictions as standard quantum theory. The microscopic trajectories are not directly observable, but provide a general, simple and intuitive microscopic interpretation of macroscopic phenomena in quantum statistical mechanics. In particular, we discuss how various notions of entropy, proper and improper mixtures, and thermodynamics are understood from the Bohmian perspective.

On the motion of a viscous liquid with a free boundary

The problem of the non-stationary flow of a viscous liquid with an external free boundary around a moving solid cylindrical body was formulated and solved. The liquid is subject to periodic impacts with or without the predominant direction in space. To formulate the problem, the Navier—Stokes equation, the continuity equation, and the equation of conditions at both the solid and free boundaries of the liquid were used. New hydro-mechanical effects were discovered.

Effects of lubricant rheology and flow turbulence on the dynamic properties of hydrodynamic bearing

The present numerical investigation, which takes into account the coexisted consequence of the turbulence domain and non-Newtonian flow rheology, predicts the dynamic properties of a finite hydrodynamic bearing. Under the proper flow boundary conditions, the turbulence (linear) and non-Newtonian hypotheses have revised the Continuity and Navier–Stokes equation. The finite element approach is acclimated to calculate the clearance extent of the finite hydrodynamic bearing using the Galerkin approach and a sturdy iteration strategy. The cylindrical coordinate's versions of continuity and momentum equations are bestowed for the lubricant field of the finite hydrodynamic bearing, postulating the turbulence domain and non-Newtonian flow rheology. Dynamic properties of a finite hydrodynamic bearing have been enumerated using cross-coupled and direct lubricant-film bearing coefficients, whirl ratio and critical mass at various eccentricity ratios for different ranges of Reynolds numbers and different values of flow behavior index of the non-Newtonian rheological model. In comparison to perfect turbulence and non-Newtonian rheological status, the mixed regime demonstrated improved properties. The direct and cross-coupled lubricant-film bearing coefficients are improved by 12.28% and 20.85% respectively while critical mass is enhanced by 20.55% at selected values of Reynolds number and power-law index. The proposed severe fluid film domain improves the stability of finite hydrodynamic bearings significantly.

A compact Non-Quasi-Static small-signal model for GaN HEMT

In this work, we introduce a compact Non-Quasi-Static (NQS) model for the long-channel Gallium nitride high-mobility field effect transistor based on charge-based EPFL (École Polytechnique fédérale de Lausanne) HEMT model implemented in Verilog-AMS modeling language. The proposed model has been validated for the low-frequency (quasi-static) operating region as well as high frequencies up to sub-terahertz. This model has been derived from the analytical solution of the drift–diffusion and continuity equations based on physically valid assumptions. Polynomial approximation was used to make the model design-oriented and compatible with conventional circuit simulators for designing microwave circuits. Intrinsic trans-admittance parameters, obtained from the model for two devices with channel lengths of 1μm and 10μm has been validated with TCAD simulation results. Besides, the scattering parameters for the GaN HEMT equivalent circuit, have been validated with TCAD simulations and with our previous work, which already validated with experimental data. These results confirm the effectiveness of the proposed compact model for circuit simulations.

DJM solution of MHD flow of ternary hybrid nanofluid between nonparallel a porous media channels with velocity slip and radiation effects

This investigation on heat transfer and Jeffery Hamel ternary hybrid nanofluid flow across nonparallel a porous media channels while accounting for radiation, velocity slip, and mixture fluid ( H 2 O − C 2 H 6 O 2 ) effects is presented in this work. The nonlinear PDEs such as continuity, momentum and heat equations transformed to a system of nonlinear ODEs using similarity transformations then solved numerically and analytically, the analytical solution has been constructed using by Daftar Dar–Jafari method (DJM), the present results in particular cases are compared to results obtained by the HAM-based Mathematica package and by the Runge–Kutta Fehlberg fourth–fifth order (RKF-45) for validation. The effects of active parameters such as the nanofluid volume fraction, velocity slip parameter, Darcy number (Da), and thermal radiation parameter (Rd), are investigated on the velocity and temperature, skin friction, and Nusselt numbers. It is found that the velocity of ternary hybrid nanofluid upsurges for both convergent and divergent channel with the increment in the Hartman number (Ha). Also, results obtained reveal that the temperature profile experiences an upward shift with an increase in the Ha, and a downward shift with an increment of the Da. Furthermore, new modeling and the reliable analytic treatment via DJM approach for the ternary hybrid nanofluid is a sizable accomplishment of the current analysis.

Simulation of dynamic pulsing fracking in poroelastic media by a hydro-damage-mechanical coupled cohesive phase field model

This study develops a hydro-damage-mechanical fully coupled numerical method capable of modelling complex fluid-driven transient dynamic crack propagation in quasi-brittle poroelastic media. In this method, the fluid flow in both fractures and porous media is described by a fluid continuity equation with the modified Darcy-Poiseuille law based on the Biot's poroelastic theory. The fluid pressure and the inertial force of solids are coupled by governing equations of the mesh-insensitive phase-field regularized cohesive zone model that can simulate quasi-brittle multi-crack initiation, propagation, branching and merging without remeshing or crack tracking. The resultant displacement-pressure-damage coupled multiphysics system of equations is solved using an alternative minimization Newton-Raphson iterative algorithm with an implicit Newmark integration scheme within the finite element framework. The new method was first validated by a few 2D problems, including crack branching and deflecting in solids, fracking in a concrete cube, and consolidation and stress wave propagation in poroelastic media, subjected to various impulsive loadings. It was then applied to pressure pulsing fracking of a 40 m granite rock reservoir with extensive parametric studies of fluid viscosity and pulsing injection rate, mode and period. It was found that pulsing injection with higher fluid rates and lower fluid viscosities resulted in more developed crack patterns, and in particular, there existed an optimal pulsing injection period that could promote fracking under relatively low injection pressures. 3D horizontal well problems with multiple non-planar crack propagation and bifurcation were also successfully simulated to demonstrate the capacity and potential of the new method for engineering design and optimization of pressure pulsing fracking.

Perturbation solution of multiphase flow of Williamson fluid through convergent and divergent conduits: Electro-osmotic effects

Problem Statement: The hafnium particles are suspended through carrier fluid (Williamson fluid) to discuss the momentum analysis in multiphase flow in two different types of configurations. Research gap: The analysis of the interaction of hafnium nanoparticles with the Williamson fluid model through the convergent and divergent conduits has not been discussed before. Methodology: The equation of continuity and momentum equations are used for this analysis. The solution of both fluid and particle velocities is obtained through the perturbation analytical technique. The perturbation solution is also compared with the numerical solution. Computational results: The Weissenberg number decays the velocity distribution. The suspension of hafnium particles updates the flow distribution through the conduits. The magnitude of the stream function decreases via the Weissenberg number. Applications: This study can help develop a new approach to cancer therapy by using a high atomic number of nanoparticles. Originality: This analysis is original and has neither been submitted nor published before.

On viscoelastic blood in a locally narrow artery with magnetic field: application of distributed-order time fractional Maxwell model

Purpose. The study of hemorheology of heterogeneous vessels is of importance for the therapy of cardiovascular diseases. The present paper aims to investigate the effects of the height parameter of the stenosis, the Reynolds number, the relaxation time and the Hartmann number on the blood flow by applying distributed order time fractional Maxwell model. Methodology. Distributed order time fractional Maxwell constitutive fluid model is used to simulate viscoelastic blood flowing in a narrowing blood vessel with uniform magnetic field. The continuity and momentum equations are established in two-dimensional cylindrical coordinate system for calculation. With vorticity-stream function method, the finite difference technique, combined with the fractional derivative L1 interpolation approximation, is utilized to obtain numerical solutions of velocity and wall shear stress. The effectiveness of the numerical method is verified. Outcomes. It reveals the choice of relaxation time in the governing equation has an influence on the results. The numerical results show that an appropriate increase of Reynolds number can reduce the possibility of vessel damage. Furthermore, when the height parameter of the stenosis increases, the damage to the vessel wall is getting serious. Besides, the wall shear stress along the blood vessel increases with the increase of the magnetic field, but the wall shear stress in the narrowest part of the blood vessel changes little.

Conserved energy–momentum tensor for real-time lattice simulations

We derive an expression for the energy–momentum tensor in the discrete lattice formulation of pure glue QCD. The resulting expression satisfies the continuity equation for energy conservation up to numerical errors with a symmetric procedure for the time discretization. In the case of the momentum conservation equation, we obtain an expression that is of higher accuracy in lattice spacing (O(a2)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\\mathcal {O}}(a^2)$$\\end{document}) than the naive discretization where fields in the continuum expressions are replaced by discretized counterparts. The improvements are verified by performing numerical tests on the derived expressions using classical real-time lattice gauge theory simulations. We demonstrate substantial reductions in relative error of one to several orders of magnitude compared to a naive discretization for both energy and momentum conservation equations. We expect our formulation to have applications in the area of pre-equilibrium dynamics in ultrarelativistic heavy ion collisions, in particular for the extraction of transport coefficients such as shear viscosity.

Boosting the efficiency up to 33 % for chalcogenide tin mono-sulfide-based heterojunction solar cell using SCAPS simulation technique

In the present article, an FTO/n-ZnO/SnS/Sb2S3/Au heterojunction photovoltaic cell structure was modeled and the cell performance in terms of output parameters viz. open circuit voltage (Voc), short-circuit current density (Jsc), efficiency (η) and fill factor (FF) by varying carrier concentration, and thickness of different layers involved, was studied at ambient conditions. The overall theoretical performance of the designed photovoltaic cell was examined using SCAPS 1-D simulation programme. The simulation programme operates by resolving semiconductor equations such as the Poisson equation, drift-diffusion equation, and continuity equation for both electrons and holes. The equations are generally solved in one-dimension (1D) in a steady state condition. The role of Sb2S3 HTL on the performance study of tin sulfide chalcogenide-based heterojunction photovoltaic cell (HPVC) was also analyzed. Moreover, the impact of functioning temperature between 273 K and 350 K on the performance of photovoltaic cell was observed, and an increase in temperature resulted in poor performance. The impact of different metal back contact work functions and some more electrical parameters viz. series-shunt resistances, defect concentration of layers and interfaces, and radiative recombination coefficients were also observed on HPVC. The simulation results revealed that the HTL Sb2S3 forms proper band alignment with the SnS chalcogenide absorber layer and enhanced the efficiency, η of HPVC by lessening the carrier recombination at the rear contact side. The dislocation and interface defect reduced significantly which leads to the smooth conduction of hole carriers, therefore, preventing of minority carriers to recombine within the absorber. A significant increase in the efficiency ∼33.24 % together with performance cell parameters Jsc ∼34.38 mA/cm2, Voc ∼1.09 V, and FF ∼88.49 % was observed.

A Generalized Townsend's Theory for Paschen Curves in Planar, Cylindrical, and Spherical Geometries in Planetary Atmospheres

AbstractIn this work, we focus on plasma discharges produced between two electrodes with a high potential difference, resulting in the ionization of the neutral particles supporting a current in a gaseous medium. At low currents and low temperatures, this process can create luminescent emissions: glow and corona discharges. The parallel plate geometry used in Townsend's theory lets us develop a theoretical formalism, with explicit solutions for the critical voltage effectively reproducing experimental Paschen curves. However, most discharge processes occur in non‐parallel plate geometries, such as discharges between particles in multiphase systems and between cylindrical conductors. Here, we propose a generalization of the classic parallel plate configurations to concentric spherical and coaxial cylindrical geometries in Earth, Mars, Titan, and Venus atmospheres. In a spherical case, a small radius effectively represents a sharp tip rod, while larger, centimeter‐scale radii represent blunted tips. In cylindrical geometries, small radii resemble thin wires. We solve continuity equations in the gap and estimate a critical radius and minimum breakdown voltage that allows the formation of a glow discharge. We show that glow coronæ form more easily in Mars's low‐pressure, CO2‐rich atmosphere than in Earth's high‐pressure, N2‐rich atmosphere. Additionally, we present breakdown criteria for Titan and Venus, two planets where discharge processes have been postulated. We further demonstrate that critical voltage minima occur at 0.5 cm⋅Torr for all three investigated geometries, suggesting easier initiation around millimeter‐size particles in dust and water clouds. This approach could be readily extended to examine other multiphase flows with inertial particles.

Hybrid lattice-Boltzmann-finite-difference approach for the simulation of micro-phase-change-material slurry in convective flow.

In this paper, we present a hybrid numerical scheme that couples the lattice Boltzmann method (LBM) with the finite difference method (FDM) to model micro-phase-change-material (MPCM) suspensions in a minichannel. Within this framework, the LBM was employed to solve the continuity, momentum, and energy equations for the fluid domain, while a Lagrangian scheme replicates the motion of MPCM particles. The LBM is coupled with an FDM solver which operates under the lumped capacitance assumption to address the phase-change phenomena within the microparticles. This hybrid coupling eliminates the necessity for any specific treatment in handling phase transitions and tracking phase interfaces. The proposed method is first evaluated on classic particle cases, demonstrating its ability to achieve four-way coupling. Furthermore, the current model effectively adapted viscosity changes when integrating the microparticles, obviating the need for homogenous viscosity models. Subsequently, the potential of this approach is demonstrated by examining the influence of the near-wall thermal interaction of MPCM particles considering three scenarios based on particle density: light (ρ_{p}<ρ_{f}), neutrally buoyant (ρ_{p}≈ρ_{f}), and dense (ρ_{p}>ρ_{f}) microparticles. The hybrid approach further revealed insights into the impact of the volume fraction on the heat transfer coefficient as well as on the overall heat transfer coefficient and performance index from a Lagrangian perspective.

The particle paths of hyperbolic conservation laws

Nonlinear scalar conservation laws are traditionally viewed as transport equations. We take instead the viewpoint of these PDEs as continuity equations with an implicitly defined velocity field. We show that a weak solution is the entropy solution if and only if the ODE corresponding to its velocity field is well-posed. We also show that the flow of the ODE is 1/2-Hölder regular. Finally, we give several examples showing that our results are sharp, and we provide explicit computations in the case of a Riemann problem.

Magnetohydrodynamic influence on fractional nanofluid convection in infinite vertical porous media with constant heat flux

This study investigates the complex dynamics governing free convection flow in nanofluids under Magnetohydrodynamics (MHD) within a porous medium along an infinite vertical surface. We specifically focus on scenarios involving constant and uniform heat flux as well as heat sources. The governing equations, incorporating the Boussinesq approximation, continuity, and momentum equations, are formulated and solved using Caputo-Fabrizio derivatives and Laplace transforms. This comprehensive approach integrates the Boussinesq approximation and Caputo-Fabrizio fractional derivatives, while the use of Laplace transforms enables precise analytical solutions. Additionally, our investigation highlights the significant impact of nanometer-sized copper particles on fluid properties, transforming the behavior from Newtonian (water-like) to non-Newtonian. This transformation profoundly affects thermal conductivity and velocity behavior within the system. Overall, this research establishes a robust foundation for future studies and provides a framework for exploring non-Newtonian nanofluid systems in the context of MHD-driven free convection flow.

Dynamical analysis of coupled curvature-matter scenario in viable f(R) dark energy models at de Sitter phase

We explore the interaction between dark matter and curvature-driven dark energy within viable f(R) gravity models, employing the phase-space analysis approach of linear stability theory. By incorporating an interacting term, denoted as Q=αHρ~m(κ23H2ρcurv+1) , into the continuity equations of both sectors, we examine dynamics of two f(R) gravity models that adhere to local gravity constraints and fulfill cosmological viability criteria. In de Sitter phase, our investigation reveals modifications of critical points compared to the conventional form, attributed to the introduced interaction. Through a comprehensive phase-space analysis, we illustrate the trajectories near critical points and outline the constraints on the phase space based on cosmological and cosmographic parameters. Furthermore, we can also observe stable behavior of this interacting system during the de Sitter phase of the Universe.

Mass transport at the bottom of propagating surface waves over a rippled bottom

The sea surface can be described by means of the superposition of many sinusoidal functions. However, quite often the amplitude of each component turns out to be much smaller than its wavelength, and any component evolves independently of the others. Hence, it is common to investigate the dynamics of a simple monochromatic surface wave. Hereinafter, the flow generated by a monochromatic surface wave within the bottom boundary layer over a rippled sea bed is determined by means of the numerical integration of vorticity and continuity equations. The forcing term that drives the fluid motion within the boundary layer is evaluated assuming that the steepness of the monochromatic surface wave is much smaller than one and considering the first term of the Stokes expansion. Even though the irrotational flow that forces the viscous rotational flow near the sea bottom is symmetric with respect to the ripple crests, Blondeaux and Vittori [“A route to chaos in an oscillatory flow: Feigenbaum scenario,” Phys. Fluids A 3(11), 2492–2495 (1991a)] showed that the symmetry of the flow field is broken when the Reynolds number becomes larger than a threshold value Rδ,t1 that depends on the geometrical characteristics of the ripples. The results of Blondeaux and Vittori [“A route to chaos in an oscillatory flow: Feigenbaum scenario,” Phys. Fluids A 3(11), 2492–2495 (1991a)] suggest that, when the Reynolds number is larger than Rδ,t1 but not too far from it, a steady current is also generated. Hereinafter, the steady velocity component is determined as a function of the ripple characteristics.

Secondary estuarine circulation and the related vertical mixing at the sill of Ambon Bay, eastern Indonesia

Secondary estuarine circulation at the sill of Ambon Bay during wet season (July 2019) was investigated to add important insights into the existing knowledge on primary estuarine circulation done by a recent study. The datasets from the previous study were employed to calculate tidal-mean vertical advection as secondary estuarine circulation at the sill. Vertical mixing at the sill was also quantified using the continuity equation. The vertical profiles of vertical advection at the sill of Ambon Bay formed bow-like shapes with zero value at the surface that increased to reach maximum at the mid-depth of the sill (∼6 m depth), subsequently decreasing to zero at the seabed. Vertical advection at the sill was found to be larger during spring tide (∼2.5 × 10−3 m/s) than during neap tide (∼1 × 10−3 m/s). Vertical diffusivity, indicating the magnitude of vertical mixing at the sill of Ambon Bay, showed similar characteristics to vertical advection in terms of vertical profiles (bow-like shapes) and spring/neap tidal variation (spring tide value: 8.5 × 10−3 m2/s; neap tide value: ∼5 × 10−3 m2/s). The intense vertical mixing at the sill of Ambon Bay has the potential to create turbid water linked to resuspension of seabed sediment in the location.

A STABILIZED FINITE ELEMENT METHOD FOR THE STOKES EIGENVALUE PROBLEM

A stabilized finite element method is considered for the Stokes eigenvalue problem based on the lowest equal-order element pair, which does not need to choose stabilization parameter. Stabilization terms of the stabilized finite element method include not only the term related to the momentum equation but also the continuity equation. Furthermore, combined the approximation of compact operator with the analysis of Stokes source problem, error estimates of the present method are deduced. Finally, numerical tests are made to confirm effectively.

An improved deviation model for transonic stages in axial compressors

Deviation model is an important model for through-flow analysis in axial compressors. Theoretical analysis in classical deviation models is developed under the assumption of one-dimensional flow, which is controlled by the continuity equation. To consider three-dimensional characteristics in transonic flow, this study proposes an improved theoretical analysis method combining force analysis of the blade-to-blade flow with conventional analysis of the continuity equation. Influences of shock structures on transverse force, streamwise velocity and streamline curvature in the blade-to-blade flow are analyzed, and support the analytical modelling of density flow ratio between inlet and outlet conditions. Thus, a novel deviation model for transonic stages in axial compressors is proposed in this paper. The empirical coefficients are corrected based on the experimental data of a linear cascade, and the prediction accuracy is validated with the experimental data of a three-stage transonic compressor. The novel model provides accurate predictions for meridional flow fields at the design point and performance curves at design speed, and shows obvious improvements on classical models by Carter and Çetin.