Transport Equation Research Articles

Nonpolynomial Spline for Numerical Solution of Singularly Perturbed Convection-Diffusion Equations with Discontinuous Source Term

Nonpolynomial Spline for Numerical Solution of Singularly Perturbed Convection-Diffusion Equations with Discontinuous Source Term

Efficient computational hybrid method for the solution of 2D multi-term fractional order advection-diffusion equation

Abstract Multi-term time-fractional advection diffusion equations are vital for simulating a wide range of physical phenomena, including fluid dynamics and environmental transport processes. However, due to their natural complexity, these equations pose challenges for conventional numerical approaches. In this article, we develop a high order accurate method to solve the multi-term time-fractional advection diffusion equations. We combine the Laplace transform (LT) to integrate the considered equations in time, with Chebyshev spectral method (CSM) for spatial terms The proposed method produces highly accurate solutions with remarkably low computational cost as compared to finite difference method. The propose numerical scheme first employs the LT which reduces the considered problem into a finite set of elliptic equations which may be solved in parallel. Then, the CSM is employed for the disctrezation of spatial operators, which makes it possibly to accurately represent the solution chebyshev grid. Finally, numerical inversion of LT is used to convert the obtain solution from the Laplace domain into the real domain. This work utilizes the modified Talbot’s method and Stehfest’s method for numerical inversion of the LT. To measure the performance, efficiency, and accuracy of the suggested approach, numerical approximations of three models are acquired and verified against the exact solution. The outcomes presented in tables and figures demonstrate that the modified Talbot’s method performed better as compared to Stehfest’s method.

Inversion of the attenuated momenta ray transform of planar symmetric tensors

We present a reconstruction method that stably recovers the real valued, symmetric tensors compactly supported in the Euclidean plane, from knowledge of their attenuated momenta ray transform. The problem is recast as an inverse boundary value problem for a system of transport equations, which we solve by an extension of Bukhgeim’s A-analytic theory. The method of proof is constructive. To illustrate the reconstruction method, we present results obtained in the numerical implementation for the non-attenuated case of one-tensors.

Thermodynamics of Irreversible Processes: Fundamental Constraints, Representations, and Formulation of Boundary Conditions

Starting from the analysis of the lack of positivity of the Cattaneo heat equation, this work addresses the thermodynamic relevance of the positivity constraint in irreversible thermodynamics, that is at least as significant as the entropic constraints. The fulfillment of this condition in hyperbolic models leads to the parametrization of the concentration fields with respect to internal variables associated with the microscopic dynamics. Using Brownian motion theory as a landmark example for deriving macroscopic transport equations from the equations of motion at the particle/molecular level, we discuss two typical problems involving hydrodynamic interactions at the microscale: surface chemical reactions at a solid interface of a diffusing reactant, and mass-balance equations in a complex viscoelastic fluid, in which the physics of the interaction leads either to overcoming the parabolic diffusion model or to considering the parametrization of the concentration with respect to the degrees of freedom associated with the relaxation dynamics of the solvent fluid.

The varying viscosity impact in an inclined peristaltic channel with diffusion‐thermo and thermo‐diffusion

AbstractA mathematical model is constructed to investigate the behavior of peristaltic flow of Jeffrey fluid in an inclined tapered asymmetric porous channel. The fluid viscosity is taken as space dependent variable quantity. Heat absorption, Soret and Dufour effects are also retained in the current scrutiny. These preferences have broad applications in engineering, biology and industry. We began our investigation by taking into account the two‐dimensional inclined asymmetric porous channel. In the context of mathematical modeling, the appropriate dimensional nonlinear equations for momentum, heat and mass transport are simplified into dimensionless equations by applying the essential estimation of long wavelength and low Reynolds number. The solution of the governing equations is executed numerically. A graphical depiction of many crucial physical characteristics on velocity, temperature, concentration, heat transfer rate, Nusselt number and Streamlines have been reported in ending section. Temperature profile exhibits an escalation with the augmentation of Brinkman number and Dufour number . For the growing values of Prandtl number , an increment in temperature profile is observed whilst a reverse tendency is captured for concentration profile. It is noted that concentration profile falls down owing to the enhancement in Soret number and Schmidt number . An oscillatory outlook is noticed for heat transfer rate and Nusselt number. The novelty of this proposed model in the research domain specifically depends on considerations of the combined study of the Variable viscosity, Darcy resistance, Viscous dissipation, Mixed convention, Heat absorption, Soret and Dufour effects in peristaltic flow of non‐ Newtonian Jeffrey fluid in an inclined Asymmetric tapered channel under the influence of convective boundary conditions.

Generalized transport characterizations for short oceanic internal waves in a sea of long waves

Wave turbulence provides a conceptual framework for weakly nonlinear interactions in dispersive media. Dating from five decades ago, applications of wave turbulence theory to oceanic internal waves assigned a leading-order role to interactions characterized by a near equivalence between the group velocity of high-frequency internal waves with the phase velocity of near-inertial waves. This scale-separated interaction leads to a Fokker–Planck (generalized diffusion) equation. More recently, starting four decades ago, this scale-separated paradigm has been investigated using ray tracing methods. These ray methods characterize spectral transport of energy by counting the amplitude and net velocity of wave packets in phase space past a high-wavenumber gate prior to ‘breaking’. This explicitly advective characterization is based on an intuitive assignment and lacks theoretical underpinning. When one takes an estimate of the net spectral drift from the wave turbulence derivation and makes the corresponding assessment, one obtains a prediction of spectral transport that is an order of magnitude larger than either observations or reported ray tracing estimates. Motivated by this contradiction, we report two parallel derivations for transport equations describing the refraction of high-frequency internal waves in a sea of random inertial waves. The first uses standard wave turbulence techniques and the second is an ensemble-averaged packet transport equation characterized by the dispersion of wave packets about a mean drift in the spectral domain. The ensemble-averaged transport equation for ray tracing differs in that it contains the intuitively motivated advective term. We conclude that the aforementioned contradiction between theory, numerics and observations needs to be taken at face value and present a pathway for resolving this contradiction.

Modeling heat and mass transfer in metal hydride hydrogen storage systems: Impact of operating parameters and reactor geometry

This work introduces a transient 2D axis-symmetrical model for hydrogen absorption in a cylindrical LaNi5 reactor employing finite volume method with a fully implicit Euler's time integration scheme, coupling equations for heat, mass, and momentum transport. The validated model is used to investigate the impacts of pressure, cooling fluid temperature, and variations in reactor geometry and size on temperature and reacted fraction profiles. The findings reveal that the charging pressure affects both peak temperature and reaction kinetics, whereas cooling fluid temperature predominantly impacts the absorption kinetics. A comparative analysis of two models, one incorporating Darcy's velocity and one without, demonstrates that while Darcy's law introduces numerical instability in the coupled equations, its impact on the model outcomes is negligible. The effect of changing the non-homogeneous Neumann to Dirichlet boundary condition is also demonstrated to anticipate the utilization of phase change materials (PCM) instead of the cooling fluid.

A Semi-implicit Finite Volume Scheme for Incompressible Two-Phase Flows

AbstractThis paper presents a mass and momentum conservative semi-implicit finite volume (FV) scheme for complex non-hydrostatic free surface flows, interacting with moving solid obstacles. A simplified incompressible Baer-Nunziato type model is considered for two-phase flows containing a liquid phase, a solid phase, and the surrounding void. According to the so-called diffuse interface approach, the different phases and consequently the void are described by means of a scalar volume fraction function for each phase. In our numerical scheme, the dynamics of the liquid phase and the motion of the solid are decoupled. The solid is assumed to be a moving rigid body, whose motion is prescribed. Only after the advection of the solid volume fraction, the dynamics of the liquid phase is considered. As usual in semi-implicit schemes, we employ staggered Cartesian control volumes and treat the nonlinear convective terms explicitly, while the pressure terms are treated implicitly. The non-conservative products arising in the transport equation for the solid volume fraction are treated by a path-conservative approach. The resulting semi-implicit FV discretization of the mass and momentum equations leads to a mildly nonlinear system for the pressure which can be efficiently solved with a nested Newton-type technique. The time step size is only limited by the velocities of the two phases contained in the domain, and not by the gravity wave speed nor by the stiff algebraic relaxation source term, which requires an implicit discretization. The resulting semi-implicit algorithm is first validated on a set of classical incompressible Navier-Stokes test problems and later also adds a fixed and moving solid phase.

Design Principles Guided by DFT Calculations and High-Throughput Frameworks for the Discovery of New Diamond-like Chalcogenide Thermoelectric Materials.

Rational design principles are one pathway to discovering new materials. However, technological breakthroughs rarely occur in this way because these design principles are usually based on incremental advances that seldom lead to disruptive applications. The emergence of machine-learning (ML) and high-throughput (HT) techniques has changed the paradigm, opening up new possibilities for efficiently screening large chemical spaces and creating on-the-fly design principles for the discovery of novel materials with desired properties. In this work, the approach is used to discover novel thermoelectric (TE) materials based on quaternary diamond-like chalcogenides. A HT framework that integrates density functional theory calculations, ML, and the solution of the Boltzmann transport equation is used to efficiently rationalize the transport properties of these compounds and identify those with potential as TE materials, achieving ZT values above 2.

Green-Kubo Paralell Tempering molecular dynamics determination of the thermal conductivity for the wurtzite ZnO: Statistical problems of the method

Abstract The thermal conductivity for the wurtzite ZnO is determined in the temperature range from 300 to 1100 K by using parallel tempering molecular dynamics within the Green-Kubo approach and a classical Morse-Born-Mayer-Coulomb hybrid interaction potential. Compared to other previous calculations for the thermal conductivity of common crystals within the same Green-Kubo and molecular dynamics approach, the used parallel tempering scheme shows some appealing improvements in the calculation of the time self-correlation of the heat flux vector, although at the price of using a relatively large number of total computational steps. However, in spite of the the found improvements for the calculation of the self-correlation of the heat flux vector, some statistical problems on this point remain on the particular application of the method. Finally, despite the presence of a clear statistical noise, the obtained values and temperature trend of the calculated thermal conductivity shows the classical 1/T decaying behavior reported in previous works for wurtzite ZnO and other related semiconductor systems using the alternative Boltzmann transport equation theory.

Effects of Reynolds Number on the Wake Characteristics of a Notchback Ahmed Body

Abstract This paper investigates the effects of Reynolds number on the wake characteristics of a notchback Ahmed body with effective backlight angle, βe=17.8 deg. The Reynolds number based on the body height was varied from 5×103 to 5×104. Prior to the Reynolds number investigation, a Reynolds-averaged Navier–Stokes (RANS) model assessment was performed using nine turbulence models consisting of one- and two-equation eddy-viscosity models and second moment closure models. The standard Spalart–Allmaras model was the only model that accurately predicted the asymmetric time-averaged wake topology, as reported in previous studies, for the βe=17.8 deg notchback Ahmed body at Re=5×104. The drag coefficient decreased with increasing Reynolds number, while the lift coefficient remained constant for Re≥1×104. The wake structure exhibited three regimes: symmetric (Re≤1×104), transitionally asymmetric (1×104<Re≤3.5×104), and fully asymmetric (Re>3.5×104) states. The wake asymmetry was attributed to an imbalance in entrainment from the sides and asymmetric separation from the roof and the C-pillars of the body. The tilting and stretching terms in the vorticity transport equation were used to provide insight into the source of asymmetry in the vorticity field around the body.

High-order Lagrange-Galerkin methods for the conservative formulation of the advection-diffusion equation

Abstract We introduce in this paper the numerical analysis of high order both in time and space Lagrange-Galerkin methods for the conservative formulation of the advection-diffusion equation. As time discretization scheme we consider the Backward Differentiation Formulas up to order $q=5$. The development and analysis of the methods are performed in the framework of time evolving finite elements presented in C. M. Elliot and T. Ranner, IMA Journal of Numerical Analysis41, 1696–1845 (2021). The error estimates show through their dependence on the parameters of the equation the existence of different regimes in the behavior of the numerical solution; namely, in the diffusive regime, that is, when the diffusion parameter $\mu $ is large, the error is $O(h^{k+1}+\varDelta t^{q})$, whereas in the advective regime, $\mu \ll 1$, the convergence is $O(\min (h^{k},\frac{h^{k+1} }{\varDelta t})+\varDelta t^{q})$. It is worth remarking that the error constant does not have exponential $\mu ^{-1}$ dependence.

TauFlowNet: Revealing latent propagation mechanism of tau aggregates using deep neural transport equations

Mounting evidence shows that Alzheimer's disease (AD) is characterized by the propagation of tau aggregates throughout the brain in a prion-like manner. Since current pathology imaging technologies only provide a spatial mapping of tau accumulation, computational modeling becomes indispensable in analyzing the spatiotemporal propagation patterns of widespread tau aggregates from the longitudinal data. However, current state-of-the-art works focus on the longitudinal change of focal patterns, lacking a system-level understanding of the tau propagation mechanism that can explain and forecast the cascade of tau accumulation. To address this limitation, we conceptualize that the intercellular spreading of tau pathology forms a dynamic system where each node (brain region) is ubiquitously wired with other nodes while interacting with the build-up of pathological burdens. In this context, we formulate the biological process of tau spreading in a principled potential energy transport model (constrained by brain network topology), which allows us to develop an explainable neural network for uncovering the spatiotemporal dynamics of tau propagation from the longitudinal tau-PET scans. Specifically, we first translate the transport equation into a GNN (graph neural network) backbone, where the spreading flows are essentially driven by the potential energy of tau accumulation at each node. Conventional GNNs employ a l2-norm graph smoothness prior, resulting in nearly equal potential energies across nodes, leading to vanishing flows. Following this clue, we introduce the total variation (TV) into the graph transport model, where the nature of system's Euler-Lagrange equations is to maximize the spreading flow while minimizing the overall potential energy. On top of this min-max optimization scenario, we design a generative adversarial network (GAN-like) to characterize the TV-based spreading flow of tau aggregates, coined TauFlowNet. We evaluate our TauFlowNet on ADNI and OASIS datasets in terms of the prediction accuracy of future tau accumulation and explore the propagation mechanism of tau aggregates as the disease progresses. Compared to the current counterpart methods, our physics-informed deep model yields more accurate and interpretable results, demonstrating great potential in discovering novel neurobiological mechanisms through the lens of machine learning.

A numerical validation between the neutron transport and diffusion theories for a spatial kinetics problem

In this paper, a comparative analysis of numerical results of the neutron transport and diffusion theories for steady-state and transient multigroup problems is presented. The neutron transport equation is known as the one that best describes the behavior of the neutron population in a nuclear reactor. However, due to the difficulty of working with its complete form, other models are considered as approximations to this equation. One such approximation is the neutron diffusion equation, which uses the Fick's Law. It is well known, however, that the diffusion model may not work well under specific conditions. The objective of this work is to establish a quantitative comparison of numerical results obtained for the K dominant eigenvalue and for the scalar fluxes from the two theories and to analyze the influence of the model on the results.

Investigation of Non-Uniform Inflow Effects on Impeller Forces in Axial-Flow Pumps Operating as Turbines

Due to the existence of an inlet elbow, transmission shaft, and other structural components, the inflow of axial-flow pumps as turbines (PATs) becomes non-uniform, resulting in the complexity of internal flow and adverse effects such as structural vibration. In this paper, numerical methods were employed to explore the non-uniform inflow effects on impeller forces and internal flow field characteristics within an axial-flow PAT. The study results indicated that non-uniform inflow caused uneven pressure distribution inside the impeller, which leads to an imbalance in radial forces and offsetting the center of radial forces. With an increasing flow rate, the asymmetry of radial forces as well as the amplitude of their fluctuations increased. Non-uniform inflow was found to induce unstable flow structures inside the impeller, leading to low-frequency, high-amplitude pressure fluctuations near the hub. Using the enstrophy transport equation, it was shown that the relative vortex generation term played a major part in the spatiotemporal evolution of vortices, with minimal viscous effects.

Physical and Numerical Experimentation of Water Droplet Collision on a Wall: A Comparison between PLIC and HRIC Schemes for the VOF Transport Equation with High-Speed Imaging

Liquid films are often found in engineering applications with thicknesses ranging from micrometer scales to large scales with a wide range of applications. To optimize such systems, researchers have dedicated themselves to the development of new techniques. To further contribute to this development, the objective of this work is to present the results of the collision of water droplets on a wall by means of experimentation and numerical simulations. For physical experimentation, an injector is used to generate a chain of water droplets that collide with the opposite wall, forming a liquid film. Images of the droplets were obtained using two high-speed recording cameras. The results for different droplet sizes and impact angles are presented and the relationship between the momentum parameter and non-dimensional pool size was established. Modeling such processes is a common challenge in engineering, with different techniques having their advantages and limitations. The simulations in this work were run using the volume of fluid method, which consists of solving a transport equation for the volume fraction of each considered fluid. A correlation was found between the surface tension to momentum transport ratio, Scd, and the non-dimensional pool size for different droplet sizes and impact angles. Regions where partial depositions were most likely to occur were found via physical experiments.

Functional renormalization group study of thermodynamic geometry around the phase transition of quantum chromodynamics

We investigate the thermodynamic geometry of the quark-meson model at finite temperature, T, and quark number chemical potential, μ. We extend previous works by the inclusion of fluctuations exploiting the functional renormalization group approach. We use recent developments to recast the flow equation into the form of an advection-diffusion equation. We adopt the local potential approximation for the effective average action. We focus on the thermodynamic curvature, R, in the (μ,T) plane, in proximity of the chiral crossover, up to the critical point of the phase diagram. We find that the inclusion of fluctuations results in a smoother behavior of R near the chiral crossover. Moreover, for small μ, R remains negative, signaling the fact that bosonic fluctuations reduce the capability of the system to completely overcome the fermionic statistical repulsion of the quarks. We investigate in more detail the small μ region by analyzing a system in which we artificially lower the pion mass, thus approaching the chiral limit in which the crossover is actually a second order phase transition. On the other hand, as μ is increased and the critical point is approached, we find that R is enhanced and a sign change occurs, in agreement with mean field studies. Hence, we completely support the picture that R is sensitive to a crossover and a phase transition, and provides information about the effective behavior of the system at the phase transition. Published by the American Physical Society 2024

Theoretical study of the effect of ionization effect on the thermal conductivity of stannous oxide

Effectively modulating the thermal conductivity of materials is critical for meeting the various requirements of thermal-management devices. In this work, the influence of ionization on the phonon-transport properties of stannous oxide (SnO) was systematically investigated using first-principles calculations combined with the Boltzmann transport equation. The results show that ionization has a positive effect on the thermal conductivity of SnO, and this phenomenon can be further enhanced with increased ionization magnitude. Specifically, it was found that the thermal conductivities of SnO along the x (y) and z directions could be increased by 35%/200% and 65%/300% after the removal of four and eight electrons, respectively, from neutral SnO. The phonon mode information implies that the enhancement of thermal conductivity mainly originates from the suppression of anharmonicity in the ionized SnO. This behavior was further demonstrated by analyzing the root mean square displacement and potential-well structure. More in-depth examination suggested that the enhancement of the thermal conductivity of SnO does not originate from the ionization itself, but from the internal strain in the lattice caused by the ionization. The findings presented in this work elucidate how ionization can impact thermal conductivity, providing theoretical guidance for modulating thermal conductivity at the electron level.

Anisotropic Fourier Heat Conduction and phonon Boltzmann transport equation based simulation of time domain thermo-reflectance experiments

The anisotropic Fourier Heat Conduction Equation (FHCE) and the multidimensional phonon Boltzmann Transport Equation (BTE) were solved numerically in cylindrical coordinates and in time domain to simulate a Time Domain Thermo-Reflectance (TDTR) experimental silicon/aluminum substrate/transducer setup. The out-of-phase response of the probe laser was predicted at various beam offset distances for a pump laser pulse frequency of 80 MHz and modulation frequency of 10 MHz and compared against experimental measurements for a silicon substrate. The isotropic FHCE was also solved for comparison. Results show that the isotropic FHCE with bulk thermal conductivity of 145 W/m/K significantly underpredicts the out-of-phase temperature difference, particularly at smaller beam offsets. With an isotropic thermal conductivity of 105 W/m/K, the computed results match experimental data at smaller beam offsets well, but overpredicts the experimental data at larger beam offsets. An almost-perfect match is obtained by using an anisotropic thermal conductivity wherein the radial (in-plane) thermal conductivity is set to 85 W/m/K and the axial (through-plane) conductivity is set to 130 W/m/K. The multidimensional frequency and polarization dependent phonon BTE is next solved. The BTE results for the out-of-phase temperature difference match experimental observations well at small and intermediate beam offsets, but overpredicts the experimental data at larger beam offsets. FHCE results are fitted to the BTE predictions, and the extracted (best fit) thermal conductivity is found to be 110 W/m/K.

Возбуждение релаксационных колебаний на искривленной межфазной границе в условиях внутренней задачи

The oscillatory mode of solutal Marangoni convection during the absorption of a surfactant from a homogeneous external solution into a water droplet is studied numerically. This is caused by the effect of gravity, which promotes the sedimentation of surfactant molecules in an aqueous medium. This version of oscillatory convection arising under the conditions of an internal problem was recently discovered experimentally. In the present paper, we consider the case of a chemically inert system, in which there are no reactions. The effects of interfacial deformation are assumed to be insignificant and thus they are neglected. The mathematical model includes the Navier—Stokes equations written in the Hele-Shaw and Boussinesq approximations, and the equations of surfactant transport in the system. We assume that the characteristic time of surfactant adsorption is shorter than the time of its diffusion in both solutions, which makes it possible to ignore the formation of a surface phase. The boundary value problem includes the equilibrium condition of the system, which takes into account different values of the chemical potential in the phases. It is shown that a water droplet is a surfactant accumulator that diffuses from the organic phase. The problem is solved in dimensional form using the COMSOL Multiphysics package and based on a set of physical constants for acetic acid which, like many other members of the carboxylic acid family, has the properties of surfactant in water. It was found that direct numerical simulation of the system is able to reproduce the relaxation oscillations observed in the experiment only under the additional phenomenological assumption of non-Newtonian rheology of the interface, which was previously proposed for the external problem. The physical mechanism which may be responsible for the delayed onset of Marangoni instability is discussed. We demonstrate that periodic oscillations are generated inside the drop due to the competition between the Marangoni effect and the gravity-dependent convective instability of the solution. Using direct numerical simulation, we identified the structures of convective motion at the interface and in its neighborhood, determined the flow intensity as a function of time, and obtained the range of change in the oscillation period.