Transport Equation Research Articles

The asymptotic stability of diverging traveling waves for reaction–advection–diffusion equations in cylinders

The asymptotic stability of diverging traveling waves for reaction–advection–diffusion equations in cylinders

Anomalous diffusion of self-propelled particles

Abstract The transport equation of active motion is generalised to consider time-fractional dynamics to describe the anomalous diffusion of self-propelled particles. In the present study, we consider an arbitrary active motion pattern modelled by a scattering function that defines the dynamics of the change of the self-propulsion direction. The exact probability density of the particle positions at a given time is obtained. From it, the time dependence of the firsts moments, i.e. the mean square displacement and the kurtosis for an arbitrary scattering function, are derived and analysed. Anomalous diffusion is found with a crossover of the scaling exponent from 2α in the short-time regime to α in the long-time one, 0 < α < 1 being the order of the fractional derivative considered. It is shown that the exact solution found satisfies a fractional diffusion equation that accounts for the non-local and retarded effects of the Laplacian of the probability density function through a coupled temporal and spatial memory function. Such a memory function holds the complete information of the active-motion pattern. In the long-time regime, space and time are decoupled in the memory function, and the time fractional telegrapher’s equation is recovered. The theoretical framework presented here can be applied as model of active motion that exhibits anomalous diffusion.

Convection-Diffusion Equations in Uniformly Local Lebesgue Spaces

Convection-Diffusion Equations in Uniformly Local Lebesgue Spaces

Dynamics of Fourier's and Fick's laws on the convectively heated oscillatory sheet under Arrhenius kinetics: The finite-difference technique

The significance of chemical reaction with activation energy and convective boundary conditions on the fluid flow via an oscillatory stretchy surface in the existence of permeable media and radiation is analyzed in this study. This inspection presents Fourier and Fick's laws-based equations for heat, mass transport, and liquid flow through an oscillating stretchy sheet. Understanding these dynamics aids in the optimisation of catalytic reaction settings, where gradients greatly influence reaction rates in concentration and temperature. The governing differential equations of the current study are modelled and changed into their non-dimensional form by employing suitable similarity variables. The finite difference method (FDM) is also used to numerically solve the obtained dimensionless equations. The influence of many factors on the several profiles is portrayed with graphical representations. The outcome of the unsteadiness and porosity parameters on the velocity profile with time coordinate is depicted. The increase in the radiation parameter and Biot number upsurges the thermal profile. The temperature reduces as the unsteadiness parameter and temperature relaxation time parameter grow. The upsurge in the activation energy parameter intensifies the mass transport. The rise in concentration relaxation time parameter diminishes the concentration profile.

Homogenization for convection-enhanced thermal transport in sea ice

Sea ice regulates heat exchange between the ocean and atmosphere in Earth’s polar regions. The thermal conductivity of sea ice governs this exchange, and is a key parameter in climate modelling. However, it is challenging to measure and predict due to its sensitive dependence on temperature, salinity and brine microstructure. Moreover, as temperature increases, sea ice becomes permeable, and fluid can flow through the porous microstructure. While models for thermal diffusion through sea ice have been obtained, advective contributions to transport have not been considered theoretically. Here, we homogenize a multiscale advection–diffusion equation that models thermal transport through porous sea ice when fluid flow is present. We consider two-dimensional models of convective flow and use an integral representation to derive bounds on the thermal conductivity as a function of the Péclet number. These bounds guarantee enhancement in the thermal conductivity due to the added flow. Further, we relate the Péclet number to temperature, making these bounds useful for global climate models. Our analytic approach offers a mathematical theory which can not only improve predictions of atmosphere–ice–ocean heat exchanges in climate models, but can provide a theoretical framework for a range of problems involving advection–diffusion processes in various fields of application.

Two-level Arrow–Hurwicz iteration methods for the steady bio-convection flows

To avoid solving a saddle-point system, in this paper, we study two-level Arrow–Hurwicz finite element methods for the steady bio-convection flows problem which is coupled by the steady Navier–Stokes equations and the steady advection–diffusion equation. Using the mini element to approximate the velocity, pressure, and the piecewise linear element to approximate the concentration, we use the linearized Arrow–Hurwicz iteration scheme to obtain the coarse mesh solution and use three different one-step Stokes/Oseen/Newton linearized scheme to obtain the fine mesh solution. The optimal error estimate O(h+H2+χm/2) of the velocity and concentration in the H1-norm and the pressure in the L2-norm are derived, where h and H are fine and coarse mesh sizes, respectively, and χm/2 denotes the iteration error with 0<χ<1. Numerical results are given to support the theoretical analysis and confirm the efficiency of the proposed two-level methods.

Interpolated coefficients stabilizer-free weak Galerkin method for semilinear parabolic convection–diffusion problem

We continue our effort in Li et al. (2024) to explore an interpolated coefficients stabilizer-free weak Galerkin finite element method (IC SFWG-FEM) to solve a one-dimensional semilinear parabolic convection–diffusion equation. Due to the introduction of interpolated coefficients and the design without stabilizers, this method not only possesses the capability of approximating functions and sparsity in the stiffness matrix, but also reduces the complexity of analysis and programming. Theoretical analysis of stability for the semi-discrete IC SFWG finite element scheme is provided. Moreover, numerical experiments are carried out to demonstrate the effectivity and stability.

Phonon modal analysis of thermal transport in ThO2 with point defects using equilibrium molecular dynamics

Defects can significantly degrade the thermal conductivity of ThO2, an advanced nuclear fuel material as well as a surrogate for other fluorite-structured materials. We investigate how point defects in ThO2 impact phonon mode-resolved thermal transport. By incorporating phonon modes from lattice dynamics, we decompose the trajectory and heat flux to phonon normal mode space and extract key phonon properties, including phonon relaxation times and their contributions to thermal conductivity. We implement two methods. The first method is based on the Green Kubo formalism to resolve the contribution of each phonon mode to thermal conductivity. The second resolves the lifetime of individual phonon modes and the thermal conductivity is calculated using the Boltzmann transport equation within relaxation time approximation. Notably, a lower contribution of acoustic modes is revealed compared to perturbative approaches considering only three-phonon scattering processes. The effects of four types of point defects are evaluated. The strongest impact on a reduction in thermal conductivity is from Th interstitials, followed by Th vacancies. O interstitials/vacancies have a similar impact, albeit smaller than defects on the thorium sublattice. These observations are consistent with previous studies.

Enhancing Transport Processes in Stratified Fluids: Exploring Rotational and Stretching Effects Coupled with Magnetic Fields, Heat, Mass Transfer and Thermal/Solutal Slip

This research paper presents an in-depth investigation into the complex interactions governing stratified fluid flow in the presence of rotational and stretching phenomena, magnetic field influences, heat and mass transfer effects, and the incorporation of thermal and solutal-slips at fluid interfaces. Stratified fluid systems are commonly encountered in various natural and industrial processes, ranging from geophysical flows to chemical engineering applications, making a thorough understanding of these phenomena paramount.The study commences by formulating a comprehensive set of governing equations that account for the unique interplay of these multifaceted physical phenomena. Integrating thermal and solutal slip conditions at fluid interfaces introduces innovative prospects for precisely controlling and manipulating mass and heat transfer in these systems. This research paper extensively explores layered fluid dynamics, encompassing phenomena such as rotation and stretching, influences from magnetic fields, effects of heat and mass transfer, and the examination of thermal and solutal slip conditions. The insights obtained from this study not only enhance our comprehension of multi-physics fluid dynamics but also offer practical implications for streamlining processes across diverse fields. The principal findings and implications of this research can be summarized as follows: Various factors that affect fluid flow exhibit evident and discernible patterns.The mathematical model includes the Navier-Stokes equations with rotational and stretching terms, the Maxwell equations describing magnetic field effects, and the transport equations for heat and mass transfer. Additionally, thermal and solutal slip conditions at the fluid interfaces are incorporated to capture boundary layer dynamics.Numerical simulations are conducted to explore the intricate dynamics arising from this amalgamation of physical effects. The results reveal complex flow patterns, including stratification, boundary layer development, and vortical structures, offering insights into the behavior of fluids under the joint influence of rotational and stretching phenomena, magnetic fields, and thermal and solutal gradients. Furthermore, the impact of key parameters, such as rotation rate, stretching rate, magnetic field strength, and concentration gradients, is systematically analyzed to elucidate their effects on flow characteristics, heat and mass transfer rates, and concentration profiles within the stratified fluid layers. The Soret effect, or thermophoresis, causes nanoparticles to migrate away from hot regions due to fluid molecule flow and high energy levels. Elevating ξNt results in decreased heat and mass transfer. The Prandtl number shows minimal influence on heat and mass transfer when increased. The disk separation, and fluid viscosity, significantly affect the rotational spectrum. As the rotational parameter increases, heat and mass transfer decrease.

A numerical model for the simulation of complex planar Newtonian interfaces

We present a numerical model for the simulation of complex planar interfaces at which moving solid objects can be immersed, reproducing a wide variety of experimental conditions. The mathematical model consists of the Navier-Stokes equations governing the incompressible viscous flow in the liquid subphase, the transport equation for the evolution of the surfactant concentration at the interface, and the interfacial stress balance equation. The equations are simplified by treating the problem as isothermal and the surfactant as insoluble. The bulk flow equations are discretized using a collocated finite volume method, while the interfacial flow equations are discretized using a finite area method. The Boussinesq-Scriven interface constitutive model and a variant form accounting for extensional viscosity are used to describe the extra surface stress tensor. The coupling between surfactant concentration, interfacial velocity, and bulk velocity is treated implicitly by solving the interfacial and bulk equations sequentially at each time step until a stopping criterion is satisfied. The motion of the solid is treated by an arbitrary Lagrangian-Eulerian method. The model has been implemented in the OpenFOAM framework and allows the incorporation of new interface models and solvers, making the developed new package a versatile and powerful tool in the field of computational rheology. Applications of the model include the numerical simulation of flow around objects, such as probes, immersed at a complex interface, reproducing given experimental conditions, and its use as a tool in the analysis and design of interfacial stress rheometers. Several test cases have been performed to validate the model by comparing the results obtained with analytical solutions and with numerical and experimental results available in the literature.

First-principles study of the effect of Dirac phonons on the thermoelectric properties in monolayer Ge2H2

The Dirac phonons have been widely attention due to their unique edge or surface states. However, research on their influence in the realm of thermoelectric transport in semiconductor materials remains limited. Here, we have systematically investigated the geometric structure, stability, electronic, thermal conductivity, and thermoelectric properties of the monolayer Ge2H2 and Ge2 based on density functional theory and the Boltzmann transport equation (BTE). The phonon spectrum of monolayer Ge2H2 exhibits both Hourglass-type and Dirac-type phonon dispersion curves. The band structure of monolayer Ge2H2 reveals a relatively large band gap of 1.02 eV (1.75 eV for HSE), indicating that it is a direct band gap semiconductor. These two features make monolayer Ge2H2 an ideal material for exploring the effect of Dirac phonons on thermoelectric transport properties. In this paper, we determined that the lattice thermal conductivity (κl) in Ge2H2 is 0.59 Wm-1K-1 at 300 K, but undergoes a sharp increase upon removal of H atoms. This is mainly attributed to the disappearance of the Dirac phonons and the change of lattice vibration modes by removing the H atoms. While also exhibiting high carrier mobilities (134.86 cm2/Vs for holes) and extended scattering time (4.22 × 10−14 s for holes), indicative of excellent thermoelectric performance. With increasing temperature, the ZTmax of p-type doping reaches 0.63 at 700 K in Ge2H2. Remarkably, at 300K, the Hourglass-type phonon contributes only 5.94% to κl, while the Dirac phonon contributes 3.44%. Further analysis reveals that the Dirac phonon within Hourglass-type in monolayer Ge2H2 contributes 2.5% to the ZTmax at 300K. Our investigation elucidates the impact of Dirac phonons on thermoelectric properties, providing valuable insights for future design and exploration of high-ZT thermoelectric materials.

Modelling and simulation on compressible multi-component gas-liquid flows with chemical reaction and phase transition effects

In this paper, mathematical models and numerical algorithm are conducted for simulating compressible two-phase multi-component reactive flows, coupling with the finite-rate-based chemistry and the instantaneous-equilibrium phase transition model. Distinctive from previous multi-component two-phase governing equations, all volume fraction transport equations, with non-conservative characteristics, of gaseous components are substituted with a thermal equilibrium assumption inside the gaseous mixture for closure. Moreover, based on the temperature polynomial (NASA-7) fitted gas properties and the stiffened gas equation of state (SG-EOS) described liquid properties, the instantaneous-equilibrium phase transition model is re-derived, and the solving strategy of the transient phase transition model is detailly provided using an algebraical approach rather than the iterative process. Having included these elements, the proposed model is validated on three canonical tests, first for pure reactive flows, second for pure two-phase flows, and third for complex compressible multi-component two-phase reactive flows with phase transition effects: hydrogen-oxygen detonation interacting with a water droplet. The cases have well demonstrated the proposed model capability and the expected fluid physics in both the multiphase and reactive flows are reproduced.

Periodic wave solutions for a generalized reaction–convection–diffusion equation of high-order

In this paper, we investigate the uniqueness of periodic wave solutions for a generalized reaction–convection–diffusion equation with arbitrarily high-order reaction term or convection term. The main technique is to prove the monotonicity of the ratio of two Abelian integrals by a new criterion and Descartes’ rule of signs. A positive answer to a conjecture stated in Wei and Chen (2023) is given and some numeric simulations are carried out to illustrate the obtained theoretical results.

Nano Thermal Simulation of Graphene Field Effect Transistor Based on Ballistic Diffusive Model

This paper investigates temperature evolution in graphene field-effect transistors. We present a refined Ballistic-diffusive equation model (BDE) tailored for scrutinizing the phonon transport within GNRFET transistors. Furthermore, we examine the influence of the channel length on thermal characteristics. COMSOL Multiphysics is employed to compute the phonon transport and temperature distribution in nano-GNRFET. Our results reveal that the phonon transport anticipated by the suggested BDE model closely corresponds to that derived from the Boltzmann transport equation. The proposed model proficiently gauges the thermal performance of GNRFET, with the outcomes suggesting that the characteristic length of the GNR has a negligible impact on the temperature increase.

First principle investigation of structural, optoelectronic and thermoelectric properties of lead-free Cs2AgAuX6 (X= Br, I) for energy harvesting applications

In this research work, the properties of double perovskites (DPs) are investigated by using the full potential linear augmented plane-wave (FP-LAPW) technique. To compute structural characteristics, we used the generalized gradient approximation (GGA) functional. On the other hand, the optoelectronic and thermoelectric properties of Cs2AgAuX6 (X = Br, I), were calculated using the modified Becke and Johnson (mBJ) potential functional. According to our work, these materials have band gaps of 1.13 eV for X = I and 1.24 eV for X = Br. These materials' optical properties have been investigated through the use of reflectivity, absorption, refractive index, and dielectric constants. With the highest transition values versus photon energy in the visible spectrum, the optical properties point to the potential application of these double perovskites in solar cells. In addition, our studies of the transport properties using the Boltzmann transport equation suggest that our understudy DPs is optimal for thermoelectric uses.

Stability of Left and Right Weyl Essential Spectra

In this work, we use the concept of demicompact linear operators to provide a characterization of the left and right Weyl essential spectra. Additionally, we study the stability of these spectra in Banach space. We illustrate our theoretical results by investigating this type of essential spectra in transport equations.

The theory of scaled electromechanics

A new scaling theory called finite similitude has appeared in the open literature for the scaling of physical systems. The theory is founded on the metaphysical concept of space scaling and consequently can in principle be applied to all physics. With regard to the application of the theory to multi-physics however, an obstacle is dissimilar mathematical formulations, that are preferred and applied in practice. This paper looks to combine electrical and mechanical physics under the rules of the scaling theory for the analysis of scaled electromechanical systems. To facilitate this the physics of electromechanics is described using transport equations on a projected space termed the scaling space. It is shown that this approach unifies the mechanical and electrical descriptions and allows the scaling theory to be applied and for scaling identities to be established. Additionally, on confirming that the scaling space possesses all the attributes of a real physical space (despite being a mere projection), mathematical modelling (to great advantage) is performed directly and integrated with the scaling theory. To showcase the concepts, mathematical models for previously researched electromechanical systems are directly analysed in the new scaling space. It is demonstrated how such models automatically account for scale dependencies in the electromechanical systems they represent. The huge potential of the new approach is revealed providing the means for formulating (for the first time) realistic representative scaled-mathematical models.

Amplification factor transport equation modelling of Mack mode disturbances in hypersonic boundary layers

In the hypersonic boundary layer, the streamwise instability modes are mainly the Mack first mode and the Mack second mode. After the Mach number exceeds 4.0, the Mack second mode becomes the dominant instability mode for boundary layer transition. Currently, there is no complete analytical method to establish an amplification factor transport equation for the second mode. In order to improve the application of the amplification factor transport (AFT) model in hypersonic boundary layer transition prediction, this paper employs boundary layer similarity solutions to conduct stability analysis on the second mode and introduces a function to calculate the growth rate of disturbances. Additionally, a transport equation for the amplification factor of the second mode is formulated. For the non-local variables appearing in the equation, a local calculation method is provided. Combined with the intermittency factor transport equation, a transition prediction model for the second mode is formed. Hypersonic flat plate, wedge, and flared wedge are selected for model validation. The computed results show good agreement with standard stability analysis or experimental results, demonstrating the rationality of the transition model and its high prediction accuracy and reliability.

ATEP: an advanced transport model for energetic particles

In this paper we report on the implementation and verification of a phase-space resolved energetic particle (EP) transport model. It is based on a first-principle theoretical framework, i.e. the system of non-linear gyrokinetic equations and the related transport equations. Its focus is primarily directed toward understanding the meso-scale character of EPs and its consequences. Compared to the conventional description of thermal radial transport via a one-dimensional radial diffusion equation, the newly developed model is three-dimensional using canonical constants-of-motion (CoM) variables. The model does not assume diffusive processes to be dominant a priori, instead the EP fluxes are self-consistently calculated and directly evolved in CoM space. We use the EP-Stability workflow and the HAGIS code to determine the phase space fluxes explicitly either in the limit of constant mode amplitudes or an energy-conserving quasi-linear model. As an application of the model the transport of neutral-beam-generated EPs due to a toroidal Alfvén eigenmode in an ITER plasma is investigated. As there are no sources and collisions taken into account so far (for an extension of the model see the companion paper (Meng et al 2024 Nucl. Fusion accepted)), the results cannot be considered as an exhaustive study, but rather as a practical demonstration of the conceptual framework on the way to a comprehensive reduced description of burning plasmas.

A modified algebraic turbulent heat flux model for non-equilibrium and rotating flow and its application in film cooling simulation of a rotating turbine blade

Thermal fields and cooling flows over rotating turbine blades are usually a non-equilibrium anisotropic flow, and most turbulence models in RANS simulations do not provide acceptable accuracy. A modified model is proposed and assessed here. This is based on an explicit algebraic heat flux model that is a low-Reynolds-number version of the higher order generalized gradient diffusion hypothesis turbulent heat flux model. The presented model, called “Non-equilibrium and Sensible to Rotation Algebraic Heat Flux Model” or “NSR-AHFM” includes additional production terms added to the second moment closure of turbulent heat flux transport equation that considers system rotation and non-equilibrium effects in velocity field, while maintaining the frame invariance during transformation to a rotating coordinate system. The model was initially tested through a-priori analysis, comparing the model to corresponding quantities obtained from Direct Numerical Simulations, allowing for the identification of suitable model parameters. Subsequently, the proposed modified model was implemented in a CFD code to a representative configuration (i.e., rotating turbine blade) and results were compared against available experimental data from literature. The present model was undergone a comparative analysis with other similar models in both stationary and rotating channel flow scenarios. Also, the present model was applied to a film cooling simulation of a rotating turbine blade in different rotational speeds. The findings indicate that the present model exhibits superior performance when compared to other models. Moreover, its behavior aligns consistently with the underlying physics governing the flow dynamics of the rotating channel.