Transport Equation Research Articles

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.

Investigation of equivalent spherical bubble diameter at high inlet velocity pool scrubbing conditions

This study investigates Equivalent Spherical Diameter (ESD) estimation at high inlet velocity pool scrubbing conditions using the Interfacial Area Transport Equation (IATE) diameter model including bubble-induced turbulence and interphase modeling. The compatibility of area-averaged Sauter Mean Diameter (SMD), area-averaged Local Equivalent Diameter (LED) and void-weighted area-averaged LED approaches to estimate the ESD are explored and the proposed model is validated against available experimental data. The study reveals that the prevalent constant ESD assumption in pool scrubbing codes is not universal by showcasing a decreasing trend along the column due to intensive bubble breakup. The area-averaged LED approach fails to capture this trend, while the area-averaged SMD and void-weighted area-averaged LED approaches provide accurate estimations aligned with experimental data. Turbulence parameters, interfacial forces, and diameter modeling are identified as crucial for accurate predictions of flow and geometrical variables by setting up the OpenFOAM framework. A sensitivity analysis indicates that the inlet velocity has an acceptable effect on the ESD along the column. The ESD increases near the exit and decreases in the swarm region by increasing the inlet velocities. Turbulent intensity reduces ESD across all column sections while changes in aspect ratio minimally impact ESD. The study shows promise in developing correlations that take into account the spatial variation of ESD in pool scrubbing conditions.

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.

Estimating hydraulic diffusivity in coastal confined aquifer under tidal fluctuation using a frequency domain model

Oceanic tidal propagation into the coastal aquifers is a cost-effective method for estimating hydraulic parameters in coastal aquifers, and identifying the most suitable parameter estimation method for coastal aquifers based on the observations has become an intensively studied subject. In this study, the frequency domain model, LPMLE3 (combination of a local polynomial (LP) signal processing technique with a maximum likelihood estimator (MLE)), was applied to estimate the hydraulic diffusivity (λ) of the coastal aquifer by adjusting the mathematical model in LPMLE3. The mathematical model in LPMLE3 has been adjusted by removing the convective term from the convective diffusion equation within LPMLE3, and a newly developed frequency-domain analytical solution that describes the head changes due to tidal fluctuations in coastal confined aquifers was used to replace the diffusion term. For comparison, an analytical solution in the time domain was also derived by Green’s functions method. After that, the particle swarm optimization algorithm and Markov Chain Monte Carlo method were used to estimate λ. Results show that LPMLE3 performed well in estimating homogeneous or heterogeneous aquifers with complicated boundary conditions. The estimated error of λ is within 10 % in a heterogeneous aquifer when the variance of the natural logarithm of the aquifer hydraulic conductivity is less than 0.05 (σlnK2≤0.05). If the coastal aquifer has a sloping beach, the estimated λ by LPMLE3 method accurately when the sloping beach angle is greater than 65°. As for an inclined coastal aquifer, the dip angle has a greater influence on the offshore areas than the nearshore areas and the estimation errors are relatively small (less than 15 %) when the dip angle is within a range of 0- 15°.

Reduction of Taylor dispersion in a capillary by spin-up flow—Theoretical insights

We have developed a theoretical framework to characterize the transport and mixing of a passive scalar in a capillary tube. In this configuration, a suspension of magnetic nanoparticles undergoes Poiseuille flow, while a rotating magnetic field is applied around the tube’s revolution axis, inducing a secondary flow in the azimuthal direction. This secondary flow facilitates the mitigation of concentration gradients and radial dispersion associated with the axial parabolic Poiseuille profile. The improvement in mixing is emphasized by a new dimensionless parameter, the mixing factor, which is incorporated into the scalar transport equation. Such a factor acts as a quantitative measure of the effect of the tangential motion induced by the spin-up flow on the overall mixing efficiency of the liquid and the observed reduction of the Taylor dispersion in the measured residence time distributions. Recognizing the mixing factor as a crucial parameter advances our understanding of the mechanisms governing scalar transport and provides a valuable tool for predicting and optimizing mixing in laminar capillary flows subjected to spin-up motion.

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.

Numerical solutions of boundary problems in partial differential equations: A deep learning framework with Green's function

This work introduces Green Networks (GreenNets), a novel class of neural network architectures designed to address a wide spectrum of boundary problems in partial differential equations (PDEs). Demonstrated versatility includes solving the Poisson equation, convection-diffusion equation, and unsteady incompressible Navier-Stokes equations. GreenNets comprises two specialized variants: Boundary Transformation Green Network (BT-GreenNet), focusing on transforming Dirichlet boundary conditions into source terms, and Boundary Integral Green Network (BI-GreenNet), considering boundary integral terms with shared weights between the boundary and source integrals. Comparisons with existing deep learning and traditional numerical methods underscore their accuracy and efficiency. BT-GreenNet excels in efficiency, and BI-GreenNet offers enhanced flexibility. The research pinnacle, Boundary Unified Green Network (BU-GreenNet), merges BT-GreenNet and BI-GreenNet to address coupled PDEs, specifically unsteady incompressible Navier-Stokes equations. BU-GreenNet stands out by surpassing traditional numerical approaches, offering remarkable computational efficiency while faithfully capturing the intricate evolution of fluid dynamics phenomena. This research redefines the means of solution to boundary problems in PDEs through innovative neural network architectures, emphasizing practical utility and computational efficiency.

On the use of spherical harmonic approximations in nonclassical particle transport problems

Presented here is the application of the spherical harmonic expansion to the nonclassical transport equation to derive a system of equations for the nonclassical flux moments, namely nonclassical spherical harmonic approximations (NSHA). The nonclassical transport equation is a recently developed mathematical model that allows the modeling of transport problems wherein the particle flux does not undergo exponential attenuation. We employ a Spectral Approach technique to depict the nonclassical flux by expressing it as a truncated series of Laguerre polynomials with respect to the free-path variable s. This yields a system of equations for the nonclassical flux moments structured similarly to the classical PN equations. We show that the NSHA is reduced to the classical PN equation in a special case. Numerical results for slab-geometry test problems are given to verify the derivation.

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.

Ion detection in a DNA nanopore FET device.

An ion detection device that combines a DNA-origami nanopore and a field-effect transistor (FET) was designed and modeled to determine sensitivity of the nanodevice to the local cellular environment. Such devices could be integrated into a live cell, creating an abiotic-biotic interface integrated with semiconductor electronics. A continuum model is used to describe the behavior of ions in an electrolyte solution. The drift-diffusion equations are employed to model the ion distribution, taking into account the electric fields and concentration gradients. This was matched to the results from electric double layer theory to verify applicability of the model to a bio-sensing environment. The FET device combined with the nanopore is shown to have high sensitivity to ion concentration and nanopore geometry, with the electrical double layer behavior governing the device characteristics. A logarithmic relationship was found between ion concentration and a single FET current, generating up to 200 nA of current difference with a small applied bias.

Assessment of Environmental Risks of a Shallow Water Body during Dredging Works

Introduction. The increasing anthropogenic impact on water bodies necessitates integrated solutions to assess environmental risks. Literature describes the stages of risk assessment, the possibilities of environmental management, and expert analysis, while risk modeling in this field is being investigated. However, the potential for predicting risks to water quality and biodiversity during frequently performed hydraulic engineering works such as dredging has not been fully explored. The relevance and practical significance of such an approach are evident. This study aims to develop a mathematical model and software package that can assess risks to species diversity of the ecosystem of a shallow reservoir ecosystem during work in its water area.Materials and Methods. The starting point for the simulation was a description of the movement of water masses based on the Navier-Stokes equations and the continuity equation at variable density. We used the diffusion-convection equation to predict the transfer of suspended and dissolved particles, as well as to assess the impact of impurities during eutrophication. To create the algorithm, we utilized the terms and definitions defined by the state standard for risk management in emergency situations.Results. To test the solution, we took data on hydro-mechanical work in the port area of Arkhangelsk. We visualized the concentration fields of suspended particles 0, 15, 30 and 45 minutes after the soil was unloaded. It was found that during the settling of the suspension, the area of its distribution expanded significantly, and this was fully consistent with the data of field experiments during dredging. We calculated and tabulated the volumes of contaminated water at soil dumps in three sites (with a single discharge and in total). To assess the risks to the Sea of Azov, we used the maximum concentrations of pollutant (copper) obtained through measurements, modeling and remote sensing of the Earth. In tests to determine the potential danger of the substance, we assumed that its concentration caused a reaction in 50% of organisms. For fish, the potentially dangerous concentration was 4 mg/l with a duration of 96 hours of exposure. For zooplankton — 50 mg/l and 48 hours. For microalgae, 20 mg/l and 72 hours. The normalized risk value Rn ≈ 0.52 was obtained. The risk of copper concentration of 80 µg/l in the waters of the Azov Sea was recognized as significant. A tendency towards increasing salinity and stratification of water masses in terms of oxygen content has been identified, consistent with the findings of expeditionary research.Discussion and Conclusion. The developed approach has allowed us to assess the change in the quality of the waters of the Azov Sea and describe some transformations of the water area. Specifically, we are talking about the distribution of suspended particles and areas of their deposition. These processes can lead to changes in the bottom topography, which in turn can reduce the species diversity of the ecosystem.

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.

On the spatio-temporal dynamics of cavitating turbulent shear flow over a microscale backward-facing step: A numerical study

The influence of cavitation on the mean characteristics and unsteady behavior of turbulent separated flows was comprehensively investigated in this study over a microscale backward-facing configuration at the Reynolds number (ReD) of 7440. The computational approach took both compressibility and finite mass transfer (Thermodynamic non-equilibrium) into account, to accurately capture the effects of shock waves, as well as to capture baroclinic phenomena on vortex dynamics within the turbulent separated flow. The compressibility effects were handled by using appropriate equation of states for each phase and for the mixture. Phase-change was considered through a transport equation for the vapor volume fraction, allowing for finite mass transfer contributions. Additionally, a wall adaptive large eddy simulation (LES) approach was utilized for simulating turbulent structures and their effects. The findings reveal that vapor development diminishes the mean growth rate of the shear layer and delays its reattachment to a longer distance from the step. Moreover, analysis of Reynolds normal and shear stresses, as well as the root mean square (RMS) of pressure fluctuations, demonstrates that the formation and collapse of vapor packets significantly influence turbulence decay and production in the second half of the shear layer and reattachment. It was also observed that both mean pressure and pressure fluctuations increased in vicinity of the reattachment region when cavitation was present, which was attributed to the condensation and collapse events. Spectral analysis further indicates the emergence of two dominant low frequency modes, linked to the displacement of the reattachment point. In the presence of cavitation, the frequencies associated with dominant Power Spectral Densities (PSDs) were smaller than those in the absence of cavitation. Additionally, each of these low frequencies corresponded to a specific vapor transport mechanism within the Turbulent Separation Bubble (TSB). Furthermore, it is shown that cavitation leads to a significantly higher spectral energy of high frequency fluctuations within the reattachment region in comparison to the condition where cavitation is absent. This can be attributed to the frequent collapse of bubbles in this region. At the end, we employed Spectral Proper Orthogonal Decomposition (SPOD) for modal analysis. This method offers valuable insights into the coherent structures and associated frequencies that arise in both the presence and absence of cavitation, which provides a deeper understanding of the effect of cavitation on the coherent structures and their dynamics.

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.

A finite element framework for fluid–structure interaction of turbulent cavitating flows with flexible structures

We present a finite element framework for the numerical prediction of cavitating turbulent flows interacting with flexible structures. The vapor–fluid phases are captured through a homogeneous mixture model, with a scalar transport equation governing the spatio-temporal evolution of cavitation dynamics. High-density gradients in the two-phase cavitating flow motivate the use of a positivity-preserving Petrov–Galerkin stabilization method in the variational framework. A mass transfer source term introduces local compressibility effects arising as a consequence of phase change. The turbulent fluid flow is modeled through a dynamic subgrid-scale method for large eddy simulations. The flexible structure is represented by a set of eigenmodes, obtained through a modal decomposition of the linear elasticity equations. While a partitioned iterative approach is adopted to couple the structural dynamics and cavitating fluid flow, the deforming flow domain is described by an arbitrary Lagrangian–Eulerian frame of reference. We establish the fidelity of the proposed framework by comparing it against experimental and numerical studies for both rigid and flexible hydrofoils in cavitating flows. Under unstable partial cavitating conditions, we identify specific vortical structures leading to cloud cavity collapse. We further explore features of cavitating flow past a rigid body such as re-entrant jet and turbulence-cavity interactions during cloud cavity collapse. Based on the validation study conducted over a flexible NACA66 rectangular hydrofoil, we elucidate the role of cavity and vortex shedding in governing the structural dynamics. Subsequently, we identify a broad spectrum frequency band whose central peak does not correlate to the frequency content of the cavitation dynamics or the natural frequencies of the structure, indicating the induction of unsteady flow patterns around the hydrofoil. Finally, we discuss the coupled fluid–structure dynamics during a cavitation cycle and the underlying mechanism associated with the promotion and mitigation of cavitation.

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.