Variable Diffusion Research Articles

Numerical analysis of Marangoni convected dusty second-grade nanofluid flow in a suspension of chemically reactive microorganisms

The present work investigates the Magneto-Marangoni convective flow of dusty second-grade nanofluid flow over a porous sheet in the presence of stagnation-point. The effects of chemical reaction, gyrotactic microorganisms are taken into account along with variable mass diffusivity, thermal radiation, and non-uniform heat source. The thermal reflection of nanoparticle molybdenum disulfide ([Formula: see text]) when it interacts with kerosene oil (KO) base fluid is determined. Nanoparticles and dust particles are presumptively spherical. The arising system of nonlinear PDEs is transformed into nonlinear ODEs by applying similarity transformations. Conservation of momentum, mass, concentration, energy, and density of microorganisms are used in to model the flow equations. The numeric expressions are computed by using the RKF-45th based shooting scheme. The outcomes revealed that the microorganisms, concentration, and temperature profiles decay in both phases, but velocity profiles of dust and fluid phases rise as the Marangoni ratio parameter grows.

Entropy optimized flow subject to variable fluid characteristics and convective conditions

Background and objectiveNanofluids possess significant enhancement in nanomaterial properties such as convective heat transfer coefficient, thermal conductivity (which increases volumetric fraction of nanoparticles) and diffusivity at reasonable nanoparticle concentration. Nanofluids provide interesting new opportunities to develop nanotechnology-based operative coolants for different kinds of innovative applications. Hydromagnetic radiative flow of nanomaterial by curved stretched sheet is addressed in presence of Darcy-Forchheimer relation. Heat generation and dissipation are deliberated. Variable mass diffusivity and thermal conductivity behaviors are under consideration. Convective conditions are taken into account. Random diffusion and thermophoresis behaviors are considered. Dufour and Soret behaviors are taken. Binary chemical reaction and thermal radiation are explored. Entropy rate is taken into consideration. MethodologyNonlinear formulation is reduced to ordinary differential system. Convergent solutions for nonlinear ordinary systems are developed invoking Optimal homotopy analysis method (OHAM). ResultsImpacts for influential parameters regarding fluid flow, pressure, thermal distribution, entropy and concentration are discussed. Larger magnetic variable has reverse behavior on entropy rate and pressure. Higher magnetic field intensifies thermal distribution whereas reverse effect noted for liquid flow. Thermal distribution has increasing trend for Dufour number and variable thermal conductivity. Concentration reduces for random motion and Schmidt number. Concentration has increasing behavior for Soret number and variable mass diffusivity parameter. Larger estimation of thermal Biot number has same impact on thermal distribution and entropy. Increasing behaviors of entropy rate and concentration for solutal Biot number are observed. Bejan number reduces against Brinkman number whereas opposite trend holds for radiation parameter.

Numerical Investigation of Diffusion Thermo, Hall Current, Activation Energy and Soret of MHD Darcy-Forchheimer Casson Fluid Flow with Inclined Plates

The present article is about the study of Darcy-Forchheimer flow of Casson fluid over a linear stretching surface. Effects like variable Hall current, Diffusion thermo, activation energy and Soret, is also incorporated for the analysis of heat and mass transfer. The governing nonlinear partial differential equations (PDEs) with convective boundary conditions are first converted into the nonlinear ordinary differential equations (ODEs) with the help of similarity transformation, and then the resulting nonlinear ODEs are solved with the help of shooting method. The impact of different physical parameters like Activation energy, Diffusion thermo, Brownian motion, thermophoresis parameter, hall current, magnetic parameter, nonlinear radiative heat flux, Prandtl number, Lewis number, reaction rate constant, on Nusselt number, velocity, temperature and concentration profile has been discussed. It is viewed that the fluid velocity and concentration profile enhance with increasing hall parameter and Activation energy respectively.

Physics informed neural networks reveal valid models for reactive diffusion of volatiles through paper

Predictive models for the transport of volatile organic compounds in paper need to consider the complex interplay of diffusion, adsorption, desorption, or chemical reactions. The relative importance of each of these processes is determined by the polarity of the volatile. Hence, it is challenging to pick a valid theoretical model that correctly predicts transport regardless of the polarity. Here, physics-informed neural networks (PINNs) assess which of five different models correctly describe transport of DMSO as polar and n-tetradecane as apolar model compound: (i) a pseudo first-order adsorption model for an irreversible sorption process, (ii) a first-order kinetics model allowing reversible sorption, (iii) a second-order model with a reversible process, and an effective diffusion model accounting for a constant (iv) and for a variable effective diffusivity (v). Each tested model is given as set of partial differential equations (PDE). Considering the model under testing and experimentally obtained spatially and temporally resolved concentration profiles through stacks of paper sheets, PINNs predict concentration of the volatiles and associated material constants such as sorption constants and effective diffusion coefficients by solving the inverse problem. Our PINNs revealed two models, pseudo first-order sorption and second-order reversible sorption, that correctly predict concentration profiles and polarity-driven differences in sorption times. While a PINN-based picking of valid transport models has important implications for the development of effective methods for controlling emission of volatiles from paper materials, PINNs represent a versatile mathematical tool to validate or refute the capability of PDE-based theoretical models to describe experimental data.

A comparative analysis of fractional model of second grade fluid subject to exponential heating: application of novel hybrid fractional derivative operator

In this article, a new approach to study the fractionalized second grade fluid flow is described by the different fractional derivative operators near an exponentially accelerated vertical plate together with exponentially variable velocity, energy and mass diffusion through a porous media is critically examined. The phenomenon has been expressed in terms of partial differential equations, then transformed the governing equations in non-dimentional form. For the sake of better rheology of second grade fluid, developed a fractional model by applying the new definition of Constant Proportional-Caputo hybrid derivative (CPC), Atangana Baleanu in Caputo sense (ABC) and Caputo Fabrizio (CF) fractional derivative operators that describe the generalized memory effects. For seeking exact solutions in terms of Mittag-Leffler and G-functions for velocity, temperature and concentration equations, Laplace integral transformation technique is applied. For physical significance of various system parameters on fluid velocity, concentration and temperature distributions are demonstrated through various graphs by using graphical software. Furthermore, for being validated the acquired solutions, accomplished a comparative analysis with some published work. It is also analyzed that for exponential heating and non-uniform velocity conditions, the CPC fractional operator is the finest fractional model to describe the memory effect of velocity, energy and concentration profile. Moreover, the graphical representations of the analytical solutions illustrated the main results of the present work. Also, in the literature, it is observed that to derived analytical results from fractional fluid models developed by the various fractional operators, is difficult and this article contributing to answer the open problem of obtaining analytical solutions the fractionalized fluid models.

Computational study on the influence of variable diffusion coefficients on non-fourier transport in shear-rate-dependent fluid with solute particles

Heat transfer in fluids occurs in numerous industrial processes. It is a fact that diffusion coefficients during heat transfer in fluids do not remain constant. Therefore, consideration of the constant diffusion coefficient is not a good approximation. In view of this, generalized modelling of heat and mass transfer in fluids of variable properties is done, and these generalized problems are solved by finite element method (FEM). After pre- and post-calculations, numerical runs are executed to observe the influence of variation in conductance on transport mechanisms in the fluid. The impact of other parameters is also studied. The outcomes are recorded in the form of graphs and numerical data. Ohmic dissipation results in an increase in the temperature of the fluid for variable and constant viscosity. The thermal relaxation time has shown significant effects on the flow and thickness of the thermal boundary layer region. The temperature of the fluid decreases with thermal relaxation time. The thermal conductance of the fluid varies with an increase in temperature. Once the temperature rises, the ability of the fluid to conduct heat is increased, due to which more heat from the heated surface diffuses into the fluid. Hence, heat flux decreases.The behavior of the variable viscosity of the fluid on the diffusion of momentum into the fluid is studied, and it is now known that the diffusion of wall momentum into the fluid is compromised because of a decrease in the viscosity, which is inversely proportional to the temperature. Hence, the diffusion of wall momentum into the fluid in the isothermal case is greater than that in the temperature-varying case.

Projection-Grid Schemes on Irregular Grids for a Parabolic Equation

A family of projection-grid schemes has been constructed for approximating parabolic equations with a variable diffusion coefficient in tensor form. The schemes are conservative and retain the self-adjointness of the original differential operator and are destined for calculations on 3D irregular difference grids, including tetrahedral, mixed (grids of arbitrary polyhedra), and locally adaptive (octal-tree type).

Dynamic modeling of airflow rate through window openings based on CO2 data

Better estimation of air change rate in naturally ventilated buildings is a key for supporting passive summer thermal comfort strategies. In window aired configurations, this estimation is challenging either with measurements or building simulation methods. In this study, we describe and apply the state space modeling methodology based on the carbon dioxide concentration (CO2) mass balance equation to a simple test case with numerically generated data. We show that a two-state CO2 / Air Change Rate model is suitable. As a modeling novelty, we demonstrate the benefit of improving the formulation with a variable diffusion term for the air change rate state equation. From the test case study findings, we emphasize that a lower interior-exterior CO2 difference results in weaker performance and we list some prospects for future work.

Hall effect on unsteady MHD flow along vertical plate with variable temperature and mass diffusion in the presence of porous medium and chemical reaction

The effects of chemical reaction and Hall current on the unsteady magneto hydrodynamic flow along a vertical plate with variable temperature and mass diffusion in the presence of a porous medium is studied hare. The fluid flow model of this paper contains governing partial differential equations of the motion, energy, and diffusion equation. To simplify the analysis, the governing equations are transformed into dimensionless form using non-dimensional variables. The Laplace-transform technique is employed to obtain an exact solution for the flow equations of the MHD model. The results of the analysis are presented using graphical representations of the velocity profiles. With the help of graphs, we discussed the behavior of fluid velocities with different parameters, including the chemical reaction parameter, Hall currents parameter, accelerated parameter, magnetic field parameter, and permeability parameter, and the numerical values of Sherwood number have been tabulated. We found that the values obtained for velocity, concentration and temperature are in concur-rence with the actual flow of the fluid

Rotating MHD Williamson nanofluid flow in 3D over exponentially stretching sheet with variable thermal conductivity and diffusivity

Nanofluids are a topic of great interest for researchers due to their remarkable performance in most heat transport applications. Williamson fluids have also gained importance due to their numerous uses in many fields. Exponentially stretching sheets with variable thermal conductivity and diffusivity is another topic of importance in fluid mechanics. Moreover, rotating MHD fluids have various applications in the industry. So, in particular, we would study rotating MHD Williamson nanofluid having variable diffusivity and thermal conductivity flowing above an exponentially stretching sheet. In this work, we proposed modified wavelets method to find the solutions of the developed mathematical model of the considered problem. The effectiveness of the developed scheme is certified by the help of tables and graphs. It is worthy to point out that the skin friction coefficient in x and y direction increases gradually against the selection of magnetic field effects and Williamson parameter. Tabular study presented to show that the suggested algorithm is convergent, and it can be extended to more physical models on non-Newtonian type.

Effect of concentration-driven magnetic phase changes on adsorption and diffusion in VSe2 monolayers: Implications for lithium-ion batteries

A vanadium diselenide (VSe2) monolayer is a two-dimensional (2D) magnetic material that exhibits ferromagnetic ordering at room temperature and exceptional metal-ion storage capacity, making it useful in spintronics and energy storage applications. However, a robust correlation between the magnetic and electrochemical properties of VSe2 remains to be established. In this study, first-principles density functional theory calculations were performed to investigate the effect of increasing Li-ion concentrations on the magnetic properties, particularly the magnetic ground state of the VSe2 monolayer. The results indicate that, as the concentration of Li ions on the surface of VSe2 monolayer increases, magnetic phase transitions occur, leading to a shift from the intrinsic ferromagnetic (FM) state to antiferromagnetic (AFM) and non-magnetic ground states. Analyses of the diffusion properties of ferromagnetic and antiferromagnetic VSe2 monolayers revealed a considerable (∼71%) increase in the Li-ion diffusion energy barrier for the AFM state compared to the FM state. This implies that FM-VSe2 facilitates relatively faster diffusion of Li ions than AFM-VSe2. Therefore, the Li-ion concentration-induced phase change in the VSe2 monolayer leads to variable adsorption and diffusion characteristics, which will have significant implications for its use in Li-ion battery anodes.

Fractional diffusion model generalised by the distributed-order operator involving variable diffusion coefficients

The diffusion process plays a crucial role in various fields, such as fluid dynamics, microorganisms, heat conduction and food processing. Since molecular diffusion usually takes place in complex materials and disordered media, there still exist many challenges in describing the diffusion process in the real world. Fractional calculus is a powerful tool for modelling complex physical processes due to its non-local property. This research generalises a fractional diffusion model by using the distributed-order operator in time and the Riesz fractional derivative in space. Moreover, variable diffusion coefficients are introduced to better capture the diffusion complexity. The fractional diffusion model is discretised by the finite element method in space. The approximation of the distributed-order operator is implemented by Simpson’s rule and the L2-1σ formula. A numerical example is provided to verify the effectiveness of the proposed numerical methods. This generalised fractional diffusion model may offer more insights into characterising diffusion behaviours in complex and disordered media.

Parameterizing Vertical Mixing Coefficients in the Ocean Surface Boundary Layer Using Neural Networks

AbstractVertical mixing parameterizations in ocean models are formulated on the basis of the physical principles that govern turbulent mixing. However, many parameterizations include ad hoc components that are not well constrained by theory or data. One such component is the eddy diffusivity model, where vertical turbulent fluxes of a quantity are parameterized from a variable eddy diffusion coefficient and the mean vertical gradient of the quantity. In this work, we improve a parameterization of vertical mixing in the ocean surface boundary layer by enhancing its eddy diffusivity model using data‐driven methods, specifically neural networks. The neural networks are designed to take extrinsic and intrinsic forcing parameters as input to predict the eddy diffusivity profile and are trained using output data from a second moment closure turbulent mixing scheme. The modified vertical mixing scheme predicts the eddy diffusivity profile through online inference of neural networks and maintains the conservation principles of the standard ocean model equations, which is particularly important for its targeted use in climate simulations. We describe the development and stable implementation of neural networks in an ocean general circulation model and demonstrate that the enhanced scheme outperforms its predecessor by reducing biases in the mixed‐layer depth and upper ocean stratification. Our results demonstrate the potential for data‐driven physics‐aware parameterizations to improve global climate models.

Two fast and unconditionally stable finite difference methods for Riesz fractional diffusion equations with variable coefficients

In this paper, for variable coefficient Riesz fractional diffusion equations in one and two dimensions, we first design a second-order implicit difference scheme by using the Crank-Nicolson method and a fractional centered difference formula for time and space variables, respectively. With the compact operator acting on, a novel fourth-order finite difference scheme is subsequently constructed. Solvability, stability and convergence of these schemes are theoretically analyzed. For these discretized linear systems, the fast implementation with preconditioners based on sine transform is proposed, which has the computational complexity of O(nlog⁡n) per iteration and the memory requirement of O(n), where n represents the total number of the spatial grid nodes. Finally, numerical experiments are performed to illustrate the preciseness and effectiveness of these new techniques.

Heat extraction calculations for deep coaxial borehole heat exchangers: matrix analytical approach

SUMMARY Deep boreholes represent a source of clean energy. Therefore, effective calculations of potential extraction of heat from boreholes for realistic models of the Earth’s crust with variable thermal conductivity and diffusivity are needed. We deal with heat extraction in a quasi-steady state from coaxial boreholes where downward and upward flows of pumped fluid (water) are separated by an inner pipe and connected only at the bottom. We first obtain theoretical estimates of heat extraction for a thermally isolated inner pipe and a model of the ground with constant thermal diffusivity and conductivity. Then, we develop a new analytical matrix method for a general layered ground model that enables us to include depth-dependent ground properties as well as heat exchange between the downward and upward flows of fluid in the borehole. Our straightforward and fast approach is thus suitable for various parametric studies or as a tool for benchmarks of numerical software. A key role in heat extraction from coaxial boreholes is played by the inner-pipe thermal resistance. We apply our method to the parametric study showing the dependence of pumped water temperature and total heat extraction from the borehole on realistic borehole geometries under different amounts of water pumping. The calculations are performed for a 3 km deep borehole as the representative of present deep boreholes used for extraction of geothermal energy and for a 10 km deep borehole. Drilling of such a superdeep borehole has just started in China and our results demonstrate potential limits of geothermal energy extraction from such great depths.

Fractal derivative model with time dependent diffusion coefficient for chloride diffusion in concrete

More research and experiments have indicated that, under different water cement ratios, the binding effect of concrete on chloride ions and the blocking effect of hydration on chloride ions are clearly dominant. During this time period, the diffusion rate of chloride ions will increase or decrease with time, resulting in the classical Fickian diffusion law no longer being applicable. This study establishes an improved fractal model with a time-dependent diffusion coefficient for chloride diffusion in concrete. An explicit solution is derived in terms of the complementary error function. The proposed model is verified by fitting the chloride ion concentration data from different concrete immersion tests. The results show that the fractal model can fit the data in different spatial scales, and the corresponding prediction accuracy is rather high when compared to the three integer order models with constant, variable diffusion coefficient, and curing age. Therefore, the fractal derivative model is feasible in describing the anomalous diffusion of chloride ions in concrete.

Significance of Hall current and Ion slip in a three-dimensional Maxwell nanofluid flow over rotating disk with variable characteristics and gyrotactic microorganisms

In varied practical situations, the assumption of constant values are considered for approximations or simplifications to make computations more adaptable. Nevertheless, when there is a requirement for precision, accuracy, and exact results, the consideration of variable characteristics becomes essential. Considering the same, the article’s uniqueness lies in studying variable thermal conductivity, variable viscosity, and variable mass diffusivity as governing factors. This study analyzes the movement of a three-dimensional Maxwell nanofluid across a rotating deformable disk deliberating the Hall current and Ion slip impact. In addition, the analysis seeks to boost the stability of the nanofluid flow by incorporating the impact of bio-convection. The velocity slip and convective constraints are enforced at the boundary. The Von Karman transformations are engaged, and accordingly, a numerical solution is computed. Graphical illustrations are used to highlight the significant effects of the problem. The findings revealed that the fluid velocity decreases in both the azimuthal and radial directions for different estimations of the variable viscosity and Deborah number. Additionally, the fluid velocities in the azimuthal and radial directions exhibit opposite trends for Hall current and Ion slip. The endorsement of the presented results is also provided in this study.

Stability analysis of radiative-magnetive Ag −MoS 2/water hybrid nanofluid flow due to a permeable exponential surface with variable thermal conductivity, mass diffusivity and binary chemical reaction: an entropy generation

ABSTRACT This investigation concentrated on the /water hybrid nanofluid to solve the problem of heat-mass transmission across a permeable stretching/contracting sheet with the impacts of radiant heat and inclined magnetic force field. Here, we additionally take into account hybrid nanofluids with variable thermal conductivity, variable mass diffusivity, Joule dissipation, binary chemical reactions with velocity, and thermal and solutal slip conditions. Before being numerically solved by the MATLAB function bvp4c, the controlling PDE is converted into nonlinear coupled ordinary differential equations by appropriate similarity transformation. The findings also point to the presence of dual solutions in the stretching/contracting sheet region for a given value of the mass suction parameter, with stable upper branch solution and unstable lower branch solution. As a result of the velocity and temperature gradients, the entropy generation equation is constructed. The effects of the selected parameters are discussed and visually displayed for the velocity, temperature, concentration, skin friction coefficient, local Nusselt number, and Sherwood number. Due to some selected flow parameters, the pattern of entropy formation and assessment of the Bejan number are anticipated. Furthermore, it is discovered that when the volume percentage of nanoparticles increases, the Sherwood number decreases, and the skin friction coefficient increases with the rate of heat transmission. The Bejan number and entropy generation rate both increase with the volume proportion of nanoparticles and the radiant heat parameter, according to the visual simulations created using the present model.

On the Control of Soil Heterogeneity, Péclet Number and Spatially Variable Diffusion Over Unsaturated Transport

AbstractPhysical properties of soils are ubiquitously heterogeneous. This spatial variability has a profound, yet still partially understood, impact on conservative transport. Moreover, molecular diffusion is often a disregarded process that can have an important counter‐intuitive effect on transport: diffusion can prevent non‐Fickian tailing by mobilizing mass otherwise trapped in low velocity zones. Here, we focus on macroscopically homogeneous soils presenting small scale heterogeneity, as described by the Miller‐Miller theory. We then analyze the dynamic control of soil heterogeneity, advection and diffusion on conservative transport. We focus especially on the importance of diffusion and of its tortuosity‐dependent spatial variability on the overall transport. Our results confirm previous finding that relatively high Péclet number systems are highly sensitive to the degree of heterogeneity, which promotes non‐Fickian transport. Also, diffusion appears to have a profound impact on transport, depending on both the degree of heterogeneity and the Péclet number. For a high Péclet number and a very heterogeneous system, diffusion leads to the counter‐intuitive decrease of non‐Fickian spreading described previously. This is not observed for a low Péclet number due to the impact of the spatial variability in the diffusion coefficient, which appears to be a significant controlling factor of transport by promoting or preventing the accumulation of mass in low velocity zones. Globally, this work (a) highlights the complex, synergistic effect of soil heterogeneity, advective fluxes and diffusion on transport and (b), alerts on potential upscaling challenges when the spatial variability of such key processes cannot be properly described.

A fast ADI based matrix splitting preconditioning method for the high dimensional space fractional diffusion equations in conservative form

In this paper, we consider the high-dimensional two-sided space fractional diffusion equations with monotonic variable diffusion coefficients, which are derived from the fractional Fick's law. We apply the implicit Euler scheme to discretize the temporal derivative and certain difference operator to discretize the space fractional derivatives. For the discretized linear systems, we apply the alternating direction implicit (ADI) scheme to split the linear systems into two sub-systems of linear equations. For both coefficient matrices of the sub-systems of linear equations, we propose the lopsided scaled diagonal and Toeplitz splitting preconditioner. The generalized minimal residual (GMRES) method combined with the proposed preconditioner is applied to solve both sub-systems of linear equations. The spectral distributions of the preconditioned matrices are analyzed, and the theoretical results are given. Numerical results demonstrate that the proposed preconditioner is efficient in accelerating the convergence rate of the GMRES method for solving the discretized system of linear equations.