Implicit Finite Difference Method Research Articles

Effect of Suction/Injection on Unsteady MHD Natural Convection Flow of Heat Mass Transfer in Porous Channel in the Presence of Soret Term

This research paper explores the effect of suction/injection on unsteady MHD natural convection flow of heat mass transfer in porous channel in the presence of Soret term. The governing partial differential equations are converted to non- dimensional forms and solved numerically by using unconditionally stable and convergent implicit finite difference method. A parametric study illustrating the influence of various physical parameters is performed. It is reported that the velocity profile increases as Soret term, Grashof number, Solutal Grashof number and Porous parameters values increase, while Magnetic field parameter decreases the velocity profile. The temperature profile rises by the influence in increasing values of Variable Thermal Conductivity and decreases by increasing values of Prandtl number and Radiation parameters. While concentration profile increase by the increasing values of Soret term and Chemical reaction. The dependence of the skin friction coefficient, rate of heat transfer and mass transfer on these parameter has been discussed.

Optimization of coupling phase-change materials and thermal screens in façade-integrated hybrid photovoltaic collectors for optimal energy production and thermal comfort in buildings

The operation of building-integrated photovoltaic (BIPV) systems gives rise to a significant proportion of the solar radiation absorbed by the cells being unable to be converted into electricity. This phenomenon consequently increases the temperature. This temperature increase impacts the cells' electrical efficiency, leading to a reduction in their performance and accelerating their degradation. The combination of phase-change materials with insulating fluid blades, situated behind photovoltaic cells, represents a passive cooling solution that optimizes the performance of hybrid photovoltaic systems when incorporated into facades. The present study assesses a system that incorporates paraffin as a PCM and an argon layer in a PV-PCM-argon layer physical model (PVT/ArPCM), in comparison with a PV-PCM system (PVT/PCM), to enhance the thermoelectric performance of photovoltaic systems mounted on façades while ensuring optimal thermal comfort within buildings. The discrete heat transfer equations were solved using the Thomas algorithm and the iterative Gauss-Seidel method in conjunction with the implicit finite difference method. The findings illustrate that the electrical efficiency experienced only a slight increase, estimated at 0.01%, while there was a notable enhancement in the indoor thermal comfort experienced by occupants, with a 65% improvement observed due to the incorporation of an argon-filled thermal screen. The incorporation of an argon layer led to a minor reduction in temperature of 0.01°C in the photovoltaic cells, resulting in a minimal improvement of 0.014% in electrical power production. The phase-changing material incorporated into PVT/ArPCM demonstrated superior thermal management capabilities in comparison to the same material employed in PVT/PCM.

Convergence Analysis of an Implicit Finite Difference Method for the Inertial Landau–Lifshitz–Gilbert Equation

Convergence Analysis of an Implicit Finite Difference Method for the Inertial Landau–Lifshitz–Gilbert Equation

Cilia-assisted flow of electrically conducting micropolar fluid in a heated curved channel under Soret and Dufour effects

Cilia flow plays a crucial role in biological systems such as the movement of mucus in the respiratory tract, circulation of cerebrospinal fluid in the brain, and propulsion of sperm cells. Understanding the cilia flow of viscous fluids is essential for elucidating the biomechanics of these processes and their implications for health and disease. Motivated by such numerous biomedical applications, this article aims to exhibit the influence of heat and mass transfer on the ciliary flow of electrical conducting micropolar fluid in a curved channel. The energy and concentration equations are modulated with viscous dissipation, Soret, and Dufour effects. The constitutive equations are simplified by the hypothesis of lubrication approximation theory and then solved numerically using the implicit finite difference method (FDM). Results for velocity, pumping phenomenon, concentration, thermal field, rate of heat transfer, streamlines, skin friction coefficient, Nusselt number, and Sherwood number are analyzed subject to pertinent parameters. The study reveals that velocity is enhanced via both the micropolar parameter and curvature parameter. Temperature enhances for larger values of Dufour and Brickman numbers. Furthermore, the radial magnetic field plays a resistive role in the trapped bolus. The findings presented in this study should prove beneficial for researchers in the domains of medicine, engineering, science, and fluid mechanics. Further, it is found that the micropolar fluid model is more suitable for biofluids like blood.

Numerical study on the heat transfer features of concurrent forced nano-PCM emulsion/water flows in a double concentric-tube configuration

To provide a feasible heat dissipation solution for a high heat flux, in this study, the internal flow and heat transfer characteristics of an externally heated concentric double tube were numerically studied via the implicit finite difference method. The outer ring and inner tube were filled with a phase change nanofluid and water, respectively. The parameter ranges were as follows: the mass fractions of the phase change material (PCM) emulsions were 2.04 %, 2.91 %, and 7.11 %, the relative flow ratio was 0.33–4.0, the heat power inputs were 70 W and 150 W, and the total tube flow rates were 304.83 cm3/min and 465.98 cm3/min. The results showed that incorporating PCM emulsions can effectively improve heat transfer performance; however, the PCM concentrations, heating conditions, and outer-annulus/inner-tube flow rates (relative flow ratios) must be properly matched so that the phase change of the working fluid occurs in the expected section. Within these parameter ranges, in the heated section, the latent heat utilization of the PCM emulsion is lower than that of the sensible heat, and only when the flow ratio is greater than 1 is the latent heat utilization of the 7.11 % PCM emulsion similar to that of the sensible heat.

Groundwater pollution equation: Lie’s symmetry analysis and numerical consideration

The current study modeled groundwater pollution through the utilization of the advection–diffusion equation - a versatile differential equation that is capable of modeling a variety of real-life processes. Indeed, various methods of solutions were then proposed to examine the governing model after being transformed, starting with Lie’s symmetry, semi-analytical, and numerical methods, including the explicit and implicit finite difference method and the finite element method. Further, the proposed methods were demonstrated on some test models; featuring forced and unforced scenarios of the forcing function. Analytically, Lie’s symmetry method failed to unswervingly reveal the required solution to the problem; however, with the imposition of certain restrictions, a generalized closed-form solution for the forced model was acquired. This fact indeed triggered the quest for the deployment of more methods. Thus, semi-analytically, the adopted decomposition method swiftly gave the resultant closed-form solutions. Numerically, the efficiency of the sought methods was assessed using the L2−norm and CPU time, upon which the implicit schemes were found to win the race. All-in-all, the beseeched semi-analytical method is highly recommended for such investigation; at the same time advocating the effectiveness of the implicit finite difference schemes on advection–diffusion-related equations.

Modeling a poiseuille flow through a rectangular pipe in a floor slab (PSD)

This study was devoted to the modeling of a laminar fluid flow in a pipe of the rectangular section located in a floor slab. This theme was raised because of its relevance in the large business sector as well as in the areas of energy saving and storage at the residential scale. The equations of motion quantity and heat transport are the classical equations of forced convection. However, to describe the movement amount within the heating slab pipeline, Prandtl simplified model was most appropriate. In addition, this model retains the most important terms for a laminar regime flow. In addition, the modeling of fluid flow conservation in the pipeline was taken into account in the equation system. To achieve the above-mentioned objective, a numerical resolution was made for the system resolution of the Navier-Stokes equations that govern this fluid flow. These equations were discretized by an implicit finite difference method. The system of algebraic equations as well as obtained was solved by the algorithm of Gauss and the algorithm of Thomas. The results obtained such as: the evolution of the fluid velocity profile within the slab according to pipeline to the Reynolds number variation, the fluid temperature have clearly demonstrated the Poiseuille effect on fluid flow. Moreover, the stabilization of the Gauss algorithm and the Thomas algorithm in this numerical modeling was well verified against the tricks that were taken into consideration. The result of this study was that Thomas algorithm and Gauss algorithm are the best in this type of numerical modeling.

Heat Transfer and Flow of Natural Convection Past a Semi-Infinite Vertical Plate

This manuscript presents a numerical investigation of the flow of an electrically conducting, incompressible, viscous fluid past a heated vertical channel with thermal radiation influence. The study uses the Rosseland approximation to describe the radiative heat flux and employs Crank Nicolson's implicit finite difference method utilizing the MATLAB programming tool for solving the non-dimensional system of equations. The results demonstrate that an increase in the thermal radiation parameter leads to a decrease in both the velocity and temperature of the fluid. The main findings of the study highlight the impact of thermal radiation on natural convection and heat transfer processes. This research is significant as it contributes to the understanding of convective heat transfer phenomena and provides valuable insights for applications in science and technology.

Implicit finite difference simulations for unsteady oscillating flow of Walters-B nanofluid with microbes using the Cattaneo–Christov model

The thermal importance of nanoparticles in different fields of engineering is still a developing issue that necessitates additional focused consideration. The suspension of non-Newtonian fluids with nonmaterials has various applications in areas such as enhanced thermal systems, energy processes, aerospace engineering, and chemical industries. Following such motivations, this work aims to analyze the heat and mass transfer flow of Walters-B nanofluid in the presence of microorganisms. The fluid motion is presumed to be nonsteady over a surface that undergoes periodic acceleration. The motivations for studying nanofluid’s oscillating flow are its use in effective mixing processes, combustion stability, and process efficiency. The text provides a detailed explanation of the saturation of porous media. A modified theory for heat and mass fluxes, known as the Cattaneo–Christov approach, suggests an extension in heat equations. Moreover, it supports the interaction of radiation phenomena for enhancing heat transfer. It is noteworthy that the entire problem is represented using extremely nonlinear partial differential equations (PDEs). The CFD study utilizing the renowned implicit finite difference method (FDM) has been effectively employed to propose a numerical solution for the problem. The numerical results are deemed valid and confirmed within certain limiting cases, based on the available data. The evaluation of velocity, skin friction coefficient, Nusselt number, and Sherwood number as a function of time has been demonstrated for various parameters. It is asserted that the wall shear force exhibited an oscillating pattern with a greater magnitude as a result of the viscoelastic parameter. The Nusselt number and Sherwood number exhibit slight acceleration as a result of changes in the Prandtl number and Schmidt number, respectively. The proposed model is specifically designed for use in heat exchangers, vibrational systems, compact designs, and vibration control.

Controlling numerical diffusion in solving advection-dominated transport problems

Numerical schemes for advection-dominated transport problems are are evaluated in a comparative study. Explicit and implicit finite difference methods are analyzed together with a global random walk algorithm in the frame of a splitting procedure. The efficiency of the methods with respect to the control of the numerical diffusion is investigated numerically on one-dimensional problems with constant coefficients and two-dimensional problems with variable coefficients consisting of realizations of space-random functions.

Composition tracking of natural gas–hydrogen mixtures in pipeline flow using high-resolution schemes

A transient pipeline flow model with gas composition tracking is solved for studying the operation of a natural gas pipeline under nonisothermal flow conditions in a hydrogen injection scenario. Two approaches to high-resolution pipeline flow modeling based on the WENO scheme are presented and compared with the implicit finite difference method. The high-resolution models are capable of capturing fast fluid transients and tracking the step changes in the composition of the transported mixture. The implicit method assumes the decoupling of the flow model components in order to enhance calculation efficiency. The validation of the composition tracking results against actual gas transmission pipeline indicates that both models exhibit good prediction performance, with normalized root mean square errors of 0.406% and 1.48%, respectively. Under nonisothermal flow conditions, the prediction response of the reduced model against a high-resolution flow model, with respect to the mass and energy linepack, is at most 3.20%.

Analysis of the effect of inert gas on alveolar/venous blood partial pressure by using the operator splitting method.

This study aims to investigate how inert gas affects the partial pressure of alveolar and venous blood using a fast and accurate operator splitting method (OSM). Unlike previous complex methods, such as the finite element method (FEM), OSM effectively separates governing equations into smaller sub-problems, facilitating a better understanding of inert gas transport and exchange between blood capillaries and surrounding tissue. The governing equations were discretized with a fully implicit finite difference method (FDM), which enables the use of larger time steps. The model employed partial differential equations, considering convection-diffusion in blood and only diffusion in tissue. The study explores the impact of initial arterial pressure, breathing frequency, blood flow velocity, solubility, and diffusivity on the partial pressure of inert gas in blood and tissue. Additionally, the effects of anesthetic inert gas and oxygen on venous blood partial pressure were analyzed. Simulation results demonstrate that the high solubility and diffusivity of anesthetic inert gas lead to its prolonged presence in blood and tissue, resulting in lower partial pressure in venous blood. These findings enhance our understanding of inert gas interaction with alveolar/venous blood, with potential implications for medical diagnostics and therapies.

Analyzing the thermal performance of transient heat transfer mechanism due to the combined effects of solar energy and variable surface temperature: Growing applications of solar energy in co-axial pipes

The current analysis is carried out to study the laminar convective heat transfer characteristics that change with time in the co-axial pipes along the impact of heat radiation and variable surface temperature. The flow is assumed along the axial direction of the pipe, and variable boundary condition is assumed at the surface of the pipe due to the variable surface temperature. A two-dimensional mathematical model made up of non-linear partial differential equations is solved using the implicit finite difference method. The project involves predicting the thermal efficiency of a pipe's time dependent flow across a number of flow model-relevant parameter ranges. Graphical representations highlighted the derived predictions. After separating the numerical solutions into the time independent and the time dependent components, the time dependent energy and surface shearness were found using the data from the time independent component. Comprehensive detail of the obtained results for the non-dimensional parameters included in the flow formulation is predicted for steady state velocity, temperature distribution, time dependent surface sheerness, and time dependent energy sheerness, which is given in results and discussion section of the manuscript. Major focus is given on the influence of the radiation parameter on the above-mentioned primary measures. Furthermore, it is concluded that in all cases, the steady state flow is as R→∞ and velocity profile as U→1 and θ→0, which ensured the accuracy of the obtained results by satisfying the boundary conditions. For various values of the fluid's absorption parameter D = 1.0, 3.0, 5.0, and 10.0, the flow profile is increased and U→1 as R→∞. Simultaneously, thermal distribution also increases and θ→0 as R→∞.

Finite difference simulation of natural convection of two-phase hybrid nanofluid along a vertical heated wavy surface

This study employs finite difference modelling to investigate the natural convection of a two-phase hybrid nanofluid consisting of Al 2 O 3 and Cu nanoparticles dispersed in water along a vertically heated wavy surface. The hybrid nanofluid has unique features that influence convective heat transfer. Numerical solutions analyze fluid flow patterns and temperature distributions, providing insights into heat exchange and thermal management. The research highlights the importance of hybrid nanofluids, notably those containing Al 2 O 3 and Cu in a water base, in improving overall heat transfer efficiency. The mathematical model accounts for laminar and incompressible fluid flow with a Prandtl number of Pr = 6.2, Lewis number Le = 10, and maximum 10 % concentration of hybrid nanoparticles. After transforming the governing equations into a non-dimensional form, the implicit finite difference method is used to solve them. The solution employs the in-house FORTRAN 90 code and compares its outcomes with the benchmark results. Various parameters, such as the Schmidt number (Sc = 1 to 10), volume fraction of nanoparticles ( ϕ = 0.0 to 0.1), wavy amplitude (A = 0.0 to 0.3), and N BT = (0.05 to 0.2), are investigated regarding temperature, velocity, local skin friction coefficient ( C f ) , local Nusselt number (Nu), streamlines, and isotherms. Elevated volume fraction generally decreases skin friction, increases temperature, and may reduce velocity. Meanwhile, higher skin friction, temperature, velocity, and local Nusselt numbers often correlate with elevated Schmidt numbers. Changes in amplitude and N BT affect skin friction, temperature, velocity, and local Nusselt numbers, providing information on the dynamics of fluid systems. For example, when the volume fraction (ϕ) increases from 0 to 0.1 at Y = 2, the temperature rises by 75 % , while the velocity gradually decreases by 11.11 % . This is in contrast to findings according to when the N BT rises from 0.05 to 0.2 at Y = 1. In this case, the temperature drops by 18.75 % , followed by a 9.68 % fall in fluid velocity. Understanding these interactions enhances comprehension of heat transfer and fluid dynamics. By investigating these interactions, we not only improve our understanding of heat transfer and fluid dynamics, but also open the door to new strategies for improving thermal systems and manufacturing procedures.

Finite element modeling of thermal‐hydro‐mechanical coupled processes in unsaturated freezing soils considering air‐water capillary pressure and cryosuction

AbstractThis paper presents a comprehensive computational model for analyzing thermo‐hydro‐mechanical coupled processes in unsaturated porous media under frost actions. The model employs the finite element method to simulate multiphase fluid flows, heat transfer, phase change, and deformation behaviors. A new soil freezing characteristic curve model is proposed to consider the suctions from air‐water capillary pressure and water‐ice cryosuction. A total pore pressure with components from liquid water pressure, air pressure, and ice pressure is used in the effective stress law. Vapor and dry air are considered miscible gases, utilizing the ideal gas law and Dalton's law. The governing equations encompass the linear momentum balance equation, the energy balance equation, and mass conservation equations for water species (ice, liquid, and vapor) and dry air. Weak forms are formulated based on primary variables of displacement, water pressure, air pressure, and temperature. The spatial discretization is achieved through the finite element method, while temporal discretization employs the fully implicit finite difference method, resulting in a system of fully coupled nonlinear equations. To verify the proposed computational model, a numerical implementation is developed and validated against a set of experimental data from the literature. The successful verification demonstrates the robustness of the model. A detailed discussion of the contributions from phase change strain and different sources of pore pressure is also addressed.

Finite difference analyses of bioconvective Williamson nanofluids stream over irregular plates with effects of nonlinear radiation and viscous dissipation

The purpose of this work is to provide a comprehensive analysis of the bioconvective Williamson nanofluid stream while considering the effects of radiation flux and viscous dissipation. The two-phase nanofluids model is applied where the impacts of the Brownian motion and thermophores are significant. This study starts with introducing the governing system in case of the irregular plane then it transforms to the regular plane. The solution methodology is based on the fully implicit finite difference method. The major results disclosed that maximizing the amplitude parameter causes a diminishing in the velocity and skin-friction coefficients. Also, lower behaviors are noted for the velocity in the case of Williamson nanofluids compared to a Newtonian nanofluid case (We = 0). Furthermore, the considered range of Biot number (1 ≤ Bi ≤ 5)causes an improvement in the surface temperature up to 71.59%.

Nonlinear Convection Flow of a Micropolar Nanofluid Past a Stretching Sphere with Convective Heat Transfer

An incompressible, steady combined nonlinear convective transport system on a micropolar nanofluid through a stretching sphere with convective heat transfer was investigated. The conservation equations corresponding to momentum, microrotation, thermal energy, and concentration particles have been formulated with suitable boundary constraints. By using the required non-dimensional variables, the conservation equations have been turned into a set of high-order standard differential equations. Then, an implicit finite difference method, also known as the Keller-Box Method (KBM), was used to numerically solve the flow problem. The obtained outcomes are displayed through graphs and tables to explain the impact of various governing variables over velocity, temperature, concentration, number of skin friction, wall coupled stress, Nusselt number, and Sherwood number. The findings demonstrate that increasing the convective heat parameter Bi enhances the factor of skin friction, local Nusselt number, Sherwood number, velocity field, and temperature profile while lowering the wall-coupled stress. It is observed that for high values of the material parameter β, the fluid velocity and the spin of the micro-elements both increase, which causes the dynamic viscosity and microrotation velocity to decrease. In addition, as the rates of magnetic constant Ma, thermophoresis Nt and Brownian movement Nb rise, the thermal distribution and its thickness of boundary layer increase. However, it decline along the enlarging quantities of nonlinear convection parameter λ, Prandtl number Pr, material parameter β, and solutal Grashof number Gm, which agrees to increase fluid density. When the range of thermophoresis Nt surges, it causes an increment in the nanoparticle species, but the opposite behavior have seen in the case of Brownian number Nb, and Lewis number Le. The comparison made with the related published paper achieves a significant agreement. The numerical result is generated through the implementation of the computational software MATLAB R2023a.

The Generalized Fractional-Order Fisher Equation: Stability and Numerical Simulation

This study examines the stability and numerical simulation of the generalized fractional-order Fisher equation. The equation serves as a mathematical model describing population dynamics under the influence of factors such as natural selection and migration. We propose an implicit exponential finite difference method to solve this equation, considering the conformable fractional derivative. Furthermore, we analyze the stability of the method through theoretical considerations. The method involves transforming the problem into systems of nonlinear equations at each time since our method is an implicit method, which is then solved by converting them into linear equations systems using the Newton method. To test the accuracy of the method, we compare the results obtained with exact solutions and with those available in the literature. Additionally, we examine the symmetry of the graphs obtained from the solution to examine the results. The findings of our numerical simulations demonstrate the effectiveness and reliability of the proposed approach in solving the generalized fractional-order Fisher equation.

Transient magnetohydrodynamic heat and mass transfer analysis of chemically reacting fluid flow over oscillatory permeable media

This study focuses on the numerical observation of the convective motion of chemically active magnetohydrodynamic (MHD) fluid through a vertically oriented permeable medium, incorporating variations in mass and heat transfer. The fluid type is assumed to be incompressible, chemically strongly ionized and viscous with some mass infusibility. The model associated with this problem is solved by a highly stable Implicit Finite Difference Method (IFDM). The method is used for small and large deflection of the physical parameters, which results in a noticeable fluid flow behavior. Numerical configuration is graphically depicted to scrutinize the fluid behavior. The momentum, energy, concentration diffusion, skin friction, Nusselt number and Sherwood number are investigated for numerous factors such as magnetic field, permeability and chemical reaction rate. The current study unveils significant findings, demonstrating that a heightened rate of chemical reaction in the presence of magnetic effects, coupled with specific porosity, diminishes ionization energy, resulting in a concurrent decrease in the concentration and momentum profiles of the fluid flow. The rise in the viscous diffusion rate is attributed to escalating values of the Schmidt number, causing an augmentation in dynamic viscosity and consequently resulting in an overall reduction in the momentum of the fluid flow.

A novel finite difference scheme for numerical solution of fractional order population growth model

In this paper, we propose a new scheme based on the implicit finite difference method for solving the fractional population growth model (FPGM). We use the well-known L1 finite difference method to approximate the Caputo fractional derivative of order 0 < α ≤ 1, and the linear interpolation to approximate the integral part. We provide a study on the stability and convergence of the scheme. We present the numerical solution of the proposed method and compare it with three other methods to demonstrate its validity and efficiency.