Runge Kutta Fehlberg Research Articles

Stagnation Point Flow of CoF e 2 O 4 / Ti O 2 - H 2 O -Casson Nanofluid past a Slippery Stretching/Shrinking Cylindrical Surface in

Mounting temperatures in electronic devices during operation may damage sensitive internal components if too much thermal energy accumulates inside the system. The advent of an innovative ultrahigh-performance thermal management technology known as nanofluid has provided a veritable platform to improve the system performance and reliability by removing the high heat flux generated in the engineering and industrial devices. This paper examines the combined effects of Darcy–Forchheimer porous medium-resistant heating and viscous dissipation on stagnation point flow of a Casson nanofluid ( CoF e 2 O 4 - H 2 O and Ti O 2 - H 2 O ) towards a convectively heated slippery stretching/shrinking cylindrical surface in a porous medium. The governing nonlinear model equations are obtained, analysed, and tackled numerically via the shooting technique with the Runge–Kutta–Fehlberg integration scheme. A unique solution is obtained when the surface is stretching. For shrinking cylindrical surface, the model exhibits nonunique dual solutions for a defined range of parameter values, and a temporal stability analysis is conducted to ascertain the stable and physically achievable solution. The effects of emerging thermophysical parameters on the overall flow structure and thermal management such as velocity and temperature profiles, skin friction, and Nusselt number are quantitatively discussed through graphs and in tabular form. It is found that the thermal performance heat transfer enhancement capability of Ti O 2 - H 2 O is higher than that of CoF e 2 O 4 - H 2 O . Moreover, the nanofluid thermal performance is enhanced with nanoparticles volume fraction, Casson nanofluid parameter, and Biot number but lessened with porous medium permeability.

Traveling wave in an eco-epidemiological model with diffusion and convex incidence rate: Dynamics and numerical simulation

This work aims at studying an epidemic model for infections in the predator–prey interaction with diffusion. We inspected the stability of the model without a diffusion case. The incidence rate is assumed to be convex relative to the infectious class. Traveling wave solutions and the minimum wave speed are also obtained using the ”linear determinacy”. Furthermore, a combined scheme based on the finite difference method and the Runge–Kutta–Fehlberg method is applied to obtain the numerical simulations. Numerical results show that by selecting the appropriate conditions and embedding parameters in the governing equations, the traveling wave solutions in the proposed eco-epidemiological model are consistent with the minimum wave speed.

A numerical investigation of cross‐diffusion on magnetohydrodynamic Cu‐Al2O3/H2O hybrid nanofluid flow over a stretching sheet with chemical reaction

AbstractHybrid nanofluids (HNFs) are a promising type of nanofluid (NF) that combines two different nanoparticles to enhance their thermal and mechanical properties. They have gained significant attention in various engineering and scientific applications. This research focuses on studying the temperature‐dependent viscosity of a magnetohydrodynamic copper‐alumina‐water‐based HNF with thermal radiation, diffusions, and chemical reaction on a stretching sheet embedded in a porous medium. To achieve this, we transform the partial differential equations (PDEs) governing the system into a system of nonlinear ordinary differential equations (ODEs) using systematic similarity transformations. The resulting ODEs are solved using the shooting mechanism in combination with the Runge–Kutta–Fehlberg methodology. The results for the impact of the nanoparticle reactions on thermal, concentration, and velocity profiles, surface drag force, as well as mass and heat transfer rates are presented through graphs and tables. The key findings of this study demonstrate that the HNF accelerates velocity by 6.62%, but the magnetic field weakens it by 9.26%. Chemical reaction and Schmidt number affect nanoparticle concentration inversely. Radiation and heat source parameters increase the temperature by 14.89%, but the Prandtl number decreases it by 2.2%. Moreover, the Cu‐Al2O3/H2O HNF exhibits better thermal efficiency by 17.75% compared with the copper‐water NF. This research holds potential for applications in heat transduction, energy production, biomedical research, the manufacturing sector, aerospace technology, and more.

Numerical approach for the modified heat transport process under improved Darcian and nonlinear mixed convective effects

Analysis regarding heat transport subjected to non-Darcian media has increased in science and technological utilization. Typical applications comprise grain storage, nuclear waste discarding, production of crude oil, pollution in groundwater, porous insulation, and several others. This motivates us to demonstrate the heat transport phenomenon in the viscous fluid's flow deformed by a linearly stretchable surface submerged in Darcy-Forchheimer porous medium. Properties of heat transfer are elaborated with nonlinear mixed convection and Cattaneo-Christov heat model. Slip phenomenon is implemented at the wall surface. The governing equations are achieved in dimensionless form through appropriate transformations. Numerical solutions are constructed through shooting scheme along with 4–5 order Runge-Kutta Fehlberg technique. Graphical behaviors of velocity field and liquid temperature are constructed through various physical parameters. Results witnessed that decrement occurs in velocity field for dominant behavior of velocity slip parameter while it enhances for dominant nonlinear mixed convection parameter.

A numerical study on model-based comparative analysis for MHD Magnetite (Fe3O4) and Cobalt Ferrite (CoFe2O4) flow past a heated shrinking Riga surface with radiative heat flux

Magnetic nanofluids exhibit remarkable ideas from their inherent ability to be externally controlled and to undergo changes in their physical characteristics. These changes, influenced by several factors such as volume concentration of nanoparticles and the strength of the applied magnetic field, those have a range of potential applications. With this understanding in mind, the current study adopts a model-based approach to probe the behavior of magnetohydrodynamic (MHD) stagnation point flow. This flow involves a hybrid nanofluid consisting of Fe3O4 (magnetite) and CoFe2O4 (cobalt ferrite) nanoparticles dispersed in water (H2O), through a heated shrinking Riga surface with velocity slip. The investigation integrates the influential conductivity models such as Hamilton-Crosser and Yamada-Ota models. Employing a numerical methodology, the dimensionless system is solved those are derived by implementing appropriate similarity transformation. Specifically, the Runge-Kutta-Fehlberg (RKF) procedure is synergistically employed in conjunction with the shooting method. The outcomes of the study reveals that; the enhanced volume concentration that enriches the physical quantities favours in augmenting the fluid temperature and the profile also enhances for the increasing thermal radiation.

Computational assessment of MHD Carreau tri-hybrid nano-liquid flow along an elongating surface with entropy generation : A comparative study

This research looks at the incompressible, viscous, steady-state laminar stagnation-point-flow of a Carreau ternary-hybrid nano-liquid towards a convectively heated expanding surface, taking into account first-order slippage and Ohmic dissipation. Viscous dissipation and the effects of Lorentz forces are also considered. It is discussed how thermal radiation and heat absorption/generation contribute to the heat transport process. Minimizing entropy during the transport of a Carreau ternary hybrid nano-liquid has also been studied. In addition, convective circumstances are used conceptually in the numerical solution of the current model. tiny-particles of silver (Ag), molybdenum disulfide (MoS2), and multi-wall carbon nanotubes (MWCNT) are analyzed in this work’s flow study. As a base fluid, carboxymethyl cellulose (CMC-water) is used. With the use of appropriate similarity trans-formations, the standard model equations are transformed into dimensionless form. Finding solutions for momentum and heat fields using the Runge–Kutta-Fehlberg technique and a shooting strategy. The figures depict a wide range of characteristics, including fluid flow velocity, temperature, skin friction, the Nusselt number, entropy minimization, and the Bejan number. Key results from the present model include the fact that the velocity curve flattens down as Weissenberg number increases.

Heat and mass transfer analysis of assisting and opposing radiative flow conveying ternary hybrid nanofluid over an exponentially stretching surface

Access to dependable and environmentally friendly energy sources is critical to a country's economic growth and long-term development. As countries seek greener energy alternatives, the interaction of environmental elements, temperature, and sunlight becomes more critical in utilizing renewable energy sources such as wind and bioenergy. Solar power has received much attention due to extraordinary efficiency advances. under this context, the present work focus on solar radiation and chemical processes in the presence of modified ternary hybrid nanofluids (THNFs) circulating over an exponentially stretched surface in both aiding flow (A-F) and opposing flow (O-F) circumstances. The primary objective of this investigation is to dive into the complicated dynamics of these structures, which are distinguished by complex interactions involving radiation, chemical reactions, and the movement of fluids. We construct reduced ordinary differential equations from the governing equations using suitable similarity transformations, which allows for a more in-depth examination of the liquid's behavior. Numerical simulations using the Runge–Kutta Fehlberg (RKF) approach and shooting techniques are used to understand the underlying difficulties of these reduced equations. The results show that thermal radiation improves heat transmission substantially under O-F circumstances in contrast to A-F conditions. Furthermore, the reaction rate parameter has an exciting connection with concentration levels, with greater rates corresponding to lower concentrations. Furthermore, compared to the O-F scenario, the A-F scenario promotes higher heat transfer in the context of a modified nanofluid. Rising reaction rate and solid fraction volume enhanced mass transfer rate. The rate of thermal distribution in THNFs improves from 0.13 to 20.4% in A-F and 0.16 to 15.06% in O-F case when compared to HNFs. This study has real-world implications in several fields, including developing more efficient solar water heaters, solar thermal generating plants, and energy-saving air conditioners.

Framing the features of nanolayer and diameter of carbon nanotubes in the flow of blood over a stretching cylinder in presence of magnetic induction

This study explores the slip boundary layer electrically conducting flow of nanofluid over a stretching cylinder. Flow type in present study is considered to be steady and incompressible flow with mixture of SWCNTs/MWCNTs and human blood. Thermal conductivity of present flow model is measured by merging the consequence of diameter of SWCNTs/MWCNTs nanoparticles and solid–liquid interfacial-layer. Velocity slip, non-linear radiation, and induced magnetic-flux are considered in flow. The mode of heat transmission in flow is presented in appearance of dual heat flux, that is, Cattaneo-Christov heat flux and prescribed heat flux. By executing similarity exploration, we modernize the leading PDEs of the acknowledged flow model into ODEs. The consequential equations are solved numerically by recalling Runge–Kutta–Fehlberg fourth-fifth order scheme through shooting system. Deliberations on consequence of flow features on flow specific are organized accurately via graphs and charts. The professed numerical consequences are novel and distinctive. These consequences can contribute to other scholars on electing the suitable factors to enlarge the heat conduction technique in medical science, society, and the industry.

Exploring the Influence of Induced Magnetic Fields and Double-Diffusive Convection on Carreau Nanofluid Flow through Diverse Geometries: A Comparative Study Using Numerical and ANN Approaches

This current investigation aims to explore the significance of induced magnetic fields and double-diffusive convection in the radiative flow of Carreau nanofluid through three distinct geometries. To simplify the fluid transport equations, appropriate self-similarity variables were employed, converting them into ordinary differential equations. These equations were subsequently solved using the Runge–Kutta–Fehlberg (RKF) method. Through graphical representations like graphs and tables, the study demonstrates how various dynamic factors influence the fluid’s transport characteristics. Additionally, the artificial neural network (ANN) approach is considered an alternative method to handle fluid flow issues, significantly reducing processing time. In this study, a novel intelligent numerical computing approach was adopted, implementing a Levenberg–Marquardt algorithm-based MLP feed-forward back-propagation ANN. Data collection was conducted to evaluate, validate, and guide the artificial neural network model. Throughout all the investigated geometries, both velocity and induced magnetic profiles exhibit a declining trend for higher values of the magnetic parameter. An increase in the Dufour number corresponds to a rise in the nanofluid temperature. The concentration of nanofluid increases with higher values of the Soret number. Similarly, the nanofluid velocity increases with higher velocity slip parameter values, while the fluid temperature exhibits opposite behavior, decreasing with increasing velocity slip parameter values.

Active- passive controls on magneto CNTs nanofluid flow over a wavy rotating disc

ABSTRACT The target of this article is to perceive the effects of the externally applied magnetic force on an unsteady nanofluid flow in the occurrence of carbon nanotubes with water as a base fluid. The flow is caused by a rotating and wavy spinning permeable disk where the disk is continuously moving vertically upward and downward. 3D nanofluid flow is created by a spinning and vertically moving disk. A study of the passive and active mass flux of nanoparticles is presented in this work. The problem is solved by the Runge–Kutta–Fehlberg 45 (RKF 45) method in the mathematical software MAPLE. The outcomes of the present work are compared to the results communicated previously. The physical measures are exemplified pictorially in thermal and concentration fields. 2D contour phenomenon and 3D flow conceptions for some pertinent parametersare also sketched. There is a slight difference in temperature and concentration profiles between MWCNTs and SWCNTs. It is detected that in the presence of active and passive control, the thermal energy profile shows decreasing behaviour by the fluctuating parameter; however, the concentration profile exhibits dual nature under passive control.

Effectiveness of Stefan blowing on MHD slip flow of Prandtl–Eyring nanofluid over an elongating sheet with activation energy and viscous-Ohmic dissipation

This work examines the Stefan blowing effect on the magnetized stagnation point flow of Prandtl–Eyring nanoliquid along a nonlinear elongating surface under the influence of numerous slips. In this study, the impressions of internal heat sources, nonlinearly varying radiation, activation energy with binary chemical reactions, viscous dissipation, and Joule dissipation are taken into account. The thermal flux associated with thermal radiation is computed using Rosseland’s approximation for an optically thick environment. The controlling dimensional mathematical equations are transformed into nondimensional representations using appropriate similarity transmissions. Numerical temperature, velocity, and concentration solutions are obtained using a shooting method based on secant iteration and the Runge–Kutta–Fehlberg method. Using tabular and graphical displays of numerical data, the physical effects of a number of relevant parameters on the nanoliquid temperature, velocity, and concentration are examined. By contrasting the current results with information that has already been published for a few limiting circumstances, the validity of the acquired results is demonstrated. The effects of physical features on the local surface drag coefficient, rates of mass and heat transfer are demonstrated using numerical data shown in tabular form. For increasing values of the Stefan blowing parameter, fluid velocity decreased. The fluid temperature tends to rise as a result of thermal radiation.

Quadratic Thermal Convection in Magneto-Casson Fluid Flow Induced by Stretchy Material with Tiny Particles and Viscous Dissipation Effects

This investigation is based on the phenomenon of quadratic thermal convection in Magneto-Casson fluid flow induced by stretchy material with tiny particles and viscous dissipation effects. The study aims to understand and optimize the complex behaviour of fluid flow and heat transfer in this system, which has significant implications for engineering and industrial applications. The Magneto-Casson fluid, characterized by its non-Newtonian behaviour and yield stress, interacts with stretchy material and tiny particles, introducing unique flow characteristics and thermal properties. Viscous dissipation, resulting from internal friction, further influences the convective heat transfer process. Mathematical models are developed and solved by employing a numerical technique through the Runge-Kutta Fehlberg scheme coupled with the shooting method to investigate these phenomena comprehensively. The results are deliberated using several graphs on the dimensionless profiles. The results showed that there is a decline in the momentum boundary film as the magnetic field improves due to the effect of the Lorentz force. Also, an increase in the Casson fluid term raises the viscosity and thereby resists the fluid motion.

Computational Examination of Heat and Mass Transfer Induced by Ternary Nanofluid Flow across Convergent/Divergent Channels with Pollutant Concentration

Studying waste discharge concentration across a convergent/divergent channel is essential in environmental-related applications. Successful environmental administration must understand the behavior and concentration of waste contaminants released into these channels. Analyzing waste discharge concentrations aids in determining the efficacy of treatment techniques and regulatory controls in lowering pollutant scales. Because of this, the current analysis examines the ternary-based nanofluid flow across convergent/divergent channels, including non-uniform heat source/sink and concentration pollutants. The study also concentrates on understanding the movement and heat transmission characteristics in ternary-based nano-liquid systems with divergent and convergent channels and maximizing the ternary nanofluid flow’s effectiveness. The equations representing the flow, temperature, and concentrations are transformed into a system of ODEs (ordinary differential equations) and are obtained by proper similarity variables. Further, solutions of ODEs are gathered by using the Runge Kutta Fehlberg 4-5 (RKF-45) method and shooting procedure. The significant dimensionless constraints and their impacts are discussed using plots. The results mainly focus on improving local and external pollutant source variation will enhance the concentration for the divergent channel while declining for the convergent channel. Adding a solid fraction of nanoparticles will escalate the surface drag force. These findings may enhance heat management, lessen pollutant dispersion, and enhance the circulation of nanofluid systems.

Effect of Nanoparticle Diameter in Maxwell Nanofluid Flow with Thermophoretic Particle Deposition

The time-dependent Maxwell nanofluid flow with thermophoretic particle deposition is examined in this study by considering the solid–liquid interfacial layer and nanoparticle diameter. The governing partial differential equations are reduced to ordinary differential equations using suitable similarity transformations. Later, these reduced equations are solved using Runge–Kutta–Fehlberg’s fourth and fifth-order method via a shooting approach. An artificial neural network serves as a surrogate model, making quick and precise predictions about the behaviour of nanofluid flow for various input parameters. The impact of dimensionless parameters on flow, heat, and mass transport is determined via graphs. The results reveal that the velocity profile drops with an upsurge in unsteadiness parameter values and Deborah number values. The rise in space and temperature-dependent heat source/sink parameters value increases the temperature. The concentration profile decreases as the thermophoretic parameter upsurges. Finally, the method’s correctness and stability are confirmed by the fact that the maximum number of values is near the zero-line error. The zero error is attained near the values 2.68×10−6, 2.14×10−9, and 8.5×10−7 for the velocity, thermal, and concentration profiles, respectively.

Theoretical analysis of Arrhenius activation energy on 3D MHD nanofluid flow with convective boundary condition

Industry and space technology have significant issues managing heat energy and controlling mass dispersion. The purpose of this study is to develop motion caused by boundary layer thickness sheets that are increasingly being used in various engineering fields (civil engineering, mechanical, aeronautical, maritime processes and constructions). The activation energy is a critical factor in chemical reactions due to the existence of many applications in gas-cooled reactors, nuclear thermal rockets and liquid-fluoride reactors. This study presents the numerical analysis of activation energy on three dimensional (3D) nanofluid (NFs) motion via Stretching Surface (SS) with nonlinear thermal radiation effect. This is in contrast to the conventional slip condition, convective condition applied at surface. The governing basic equations are translated into nonlinear ODEs by suitable similarity transformations. The relevant boundary value problem was explored for a numerical solution for applying the MATLAB based on Runge–Kutta–Fehlberg (RKF) scheme via shooting technique. The major outcomes of current work have more concentration ([Formula: see text]) and Mass Transfer Rate ([Formula: see text]) for various numerical values of Activation Energy ([Formula: see text]). The present solutions determine very good correlation with the previously studied ones in a special case as predicted in the tables.

Magnetic dipole effects on unsteady flow of Casson-Williamson nanofluid propelled by stretching slippery curved melting sheet with buoyancy force

In particular, the Cattaneo-Christov heat flux model and buoyancy effect have been taken into account in the numerical simulation of time-based unsteady flow of Casson-Williamson nanofluid carried over a magnetic dipole enabled curved stretching sheet with thermal radiation, Joule heating, an exponential heat source, homo-heterogenic reactions, slip, and melting heat peripheral conditions. The specified flow's partial differential equations are converted to straightforward ordinary differential equations using similarity transformations. The Runge–Kutta–Fehlberg 4-5th order tool has been used to generate solution graphs for the problem under consideration. Other parameters are simultaneously set to their default settings while displaying the solution graphs for all flow defining profiles with the specific parameters. Each produced graph has been the subject of an extensive debate. Here, the analysis shows that the thermal buoyancy component boosts the velocity regime. The investigation also revealed that the melting parameter and radiation parameter had counterintuitive effects on the thermal profile. The velocity distribution of nanofluid flow is also slowed down by the ferrohydrodynamic interaction parameter. The surface drag has decreased as the unsteadiness parameter has increased, while the rate of heat transfer has increased. To further demonstrate the flow and heat distribution, graphical representations of streamlines and isotherms have been offered.

Combined effects of thermodynamic factors and external fields for nonlinear optical processes of deformed Mathieu quantum dot containing central impurity

The theoretical impact of external parameters such as the hydrostatic pressure, temperature, electric field, magnetic field, as well as the Indium (In) concentration and quantum dot width on the nonlinear optical properties of a Mathieu quantum dot (MQD) with a screw dislocation and a hydrogenic impurity at its center, formed by an InxGa1−xAs/GaAs heterojunction, is presented. The wave equation of the system is solved in cylindrical coordinates using the effective mass approximation and the Runge-Kutta-Fehlberg (RKF) method, taking into account the direction of the screw dislocation and the symmetry of the structure. In addition to the screw dislocation, thermodynamic effects, electric and magnetic fields, In concentration, and the width of the MQD have significant effects on the electronic energy levels, dipole matrix elements, and transition frequencies, resulting in important consequences for the nonlinear optical properties. The effects and alternatives of the relevant parameters on the nonlinear optical properties of the MQD with a screw dislocation are discussed in detail. The optical properties that can be altered by these parameters may also be important for experimental studies in determining the optimality of the structure.

Thermal instability in electrically conducting nanoliquid filled in Hele-Shaw cell under 3-types of rotational-speed modulation with impact of through-flow and magnetic-field

Due to the great significance of rotation-speed modulation in various areas of science and technology as well as in day-to-day life, our aim, in this study, is to perform linear/nonlinear studies to examine the impacts of sinusoidal like sine waveform and non-sinusoidal like square and triangular waveforms of rotational-speed modulation in nanoliquid (electrically-conducting) filled within a Hele-Shaw cell with through-flow and magnetic-field. The applied rotational-speed modulation has periodic components that are both constant and time-dependent and changes sinusoidally over time. The linear analysis is done using Galerkin’s approach while the nonlinear analysis is done using truncated Fourier series method. The results of Mathematica’s built-in NDSolve tool are validated using the Runge–Kutta-Fehlberg technique. Hele-Shaw number, Taylor-number, nanoliquid magnetic number and frequency of rotational-speed modulation have a stabilizing impact on the system, whereas through-flow, magnetic Prandtl-number, nanoliquid Prandtl-number and amplitude of rotational-speed modulation have destabilizing impact on the system. Additionally, it has been found that rotational-speed modulation can be utilised to efficiently increase or decrease heat and mass movement. Square-waveform is most efficient for both heat and mass transport in compression of sinusoidal and triangular waveforms.

MHD thermo-radiative heat transfer characteristics of carbon nanotubes based nanofluid over a convective expanding sheet in a porous medium with variable thermal conductivity

ABSTRACT The impacts of thermal radiation (non-linear) and thermal conductivity (variable) on magnetohydrodynamic (MHD) heat transfer over a flat stretchable surface by taking single-walled carbon nanotubes (SWCNTs) and multiple-walled carbon nanotubes (MWCNTs) inside the base liquid (water) embedded in a porous media have been analysed. We have utilized an appropriate set of similarity transformations to rehabilitate the governing partial differential equations into a non-linear ordinary differential equations system. These equations are then solved using a Runge–Kutta–Fehlberg technique with a shooting algorithm. The influences of several critical physical parameters on the velocity (V) and temperature (T) fields, coefficient of skin-friction (C f ), and local Nusselt number (Nu x ) have been examined in detail by plotting several graphs and tables for both SWCNTs and MWCNTs. The acquired outcomes show that SWCNTs in water-based nanofluid has remarkable effects on skin friction and the heat transfer rate when contrasted with MWCNTs. Likewise, the variable thermal conductivity enhancement diminishes the Nu x , whereas the inverse impacts the expanding T ratio parameter.

Computational analysis of ternary nanofluid flow in a microchannel with nonuniform heat source/sink and waste discharge concentration

The investigation of waste discharge concentration with microchannel plays a significant impact on environmental applications. Microchannels are utilized in treatment facilities for wastewater to manage the concentration of pollutants and impurities. Effective waste treatment and pollution control techniques may be created by investigating waste material’s fluid flow and dispersion in microchannels. In view of the above applications, the present investigation examines the electrically conducting and incompressible ternary hybrid nanofluid movement in a microchannel with a nonuniform heat source/sink and waste discharge concentration. The governing equations are converted into a system of ordinary differential equations using appropriate similarity constraints, and the resulting equations are numerically solved using the RKF-45 (Runge Kutta Fehlberg 4th and 5th) method. The effects of different nondimensional limitations are visually displayed and thoroughly explored on flow patterns, heat distribution, and concentration profiles. The analysis is done on important engineering variables such as skin friction, heat transfer rate, and mass transfer. Magnetic and porous media shape factors directly influenced the velocity and temperature profiles. These two factors will decline the velocity and improve the temperature. Further, the improvement in local and external pollutant concentration will improve the concentration of the fluid. The rate of mass transfer improves with increasing the local external source constraint. The present work outcomes are helpful in heat exchangers, pollution control, microfluidic chip devices, wastewater treatment, and energy conversion.