Thermal Radiation Parameter Research Articles

Entropy generation on MHD Casson and Williamson hybrid nanofluid over a curved stretching sheet with the Cattaneo–Christov heat flux model: semi-analytical and numerical simulations

This study demonstrates the results of a non-Newtonian comparative analysis of entropy generation on the MHD flow of a permeable curved stretching sheet with thermal radiation using the Cattaneo-Christov heat flux model. Mathematical modelling of hybrid nanofluid flow containing titanium dioxide (TiO2 ) and silver (Ag) as nanoparticles and blood as a base liquid through a curved stretching sheet has been investigated, which has vital application in the field of drug delivery. The controlling non-linear coupled PDEs are transformed into ODEs with similarity variables and then solved using the Homotopy Perturbation Method (HPM) and shooting technique (Runge Kutta fourth order). The velocity, temperature, entropy production, and Bejan number on various physical parameters like curvature, magnetic field, thermal relaxation, mixed convection, and thermal radiation parameters are discussed and explored through graphs. The heat transfer and skin friction coefficients are also studied and portrayed via graphs. The velocity profile decreases for Casson and Williamson fluids, as the magnetic field parameter and porosity parameters are increased. When the thermal relaxation parameter increases, the temperature profile decreases. The impact of the Titanium dioxide (TiO2 ) nanoparticle in volume fraction and silver (Ag) nanoparticles in volume fraction enhanced the temperature profile for Casson and Williamson fluids.

Analysis of Tangent Hyperbolic over a Vertical Porous Sheet of Carreau Fluid and Heat Transfer

The purpose of this study is to investigate the boundary layer of Carreau fluid and heat transfer over an exponentially stretching plate derived in a vertical porous with variable surface thermal flux. The partial differential equations that represent the momentum equation and heat equation are commuted into nonlinear ODEs by applying similarity transformations and results found numerically. The impact of several emerging dimensionless parameters labelled the Weissenberg number (We), the power-law index ( ), Velocity slip ( ), Thermal jump ( ), and Prandtl number ( ) on the velocity profile and heat transfer on the boundary layer are showed in detail. In more detail, also the influence of physical parameters on local skin friction and Sherwood number are studied. The shooting method with the explicit technique is used to find the solution and all results are illustrated graphically and numerically. We noted that by increasing power index, radiation parameter and velocity slip, the velocity profile increases, and the temperature profile decreases. Furthermore, it is deduced that rising the thermal radiation parameter reduces the local Nusselt number.

The impact of heat source and chemical reaction on MHD blood flow through permeable bifurcated arteries with tilted magnetic field in tumor treatments

In this research work, we investigate the influence of heat source and chemical reaction on electro-magneto-hydrodynamic (EMHD) blood flow through bifurcated arteries with external tilted magnetic field for treating tumor. The potential electric field applied along the bifurcated arterial wall, accurately described the Poisson-Boltzmann equation. We modeled the EMHD blood flow to obtain the non-dimensionalized form of the equations. We converted the modeled equations to ordinary differential equations by the use of suitable variables. Exact solutions of the converted equations are calculated by the method of undetermined coefficients and the results obtained with the aid of Mathcad software were simulated and presented graphically. From the graphical representation of results, we observed that increase in Joule heating and Eckert number, increases the temperature distribution in the affected tumor cells which prevent high thermal radiation exposure from killing the healthy cells within the tumor region. The curves of the wall shear stress seem to be greater in the converging region in comparison to the diverging region, but when the strength of magnetic field and thermal radiation parameters increase the wall shear stress decreases at the bifurcated wall where blockage may likely occur due to the development of boundary layers on the inner walls of the bifurcated region. By increasing the heat radiation parameter, we observed that the curves representing both velocity and temperature profiles increase rapidly from the origin and the velocity of blood flow varies directly for the converging, diverging and the tumor regions of the bifurcated arteries. Thus, thermal radiation effect is a result of the higher rate of heat transfer at the vessel walls. Further, combining the electromagnetic field and the heat radiation together gives new insights of the physical properties of blood flow in the body system network which is essential for health practitioners and clinicians.

Numerical scrutinization of micropolar and Walters-B non-Newtonian fluids motion under the influence of thermal radiation and chemical reaction

Purpose The purpose of this study is to explore numerical scrutinization of micropolar and Walters-B non-Newtonian fluids motion under the influence of thermal radiation and chemical reaction. Design/methodology/approach The two fluids micropolar and Walters-B liquid are considered to start flowing from the slot to the stretching sheet. A magnetic field of constant strength is imposed on their flow transversely. The problems on heat and mass transport are set up with thermal, chemical reaction, heat generation, etc. to form partial differential equations. These equations were simplified into a dimensionless form and solved using spectral homotopy analysis method (SHAM). SHAM uses the basic concept of both Chebyshev pseudospectral method and homotopy analysis method to obtain numerical computations of the problem. Findings The outcomes for encountered flow parameters for temperature, velocity and concentration are presented with the aid of figures. It is observed that both the velocity and angular velocity of micropolar and Walters-B and thermal boundary layers increase with increase in the thermal radiation parameter. The decrease in velocity and decrease in angular velocity occurred are a result of increase in chemical reaction. It is hoped that the present study will enhance the understanding of boundary layer flow of micropolar and Walters-B non-Newtonian fluid under the influences of thermal radiation, thermal conductivity and chemical reaction as applied in various engineering processes. Originality/value All results are presented graphically and all physical quantities are computed and tabulated.

Dual solutions and linear temporal stability analysis of mixed convection flow of non-Newtonian special third grade fluid with thermal radiation

The present study aims to investigate the steady mixed convection stagnation-point flow of a special third grade fluid over a vertical stretching/shrinking sheet with the inclusion of suction and thermal radiation effects. The governing momentum and energy equations are transformed into a system of ordinary differential equations (ODEs) using the Lie scaling group of transformations. The resulting system of highly nonlinear boundary value problems (BVPs) is solved numerically with the help of Matlab solver ‘bvp4c’. Dual similarity solutions are found for the opposing flow case within the range λc≤λ<0, where λ is the buoyancy parameter. Emphasis has been given to conducting a temporal stability analysis to determine the practically reliable solution. The analysis reveals that the primary (upper) solution branch is stable while the other solution branch is unstable. It has been observed that the range of dual solutions and the magnitude of the critical point increase with the increasing value of the suction parameter. The increment of the thermal radiation parameter enhances fluid temperature in the upper solution branch, whereas temperature diminishes far away from the sheet in the lower solution branch.

Numerical analysis of magnetohydrodynamics Casson nanofluid flow with activation energy, Hall current and thermal radiation

In this study we analyzed the flow, heat and mass transfer behavior of Casson nanofluid past an exponentially stretching surface under the impact of activation energy, Hall current, thermal radiation, heat source/sink, Brownian motion and thermophoresis. Transverse magnetic field with the assumption of small Reynolds number is implemented vertically. The governing partial nonlinear differential equations of the flow, heat and mass transfer are transformed into ordinary differential equations by using similarity transformation and solved numerically by using Matlab bvp4c package. The impact of each of the Hall current parameter, thermal radiation parameter, heat source/sink parameter, Brownian motion parameter, Prandtl number, thermophoresis parameter and magnetic parameter on velocity, concentration and temperature, is discussed through graphs. The skin friction coefficient along the x-and z-directions, the local Nusselt number and the Sherwood number are calculated numerically to look into the inside behavior of the emerging parameters. It is witnessed that the flow velocity is a diminishing function of the thermal radiation parameter and the behavior has observed in the case of Hall parameter. Moreover, mounting values of Brownian motion parameter reduce the nanoparticle concentration profile.

Study of fractional-order model on Casson blood flow in stenosed artery with magnetic field effect

In this research work, the impacts of magnetic fields on Casson blood flow with stenosed artery are studied. The flow under investigation is affected by a magnetic field and an oscillating pressure gradient. The Caputo–Fabrizio fractional-order derivative was used to express the governing system of partial differential equations with imposed boundary and initial conditions. The governing problem is solved using Laplace and Finite Hankel transformations. The exact solution for blood velocity, magnetic particle velocity, and temperature profiles are defined. For better understanding of the various physical parameter, numerical results are obtained for blood velocity, magnetic particle velocity and temperature profiles and presented through the graphs. It is seen that the magnetic fields tend to reduce the blood viscosity which can be used for controlling the blood flow rate. It is also seen that Casson fluid parameter tends to improve the blood flow rate. From the results, it is illustrated that the heat transfer process improves with thermal radiation and heat generation parameter.

Variational iteration method along with intelligent computing system for the radiated flow of electrically conductive viscous fluid through porous medium

This article aims to investigate the analytical nature and approximate solution of the radiated flow of electrically conductive viscous fluid into a porous medium with slip effects (RFECVF). In order to build acceptable accurate solutions for RFECVF, this study presented an efficient Levenberg-Marquardt technique of artificial neural networks (LMT-ANNs) approach. One of its fastest back-propagation algorithms for nonlinear lowest latency is the LMT. To turn a quasi-network of PDEs expressing RFECVF into a set of standards, the appropriate adjustments are required. During the flow, the boundary is assumed to be convective. The flow and heat transfer are governed by partial differential equations, and similarity transform is the main tool to convert it into a coupled nonlinear system of ODEs. The usefulness of the constructed LMT-ANNs for such a modelled issue is demonstrated by the best promising algebraic outputs in the E−03 to E−08 range, as well as error histogram and regression analysis measures. Mu is a controller that oversees the entire training procedure. The LMT-ANNs mainly focuses on the higher accuracy of nonlinear systems. Analytical results for the improved boundary layer ODEs are produced using the Variational Iteration Method, a tried-and-true method (VIM). The Lagrange Multiplier is a powerful tool in the suggested method for reducing the amount of computing required. Further, a tabular comparison is provided to demonstrate the usefulness of this study. The final results of the Variational Iteration Method (VIM) in MATLAB have accurately depicted the physical characteristics of a number of parameters, including Eckert, Prandtl, Magnetic, and Thermal radiation parameters.

Rheological behavior of magnetized ZnO – SAE50 nanolubricant over Riga plate: A theoretical study

[Formula: see text] nanolubricant is one of the most important nanolubricants having widespread uses in heat exchange systems and automobiles. It minimizes friction between moving parts and prevents them from corrosion and scrape, improves durability and performance of the system and also minimizes the consumption of fuel. This work investigates the rheological behavior of magnetized [Formula: see text] nanolubricant over moving/stationary Riga plate with viscous dissipation and nonlinear heat generation. The novel features of the current work are the use of [Formula: see text] nanolubricant as a flow fluid across a Riga plate and the application of the Patel model to boost the thermal conductivity of the nanolubricant. The governing system of equations is transformed to nonlinear ODEs and then treated analytically by using HAM. The augmentation in the velocity of nanolubricant [Formula: see text] is observed due to increasing values of Grashof number. The higher values of nonlinear thermal radiation and nonlinear heat generation parameters upsurge the temperature profile. The value of skin friction increases by increasing modified Hartman number and velocity slip parameter while reverse trend is observed by enhancing magnetic parameter. The radiation parameter, temperature dependent heat source parameter and solid volume fraction tend to augment Nusselt number at the Riga surface.

Simultaneous features of MHD and radiation effects on the UCM viscoelastic fluid through a porous medium with slip conditions

In this article, we analyzed upper-convected Maxwell (UCM) viscoelastic fluid flow in a permeable medium. Cattaneo–Christov (CC) heat flux theory is utilized for heat transfer phenomena. Impacts of thermal radiation, thermal slip condition and magnetohydrodynamic (MHD) are considered in modeling of the physical phenomena under investigation. The developed equations of fluid flow model are transformed to non-linear ordinary differential equations (ODEs) with some suitable similarity transformations. Numerical solution of ODEs with boundary conditions are investigated and the results are depicted through graphs and analyzed accordingly. From the present results, we concluded that velocity is declining for magnetic parameter; temperature is rising with thermal radiation and magnetic parameters while it shows reducing behavior for thermal slip parameter. Nusselt number observes decreasing behavior for larger values of porosity parameter, magnetic parameter, Deborah number, velocity slip and thermal slip parameters while opposite behavior is seen for radiation and thermal relaxation parameters. Furthermore, the accuracy of the current data was determined by comparing the heat transfer rate for several values of the porosity parameter to previously published results.

Numerical simulation on energy transfer enhancement of a Williamson ferrofluid subjected to thermal radiation and a magnetic field using hybrid ultrafine particles

In this numerical investigation, completely developed laminar convective heat transfer characteristics of a Williamson hybrid ferronanofluid over a cylindrical surface are reported. This new model in 2D is engaged to examine the effects of the magnetic field, thermal radiation factor, volume fraction of ultrafine particles, and Weissenberg number with the help of the Keller box method. The numerical calculations are implemented at a magnetic parameter range of 0.4 to 0.8, volume fraction range of 0.0 to 0.1, and a Weissenberg number range of 0.1 to 0.8. The numerical outcomes concluded that the velocity increases when the thermal radiation parameter and the volume fraction of a nanoparticle are increased, but inverse impacts are obtained for the magnetic parameter and the Weissenberg number. The rate of energy transport increases with increasing thermal radiation and volume fraction, while it declines with increasing the magnetic parameter and Weissenberg number. The drag force shows a positive relationship with the thermal radiation parameter and has an opposite relationship with the Weissenberg number and the magnetic parameter. Furthermore, even when the magnetic field, thermal radiation, volume fraction, and Weissenberg number are all present, the heat transfer rate of Williamson hybrid ferronanofluid is greater than that of mono Williamson ferronanofluid.

Unsteady MHD Flow of Casson Fluid Past Vertical Surface Using Laplace Transform Solution

The study explores the velocity of Casson fluid that is time-independent over an exponentially infinite isotherm vertical permeable sheet. The impact of magnetohydrodynamic (MHD) with Casson flow over the permeability sheet is examined. In contrast, thermal radiation and heat sink parameters have been incorporated. This study’s primary goal is to determine the significance of thermal radiation on Casson flow with MHD using an analytical solution over a permeable sheet. The flow of the fluid occurs above the sheet when [Formula: see text] is greater than zero, and the sheet extends far away in the [Formula: see text]-direction. The model of governing equations is reduced by applying a suitable set of dimensionless parameters. These dimensionless systems of equations are solved through the Laplace transformation method. The impacts of various variables over velocity, temperature, concentration, skin friction, and Nuseelt number are scrutinized. These variables contain magnetic field [Formula: see text], Casson fluid parameter [Formula: see text], Ghroshof number [Formula: see text], modified Grashof number [Formula: see text], Prandtl number [Formula: see text], thermal radiation [Formula: see text] and Scimdth number [Formula: see text]. These plots are sketched for the considerable magnitude of these variables through the Mathematica Software, and these plots are discussed in detail. Results show that the increasing value of [Formula: see text] reduces the fluid velocity but velocity of fluid is enhanced with larger values of [Formula: see text], [Formula: see text] and [Formula: see text]. The impact of Skin friction and Nusselt number is elaborated by tabular outlined.

Physical insight into magneto-thermo-migration of motile gyrotactic microorganisms over a flexible cylinder with wall slip, and Arrhenius kinetics

A numerical simulation of magneto-bioconvective DF (Darcy-Forchheimer) transport of gyrotactic microbes using the Williamson nanofluid model over a flexible cylinder under the physical effects of Arrhenius activation energy, thermal radiation, triple stratifications and wall slip is performed in this research communication. The flow dynamics also take into consideration the thermo-migration and random (haphazard) motion's physical effects. The similarity transformations are opted to translate the governing system of non-linear coupled PDEs into ODEs, which are then numerically tackled using the sophisticated MATLAB function named bvp4c. The significant effects of developing emergent physical factors on the accompanying fields are exploited via graphical sketches and numerically constructed tables. It is determined that strengthening the Williamson, porosity, magnetic parameters, and Forchheimer number causes considerable slowing of transport profiles. Thermal enrichment can be seen by increasing thermal radiation and thermophoresis parameters. Microbe concentration rises as a response to activation energy and reaction parameters. The current model may be used to solve various biological, biomedical, bioengineering, architectural thermal insulation, geophysical activities, and ecological problems.

Natural convection of NEPCM in a partial porous H-shaped cavity: ISPH simulation

PurposeThis paper aims to investigate the conformable fractal approaches of unsteady natural convection in a partial layer porous H-shaped cavity suspended by nano-encapsulated phase change material (NEPCM) by the incompressible smoothed particle hydrodynamics method.Design/methodology/approachThe partial hot sources with variable height L_Hot are in the H-cavity’s sides and center. The performed numerical simulations are obtained at the variations of the following parameters: source of hot length L_Hot = (0.4–1.6), conformable fractal parameterα(0.97–1), fusion temperatureθf(0.05–0.9), thermal radiation parameterRd(0–7), Rayleigh numberRa(103–106), Darcy parameterDa(10−2to 10−5) and Hartmann numberHa(0–80).FindingsThe main outcomes showed the implication of hot source length L_Hot, Rayleigh number and fusion temperature in controlling the contours of a heat capacity within H-shaped cavity. The presence of a porous layer in the right zone of H-shaped cavity prevents the nanofluid flow within this area at lower Darcy parameter. An increment in the thermal radiation parameter declines the heat transfer and changes the heat capacity contours within H-shaped cavity. The velocity field is strongly enhanced by an augmentation on Rayleigh number. Increasing the Hartmann number shrinks the velocity field within H-shaped cavity.Originality/valueThe novelty of this work is solving the conformable fractal approaches of unsteady natural convection in a partial layer porous H-shaped cavity suspended by NEPCM.

Effect of porous dissipation on nonlinear radiative flow of viscous fluid over a stretching sheet

This paper depicts the fully developed natural convective flow on a conducting viscous fluid towards a nonlinearly stretching sheet. Furthermore, the porous dissipation, thermal radiation and heating parameter effects are implemented on both the vertical walls of the stretchy channel. To model the stretchy flow equations, the Cartesian coordinates’ system is utilized. Through the utilization of similarity variables, the nonlinear partial differential equations that describe the flow (mass, momentum and energy conservation) are converted into nonlinear ordinary differential equations. With the help of the MAPLE, a well-known fourth-order Runge–Kutta procedure is used to do a numerical evaluation of the stated nonlinear and non-dimensional set of equations. For each of the several nonlinear radiative parameters regulating the flow regime, the velocity and temperature distribution functions are determined, viz the nonlinear heating parameter [Formula: see text], Eckert number [Formula: see text], Prandtl number [Formula: see text], porosity variable [Formula: see text] and thermal radiation parameter [Formula: see text]. Graphic representations are provided for every outcome. Furthermore, skin friction and Nusselt number are also computed to give an approximation of the surface shear stress and cooling rate, respectively. A remarkable compaction is obtained between computed numerical data and published results. It has been demonstrated that an increase in the value of the nonlinear parameter [Formula: see text] outcomes creates a reduction in the dimensionless translational velocity [Formula: see text] of both viscous and Newtonian fluids. Dimensionless temperature mostly upsurges with growth in nonlinear parameters [Formula: see text], [Formula: see text], [Formula: see text] and decreases with an intensification in convective parameters, [Formula: see text], [Formula: see text]. There is a detailed discussion on the implications of all embedded stretching sheet variables on the flow. The flow regime is extremely useful in the technology of polymer processing as well as in the field of materials science.

Impact of magnetohydrodynamics in bidirectional slip flow of Maxwell fluid subject to stretching, radiation, and variable properties

The viscosity of fluids varies according to changes in pressure, temperature, and shear rate. As a consequence, it is essential to look at specific qualities in the paper industry, plastics extrusion in Rayon and Nylon production, the textile sector, and other industries. In such a way, the model is more precisely anticipated when the flow configuration comprises the viscosity and thermal conductivity of the fluid as a function of temperature. Owing to the importance of variable viscosity and thermal conductivity, this article highlights the importance of the magnetohydrodynamic (MHD) flow of Maxwell fluid over a bidirectional stretched surface. Fluid is electrically conducting in the presence of an applied magnetic field. Heat transfer analysis is carried out in the presence of thermal radiation and heat generation/absorption. A system of ordinary differential equations is obtained by appropriate transformations. Numerical solutions are computed for the resulting nonlinear differential system by applying the shooting method with the RK-5 scheme. The impact of different parameters on the velocity, temperature, and concentration profiles is examined. Computations for surface drag force, heat, and mass transfer rates are presented and examined for the influences of pertinent parameters. It is observed that fluid velocity reduces for larger Maxwell fluid parameter. Increasing values of thermal radiation parameter correspond to the enhancement in temperature. Mass transfer rate enhances for larger Schmidt number while heat transfer rate is increasing function of Prandtl number.

Boundary Layer Flow of Williamson Hybrid Ferrofluid over A Permeable Stretching Sheet with Thermal Radiation Effects

This research investigated the convective boundary layer flow and heat transfer of Williamson hybrid ferrofluid over a permeable stretching sheet with thermal radiation effects. Human blood is employed as a based fluid while magnetite (Fe3O4) and copper (Cu) are taken as the hybrid ferroparticle. The study started with transforming the non-linear partial differential equation system that governed the model to a more convenience non-linear dimensionless ordinary differential equations using the similarity transformation. The transformed equations obtained then are solved numerically using the Runge-Kutta-Fehlberg (RKF45) method in Maple software. The characteristics and effects of stretching parameter, permeability parameter, thermal radiation parameter as well as the ferroparticle volume fraction in the Williamson hybrid ferrofluid towards the temperature profiles, velocity profiles as well as the Nusselt number and the skin friction coefficient are analysed and discussed. The result of this research for various pertinent parameter varies differently. It can be concluded that the increase in magnetic parameter, the Williamson parameter, the stretching parameter, and the permeability rate parameter increase the skin friction and reduced the velocity profile. Furthermore, the increase in stretching parameter, thermal radiation parameter and the permeability rate results to the increase in the Nusselt number.

MHD Mixed Convective Non-Newtonian Stagnation Point Flow Over an Inclined Stretching Sheet: Numerical Simulation

This paper includes numerical simulation upon MHD mixed convective heat transfer properties of stagnation point flow across an angled stretched sheet. Boundary value problem is solved using similarity transformation approach with shooting technique. The impact of different corporeal constraints like mixed convection parameter, thermal radiation parameter, chemical reaction parameter, Brownian motion and thermophoresis, Casson parameter upon velocity and temp profile as well as skin-friction coefficient , Nusselt number, Sherwood number on velocity, temp concentration profile are shown graphically. Casson parameter increases velocity and diminishes temperature profile. Chemical reaction term decreases Sherwood number and increases concentration profile.

Stability analysis of electrical magneto hydrodynamic stagnation point flow of Ag-Cu/water hybrid nanofluid over a permeable stretching/shrinking slendering sheet: Entropy generation

The current study emphasises the stability test of electrical magneto hydrodynamic stagnation point flow of hybrid nanofluids with entropy generation in the existence of velocity and thermal slips over a porous slendering stretchable sheet. A mathematical modelling of [Formula: see text]/water hybrid nanofluid has been explored. Heat transfer proficiency of airplane wings is evaluated with the inclusion of distinguished effects like viscous dissipation and solar-based thermal radiation, so we included them in this article. Using suitable self-similarity transformations, the system of the fluid transport equations are converted into an ordinary differential system, which are determined by utilising the bvp4c in MATLAB software. The achieved outcomes are expressed as the skin friction, Nusselt number, velocity, temperature, Bejan number, entropy generation and streamlines with an influence of diverse physical parameters. The boundary-layer separation happens when the number of results fails to happen ahead of the critical value of the contracting parameter. To show the effect on flow and heat transfer of appropriate parameters, graphs are used. Stability is shown to be better in solution one than in the second, according to the results. The wall thickness variable augmented the velocity profile of the hybrid nanofluid for the first solution and a contrary for the second solution. Higher values of the thermal radiation parameter resulted in enhanced temperatures in the first and second solutions. It is also observed that the rate of heat transfer in the first solution increased as the electric field value increased.

Significance of thermal conductivity and heat transfer mechanism through copper nanofluid with convective condition via heated Riga plate

This study investigates the heat and mass transfer of magnetohydrodynamic nanofluids via the Riga plate subjected to convective boundary conditions, including the thermal radiation parameter. Utilizing variable thermal conductivity and mixed convection effects, the heat transfer process is investigated. The peculiarity of the flow model enables us to investigate the importance of thermophoresis and Brownian motion to the kinetics of Newtonian fluids. The governing partial differential equations are converted into non-dimensional ordinary differential equations using proper similarity transformations and numerically solved using bvp4c in Matlab. The graphs explore the effects of pertinent parameters on the transference of heat, mass and velocity profiles. As the Biot number [Formula: see text] increases, so does the thermal boundary layer. The Nusselt number intensifies with radiation parameter but decreases with intensifying magnetic field parameter [Formula: see text], Eckert number [Formula: see text] and Brownian motion parameter [Formula: see text]. The influence of several physical quantities is illustrated and displayed through graphs and tables. Furthermore, it is observed that the thickness of the temperature profile is increased by combining the Biot number, the thermal radiation parameter, the thermal conductivity parameter and the modified Hartmann number.