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This paper presents the numerical study for heat generation and electro-osmotic effects of Carreau model of a fluid in an unsteady channel. This prepared model has many useful applications in microfluidic biological and mechanical research. Modified Darcy’s law, viscous dissolution and ciliated walls are also presented. Non-Newtonian fluids play a key role in biological fluids which include the motion of foods within digestive zone, blood circulation, including small bowls and migration of urine. The main findings for the research is to analyze the flow rate change and check the behavior of velocity and temperature by using different parameters. The modeled equations of Carreau fluid flow are depressed into a simple form by employing the long wavelength approximation and small Reynolds number, then the resulting equations are calculated numerically by using MATHEMATICA. Impact of physical factors are graphically figured for velocity, temperature field and streamline distribution. The results show that velocity modifies by applying different values of flow rate and Darcy’s number. Temperature profile increases for advanced values of Prandtl number, Eckert number and heat source parameter. For heat source parameter (γ=0.5, 0.6 and 0.7) and Eckert number (Ec=1.0, 1.5 and 2.0) the temperature profile expands at the region (y

The main aim behind this work is that we have to determine the enhancement that happens in thermal conductivity by using the Buongiorno model of nanofluid when the nanofluid particles are added to the tangent hyperbolic base fluid moving above the stretched surface. The viscosity of tangent hyperbolic nanofluid is assumed to be temperature-dependent. The modified form of the Fourier law of heat conduction motivated us; therefore, the Cattaneo-Christov heat and mass flux model is considered to observe its significance for heat and mass transport. The chemical reaction and heat generation are also considered to determine their influence on temperature and concentration gradients. This study is crucial because of its application in industrial manufacturing processes such as the coating of wire, the thinning of copper, the production of paper, photographic films, hot rolling, and the purification of crude oil. In electronic cooling systems, the efficient dissipation of heat in smartphones, computers, and data centers is critical to preventing overheating. Nanofluids, with their enhanced thermal properties, can significantly improve heat transfer rates, ensuring better performance, stability, and energy savings. The partial governing equations arising from fluid flow, mass, and heat transfer are transformed into ordinary differential equations via appropriate similarity variables. The obtained ordinary differential equations are solved numerically by using the Runge-Kutta Fehlberg method in the MATLAB software. The effects of the emerging parameters are represented through graphs. According to the results, the velocity profile upsurges due to the natural convection parameters and curvature parameter, while the power law index declines the velocity profile. As the Brownian motion coefficient rises, the concentration profile diminishes while the temperature profile increases. Both the concentration and temperature profiles increase for larger values of the thermophoresis parameter. The concentration profile declines for larger values of chemical reaction parameters, and enhancement happens in the temperature region.

The current approach aims to investigate the behavior of Williamson fluid model convoyed with thermal radiation, viscous dissipation and activation energy between sinusoidal walls. The electro-osmotic phenomenon is modeled through Poisson-Boltzmann equation. Electro-osmotic fluid has many applications such as, electro-osmotic fluid pumps, liquid medicine delivery, micro-fabricated fluid devices and lab-on-a-chip devices, etc. The model is produced by using a very relatively small Reynolds number and the long-wavelength approximation. Dimensionless transformations are used to reduce the system into dimensional form. DSolve command in MATHIMATICA and perturbation approach is used to simplify the governing equations. After applying the perturbation technique, graphs are used to illustrate the physical effects of temperature, velocity, concentration, pressure gradient, and streamline distribution.

In this article, we consider the boundary layer two-dimensional Carreau fluid flow induced by a heated inclined flat plate. The flow having variational viscosity of Carreau fluid is considered. The dependence of thermo-physical properties on temperature and concentration is also discussed. Using the Falkner-Skan solution, the governing equations are converted into a system of ordinary differential equations, which have been solved by RK-five integration scheme along with a shooting method. Solutions are calculated for fluid with thermo-physical properties of shear-thinning (viscosity decreases with increasing shear-rate) and thickening (viscosity increases with increasing shear-rate) effects. Physical interpretation and discussion are presented regarding the base flow, temperature and concentration profiles. Boundary layer thickness engages inversely with pressure gradient m, whereas temperature profile shows a significant increase for higher values of ε1.

The Williamson nanofluid flow along with the thermal and solutal transport effects like enthalpy and viscous dissipation, has lots of applications in numerous engineering fields such as heat exchangers, automotive engineering, cooling systems, aerospace, and biomedical applications. The temperature-dependent viscosity has implications in the polymer processing, chemical processing, microfluidic devices, environmental fluid dynamics, and oil and gas industries. The current research deals with the numerical study of the magnetohydrodynamic flow of 2-dimensional Williamson nanofluid flowing on a stretched cylinder. The thermophysical property (viscosity) is assumed to depend on temperature for this flow system. The transmission analysis of solutal and heat processes, along with enthalpy and viscous dissipation, is also considered here. The boundary layer approach is used to address the main leading equations, and ODEs are generated from the main equations by assuming similarity transformations. The main novelty of this problem is the use of a multistep technique, which is the Adams-Bashforth Predictor and Corrector method, along with the RK-4 scheme and the secant method. The key parameters are used to illustrate the graphical and numerical behavior of the velocity, temperature, and concentration regions. The variations in the Williamson fluid coefficient cause the fluid velocity to increase, whereas alterations in the curvature parameter cause the velocity region to decrease. The temperature in a given area declines due to the impact of the Prandtl number. The fluids temperature shows augmented values in the occurrence of variation in the thermopherosis coefficient, Brownian motion coefficient, curvature parameter, and heat generation parameter. The curvature parameter, thermopherosis coefficient, and activation energy lead to an increase in the fluid concentration region, whereas a decrease in concentration is noted due to the Brownian motion coefficient, Damkholer number and Schmidt number.

In this paper, the stagnation point and boundary layer flow of two-dimension Williamson fluid flow near a stretching cylinder along porous medium is examined. The transference analysis of mass and heat in the occurrence of thermal radiation, heat absorption and activation energy are investigated. The double stratification is implemented to analyze its impact on the thermal and solutal fields. The main leading equations are solved by applying the boundary layer approximation. By assuming the similarity variables, the main equations of the Williamson fluid are transformed into ordinary differential equations. In order to solve numerical solutions, a well-known numerical technique is adopted on MATLAB software namely ‘Bvp4c’. The fluid's velocity experiences an increment with changes in the buoyancy parameter, curvature parameter and the Williamson fluid coefficient, whereas the velocity distribution diminishes due to variations in the porosity parameter, stretching parameter and Forchheimer number. The temperature in a given region decreases due to the influence of both the Prandtl number and curvature parameter. The fluid temperature exhibits incrementing outputs in the presence of changes in the curvature parameter, thermal stratification parameter and heat generation. The solutal stratification parameter, temperature difference parameter and Schmidt number lead to a reduction in the fluid concentration. The concentration profile increases under the influence of curvature parameter. The validation of numerical finding is also included here which show similar trend with published work.

Heat and mass transfer having variable fluid properties has wide range of applications in industrial and engineering processes.The transference of species arises in several processes like oil transport phenomenon, food processing and diffusion of assured medicines in blood (drug delivery). Current research deals with the mixed convection and radiation effects on flow of a non-Newtonian tangent hyperbolic fluid with variable thermo-physical properties and activation energy. Arrhenius kind of chemical reaction is taken along an exponentially stretchable surface. The main focus of the current exploration is to execute shear thinning tangent hyperbolic fluid past an exponentially stretchable surface under the influence of variable viscosity, mixed convection, thermal radiation and activation energy. Transformation approach is opted to attain the non-linear ODE's based on standard conservation laws of mass, linear momentum and energy. Numerical solutions of the governing problem are computed via the shooting technique. Characteristics of the parameters appearing in modeling like radiation parameter, variable viscosity parameter, mixed convection parameter, Williamson parameter, variable conductivity and diffusivity are comprehensively analyzed through graphical behavior.

Fluids like hybrid nanofluids have great potential that present exceptional thermal behavior and thermophysical properties as compare to ordinary nanofluids. Hybrid nanofluids are obtained by the mixture of two different nanoparticles in a base fluid. Many scholars stated that hybrid-nanofluids could be replaced by conventional coolants, especially fluids which work at high temperature. Therefore, these kinds of nanofluids are less damaging for environmental impact and also lead to save energy. The main idea of hybrid nanofluids is to develop performance of heat transfer and its advantages has directed to comparatively good hope for its applications. The objective of present article is to examine 3D stagnation point flow of hybrid nanofluid passes along a stretchable heated wavy cylinder under the impact of variable thickness and slip conditions. For this purpose we consider alloy particles of AA7075 + AA7072 and AA7072 which are suspended in methanol liquid. The alloy particles merged in this analysis are entirely manufactured materials as well as possessing higher heat transfer characteristics. Alloy AA7072 is a combination of zinc and aluminum in ratio 1:98 with addition of metals like silicon, copper and ferrous. Similarly, alloy AA7075 is a combination of zinc, aluminum, copper and magnesium in ratio −6, −9, −1 and −3 respectively with addition of metals like silicon, magnesium and ferrous. The scenario of this model is scrutinized mathematically by captivating induced magnetic field with in the domain of thermal boundary layer. Similarity transformations are deployed to change the governing equations into nonlinear ODEs and consequently solved this system through bvp4c solver. The impact of emerging parameters on velocity, induction and energy distributions are examined through graphical depiction. Furthermore, heat transfer rates and drag forces near the boundary are designed numerically in tabular form.

Thermophoresis effect has wide range of applications in electro-static precipitators and in biology for calculating single biological macro molecules, such as genomic-length DNA and HIV virus in the micro channels. Current study deal with effects of Coriolis and buoyancy forces on the three-dimensional boundary layer flow of tangent hyperbolic fluid with thermo-migration and haphazard motion of nano-sized particles. Arrhenius kind of chemical reaction is taken along an exponentially stretchable surface. The main focus of current exploration is to execute shear thinning nano-liquid flow past an exponentially rotating stretchable surface under the influence of variable viscosity, mixed convection and activation energy. We are motivated to explore the features of three-dimensional shear thinning model combined under the features of mixed convention, variable viscosity, and activation energy. The mathematical model is designed to generate PDEs and converted them into ODEs by employing fractious transformation. The numerical outcomes are exhibited via graphs by employing Bvp4c numerical technique whereas the values of skin friction coefficient are calculated by monopolizing shooting method. Characteristics of the parameters appearing in modeling like the viscosity parameter, power-law index, local Weissenberg number, mix convection parameter, rotation parameter, Prandtl number and chemical reaction parameter are comprehensively analyzed through graphical behavior. The impact of governing parameters on skin friction, heat and mass transfer rates is illustrated through tables. The detail analysis anticipates that the elevation in Weissenberg number and porosity caused decline in velocity. Further, the temperature behaves doppositely analogous to development Prandtl besides the thermophoresis parameter.

This goal of this study is to examine the incompressible steady 2D flow of MHD Carreau Yasuda model along with the heat generation and chemical reaction near a zero velocity region. The magnetic field and thermally conducting fluid towards a stretching cylinder are very significant due to its usage in the various manufacturing sector. Chemical reactions are widely practice in everyday life as turning nutrition into energy fuel for our body, food change, fireworks expulsions, removing grimes, photosynthesis, etc. The nonlinear flow model equations and their corresponding boundary conditions are changed into non-dimensional shape using similarity variables. The role of vital parameters is discussed with the assistance of MATLAB software by BVP4C method. It is concluded that the momentum increases for rising the curvature and stretching ratio parameter. This is examined that the heat field improves for rising behavior of the magnetic force, curvature coefficient and heat generation. The fluid concentration upsurges due to curvature and magnetic field parameter while reverse results shown due to chemical reaction parameter. We graphically investigate the impression of magnetic effect and chemical force for heat and mass profiles.

This paper examines the unsteady two-dimensional boundary layer heat and mass transfer features of viscoelastic fluid flow near the neighborhood of forward and rare stagnation points. The heated circular cylinder is immersed in a viscoelastic fluid and the forward and rare stagnation points occurs near downward and upward directions of the cylinder respectively. The velocity near the surface of the cylinder is assumed to be zero and at very large distance from cylinder a uniform free stream velocity is spontaneously started in vertical direction. A non-similar transformation reduces the set of partial differential equations into non-similar partial differential equations. The transformed non-linear partial differential system is solved by using Keller box method. This technique shows well-behaved solutions for the steady-state (large time) and transient (small time) flows near the boundary layer region. Moreover, the body force present in this phenomena is in the form of convection which gives assisting (greater than zero) and opposing flows (less than zero). Parameters like the viscoelastic parameter, the convection parameter and the stagnation point parameter (forward and rare stagnation points) are found to control the flow field. Moreover, the Schmidt and the Prandtl numbers are used to control the concentration and temperature distributions. Features of several parameters on the non-dimensional concentration, temperature, local Nusselt and Sherwood numbers are viewed through graphs (2-d and 3-d) and tables. Contour images are plotted in order to visualize the three dimensional surface into two dimensional plane.

Rotating dynamical system (centrifugal and coriolis forces) is employed in this analysis to study the influence of enthalpy and rotated fluid flow in a finite three dimensional channel. The exponential varying viscosity model is used in this paper. The chemically reactive diffusion is examined with Arrhenius activation energy. The thermal conductivity and diffusivity are assumed to be linear dependent. The lower surface has velocity in negative direction but upper surface is fixed. The main equations in the three-dimensional form are converted into non-dimensional form via applicable scaling transformations. The shooting process is applied and coded in Matlab software to get the numerical solution of the non-linear ordinary differential problem. The varying shapes of velocities, fluid temperature and concentration functions are calculated to disclose the physical nature of this analysis.

A numerical study is presented to discuss the temperature dependent viscosity and the thermo-physical properties near a heated flat surface. Temperature dependent viscosity for liquids show decrease in mean velocity profile and enhancement in mean temperature profile, whereas for gasses, it shows reverse effect. Moreover, the thermo-physical properties are treated as variable as they affect the flow phenomena. The basic flow can be modeled in terms of the scaling transformation for the large Reynolds number. Using similarity solutions, the leading order equations are converted into a set of ordinary differential equations which have been solved by fifth order Runga-Kutta integration method and Newton Raphson method. All of the obtained results are investigated for the base flow, concentration and temperature profiles and physical interpretation has been provided in the form figures and tables. The impact of viscosity for gases show an increase behavior in velocity profile but for liquids opposite impact is noticed. The variable conductivity and mass diffusivity parameters rise the temperature and concentration profiles. The temperature profile increases for increasing values ofradiation parameter. Comparison has been made with past literature and the results show good agreement.

Here flow behavior of stratified Williamson nanofluid over porous surface of stretching cylinder is examined. The concept of gyrotactic miscroorganisms is implemented to control the random movement of suspended nanoparticles. Effects of magnetic field is accounted. Further chemical reaction with Arrhenius activation energy is considered for the modeling of concentration equation. Brownian motion and thermophoresis effects are further considered. The flow model is obtained by employing the boundary layer assumptions. Appropriate transformations are used to reduced the dimensional system into non-dimensional ones. NDSolve code in MATHEMATICA software is used to tackle the obtained non-dimensional flow expressions. Behavior of velocity, mass concentration, temperature and motile microorganisms versus involved variables is examined graphically. The engineering curiosity like skin friction coefficient, heat and mass transfer rates (Nusselt and Sherwood numbers) and density number are computed and analyzed. Important observations are highlighted at the end.

The present article communicates mathematical and theoretical study of three dimensional mixed convection flow of an electrical conducting third grade fluid flow near a non-uniform irregular wavy surface. The flow field is incorporated with an applied magnetic field. Heat and mass transfer characteristics are scrutinized with combined impact of enthalpy change along with energy activation and non-linear thermal radiation. The prime structure of differential equations is achieved in momentum, energy, magnetic and concentration forms. We defined some new dimensionless functions which transformed non-linear PDEs into dimensionless ODEs. Lobatto(iiiA) formula based technique Bvp4c is used for graphical and numerical solution of governing equations. Features of different dimensionless parameters are studied for velocity, concentration, magnetic and energy distributions by using graphical approach. Skin friction coefficients, rate of heat and mass fluxes are calculated by changing the values of some dimensionless parameters. Comparative study of Nusselt number with previous published results are demonstrated to investigate the accuracy of current problem. Final outcome justifies that better agreement is achieved with existing published work.

The central goal of the current research work is to develop a mathematical model by which we evaluate the influence of mixed convection on the flow rate, temperature profile and concentration rate in the occurrence of radiation and concentration impact respectively, through asymmetric divergent channel. The obtained resultant mathematical equations are transformed into system of first order differential equation by applying suitable transformation. Next we apply shooting method to calculated the numerical solutions with the help of MATLAB. Furthermore we graphically analysis the impact of geometry parameter a, Grashof number Gr and modified Grashof number Br on velocity profile. Also, we discuss the impact of heat generation parameter M and Prandtl number Pr on thermal profile. Moreover, we interpreter the consequence of activation energy E and temperature difference parameter Ω on concentration profile. Velocity profile u(y) reduces by enlarging geometry parameter a, velocity profile u(y) gives opposite performance for high value of Grashof Gr and modified Grashof numbers Br, temperature profile grows for large values of Prandtl numbers Pr and heat generation parameter M and concentration profile shows prominent impact more large values of activation energy and decreases for high values of difference parameter.

The main objective of this analysis is to present an unsteady magneto-micropolar fluid throw towards a circular object. An impulsive flow is assumed at a very large distance from object which is known as free stream region. The free stream region is perpendicular to the object in upward direction. Due to this phenomena, body forces are taken in the form of convection. In this problem, the object is assumed to be extremely heated which take us towards two main assumptions: i) assisting flow is taken and opposing flow is neglected and ii) radiation factor appears. The free stream flow is so dominant that it touches all external sides of the object which gives a good mathematical study of the stagnation regions (forward and rare) and the remaining points in which fluid particles touch the object. The different situations are assumed at different angles to analyze the flow, temperature, microrotation and concentration distributions. But our main focus will be rare and forward stagnation points. Heat generation and chemical reaction influence are also studied for such complex formulation. A magnetic field is applied to align these fluid particles motion. The so-called non-similar transformations and boundary layer approximation are used to obtain the very complex highly non-linear partial differential system depending on three independent variables. A second order convergent scheme known as Keller box is implemented to calculate the solution of the system. The results for separation time around the cylinder are calculated in tabular form. Moreover, graphical behavior is examined for forward and rare stagnation points for concentration, microrotation, velocity and temperature distributions. A comparative analysis is executed from previous data for different values of separation time.

In this paper, numerical study of nanoparticles immersed in a viscoelastic fluid is discussed. The condition of zero normal flux is assumed which means that no mass flux exists near the variable thick surface. To control the arbitrary motion of nanoparticles, magnetic field is applied at the surface of sheet. After boundary layer, the governing partial differential equations are converted into ordinary differential equations by means of similarity transformations. The following system is calculated numerically by employing shooting method in conjunction with RK-five method. The velocity profile shows an increasing behavior for large values of α and λ. The base flow temperature profile reduces for increasing values of Prnumber and Nb parameter. Also, the results for skin frictions coefficient, mass and heat transport rate are explored with the help of tables. A comparative study has been made with prior published work.

In this study, we assumed a thermal energy system with variable controlling properties which effect the entire system. One of the best applications of thermal energy system is heat exchanger. A good heat exchanger is a big challenge over the past years, due to enlarged public request on improving energy efficiency and decreasing energy. Normally, heat exchangers are used in the transmission of heat between two or more fluids [46]. Furthermore, the analysis of MHD flow of an electrically conducting fluid due to the thermal system is significant in current metallurgy and metal-working methods. Due to large amount of applications, variable thermo-physical properties of induced magnetic field in Williamson fluid flow is studied. The achieved system is partial differential, which is converted into ordinary differential equations by using the transformations. Then the attained system is solved by using shooting method. The thermal extrusion system contains velocity, concentration, and temperature fields. The associated controlling parameters contain small parameter, magnetic field, the reciprocal magnetic Prandtl number, variable viscosity parameter, Williamson parameter, small concentration parameter and Schmidt number. The outlined results deliver good thermal impact on entire system for minor and big values of these pertinent parameters. Williamson fluid model is executed to examine the properties of shear thinning fluids. It is noticed that a big reduction is noticed in velocity profile for large values of variable viscosity parameter, Williamson parameter and magnetic parameter. Moreover, the reciprocal magnetic Prandtl number reduces the magnetic profile. The temperature and concentration small parameters show enhanced impact on temperature and concentration profiles. Comparison of present thermal system is considered for limited values of these parameters.

An approximate approach is used to evaluate the effect of different shaped nano-particles on the thermal behavior and flow performance of nanofluid flow through inflexible and sinusoidal walls in the existence of heat generation effect. To achieve this phenomena we considered boehmite alumina in different shapes like, blade, platelet, brick and cylinder as nano-particles and methanol as a base fluid. The model considered here is pumping of methanol in combination with boehmite alumina through inflexible and sinusoidal walls, which inspired due to the reality that thread injection is a probable technique for assigning clinical implantation's in the humanoid frame with least clinical trauma. The interior cylinder is endoscope (stiff and uncompromising) rotating with constant velocity (V) and the external cylinder is sinusoidal (wave traveling down to its wall). The resulting mathematical relations are simplified with the help of low Reynolds number and long wavelength, the resulting non-linear partial differential equations are converted into ordinary differential equations by using perturbation approach. After finding the analytic solutions we graphically compare the different shapes (platelet, blade, cylinder and brick) of nano-particles to evaluate which shape of nano-particles give batter performance. To observe the performance we compare the friction forces on interior and exterior cylinders, pressure gradient, pressure rise, velocity and temperature profiles for different values of solid volume fraction and also presented the streamlines for all shapes of nano-particles.

Current analysis has been made to scrutinize the consequences of chemical response against magneto-hydrodynamic Carreau-Yasuda nanofluid flow induced by a non-linear stretching surface considering zero normal flux, slip and convective boundary conditions. Joule heating effect is also considered. Appropriate similarity approach is used to convert leading system of PDE’s for Carreau-Yasuda nanofluid into nonlinear ODE’s. Well known mathematical scheme namely shooting method is utilized to solve the system numerically. Physical parameters, namely Weissenberg number We, thermal slip parameter δ, thermophoresis number NT, non-linear stretching parameter n, magnetic field parameter M, velocity slip parameter k, Lewis number Le, Brownian motion parameter NB, Prandtl number Pr, Eckert number Ec and chemical reaction parameter γ upon temperature, velocity and concentration profiles are visualized through graphs and tables. Numerical influence of mass and heat transfer rates and friction factor are also represented in tabular as well as graphical form respectively. Skin friction coefficient reduces when Weissenberg number We is incremented. Rate of heat transfer enhances for large values of Brownian motion constraint NB. By increasing Lewis quantity Le rate of mass transfer declines. Keywords: Carreau-Yasuda nanofluid, Chemical reaction, Slip condition, Zero normal flux condition, MHD, Joule heating, Convective condition, Shooting method

The present analysis is devoted to explore the computational solution of the problem addressing the variable viscosity and inclined Lorentz force effects on Williamson nanofluid over a stretching sheet. Variable viscosity is assumed to vary as a linear function of temperature. The basic mathematical modelled problem i.e. system of PDE’s is converted nonlinear into ODE’s via applying suitable transformations. Computational solutions of the problem is also achieved via efficient numerical technique shooting. Characteristics of controlling parameters i.e. stretching index, inclined angle, Hartmann number, Weissenberg number, variable viscosity parameter, mixed convention parameter, Brownian motion parameter, Prandtl number, Lewis number, thermophoresis parameter and chemical reactive species on concentration, temperature and velocity gradient. Additionally, friction factor coefficient, Nusselt number and Sherwood number are describe with the help of graphics as well as tables verses flow controlling parameters. Keywords: Williamson nanofluid, Temperature depended viscosity, Inclined magnetic field, Mixed convection, Chemical reactive species, Variable viscosity, Shooting method

An analysis is executed to study the influence of heat generation/absorption on tangent hyperbolic nanofluid near the stagnation point over a stretching cylinder. In this study the developed model of a tangent hyperbolic nanofluid in boundary layer flow with Brownian motion and thermophoresis effects are discussed. The governing partial differential equations in terms of continuity, momentum, temperature and concentration are rehabilitated into ordinary differential form and then solved numerically using shooting method. The results specify that the addition of nanoparticles into the tangent hyperbolic fluid yields an increment in the skin friction coefficient and the heat transfer rate at the surface. Comparison of the present results with previously published literature is specified and found in good agreement. It is noticed that velocity profile reduces by enhancing Weissenberg number λ and power law index n. The skin friction coefficient, local Nusselt number and local Sherwood number enhances for large values of stretching ratio parameter A. Keywords: Stagnation point flow, Tangent hyperbolic nanofluid, Stretching cylinder, Heat generation/absorption, Boundary layer, Shooting method

Present work examines the effects of chemical reactive species and solar radiation on Carreau nanofluid induced by a stretching sheet with variable thickness. Heat generation effects are taken for exothermic reaction. In the boundary layer region, nanoparticles provides enhancement in the heat transfer phenomena. In the literature, most of the researchers used Runge Kutta Fehlberg method, but our main concentration here is to find the solution by improved form of Fehlberg method (Cash and Carp). The velocity, temperature and concentration profiles are plotted for different governing parameter, namely, wall thickness parameter, magnetic parameter, thermal conductivity parameter, chemical reaction parameter, Schmidt number and Prandtl number. It is found that the velocity profile increases by increasing Weissenberg numberWe, Hartmann number M and wall thickness parameter n. Moreover, temperature profile enhances by increasing heat generation parameter γ and radiative heat flux parameter R. The accuracy of the present method is checked with previously published work and excellent agreement is noticed. Keywords: Carreau nanofluid, Heat generation, Solar radiation, Chemical reaction, Stretching sheet with flexible thickness, Shooting method (Cash and Carp)

Present article explored the physical characteristics of biconvection effects on the MHD flow of Carreau nanofluid over upper horizontal surface of paraboloid revolution along with chemical reaction. The concept of the Carreau nanofluid was introduced the new parameterization achieve the momentum governing equation. Using similarity transformed, the governing partial differential equations are converted into the ordinary differential equations. The obtained governing equations are solved computationally by using implicit finite difference method known as the Keller box technique. The numerical solutions are obtained for the velocity, temperature, concentration, friction factor, local heat and mass transfer coefficients by varying controlling parameters i.e. Biconvection parameter, fluid parameter, Weissenberg number, Hartmann number, Prandtl number, Brownian motion parameter, thermophoresis parameter, Lewis number and chemical reaction parameter. The obtained results are discussed via graphs and tables. Keywords: Biconvection, Fourier’s law, Fick’s law, Carreau nanofluid, Chemical reaction, Paraboloid surface, Implicit finite difference method

Present analysis deals with the study of two dimensional MHD flow of Carreau fluid over a variable stretching sheet. The governing system partial differential equation (PDE) is reduced into ordinary differential equation (ODE) by using similarity approach. The solution of the differential equation is calculated by using a moderate and well-known numerical technique namely Keller box method. Intricate parameters, namely Hartmann number, Weissenberg number, wall thickness parameter and power law index are utilized to control the motion of fluid. Skin friction coefficient is calculated in order to examine the flow behavior near the surface of the sheet. A comparison has been made with the previous published work in order to check the accuracy of the technique. Conclusion is drawn on the basis of entire study and it is found that velocity profile reduces for large values of Hartmann number, wall thickness parameter and stretching index. Keywords: MHD flow, Carreau fluid, Variable stretching sheet, Keller box method

This article presents the two-dimensional flow of MHD hyperbolic tangent fluid with nanoparticles towards a stretching surface. The mathematical modelling of current flow analysis yields the nonlinear set of partial differential equations which then are reduce to ordinary differential equations by using suitable scaling transforms. Then resulting equations are solved by using shooting technique. The behaviour of the involved physical parameters (Weissenberg number We, Hartmann number M, Prandtl number Pr, Brownian motion parameter Nb, Lewis number Leand thermophoresis number Nt) on velocity, temperature and concentration are interpreted in detail. Additionally, local skin friction, local Nusselt number and local Sherwood number are computed and analyzed. It has been explored that Weissenberg number and Hartmann number are decelerate fluid motion. Brownian motion and thermophoresis both enhance the fluid temperature. Local Sherwood number is increasing function whereas Nusselt number is reducing function for increasing values of Brownian motion parameter Nb, Prandtl number Pr, thermophoresis parameter Nt and Lewis number Le. Additionally, computed results are compared with existing literature to validate the accuracy of solution, one can see that present results have quite resemblance with reported data. Keywords: Tangent hyperbolic fluid, MHD, Nanofluid, Stretching sheet, Shooting method (Cash and Crap)

This article concerns with mixed convection, heat and mass transfer characteristics of Eyring-Powell nanofluid over a cone and plate with chemical reactive species. The nonlinear partial differential equations (PDE’S) are normalized by means of appropriate transformations. Governing equations are solved computationally by using shooting technique with fifth order Runge-Kutta scheme. A comparison for the cone and plate is given in each graph and table. The effects of emerging physical parameters such as porosity parameter, Lewis number, Prandtl number, Brownian moment, magnetic field parameter, thermophoresis parameter and mixed convection parameter on the momentum, temperature and concentration profiles are examined through graphs and tables. Keywords: Mixed convection flow, Eyring-Powell nanofluid, Chemical reaction, Cone and plate

Present work is communicated to identify characteristics of magnetohydrodynamic (MHD) three dimensional boundary layer flow of Williamson fluid confined by a bidirectional stretched surface. Conductivity of working fluid is assumed to be temperature dependent. Generative/absorptive heat transfer is also taken into account. Mathematical model is formulated in the form of partial expressions and then transmuted into ordinary differential equations with the help of newfangled set of similarity transformations. The resulting non-linear differential system of equations is solved numerically with the aid of Runge-Kutta algorithm supported by shooting method. Flow features are exemplified quantitatively through graphs. Scintillating results for friction factor and convective heat transfer are computed and scrutinized tabularly. Furthermore, the accuracy of present results is tested with existing literature and we found an excellent agreement. It is inferred that velocity along x-direction mounts whereas along y-direction depreciates for incrementing values of stretching ratio parameter. Moreover, it is also elucidated that non-linearity index tends to decrement the velocity and thermal distributions of fluid flow. Keywords: MHD flow, Williamson fluid, Variable thermal conductivity, Heat absorption/generation, Bidirectional non-linear stretching

An analysis has been executed to investigate the generalized slip effects on MHD Carreau nanofluid induced by a linear stretching cylinder in the presence of reactive species. Adequate transformations are used to rehabilitate the system of partial differential equations into system of non-linear ordinary differential equations. Similarity equations are then solved numerically by using an implicit finite difference scheme known as Keller box method. The interesting findings for various crucial parameters on dimensionless velocity, temperature and concentration profiles are demonstrated through graphs. Plots and tables reflecting the impact on skin friction coefficient, local Nusselt number and local Sherwood number are presented and explored. Comparison has been validated with the prior published results and found to be in commendable agreement. Keywords: Generalized slip boundary conditions, MHD flow, Chemical reactive species, Carreau nanofluid, Stretching cylinder, Keller box method

Current work highlights the computational aspects of MHD Carreau nanofluid flow over an inclined stretching cylinder with convective boundary conditions and Joule heating. The mathematical modeling of physical problem yields nonlinear set of partial differential equations. A suitable scaling group of variables is employed on modeled equations to convert them into non-dimensional form. The integration scheme Runge-Kutta-Fehlberg on the behalf of shooting technique is utilized to solve attained set of equations. The interesting aspects of physical problem (linear momentum, energy and nanoparticles concentration) are elaborated under the different parametric conditions through graphical and tabular manners. Additionally, the quantities (local skin friction coefficient, local Nusselt number and local Sherwood number) which are responsible to dig out the physical phenomena in the vicinity of stretched surface are computed and delineated by varying controlling flow parameters. Keywords: MHD, Carreau nanofluid, Inclined stretching cylinder, Joule heating, Shooting technique

Present analysis is concentrating on the thermo-physical aspects of MHD tangent hyperbolic fluid flow over a non-linear stretching sheet with viscous dissipation and convective boundary conditions. Mathematical modelling yields non-linear partial differential equations. The governing equations are transformed into corresponding coupled ordinary differential equations via using local similarity variables. The accomplished boundary layer ordinary differential equations are solved with the aid of both homotopy analysis method and shooting method. The effects of flow govern parameters are visualized on velocity and temperature in both qualitative and quantitative manners. The computations of wall friction factor and local Nusselt number are performed to analyze the behavior in the vicinity of stretching sheet. The contrast between analytically and numerically computed values of wall friction factor and local Nusselt number is presented. It is worth mentioning that the both results are in excellent agreement, this favorable comparison led to confidence on computed results. Keywords: Tangent hyperbolic fluid, MHD, Viscous dissipation, Convective boundary conditions, Non-linear stretching sheet, HAM, Shooting method

Present phenomenon is dedicated to investigate the heat transfer and squeezed flow of Carreau fluid over a sensor surface. The thermal conductivity of the fluid is assumed to be temperature dependent. After assimilating these assumptions, the appropriate transformations are used to formulate the partial differential equations into non-dimensional system of ordinary differential equations. Solutions for the boundary layer momentum and heat equations are accomplished by a well-known numerical technique namely shooting method. The related essential physical parameters are visualized through graphs and tables. It is found that the velocity profile increases by increasing squeezed flow parameter b, permeable velocity parameter f0, power law index n and Weissenberg number We Similarly, temperature profile increases by increasing small parameter ∊. Keywords: Heat transfer, Squeezed flow, Carreau fluid, Sensor surface, Variable thermal conductivity, Shooting method