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An unsteady flow and melting heat transfer of a nanofluid over a stretching sheet was numerically studied by considering the effect of chemical reaction and thermal radiation. The governing non-linear partial differential equations describing the flow problem are reduced to a system of non-linear ordinary differential equations using the similarity transformations and solved numerically using the Runge–Kutta–Fehlberg fourth–fifth order method. Numerical results for concentration, temperature and velocity profiles are shown graphically and discussed for different physical parameters. Effect of pertinent parameters on momentum, temperature and concentration profiles along with local Sherwood number, local skin-friction coefficient and local Nusselt number are well tabulated and discussed.

PurposeThe purpose of this paper is to study the impact of Joule heating on boundary layer flow and melting heat transfer of Prandtl fluid over a stretching sheet in the presence of fluid particles suspension. The transformed boundary layer equations are solved numerically by RKF-45 method. The influence of the non-dimensional parameters on velocity and temperature growths in the boundary layer region is analyzed in detail and the results are shown graphically. The results indicate that the larger estimation ofαandβreduces for both velocity and temperature profile. Further, the rate of heat transfer decreases by increasing melting parameter.Design/methodology/approachThe converted set of boundary layer equations is solved numerically by RKF-45 method. Obtained numerical results for flow and heat transfer characteristics are deliberated for various physical parameters. Furthermore, the skin friction coefficient and Nusselt number are also presented.FindingsIt is found that the heat transfer rates are advanced in the occurrence of non-linear radiation camper to linear radiation. Also, it is noticed that velocity profile increases by increasing Prandtl parameter but establishes opposite results for temperature profile.Originality/valueThe authors intend to analyze the boundary layer flow and melting heat transfer of a Prandtl fluid over a stretching surface in the presence of fluid particles suspension. The governing systems of partial differential equations have been transformed to a set of coupled ordinary differential equations by applying appropriate similarity transformations. The reduced equations are solved numerically. The pertinent parameters are discussed through graphs and plotted graphs. The present results are compared with the existing limiting solutions, showing good agreement with each other.

Present communication aims to discuss the impact of viscous dissipation on MHD flow, heat and mass transfer of Casson fluid over a plate by considering mixed convection. Nonlinear partial differential systems are reduced to the ordinary ones through transformation procedure. The modelled nonlinear systems are computed implementing RKF-45 scheme. Convergent solutions for velocity and temperature and concentration fields are given diagrammatically. The obtained results are compared with published literatures and reasonable agreement is found. It is found that, temperature profile increases by increasing values of Dufour parameter, whereas on opposite trend is observed in concentration profile for increasing values Soret parameter.

This examination emphasizes the analysis of thermal transmission of Carreau fluid flow on a permeable sensor surface equipped with radiation, Joule heating, an internal heat source, and a magnetic field. With the above effects and assumptions, the equations that administer the flow are formulated. A configured system of equations is productively reduced to a system of ordinary differential equations. The reduced system is then dealt with using the Runge–Kutta-Fehlberg fourth–fifth order tool equipped by the shooting technique. Derived numerical solutions are utilized to plot graphs and tables. The conclusion of the study outlines some important findings such as the power law index, the thermal radiation parameter and the heat source parameter enhance the thermal panel whereas the Weissenberg number deescalates the same. The power law index and permeable velocity decrease the velocity panel significantly. Diagrammatic representation of streamlines of the flow has been given to strengthen the study. A detailed description has been produced about the results obtained in the study.

This article deals with the combined effects of viscous dissipation and convective condition on 3D flow, heat and mass transfer of a nanofluid over a stretching sheet by considering gyrotactic microorganism. Appropriate transformations yield the nonlinear ordinary differential systems. The resulting nonlinear system has been solved. Role of substantial parameters on flow fields as well as on heat, mass and microorganism transportation rates are determined and conferred in depth through graphs. It is found that, the larger values of bio-convection Schmidt number decreases the microorganisms profile.

The main objective of the present analysis is to study an enhancement of heat transfer in an unsteady rotating flow for the aqueous suspensions of single wall nanotubes under nonlinear thermal radiation. Appropriate transformations are implemented for the conversion of partial differential systems into a set of ordinary differential equations. The transformed expressions have been scrutinized through RKF-45 order method along with shooting technique. The impact of various pertinent parameters for the velocity and temperature fields are analyzed through graphs in detail. Also, the role of substantial parameters on the fiction factor and mass transportation rates is determined and conferred in depth through graphs. Our simulations established that the higher value of rotation rate parameter reduces the thickness of the momentum boundary layer thickness. Further, an unsteadiness parameter increases velocity and temperature profile decreases.

A study on magnetohydrodynamic mixed convection flow of Casson fluid over a vertical plate has been modelled in the presence of Cross diffusion effect and nonlinear thermal radiation. The governing partial differential equations are remodelled into ordinary differential equations by using similarity transformation. The accompanied differential equations are resolved numerically by using Runge–Kutta–Fehlberg forth-fifth order along with shooting method (RKF45 Method). The results of various physical parameters on velocity and temperature profiles are given diagrammatically. The numerical values of the local skin friction coefficient, local Nusselt number and local Sherwood number also are shown in a tabular form. It is found that, effect of Dufour and Soret parameter increases the temperature and concentration component correspondingly. Keywords: Casson fluid, Nonlinear thermal radiation, Magnetic field, Cross diffusion effect, Vertical surface

Present exploration discusses the combined effect of viscous dissipation and Joule heating on three dimensional flow and heat transfer of a Jeffrey nanofluid in the presence of nonlinear thermal radiation. Here the flow is generated over bidirectional stretching sheet in the presence of applied magnetic field by accounting thermophoresis and Brownian motion of nanoparticles. Suitable similarity transformations are employed to reduce the governing partial differential equations into coupled nonlinear ordinary differential equations. These nonlinear ordinary differential equations are solved numerically by using the Runge–Kutta–Fehlberg fourth–fifth order method with shooting technique. Graphically results are presented and discussed for various parameters. Validation of the current method is proved by comparing our results with the existing results under limiting situations. It can be concluded that combined effect of Joule and viscous heating increases the temperature profile and thermal boundary layer thickness.

The current investigation deals with melting heat transport in hyperbolic tangent fluid past a stretching sheet with fluid particle suspension and thermal radiation. Numerical solution is obtained via shooting technique after the applying of similarity variables to the governing partial equations. The effects of key parameters like magnetic parameter, fluid particle interaction parameter, Eckert number, Prandtl number, Weissenberg number and melting parameter are analyzed and discussed in detail with help of graphs and tables. Finally, the numerical values of the friction co-efficient and local Nusselt number are tabulated. Checked our results with existing work found excellent agreement. Results reveals that in the presence of melting parameter, the higher values of $W_e$ and $n$ depreciates the rate of heat transfer.

The present work has abundant applications in the branch of mechanical engineering. Heat and mass transport analysis is very important in the manufacturing of various mechanical systems like heat exchangers, domestic refrigerator, fuel cells, hybrid-powered engines and thermal management of vehicles. We explored the features of three-dimensional magnetohydrodynamic (MHD) flow and heat transport phenomenon of Carreau nanoliquid over a stretched sheet under the influence of thermal radiation. The system of ordinary differential expressions is developed through the implication of similarity variables. Solutions of governed non-linear expressions are accomplished by a well-known Runge–Kutta–Fehlberg fourth and fifth order scheme. The characterization of pertinent parameters is visualized through plotted graphs and tables. It is found that both Weissenberg number and magnetic field parameter enhance the temperature and reduce the liquid velocity. The temperature of liquid is increased when the values of Brownian motion and thermophoretic parameters enhance. A comparative benchmark is presented to justify the validity of solutions.

The paper explores the numerical study on squeezed flow of tangent hyperbolic fluid over a sensor surface with variable thermal conductivity. The governing equations are reassembling into ordinary non-linear differential equations by using suitable transformation and ensuing differential equations are solved numerically by RKF45 technique. The results of impartment parameter on velocity and temperature profiles are given diagrammatically. It is found that the variations of Weissenberg number, squeezed flow index parameter and permeable velocity parameter have strong influence on the momentum and thermal boundary layer thickness. Keywords: Squeezed flow, Tangent hyperbolic fluid, Variable thermal conductivity, Magnetic field

The present study addresses the three-dimensional flow of a Prandtl fluid over a Riga plate in the presence of chemical reaction and convective condition. The converted set of boundary layer equations are solved numerically by RKF four-fifth method. Obtained numerical results for flow and mass transfer characteristics are discussed for various physical parameters. Additionally, the skin friction coefficient and Sherwood number are also presented. It is found that, the momentum boundary layer thickness is dominant for higher values of α and solutal boundary layer is low for higher Schmidt number and chemical reaction parameter. Keywords: 3D flow, Riga plate, Prandtl fluid, Convective condition, Chemical reaction

A mathematical analysis of two-phase boundary layer flow and heat transfer of a Williamson fluid with fluid particle suspension over a stretching sheet has been carried out in this paper. The region of temperature jump and nonlinear thermal radiation is considered in the energy transfer process. The principal equations of boundary layer flow and temperature transmission are reformed to a set of non-linear ordinary differential equations under suitable similarity transformations. The transfigured equalities are solved numerically with the help of RKF-45 order method. The effect of influencing parameters on velocity and temperature transfer of fluid is examined and deliberated by plotted graphs and tabulated values. Significances of the mass concentration of dust particle parameter play a key role in controlling flow and thermal behavior of non-Newtonian fluids. Further, the temperature and concern boundary layer girth are declines for increasing values of Williamson parameter. Keywords: Two-phase flow, Williamson fluid, Nonlinear thermal radiation, Magnetic field, Temperature jump

The current study is to deliberate the flow and heat transfer of a Carreau fluid over a stretching sheet with fluid particle suspension. The temperature jump is also taken into account. The standard nonlinear system is resolved numerically via Runge-Kutta based shooting scheme. Role of substantial parameters on flow fields as well as on the fiction factor and heat transportation rates are determined and conferred in depth through graphs. It’s found that the velocity profile decreases and temperature profile increases, with an increasing the values of Weissenberg parameter. Further, the higher thermal slip parameter reduces the thermal boundary layer thickness. The thermal boundary layer thickness of fluid and dust particles decreases with the rise in Prandtl number. Keywords: Temperature jump, Thermal radiation, Carreau fluid, Fluid particle suspension

This research work explores the effect of hybrid nanoparticles on the flow over a rotating disk by using an activation energy model. Here, we considered molybdenum disulfide and ferro sulfate as nanoparticles suspended in base fluid water. The magnetic field is pragmatic normal to the hybrid nanofluid flow direction. The derived nonlinear ordinary differential equations are non-dimensionalized and worked out numerically with the help of Maple software by the RKF-45 method. The scientific results for a non-dimensionalized equation are presented for both nanoparticle and hybrid nanoparticle case. Accoutrements of various predominant restrictions on flow and thermal fields are scanned. Computation estimation for friction factor, local Nusselt number and Sherwood number are also executed. Results reveal that the reduction of the heat transfer rate is greater in hybrid nanoparticles when compared to nanoparticles for increasing values of Eckert Number and the thermal field enhances for the enhanced values of volume fraction.

Background The present exploration deliberates the effect of nonlinear thermal radiation on double diffusive free convective boundary layer flow of a viscoelastic nanofluid over a stretching sheet. Fluid is assumed to be electrically conducting in the presence of applied magnetic field. In this model, the Brownian motion and thermophoresis are classified as the main mechanisms which are responsible for the enhancement of convection features of the nanofluid. Entire different concept of nonlinear thermal radiation is utilized in the heat transfer process. Methods Appropriate similarity transformations reduce the nonlinear partial differential system to ordinary differential system which is then solved numerically by using the Runge–Kutta–Fehlberg method with the help of shooting technique. Validation of the current method is proved by having compared with the preexisting results with limiting solution. Results The effect of pertinent parameters on the velocity, temperature, solute concentration and nano particles concentration profiles are depicted graphically with some relevant discussion and tabulated result. Conclusions It is found that the effect of nanoparticle volume fraction and nonlinear thermal radiation stabilizes the thermal boundary layer growth. Also it was found that as the Brownian motion parameter increases, the local Nusselt number decreases, while the local friction factor coefficient and local Sherwood number increase.

In the present article a numerical analysis has been carried out to study the boundary layer flow behavior and heat transfer characteristics of a nanofluid over an exponential stretching sheet. By assuming the stretching sheet to be impermeable, the effect of chemical reaction, thermal radiation, thermopherosis, Brownian motion and suction parameters in the presence of uniform magnetic field on heat and mass transfer are addressed. The governing system of equations is transformed into coupled nonlinear ordinary differential equations using suitable similarity transformations. The transformed equations are then solved numerically using the well known Runge-Kutta-Fehlberg method of fourth-fifth order. A detailed parametric study is performed to access the influence of the physical parameters on longitudinal velocity, temperature and nanoparticle volume fraction profiles as well as the local skin-friction coefficient, local Nusselt number and the local Sherwood number and the results are presented in both graphical and tabular forms.

A numerical investigation has been conducted to study the laminar, boundary layer stagnation point flow of a nanofluid over a permeable, vertical stretching sheet. The model used incorporates the effects of Brownian motion with thermophoresis in the presence of uniform magnetic field and non-uniform source/sink under the influence of chemical reaction. Governing partial differential equations are transformed into a set of nonlinear ordinary differential equations using suitable similarity transformations. The transformed equations are then solved numerically using well known Runge-Kutta-Fehlberg method of fourth-fifth order with the help of symbolic software MAPLE. The influence of governing parameters on flow field, temperature and nanoparticle volume fraction profiles are provided both in graphical and tabular form. It is observed that heat source/sink with thermophoresis particle deposition in the presence of magnetic field have a substantial effect on the flow field and thus, on the heat and mass transfer rate from the sheet to the fluid. As the strength of the chemical reaction is higher than the thermophoresis particle deposition, nanoparticle volume fraction of the fluid gradually decreases. A comparative study has been conducted and is found to be in excellent agreement.Keywords: Stagnation point flow; Chemical reaction; Heat transfer; Stretching surface; Nanofluid; Numerical solution.