Magnetohydrodynamic boundary layer flow of nanofluid with variable chemical reaction in a radiative vertical plate

The nanotechnology-based nanofluid has extraordinary prospects in heat transfer engineering. Analysis of these applied nanofluids can yield the appropriate combinations of various useful physical parameters. In the present study, the incompressible boundary layer flow of a nanofluid in the presence of the variable chemical reaction, temperature-dependent viscosity, hydromagnetic force, and the radiation past an infinite vertical plate has been investigated. The governing nanofluid equations are simplified to ordinary differential equations, which are solved using the function bvp4c from MATLAB. The effects of the physical parameters including the similarity parameter, magnetic field, two dimensionless constant temperatures, Schmidt number, local Grashof number, radiation parameter, local chemical reaction parameter, kinematic diffusion parameter, and temperature-independent kinematic diffusion parameter on the velocity, temperature, concentration and the local Nusselt number are demonstrated. The results show that as the magnetic field parameter increases, the heat transfer decreases, and the increase of the radiation parameter yields the opposite effect. The kinematic diffusion and the chemical reaction parameters greatly stimulate the concentration of nanofluid and reduce the heat transfer.     

Heat and mass transfer of MHD Jeffrey nanofluid flow through a porous media past an inclined plate with chemical reaction,radiation, and Soret effects

This paper investigates the heat and mass transfer of an unsteady, magnetohydrodynamic incompressible water-based nanofluid (Cu and TiO₂) flow over a stretching sheet in a transverse magnetic field with thermal radiation Soret effects in the presence of heat source and chemical reaction. The governing differential equations are transformed into a set of nonlinear ordinary differential equations and solved using a regular perturbation technique with appropriate boundary conditions for various physical parameters. The effects of different physical parameters on the dimensionless velocity, temperature, and concentration profiles are depicted graphically and analyzed in detail. Finally, numerical values of the physical quantities, such as the local skin-friction coefficient, the Nusselt number, and the Sherwood number, are presented in tabular form. It is concluded that the resultant velocity reduces with increasing Jeffrey parameter and magnetic field parameter. Results describe that the velocity and temperature diminish with enhancing the thermal radiation. Both velocity and concentration are enhanced with increases of the Soret parameter. Also, it is noticed that the solutal boundary layer thickness decreases with an increase in chemical reaction parameters. This is because chemical molecular diffusivity reduces for higher values of chemical reaction parameter. Also, water-based TiO₂ nanofluids possess higher velocity than water-based Cu nanofluids. Comparisons with previously published work performed and the results are found to be in excellent agreement. This fluid flow model has several industrial applications in the field of chemical, polymer, medical science, and so forth.     

Radiation and Hall effects on a 3D flow of MHD Williamson fluid over a stretchable surface

Magnetohydrodynamics (MHD) three-dimensional flow of an unsteady Williamson fluid on an enlarging surface with Hall current, radiation, heat source/sink, and chemical reaction is investigated in this article. The basic governing equations are transformed into a system of ordinary differential equations by using an appropriate similarity transformation. The system is deciphered using the shooting method. The properties of influential parameters such as parameters of magnetic field, Hall current, radiation, and so forth, on the flow are discussed with the help of graphs and tables. We noticed that the increase in the magnetic field reduces the velocity in x-direction and the rate of heat and mass transfer. We also acknowledged that the growing values of Hall current parameter boost the velocity in z-direction but it reduce the temperature and concentration distributions, respectively. The results of this study represent many applications in biomedical engineering and these results are helpful for further study of non-Newtonian fluids in various circumstances.     

A two-component modeling for free stream velocity in magnetohydrodynamic nanofluid flow with radiation and chemical reaction over a stretching cylinder

This analysis explores the influence of magnetohydrodynamic (MHD) nanofluid flow over a stretching cylinder with radiation effect in presence of chemically reactive species. The thermal radiation phenomenon is incorporated in the temperature equation. The mathematical modeling of the physical problem produces nonlinear set of partial differential equations corresponding to the momentum and energy equations that can be transformed into simultaneous system of ordinary differential equations with appropriate boundary conditions by applying similarity transformations. Shooting technique is used to solve the molded equations after adoption of Runge–Kutta–Fehlberg approach and ODE45 solver in MATLAB. A parametric analysis has been carried out to investigate the impacts of physical parameters that are considered in the current study. The attractive pattern studied the consequence of Brownian motion along with thermophoresis parameter. The outcomes of prominent fluid parameters, especially heat radiation, Lewis number, free stream velocity, chemical reaction, thermophoresis, and Brownian motion on the concentration, temperature, as well as velocity have been examined and are displayed through graphs and tables. The present study reveals that the temperature phenomenon enhances with an increase in radiation parameter, while nanoparticle concentration phenomenon reduces with an increase in chemical reaction parameter.     

OHAM simulation of 3D transverse magnetic field in rotating flow of Maxwell nanofluid over a stretching sheet with chemical reaction and heat source/sink

In this article, we investigate the heat transfer characteristics of a Maxwell nanofluid along a stretching sheet with transverse magnetic field, considering the presence of heat source/sink and chemical reaction. We consider appropriate similarity transformation for transforming the governing nonlinear equations into nondimensional highly nonlinear coupled ordinary differential equations. The optimal homotopy analysis method is utilized for solving the resultant-coupled equations. The impact of all sundry parameters, like, Deborah number, Prandtl number, magnetic parameter, thermophoresis, rotation parameter, chemical reaction, velocity slip, Schmidt number, Brownian motion parameter, heat sources per sink, Biot number, and Eckert number, on the temperature, velocity, and concentration fields is reported, analyzed, and described through graphs and tables. It is noticed that higher values of magnetic parameter and Deborah number reduce the horizontal velocity field. Furthermore, it is observed that the Biot number and heat source/sink parameter enhance the temperature distribution.     

Thermochemistry and viscous heat dissipative effects on unsteady upper-convected Maxwell fluid flow past a stretching vertical plate with thermophysical variables

In this paper, unsteady upper-convected Maxwell fluid flow with variability in viscosity, thermal conductivity, and mass diffusivity is presented. The effects of chemical reaction, internal heat generation, and viscous dissipation with respect to variability properties were explored. The governing partial differential equations were transformed with the appropriate similarity transformation variables into nonlinear coupled ordinary differential equations. The spectral collocation method was used to solve the resulting ordinary differential equations. Hence, the effects of various parameters such as temperature-dependent viscosity and thermal conductivity, mass diffusivity parameters among others on velocity, temperature, concentration, skin friction, local heat and mass transfers were presented in graphs and tables. It is seen that heat and molecules of the fluid disperse faster as a result of destructive chemical reaction, while, the temperature-dependent viscosity and thermal conductivity gave increasing profiles of the momentum and thermal boundary layer. The viscous dissipative parameter generates heat and yields a buoyancy force in consequence.     

Heat and Mass Transfer Flow of a Nanofluid over an Inclined Plate under Enhanced Boundary Conditions with Magnetic Field and Thermal Radiation

This article presents the magnetohydrodynamic boundary layer flow, heat and mass transfer characteristics of a nanofluid over an inclined porous vertical plate with thermal radiation and chemical reaction. The new enhanced concentration boundary condition on the surface of the wall is considered in this analysis. The governing nonlinear partial differential equations are transformed into a system of nonlinear ordinary differential equations using the similarity variables and are solved numerically using the finite element method. The effect of key parameters such as magnetic parameter (M), buoyancy ratio (Nr), Prandtl number (Pr), thermal radiation (R), Brownian motion (Nb), thermophoresis (Nt), Lewis number (Le), and chemical reaction parameter (Cr) on velocity, temperature, and concentration distributions is discussed in detail and the results are shown graphically. Furthermore, the impact of these parameters on skin‐friction coefficient, Nusselt number, and Sherwood number is also investigated and the results are shown in tabular form. The developed algorithm is validated with works published previously and was found to be in good agreement. The thermal boundary layer thickness is elevated, whereas the solutal boundary layer thickness retards with the improving values of the Brownian motion parameter (Nb). The rates of nondimensional temperature and concentration both decelerate with higher values of the thermophoresis parameter (Nt).     

Unsteady MHD fluid flow past an inclined vertical porous plate in the presence of chemical reaction with aligned magnetic field,radiation, and Soret effects

The aim of the present paper is to investigate the Soret effect due to mixed convection on unsteady magnetohydrodynamics flow past a semi-infinite vertical permeable moving plate in the presence of thermal radiation, heat absorption, and homogenous chemical reaction subjected to variable suction. The plate is assumed to be embedded in a uniform porous medium and moves with a constant velocity in the flow direction in the presence of a transverse magnetic field. The equations governing the flow are transformed into a system of nonlinear ordinary differential equations by using the perturbation technique. Graphical results for the velocity distribution, temperature distribution, and concentration distribution based on the numerical solutions are presented and discussed. Also, the effects of various parameters on the skin-friction coefficient and the rate of heat transfer in the form of Nusselt number, and rate of mass transfer in the form of Sherwood number at the surface are discussed. Velocity distribution is observed to increase with an increase in Soret number and in the presence of permeability, whereas it shows reverse effects in the case of the aligned magnetic field, inclined parameter, heat absorption coefficient, magnetic parameter, radiation parameter, and chemical reaction parameter.     

MHD flow in a rotating vertical cone through a porous medium

In the context of advancements in both heat and mass transfer, the current study intends to analyze the impacts of thermal radiation, Soret, and Dufour on the magnetohydrodynamic boundary layer flow through a vertical spinning cone in porous media. The Dufour effect is the energy flux due to the mass concentration gradient with a reciprocal phenomenon, the Soret effect. Energy expression considers the physical aspects of heat generation and absorption. It is expected that the tangential, circumferential, and normal directions will all have velocity components in flow through a porous media. The governing equations are nonlinear partial differential equations that are rearranged into ordinary differential equations via similarity transformation, and then they are numerically solved using the Runge–Kutta method along with a proper shooting strategy. Graphs are used to examine the impacts of many parameters on flow characteristic velocity, temperature, and concentration, including magnetic parameters, porous parameters, Dufour and Soret parameters, chemical reaction parameters, and more. The numerical findings of the gradient of velocity, the Nusselt and Sherwood numbers, and the surface drag force are tabulated and compared with the current result and the one from the literature. The findings are found to be in good agreement. Circumferential and normal velocities are improved visually for greater values of the porosity parameter, but the tangential velocity behavior of the magnetic parameter exhibits the reverse behavior. In addition, the Dufour parameter and chemical reaction both exhibit diminishing behavior when the Soret parameter increases.     

Heat and mass transfer of two immiscible flows of Jeffrey fluid in a vertical channel

In this article, we examined the effect of heat and mass transfer flow of two immiscible Jeffrey fluids in a vertical channel. The highly nonlinear coupled ordinary differential equations are evaluated using regular perturbation parameters, for small values of perturbation parameter. The effect of Jeffrey's parameter on the flow and the effects of various physical parameters entering into the problem on dimensionless velocity, temperature, and concentration distribution is illustrated graphically. We observe that the Jeffrey parameter, thermal, and mass Grashof number enhance the fluid flow, while the chemical reaction parameter suppresses the fluid flow, also it is established that the Nusselt number is boosted by enhancing the thermal and mass Grashof number. We observed that the results are in very good agreement with the results obtained for a viscous fluid.     

Influences of Soret and Dufour numbers on mixed convective and chemically reactive Casson fluids flow towards an inclined flat plate

The purpose of this study is to examine the magnetohydrodynamic mixed convection Casson fluid flow over an inclined flat plate along with the heat source/sink. The present flow problem is considered under the assumption of the chemical reaction and thermal radiation impacts along with heat and mass transport. The leading nonlinear partial differential equations of the flow problem were renovated into the nonlinear ordinary differential equations (ODEs) with the assistance of appropriate similarity transformations and then we solved these ODEs with the employment of the bvp4c technique using the computational software MATLAB. The consequences of numerous leading parameters such as thermophoretic parameter, local temperature Grashof number, solutal Grashof number, suction parameter, magnetic field parameter, Prandtl number, chemical reaction parameter, Dufour number, Soret number, angle of inclination, radiation parameter, heat source/sink, and Casson parameter on the fluid velocity, temperature, and concentration profiles are discoursed upon  and presented through different graphs. Some important key findings of the present investigation are that the temperature of the Casson fluid becomes lower for local temperature Grashof number and solutal Grashof number. It is initiated that the Casson fluid parameter increases the velocity of the fluid whereas the opposite effect is noticed in the temperature profile. Higher estimation of Prandtl number and magnetic parameter elevated the Casson fluid concentration. Finally, the skin friction coefficient, Nusselt number, and Sherwood number are calculated and tabulated. It is also examined that the Nusselt number is weakened for both the Dufour number and Soret number but the skin fraction coefficient is greater for both the Dufour number and Soret number.     

Heat and mass transfer effect on an unsteady MHD radiative chemically reactive Casson fluid over a stretching sheet in porous medium

The objective of the present study is to investigate the effects of the variable magnetic field, chemical reaction, thermal radiation, Soret effect, and variable heat absorption on the fluid flow and heat and mass transfer of an unsteady Casson fluid past a stretching surface in a saturated porous medium. Velocity slip near the plate and conjugate heating boundary conditions in heat and mass transfer have been considered in this study. Due to the complexities in boundary conditions, the analytic solution of the governing equations of the present model is not possible. Thus, to overcome these issues, the coupled partial differential equations of the model are converted into a set of ordinary differential equations using similarity transformation. These equations have then been solved numerically using the fourth-order Runge-Kutta technique via the shooting method. The effects of various pertinent flow parameters on the velocity, concentration, and temperature field have been studied graphically. For the field of engineering, to get an insight into the physical quantities, especially Nusselt number, Sherwood number, and skin friction, their numerical values have been estimated against various parameters and presented in tables. From the tabulated values, it has been perceived that the shear stress increases with an increase in magnetic parameter, unsteadiness parameter, Casson parameter, and heat source parameter, whereas the Biot number shows the reverse trend. The mixture of porous media has justified that the heat transport process over a stretching sheet results in averting heat loss and accelerating the process of cooling, which is a significant outcome of the study. Furthermore, it has also been revealed that with the increase in the Soret effect and magnetic field, there is a reduction in the fluid velocity and temperature near the plate, whereas there is an increase in the species concentration. It has also been mentioned that the effects of the variable magnetic field have been widely applied in various engineering applications like magnetohydrodynamic (MHD) propulsion forces, rate of cooling, MHD power generation, and so on.     

Analysis of nonlinear thermal radiation and higher‐order chemical reactions on the non‐orthogonal stagnation point flow over a lubricated surface

A numerical study is performed to discuss the heat and mass transfer on oblique stagnation point flow over a lubricated surface with nonlinear thermal radiation and higher‐order chemical reactions. The problem is formulated using basic conservation laws of mass, momentum, energy, and mass concentration in terms of partial differential equations along with nonlinear boundary conditions. These governing equations are transformed into ordinary differential equations by means of similarity transformations. The system of resulting ordinary differential equations are solved numerically by an implicit finite difference scheme known as the Keller–box method. The quantities elaborated in the problem, such as velocity, temperature, skin friction, and local Nusselt and Sherwood numbers are analyzed for several values of involved parameters. The obtained results are presented through various graphs and tabular data and showed a good agreement with the existing results in the literature, which are the subcases of the present work. The heat transfer rate enhances with nonlinear thermal radiation and mass transfer rate decreases with increasing the order of chemical reaction.     

Silver–ethylene glycol and copper–ethylene glycol based thermally radiative nanofluid characteristics between two rotating stretchable disks with modified Fourier heat flux

In the present analysis, our aim is to investigate the mass and heat transport of silver (Ag)–ethylene glycol (EG) and copper (Cu)–EG-based nanofluids between two rotating stretchable disks under the convective boundary conditions. We have also incorporated Cattaneo–Christov heat flux, thermal radiation, and chemical reaction in the fluid flow. The system of coupled partial differential equations is transformed into ordinary differential equations by using similarity transformations. The finite element method has been accomplished to find numerical solutions to transformed equations. The behavior of radial and tangential velocity, temperature fields, and concentration fields influenced by the various parameters are sketched through graphs. The local skin friction coefficient, Nusselt number, and Sherwood number are also calculated for the pertinent parameters and displayed the results through tables. It is perceived that velocity sketches of both nanoliquids degenerate with larger values of thermal relaxation parameters. Also, the values local Nusselt number of both Ag–EG, and Cu–EG based Cattaneo–Christov nanofluid intensifies with improving values of stretching parameter at the lower disk, whereas, it impedes at the upper disk.     

Entropy generation in MHD Williamson nanofluid over a convectively heated stretching plate with chemical reaction

This investigation focuses on the influence of thermal radiation on the magnetohydrodynamic flow of a Williamson nanofluid over a stretching sheet with chemical reaction. The phenomena at the stretching wall assume convective heat and mass exchange. The novelty of the present study is the thermodynamic analysis in the nonlinear convective flow of a Williamson nanofluid. The resulting set of the differential equations are solved by the homotopy analysis method. We explored the impacts of the emerging parameters on flow, heat, and mass characteristics, including the rate of entropy generation and the Bejan number through graphs, and extensive discussions are provided. The expressions for skin friction, Nusselt and the Sherwood numbers are also analyzed and explored through tables. It is concluded that the rate of mass transfer may be maximized with the variation of the Williamson and chemical reaction parameters. Moreover, the entropy generation rate and the Bejan number are augmented via increasing the Williamson parameter.     

Entropy analysis of unsteady MHD three-dimensional flow of Williamson nanofluid over a convectively heated stretching sheet

This article addresses an investigation of the entropy analysis of Williamson nanofluid flow in the presence of gyrotactic microorganisms by considering variable viscosity and thermal conductivity over a convectively heated bidirectionally stretchable surface. Heat and mass transfer phenomena have been incorporated by taking into account the thermal radiation, heat source or sink, viscous dissipation, Brownian motion, and thermophoretic effects. The representing equations are nonlinear coupled partial differential equations and these equations are shaped into a set of ordinary differential equations via a suitable similarity transformation. The arising set of ordinary differential equations was then worked out by adopting a well-known scheme, namely the shooting method along with the Runge-Kutta-Felberge integration technique. The effects of flow and heat transfer controlling parameters on the solution variables are depicted and analyzed through the graphical presentation. The survey finds that magnifying viscosity parameter, Weissenberg number representing the non-Newtonian Williamson parameter cause to retard the velocity field in both the directions and thermal conductivity parameter causes to reduce fluid temperature. The study also recognizes that enhancing magnetic parameters and thermal conductivity parameters slow down the heat transfer rate. The entropy production of the system is estimated through the Bejan number. It is noticeable that the Bejan number is eminently dependent on the heat generation parameter, thermal radiation parameter, viscosity parameter, thermal conductivity parameter, and Biot number. The skillful accomplishment of the present heat and mass transfer system is achieved through the exteriorized choice of the pertinent parameters.     

Numerical solution for natural convection flow near a vertical porous plate having convective boundary condition with nonlinear thermal radiation

This article presents the theoretical study of the effects of suction/injection and nonlinear thermal radiation on boundary layer flow near a vertical porous plate. The importance of the convective boundary condition as regards the heat transfer rate is taken into account. The coupled nonlinear boundary layer equations are translated into a set of ordinary differential equations via a similarity transformation. The consequences of the active parameters like the suction parameter, injection parameter, convective heat transfer parameter, nonlinear thermal radiation parameters, and Grashof number dictating the flow transport are examined. The numerical result obtained shows that with suction/injection, the heat transfer rate could be increased with nonlinear thermal radiation parameter augment whereas decays with the convective heat transfer parameter and Grashof number. In the presence of suction/injection, the wall shear stress generally increases with nonlinear thermal radiation parameter, convective heat transfer parameter, and Grashof number. The suction has an increasing effect on Nusselt number and shear stress whereas a decreasing effect on Nusselt number and skin friction is seen with injection augment. The nonlinear thermal radiation is an increasing function of the temperature gradient far away from the plate whereas a decreasing function near the porous plate.     

Thermal analysis of buoyancy-motivated Casson fluid flow with time-independent chemical reaction under Lorentz forces

The present research study examines the magneto-hydrodynamic natural convection visco-elastic boundary layer of Casson fluid past a nonlinear stretching sheet with Joule and viscous dissipation effects under the influence of chemical reaction. To differentiate the visco-elastic nature of Casson fluid with Newtonian fluids, an established Casson model is considered. The present physical problem is modeled by utilizing the considered geometry. The resulting system of coupled nonlinear partial differential equations is reduced to a system of nonlinear ordinary differential equations by applying suitable similarity transformations. Numerical solutions of these reduced nondimensional governing flow field equations are obtained by applying the Runge-Kutta integration scheme with the shooting method (RK-4). The physical behavior of different control parameters is described through graphs and tables. The present study describes that the velocity and temperature profiles decreased for increasing values of Casson fluid parameter. Velocity field diminished for the increasing nonlinear parameter whereas velocity profile magnified for increasing free convection parameter. Thermal field enhanced with increasing magnetic parameter in the flow regime. The concentration profile decreased for the rising values of the chemical reaction parameter. The magnitude of the skin-friction coefficient enhanced with increasing magnetic parameter. Increasing Eckert number increases the heat transfer rate and increasing chemical reaction parameter magnifies the mass transfer rate. Finally, the similarity results presented in this article are excellently matched with previously available solutions in the literature.     

Characteristics of MHD three-dimensional flow of nanofluid over a permeable stretching porous sheet

The aim of the present work is to focus on heat and mass transfer characteristics of the magnetohydrodynamic three-dimensional flow of nanofluid over a permeable stretching porous sheet. The significance of this study is the consideration of copper-based and aluminum oxide-based nanofluids. The physical parameters like a chemical reaction, Soret effect, radiation, and heat generation, and radiation absorption being involved in this examination are novel. The nonlinear partial differential equations are transformed into ordinary differential equations by adopting suitable similarity transformations. The numerical solutions are obtained by applying the Runge–Kutta method of fourth-order with the Shooting technique using MATLAB. The results obtained are presented through graphs and tables for various parameters. A comparison with published results has been done to validate the methodology and found good coincidence. It is claimed that the increase in heat generation parameters results in increasing the temperature. With an increase in the Soret effect, the skin friction coefficient along x-axis increases and skin friction coefficient along the y-axis, Nusselt number and Sherwood number decrease.     

Thermophoresis and Brownian motion effects on Williamson nanofluid flow past a stretching surface with thermal radiation and chemical reaction

The heat transfer mechanism of nanofluids has numerous industrial applications owing to the non-Newtonian behavior and has been exercised as a thermophysical phenomena in presence of thermal radiation. The present paper deals with the thermal transfer characteristics of time-independent magnetohydrodynamics Williamson fluid past a stretching surface in presence of the reaction of chemical equilibrium is dealt. The flow constitutive nonlinear partial differential coupled equations are transmitted into ordinary differential equalities by employing relevant similarity transmutations. These deduced equations are determined by using the Runge–Kutta numerical technique with a shooting approach with the aid of MATLAB software. Influences of distinct pertinent flow parameters like an inclined uniform magnetic field, Soret number, heat generation/absorption, and Schmidt number constrained to convective boundary condition is displayed through graphs with relevant physical interpretations. Computed numerical values for the friction factor coefficient, local Nusselt parameter, and Sherwood number are tabulated.