Effect of Joule heating on MHD non‐Newtonian fluid flow past an exponentially stretching curved surface

The current endeavor examines the convective heat transfer characteristics on magnetohydrodynamic stagnation point flow of micropolar fluid past an exponential curved surface. The flow is supposed to be laminar and time‐independent. The influence of radiation, irregular heat source/sink, Joule heating, and variable thermal conductivity are supposed. Suitable similarity renovations are considered to transform the original partial differential equations as ordinary ones and then resolved by shooting and fourth‐order Runge–Kutta methods. Graphs are drawn to inspect the impacts of sundry nondimensional parameters on the distributions of velocity, microrotation, and temperature. We detect that there is an escalation in temperature with Eckert number and variable heat source/sink parameters. Also, it is motivating to comment that Biot number is an increasing function of local Nusselt number.     

Significance of heat source/sink on the radiative flow of Cross nanofluid across an exponentially stretching surface towards a stagnation point with chemical reaction

A numerical review on magnetohydrodynamics radiative motion of Cross nanofluid across an exponentially stretchable surface near stagnation point with varying heat source/sink is addressed. Brownian movement and thermophoretic impacts are assumed. The governing equations for this study are first altered as a system of ordinary differential equations by similarity transformation. With an aid of the Runge–Kutta 4th order mechanism together with the shooting procedure, the impacts of several pertinent parameters including chemical reaction on regular profiles (velocity, temperature, and concentration) are explicated. The consequences of the same parameters on surface drag force, transfer rates of heat, and mass are visualized in tables. From the analysis, it was noticed that the magnetic field parameter enhances the temperature and decreases the velocity of the Cross nanofluid. Also, fluid temperature is an increasing function with thermal radiation and nonuniform heat source/sink. The rate of heat transfer is increased with thermophoresis and diminished with Brownian motion. Sherwood's number is diminished with Brownian motion but it was boosted up with thermophoresis. The present results are compared with published results and those are in agreement.     

Second order velocity slip and thermal jump of Cu–water nanofluid over a cone in the presence of nonlinear radiation and nonuniform heat source/sink using homotopy analysis method

The purpose of the present paper is to explore the second order slip effects on nanofluid flow over a vertical cone. The effects of nonlinear thermal radiation and nonuniform heat source/sink are also taken into account. Water with copper nanoparticles is used as nanofluid in this investigation. The governing partial differential equations for the flow are converted into ordinary differential equations by using transformations and then are solved using homotopy analysis method. The influence of various important parameters on velocity, temperature, skin‐friction, and Nusselt number are presented through graphs. Results indicate that the velocity and magnitude of skin friction decrease with a rise in first and second order velocity slips. A raise in either first or second order temperature jump causes a fall in temperature. Nonlinear radiation increases the more rapidly when compared to the linear radiation case.     

Influence of thermophoresis and chemical reaction on MHD micropolar fluid flow with variable fluid properties

This work investigates the effect of thermophoresis and chemical reaction on heat and mass transfer in hydromagnetic micropolar fluid flow over an inclined permeable plate with constant heat flux and non-uniform heat source/sink in the presence of thermal radiation. It is assumed that the transverse magnetic field is a function of the distance from the origin. The analysis accounts for both temperature dependent fluid viscosity and thermal conductivity. Using the similarity transformation, the governing system of equations are transformed into non-linear ordinary differential equations and are solved numerically using symbolic software MATHEMATICA. Numerical results for the velocity, microrotation, temperature and species concentration as well as for the skin friction, heat and mass transfer are obtained and displayed graphically for pertinent parameters to show interesting aspects of the solution.     

Influence of homogeneous‐heterogeneous reactions and induced magnetic field on the nonlinear thermal convective radiative Jeffrey liquid flow with heat source/sink

An analysis has been carried out to investigate the effect of homogeneous‐heterogeneous reactions and induced magnetic field on the unsteady two‐dimensional incompressible nonlinear thermal convective velocity slip flow of a Jeffrey fluid in the presence of nonlinear thermal radiation and heat source/sink. We assumed that the flow is generated due to injection at the lower plate and suction at the upper plate. We obtained a numerical solution for the reduced nonlinear governing system of equations via the shooting technique with fourth‐order Runge‐Kutta integration. We plotted the graphs for various nondimensional parameters, like Deborah number, heat source/sink parameter, nonlinear convection parameter, nonlinear radiation parameter, magnetic Reynolds number, Strommer's number, velocity slip parameter, strengths of homogeneous, heterogeneous reaction parameters and skin friction over the nondimensional flow, temperature, concentration profiles and magnetic diffusivity fields. Also, we calculated the numerical values of boundary properties, such as the skin friction and heat transfer rate. We noticed that the temperature of the fluid is enhanced with the radiation parameter, whereas the concentration decreases with increase of the magnetic Reynolds number. The present results have good agreement with published work for the Newtonian case.     

Effect of heat source/sink on MHD flow due to porous rotating disk with variable thickness in the presence of cross diffusion

An analysis of steady magnetohydrodynamic axisymmetric flow of a viscous incompressible electrically conducting fluid due to porous rotating disk with variable thickness in the presence of heat source/sink is presented. Soret and Dufour effects (cross‐diffusion) are also considered. The governing partial differential equations are transformed into a system of nonlinear ordinary differential equations. The homotopy analysis method is used to solve the resulting coupled nonlinear equations under appropriate transformed boundary conditions. A parametric study of the physical parameters is made and results are presented through graphs and tables. The results indicate that the thermal boundary layer is thicker for the flow problems having a heat source when compared with that of the problems without a heat source. It is also found that thickness of the disk is having a high impact on fluid velocity, temperature, and concentration.     

Numerical computation on Marangoni convective flow of two‐phase MHD dusty nanofluids under Brownian motion and thermophoresis effects

The current theoretical study describes the Marangoni thermal convective flow of magnetohydrodynamic dusty nanofluids along a wavy vertical surface. The two‐phase mathematical model is developed under the influence of thermal radiation and exponentially varying space‐dependent heat source. Pure and hybrid nanoparticles together with dust particle suspension in the base fluid are taken into consideration to characterize the behavior of the flow. Brownian motion and thermophoresis mechanisms are considered, since it enhances the convection features of dusty nanofluid. Appropriate transformations are adopted to modify the flow governing equations and boundary conditions to dimensionless form. The forward finite difference scheme is implemented to illustrate the resultant coupled partial differential equations. The Newton quasi‐linearization technique is utilized to reduce the nonlinear system into a linear form, which is solved thereafter by Thomas algorithm. The responses of velocity, temperature, concentration, friction factor, and heat and mass transfer rate profiles with various governing parameters are discussed and portrayed graphically. The study evidences that the radiation and space‐dependent heat generating parameters strengthen the temperature distribution. Also, the heat transfer rate appreciably rises with the increment in Marangoni convection. The solution methodology and accuracy of the model is validated by generating the earlier outcomes for nonradiating nanofluid flow without heat source/sink.     

Soret and Dufour effects between two rectangular plane walls with heat source/sink

This study addresses the thermo‐diffusion and the diffusion‐thermo phenomena in a semi‐infinite absorbent channel whose walls are contracting/expanding, with heat source/sink effects. The governing partial differential equations with suitable boundary conditions are transformed to a system of dimensionless ordinary differential equations. An analytic solution of the problem has been found using a technique called homotopy analysis method (HAM). HAM gives consistently valid answers to the problem over an extensive variety of parameters and also provides better accuracy. To validate the analytical results, a comparison has been presented with a numerical solution calculated by using the parallel shooting method. The effects of dimensionless parameters, that is, deformation parameter, Reynolds number, Soret and Dufour numbers, and heat source/sink parameter on the expressions of velocity, temperature, and concentration profiles are analyzed graphically to understand the physics of the deformable channel. It has been noted that the velocity across the channel is higher for the expanding channel, as compared to that for the contracting channel. Also the Soret and Dufour number increases the temperature of the fluid, and decreases the concentration. The temperature profile has an increasing behavior in the case of heat source, and a decreasing behavior in the case of heat sink.     

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.     

A generalized perspective of magnetized radiative squeezed flow of viscous fluid between two parallel disks with suction and blowing

In this research article, the electrically conducting magnetized radiative squeezed flow of two‐dimensional time‐dependent viscous incompressible flow between two parallel disks with heat source/sink and Joule heating effects under the presence of an unsteady homogeneous first order chemical reaction is demonstrated numerically. The considered physical problem is studied under the influence of Lorentz forces to describe the effect of an applied magnetic field. Heat dissipation due to viscosity and Joule heating are considered in the energy equation to demonstrate the behavior of the thermal profile. Also, the thermodynamic behavior of temperature field is described by considering the concept of heat source/sink in the energy equation. The mass transport characteristics of a viscous fluid are described through the time‐dependent chemical reaction of first order type with homogenous behavior. Thus, the considered physical problem gives the time‐dependent, highly nonlinear coupled partial differential equations, which are reduced to a system of ordinary differential equations by invoking the suitable similarity transformations. The discretized first order ordinary differential equations are solved by using the Runge‐Kutta fourth order integration scheme with the shooting technique (RK‐SM) and bvp4c Matlab function. Flow sensitivity of various emerging control parameters are described with the help of tables and graphs. The axial velocity field enhanced for the suction case and suppressed for the blowing case for the increasing values of suction/injection parameter. Also, an excellent comparison between the present solutions and previously published results shows the accuracy and validity of the present similarity solutions and used numerical methods.     

Influence of Hall current on hydromagnetic natural convective flow between two vertical concentric cylinders in presence of heat source/sink

The fully developed laminar magnetohydrodynamic free convection between two concentric vertical cylinders with Hall currents and heat source/sink, in the presence of the radial magnetic field, are studied. The governing thermal energy and momentum equations are changed into ordinary differential equations whose solutions are determined in closed‐form expressions of the Bessel and modified Bessel functions of order zero. A parametric investigation illustrating the impacts of the Hall current, magnetic field, heat source, and radii ratio has been accomplished graphically to examine the changes in temperature as well as velocity while the Nusselt number, mass flux, and skin friction values have been presented in tabular forms. The results ensure that the Hall current has a strong and direct impact on the flow character, such that the influence of the Hall parameter enhances the velocity fields in the appearance of heat source and sink. The velocity remains almost constant as the Hall parameter value is greater than four. Moreover, the velocity and temperature fields have an increasing tendency due to the heat source and inversely for the heat sink.     

Radiation effect on MHD flow past a porous vertical plate in the presence of heat sink

This study investigates heat and mass transfer in MHD convective flow through a vertical plate via porous media in the presence of radiation and a heat source/sink. It is assumed that a uniform magnetic field of strength is imposed perpendicular to the plate and directed into the fluid area. The governing nondimensional equations are solved using the perturbation technique. We further derived the skin friction, Nusselt number, and Sherwood number. The computation of results is performed with the aid of mathematical software and results are presented in graphical and tabular forms for distinct flow impacting parameters. It is observed that fluid motion is retarded due to the application of the magnetic field. Furthermore, the fluid temperature comprehensively falls under the Prandtl number as well as the thermal radiation effect. It is important to note that the heat sink causes fluid velocity and fluid temperature to fall drastically.     

On heated surface transport of heat bearing thermal radiation and MHD Cross flow with effects of nonuniform heat sink/source and buoyancy opposing/assisting flow

Heat transport subject to nonlinear thermal radiation has multiple applications in physics, industry, engineering field, and space technology, such as aerodynamic rockets, solar power technology, large open water reservoirs, and gas-cooled nuclear reactors. This effort studies the magnetohydrodynamic flow of cross fluid, which is a type of non-Newtonian, along a heated surface. Furthermore, the transportation of heat in the fluid is induced by  thermal radiation. Furthermore, the behavior of opposing/assisting flow and impact of nonuniform heat sink/source is scrutinized. The reserved suitable transformations are carried out to shift the ruling equations into nondimensional class. Through reserved transformations, two nonlinear partial differential equations are altered into corresponding nonlinear ordinary differential equations. Then a scheme of integration referred to as Runge–Kutta–Fehlberg is imposed to get a numerical solution of these. The impact of parameters are mentioned concisely on temperature and velocity profiles in the absence and presence of a magnetic parameter. It is proved that the presence of a magnetic field steps up the velocity and temperature as well.     

Thermal slip and radiative heat transfer effects on electro‐osmotic magnetonanoliquid peristaltic propulsion through a microchannel

A mathematical study is described to examine the concurrent influence of thermal radiation and thermal wall slip on the dissipative magnetohydrodynamic electro‐osmotic peristaltic propulsion of a viscous nanoliquid in an asymmetric microchannel under the action of an axial electric field and transverse magnetic field. Convective boundary conditions are incorporated in the model and the case of forced convection is studied, that is, thermal and species (nanoparticle volume fraction) buoyancy forces neglected. The heat source and sink effects are also included and the diffusion flux approximation is employed for radiative heat transfer. The transport model comprises the continuity, momentum, energy, nanoparticle volume fraction, and electric potential equations with appropriate boundary conditions. These are simplified by negating the inertial forces and invoking the Debye–Hückel linearization. The resulting governing equations are reduced into a system of nondimensional simultaneous ordinary differential equations, which are solved analytically. Numerical evaluation is conducted with symbolic software (MATLAB). The impact of different control parameters (Hartmann number, electro‐osmosis parameter, slip parameter, Helmholtz–Smoluchowski velocity, Biot numbers, Brinkman number, thermal radiation, and Prandtl number) on the heat, mass, and momentum characteristics (velocity, temperature, Nusselt number, etc) are presented graphically. Increasing Brinkman number is found to elevate temperature magnitudes. For positive Helmholtz–Smoluchowski velocity (reverse axial electrical field) temperature is strongly reduced, whereas for negative Helmholtz–Smoluchowski velocity (aligned axial electrical field), it is significantly elevated. With increasing thermal slip, nanoparticle volume fraction is also increased. Heat source elevates temperatures, whereas heat sink depresses them, across the microchannel span. Conversely, heat sink elevates nanoparticle volume fraction, whereas heat source decreases it. Increasing Hartmann (magnetic) parameter and Prandtl number enhance the nanoparticle volume fraction. Furthermore, with increasing radiation parameter, the Nusselt number is reduced at the extremities of the microchannel, whereas it is elevated at intermediate distances. The results reported provide a good insight into biomimetic energy systems exploiting electromagnetics and nanotechnology, and, furthermore, they furnish a useful benchmark for experimental and more advanced computational multiphysics simulations.     

Numerical modelling of second‐grade fluid flow past a stretching sheet

The present numerical study reports the chemically reacting boundary layer flow of a magnetohydrodynamic second‐grade fluid past a stretching sheet under the influence of internal heat generation or absorption with work done due to deformation in the presence of a porous medium. To distinguish the non‐Newtonian behaviour of the second‐grade fluid with those of Newtonian fluids, a very popularly known second‐grade fluid flow model is used. The fourth order momentum equation with four appropriate boundary conditions along with temperature and concentration equations governing the second‐grade fluid flow are coupled and highly nonlinear in nature. Well‐established similarity transformations are efficiently used to reduce the dimensional flow equations into a set of nondimensional ordinary differential equations with the necessary conditions. The standard bvp4c MATLAB solver is effectively used to solve the fluid flow equations to get the numerical solutions in terms of velocity, temperature, and concentration fields. Numerical results are obtained for a different set of physical parameters and their behaviour is described through graphs and tables. The viscoelastic parameter enhances the velocity field whereas the magnetic and porous parameters suppress the velocity field in the flow region. The temperature field is magnified for increasing values of the heat source/sink parameter. However, from the present numerical study, it is noticed that the flow of heat occurs from sheet to the surrounding ambient fluid. Before concluding the considered problem, our results are validated with previous results and are found to be in good agreement.     

Effect of thermal radiation on magnetohydrodynamic three‐dimensional motion of a nanofluid past a shrinking surface under the influence of a heat source

An analytical technique known as the homotopy analysis method is used to acquire solutions for magnetohydrodynamic 3‐D motion of a viscous nanofluid over a saturated porous medium with a heat source and thermal radiation. The governing nonlinear partial differential equations are changed to ordinary differential equations employing appropriate transformations. Validation of the present result is done with the help of error analysis for flow and temperature. The influences of pertinent parameters on momentum, energy, and Nusselt number are studied and discussed. The major findings are: the velocity of the nanofluid is affected by the nanoparticle volume fraction and the thickness of the thermal boundary layer becomes thinner and thinner subject to sink, whereas the effect is revered in case of the source.     

Symmetry analysis of MHD Casson fluid flow for heat and mass transfer near a stagnation point over a linearly stretching sheet with variable viscosity and thermal conductivity

In this article, the impacts of variable viscosity and thermal conductivity on magnetohydrodynamic, heat transfer, and mass transfer flow of a Casson fluid are analyzed on a linearly stretching sheet inserted in a permeable medium along with heat source/sink and viscous dissipation. To reduce the ascendant partial differential equations into ordinary differential equations, Lie group transformation is utilized. Further, the fourth-order Runge–Kutta strategy is utilized to solve the ordinary differential equations numerically. The numerical results obtained for various parameters by employing coding in MATLAB programming are investigated and considered through graphical representation and tables. We anatomize the impacts of distinctive parameters on velocity, temperature, and concentration distributions.     

Study of an unsteady magnetohydrodynamic-free convective flow under parabolic ramped temperature and concentration in the presence of Soret and heat sink

The purpose of this paper is to investigate the effects of Soret, thermal radiation, and chemical reaction on an unsteady magnetohydrodynamic free convective flow past an impulsively initiated semi-infinite vertical plate with heat sink under parabolic ramped temperature and parabolic ramped concentration. Using some nondimensional parameters, the flow boundary equations in this case are first converted to dimensionless equations. The closed-form Laplace transform technique is employed here to solve the partial differential equations and get the solutions for fluid velocity, temperature, and concentration. The velocity, temperature, and concentration of the fluid tend to vary with the effect of various flow factors. These changes are graphically represented and analyzed. Differences in skin friction, Nusselt number, and Sherwood number for the different relevant parameters are also recorded. The Soret number hikes the fluid velocity and concentration. The rate of heat transfer, mass transfer, and momentum transfer improves due to the application of parabolic ramped conditions.     

Heat transfer in a liquid film over an unsteady stretching sheet

The effects of viscous dissipation, non-uniform heat source/sink, magnetic field, and thermal radiation on heat transfer characteristics of a thin liquid film flow over an unsteady stretching sheet are analyzed. A similarity transformation is used to reduce the governing time dependent momentum and energy equations into non-linear ordinary differential equations. The resulting differential equations with the appropriate boundary conditions are solved by an efficient shooting algorithm with fourth order Runge–Kutta technique. The effects of the physical parameters on the flow and heat transfer characteristics are presented through graphs and analyzed. The numerical results for the wall temperature gradient (Nusselt number) are calculated and presented through tables. Also, the effects of the physical parameters on the heat transfer characteristics are brought out: suggestions are made for efficient cooling. Furthermore, the limiting cases are obtained and are found to be in good agreement with the previously published results.     

Heat and mass transfer flow over a stretching surface with Ohmic heating and chemical reaction

The present study analyzes the effect of chemical reaction on an unsteady magnetohydrodynamic boundary layer viscous fluid over a stretching surface embedded in a porous medium with a uniform transverse magnetic field. A Darcy‐Forchheimer drag force model is employed to simulate the effect of second‐order porous resistance. Dissipative heat energy based on both viscous and Joule dissipation along with a heat source/sink is considered to enhance the energy equation. Similarity analysis is imposed to transform the governing differential equations into a set of nonlinear coupled ordinary differential equations. These sets of equations are solved numerically using the Runge‐Kutta fourth‐order scheme followed by the shooting algorithm. The effects of physical parameters such as magnetic field, Prandtl number, Eckert number, Schmidt number, unsteadiness parameter, and chemical reaction parameters have been discussed on velocity, temperature, and concentration fields. Computation for the coefficient of skin friction, rate of heat and mass transfer is done and presented in a table for validation of the present outcomes.