Differential transformed approach of unsteady chemically reactive nanofluid flow over a bidirectional stretched surface in presence of magnetic field

This study investigates the consequences of steady and unsteady nanofluid flow over a bidirectional stretching sheet. Herein, the magnetic field is working in the normal direction. The Brownian motion together with thermophoresis is taken into consideration. Moreover, the chemical reaction within the nanoingredients also occupies a novel corner in this study. The leading equations of the considered model are transferred to nonlinear ordinary differential equations (ODEs) by an appropriate similarity transformation. The differential transformation method is used to solve the set of ODEs. We have used MAPLE‐17 software to solve this with the desired accuracy rate. Results are portrayed with graphs and tables. The corresponding physical consignments such as the Nusselt number, skin friction, and the Sherwood number are exhibited. Excellent improvement in heat and mass transport is observed, which can be visualized through tables. Outcomes reveal that both the temperature and x‐direction velocity are reduced for the stretching parameter. Heat transport escalates for stretching factor, but higher outcomes are marked for the unsteady flow as compared with the steady flow. The mass transfer also increases for the chemical reaction factor, but the rate of increment is higher for the unsteady flow.     

Effects of heat and mass transfer on MHD nonlinear free convection non-Newtonian fluids flow embedded in a thermally stratified porous medium

This communication examines heat alongside mass transport in a nonlinear free convection magnetohydrodynamics (MHD) non-Newtonian fluid flow with thermal radiation and heat generation deep-rooted in a thermally stratified penetrable medium. The Casson and Williamson fluid considered in this communication flos simultaneously across the boundary layer and are mixed together. The model of heat alongside mass transport is set up with chemical reaction and thermal radiation alongside heat generation to form a system of partial differential equations (PDEs). Appropriate similarity variables are used to simplify the PDEs to obtain systems of coupled ordinary differential equations. An efficiently developed numerical approach called the spectral homotopy analysis method was used in providing solutions to the transformed equations. A large value of Casson term is observed to degenerate the velocity plot while the Williamson parameter enhances the velocity profile. The parameter of thermal stratification is found to enhance the rate of heat transport within the boundary layer. An incremental value of the magnetic parameter declines the velocity of the fluid and the entire boundary layer thickness. The present result was compared with previous studies and was seen to be in good agreement.     

On the mixed convective carbon nanotube flow over a convectively heated curved surface

In this article, mixed convective boundary layer stream of nanofluid flow with carbon nanotube as nanoparticles and transmission of heat over a coiled stretched surface are studied. The influence of magnetic orientation and velocity slip is also encountered in this problem. Two classes of carbon nanotubes, SWCNT and MWCNT, are considered as nanoparticles and water as a pure liquid. The foremost leading partial differential equations (PDEs) are formulated through curvilinear coordinate system subjected to proper boundary conditions. To simplify this nonlinear PDE‐associated model, we have employed a compatible similarity conversion and acquired the nonlinear dimensionless ordinary differential equations (ODEs). To determine the requisite numerical solution of the transformed problem, a shooting procedure embedded with RK‐4 technique has been applied. Various pictorial attempts have been initiated against different parametric inputs to reveal the hydrothermal scenario. Some physical quantities like skin friction and Nusselt numbers are calculated to investigate flow distribution inside the preferred system. A comparison with earlier research depicts parallel outcomes. Results assured that velocity is a cumulative function with positive increment of curvature parameter, but an opposite scenario is shown for temperature for both type of nanofluids. The amount of heat transition has been declined against the improvement of the magnetic parameter.     

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.     

Transient electromagnetohydrodynamic radiative squeezing flow between two parallel Riga plates using a spectral local linearization approach

This article addresses transient electromagnetohydrodynamic radiative squeezing flow due to convectively heated electromagnetic actuator. The transport analysis of heat and mass is explored considering the heat generation/absorption and destructive species homogeneous reaction. Suitable transformations are applied on the mathematical model developed to convert governing partial differential equations to ordinary differential equations (ODEs). Spectral local linearization method (SLLM) is employed on the resultant nonlinear coupled ODEs to compute the numerical results. Influence of sundry physical quantities on heat mass transfer of squeezing flow characteristics are determined using graphs and tabular results. SLLM results exhibit that momentum and temperature improved with rise in squeezing and heat source parameters correspondingly. Momentum enhances at lower plate and detracts with rise in modified Hartmann number. For improved heat source parameter, the rate of heat transfer diminishes and is more significant for higher Prandtl number values. This investigation has relevance in disk style magnetic clutches, rolling elements, food processing, bearings, squeezing film pressure sensors, and flow rheostats.     

Analytical study of heat and mass transfer in unsteady MHD radiant flow of Williamson nanofluid over stretching sheet with heat generation and chemical reaction

This paper presents the analytical study of heat and mass transfer in a two-dimensional time-dependent flow of Williamson nanofluid near a permeable stretching sheet by considering the effects of external magnetic field, viscous dissipation, Joule heating, thermal radiation, heat source, and chemical reaction. Suitable transformations are introduced to reformulate the governing equations and the boundary conditions convenient for computation. The resulting sets of nonlinear differential equations are then solved by the homotopy analysis method. The study on the effects of relevant parameters on fluid velocity, temperature, and concentration profiles is analyzed and presented in graphical and tabular forms. Upon comparison of the present study with respect to some other previous studies, a very good agreement is obtained. The study points out that the transfer of heat can substantially be enhanced by decreasing viscoelasticity of the fluid and the transfer of mass can be facilitated by increasing permeability of the stretching sheet.     

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.     

Influence of radiation and viscous dissipation on MHD heat transfer Casson nanofluid flow along a nonlinear stretching surface with chemical reaction

The present article investigates the influence of Joule heating and chemical reaction on magneto Casson nanofluid phenomena in the occurrence of thermal radiation through a porous inclined stretching sheet. Consideration is extended to heat absorption/generation and viscous dissipation. The governing partial differential equations were transformed into nonlinear ordinary differential equations and numerically solved using the Implicit Finite Difference technique. The article analyses the effect of various physical flow parameters on velocity, heat, and mass transfer distributions. For the various involved parameters, the graphical and numerical outcomes are established. The analysis reveals that the enhancement of the radiation parameter increases the temperature and the chemical reaction parameter decreases the concentration profile. The empirical data presented were compared with previously published findings.     

Inclined magnetized and energy transportation aspect of infinite shear rate viscosity model of Carreau nanofluid with multiple features over wedge geometry

Heat transference in fluid mechanism has a deep influence in real-life applications like hot-mix paving, recovery of energy, concrete heating, heat spacing, refineries, distillation, autoclaves, reactors, air conditioning, and so forth. In this attempt, findings related to energy exchange with features of infinite shear rate viscosity model of Carreau nanofluid by placing inclined magnetic dipole over the wedge are made. The main role in the transportation of heat is exercised by incorporating facts of r adiation, nonuniform heat sink source, Brownian motion, thermophoresis, and chemical reaction. The mathematical system of the infinite shear rate viscosity model of Carreau nanofluid gives a system of partial differential equations and furthermore, these are moved into ordinary differential equations. A numerical procedure is applied via shooting/bvp4c to obtain numerical results. Inclined magnetic dipole gives a lower velocity of Carreau nanofluid. Due to the relaxation time factor velocity of Carreau fluid gets down. A* causes to generate the heat internally, so due to this, temperature increases rapidly. The increasing rate of temperature is found very high for the growing Hartmann number. The rate of mass transport becomes low for gradual increment in the parameter of thermophoresis, wedge angle, and Prandtl. Inclined magnetic dipole gives a lower velocity of Carreau nanofluid. Due to the relaxation time factor, the velocity of the Carreau fluid goes down. The absence and presence of magnetic numbers have no influence on velocity, temperature, and concentration files for Le, R_d, θ_f, γ, We, β, Pr, Nb, Nt, A.     

Cattaneo–Christov heat flux model and multiple slip effect on carbon nanofluid over a stretching sheet in a Darcy–Forchheimer porous medium

In several biotechnological processes, multiple slips are the most paramount, such as blood pumping from the heart to different body components, endoscopy treatment, pabulum distribution, and the heat transport phenomenon regulation. In the current research, we have studied the multiple slips, Darcy–Forchheimer, and Cattaneo–Christov heat flux model on a stretching surface exposed to magnetic carbon nanotube nanofluid. We have additionally included a heat source or sink, a chemical reaction for manipulating the heat and mass transport phenomena. The resulting governing partial differential equations have been transformed into ordinary differential equations. With the Runge–Kutta–Fehlberg fourth–fifth-order procedure, the transformed governing equations are numerically solved. Numerical solutions for different parameters for velocity, temperature, and concentration profiles (Eckert number, velocity slip, thermal slip, mass slip, etc.) are highlighted. Graphical and numerical results for the various parameters in the modeled problem have been outlined. The present numerical results are compared with the published ones for some limiting cases. The slip has been found to control the flow of the boundary layer.     

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).     

Aspects of second-order velocity slip and radiation absorption on hydromagnetic stagnation point flow of Jeffrey fluid with chemical reaction

The present article describes the magnetohydrodynamic flow of a moving Jeffrey fluid along a convectively heated porous stretching surface with second-order velocity slip and radiation absorption effects. Furthermore, chemical reactions and viscous dissipation impacts are also taken into account. The governing equations are converted into dimensionless ordinary differential equations (ODEs) using appropriate similarity transformations. The highly nonlinear ODEs are solved numerically by employing a shooting technique based on the Runge–Kutta Cash–Karp formula. The figures are used to study the variations in temperature, velocity, and concentration profiles for several physical factors. The numerical values of the local skin friction, Sherwood number, and Nusselt number are explained and shown in tables. The analysis reveals that the velocity profile is enhanced for amplifying values of velocity ratio parameter and first-order velocity slip parameter. However, the temperature profile of Jeffrey nanofluid is highlighted w.r.t. Eckert number and radiation absorption parameter. This study may find significant applications in polymer production, food processing, instrumentation, combustion modeling, catalytic chemical reactors, and so on.     

Numerical investigation of MHD convective free stream nanofluid flow influenced by radiation and chemical reaction over stretching cylinder

Chemically reacting magnetohydrodynamic radiative flow of convective free stream nanofluid through a stretching cylinder using Buongiorno's model is discussed. The behavior of Brownian motion and thermophoresis is also appropriate. By adopting the similarity transformation, the partial differential equation is diminished into a first-order ordinary differential equation (ODE). Since transformed equations are highly nonlinear these ODEs are solved by using mathematical simulation. The shooting procedure has been adopted to resolve converted equations along the attendant Runge–Kutta–Fehlberg technique. The reason behind the present work is to research the effects of different parameters of fluid, namely, magnetic parameter, free stream velocity, Brownian motion, thermophoresis, chemical reaction, heat radiation, Lewis number on nanoparticle concentration, temperature, and velocity distribution. The impact of significantly participating parameters on velocity, concentration, and temperature distribution is distinguished with appropriate physical significance. The convergence of solutions for temperature, velocity, and concentration profiles is studied carefully. The measured challenges of nanofluids are scale-up capacity, increase in nanofluid viscosity, nanoparticle dispersion, and nanofluid cost. It is observed that nanoparticle temperature rises for more value of Brownian motion parameter while it declines for higher Lewis number. The current study in the cylindrical region is related to novel free stream flow in the presence of chemical reactions along with convective conditions which find applications in electronic systems like microprocessors and in a wide variety of industries and in the field of biotechnology. The current research helps control the transport phenomena, helping production companies to find the quality of the desired product.     

Radiation and slip effects on MHD Maxwell nanofluid flow over an inclined surface with chemical reaction

The numerical solutions of the upper-convected Maxwell (UCM) nanofluid flow under the magnetic field effects over an inclined stretching sheet has been worked out. This model has the tendency to elaborate on the characteristics of “relaxation time” for the fluid flow. Special consideration has been given to the impact of nonlinear velocity slip, thermal radiation and heat generation. To study the heat transfer, the modified Fourier and Fick's laws are incorporated in the modeling process. The mass transfer phenomenon is investigated under the effects of chemical reaction, Brownian motion and thermophoresis. With the aid of the similarity transformations, the governing equations in the ordinary differential form are determined and then solved through the MATLAB's package “bvp4c” numerically. This study also brings into the spotlight such crucial physical parameters, which are inevitable for describing the flow and heat transfer behavior. This has been done through graphs and tables with as much precision and exactitude as is possible. The ascending values of the magnetic parameter, the Maxwell parameter and the angle of the inclined stretching sheet cause decay in the dimensionless velocity while an assisting behavior of the thermal and concentration buoyancy parameters is noticed.     

Finite element analysis of couple stress micropolar nanofluid flow by non‐Fourier's law heat flux model past stretching surface

Numerical analysis has been done to investigate magnetohydrodynamics nonlinear convective flow of couple stress micropolar nanofluid with Catteneo‐Christov heat flux model past stretching surface with the effects of heat generation/absorption term, chemical reaction rate, first‐order slip, and convective boundary conditions. The coupled highly nonlinear differential equation governing the steady incompressible laminar flow has been solved by a powerful numerical technique called finite element method. The impacts of diverse parameters on linear velocity, angular velocity (microrotation), temperature, concentration profile, local skin friction coefficient, local wall couple stress, local Nusselt number, and Sherwood number are presented in graphical and tabular form. The result pointed out that the enhancement in material parameter β increases the velocity of the fluid while the couple stress parameter K has quite opposite effect. Heat and mass transfer rate of the fluid are enhanced by increasing material parameter while couple stress parameter shows the opposite influence. Moreover, heat and mass transfer rate are higher with the Catteneo‐Christov heat flux model than Fourier's law of heat conduction. The accuracy of the present method has been confirmed by comparing with previously published works.     

Effect of SWCNTs and MWCNTs Maxwell MHD nanofluid flow between two stretchable rotating disks under convective boundary conditions

The aim of the current analysis is to investigate heat and mass transfer characteristics of single and multi‐walled water‐based carbon nanotubes Maxwell nanofluid flow between continuously rotating stretchable disks under the sway of chemical reaction and radiation. Boundary conditions of the convective type of temperature are employed at both lower and upper rotating disks in the preparation. Similarity variables are employed to transform the governing partial differential equations into the nonlinear ordinary differential equations. The computational finite element method is applied to solve this nonlinear system of equations along with boundary conditions. The sway of different admissible parameters on the profiles of concentration, temperature, and velocity are inspected and revealed through graphs. Furthermore, the numerical solutions for rates of temperature, concentration, and rates of velocity are depicted in tabular form. It is revealed that temperature sketches deteriorate with augmented values of Deborah number at both upper and lower disks of single‐walled carbon nanotubes and multi‐walled carbon nanotubes with water‐based Maxwell nanofluids.     

Numerical simulation of heat and mass transfers radiative–convective fluid flow on a stretching porous surface with temperature-dependent thermophysical properties

This paper is focused on the analysis of heat and mass transfer radiative–convective fluid flow using quadratic multiple regression and numerical approach. The physical phenomenon is analyzed by utilizing partial differential equations (PDEs). Thermophysical properties, such as viscosity, thermal conductivity, and mass diffusivity, are varied and temperature-dependent. This study is unique because of its applications in magnetohydrodynamic power accelerators, drilling operators, and bioengineering. The governing PDEs are transformed into coupled nonlinear ordinary differential equations (ODEs). The transformed ODEs are solved numerically using the spectral homotopy analysis method. Also, a quadratic multiple regression analysis is performed on quantities of engineering interest to show the significance of the flow parameters. It is observed that the heat and mass transfer process is affected by nonlinear buoyancy impact. The Lorentz force produced by the imposed magnetic field decline the thickness of the hydrodynamic boundary layer. Findings revealed that the nonlinear convective parameter and variable thermophysical properties are greatly affected by the rate of heat and mass transfer. Previously published work was used to validate the present one, which conformed with it. The slope of linear regression through data points is adopted to show the rate of change in skin friction, Nusselt, and Sherwood numbers during the flow phenomenon.     

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.     

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.     

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.