A mathematical simulation of unsteady MHD Casson nanofluid flow subject to the influence of chemical reaction over a stretching surface: Buongiorno's model

In this study, unsteady boundary layer flow with Casson nanofluid within the sight of chemical reaction toward a stretching sheet has been analyzed mathematically. The fundamental motivation behind the present examination is to research the influence of different fluid parameters, in particular, Casson fluid $\beta (0.2\le \beta \le 0.4)$, thermophoresis $Nt(0.5\le Nt\le 1.5)$, magnetohydrodynamic $M(3.0\le M\le 5.0)$, Brownian movement $Nb(0.5\le Nb\le 2.0)$, Prandtl numberty, unsteadiness parameter $A(0.10\le A\le 0.25)$, chemical reaction parameter $\gamma (0.1\le \gamma \le 0.8)$, and Schmidt number $Sc(1.0\le Sc\le 3.0)$ on nanoparticle concentration, temperature, and velocity distribution. The shooting procedure has been adopted to solve transformed equations with the assistance of Runge–Kutta Fehlberg technique. The impact of different controlling fluid parameters on flow, heat, and mass transportation are depicted in tabular form and are shown graphically. Additionally, values of skin friction coefficient, Nusselt number, and Sherwood number are depicted via tables. Present consequences of the investigation for Nusselt number are related with existing results in writing by taking $Nb=0$ and $Nt=0$ where results are finding by utilization of MATLAB programming. Findings of current research help in controlling the rate of heat and mass aspects to make the desired quality of final product aiding manufacturing companies and industrial areas.     

Thermal performance of Ag–water nanofluid flow over a curved surface due to chemical reaction using Buongiorno's model

An analysis of heat and mass transfer is carried out under the influence of chemical reaction, friction heating, and heat generation/absorption over a curved surface. The impacts of random motion attributes of nanoparticles and thermophoresis are also applied in the expressions of energy and concentration. With the help of assigned transformations, the nonlinear partial differential equations are changed to dimensionless nonlinear ordinary differential equations. Then, the numerical solution is obtained using fourth‐fifth order Runge‐Kutta‐Fehlberg method via the shooting technique. The impacts of relevant parameters on velocity, temperature, and concentration are depicted through graphs and tables. The results illustrate that the lowest concentration distribution of nanofluid is related to the higher value of chemical reaction parameter. Moreover, it is found that thermophoresis and Brownian motion parameters have a propensity to increase the temperature profile while curvature parameter decreases the velocity profile. Also, velocity and temperature fields show a similar behavior for the increasing values of volume fraction of the nanoparticles, while a reverse trend is observed in the concentration profile under the same condition. To authenticate the results of the current study, the obtained data were compared with previously published data.     

Thermal and solutal Marangoni stagnation point Casson fluid flow over a stretching sheet in the presence of radiation,Soret and Dofour effect with chemical reaction

The Marangoni flow is involved with microgravity and earth gravity, which causes undesirable effects in crystal growth experiments. Crystal growth experiments were designed in such a manner so as to appraise MIR (space station), which is one of the best platforms for protein crystallization and radiation experiments. In this article, a model is proposed with a stagnation point and a Casson fluid flow at the interface of the plate in the presence of Marangoni convection influenced by a magnetic field and chemical reaction. Furthermore, it is considered that both temperature and concentration surface tension vary linearly with the interface. It is important to choose similarity transformations for implementing nonlinear differential equations into linear ordinary differential equations. We solved the system of differential equations using fourth order Range‐Kutta method with suitable shooting techniques, and the results are displayed through graphs. A comparison is made with the earlier existing literature, and it shows a very good agreement. The results and a detailed discussion of velocity, temperature, and concentration have been shown graphically. The favorable and unfavorable buoyancy force to Marangoni flow, the features of temperature and concentration field, have been investigated.     

Heat transport and stagnation‐point flow of magnetized nanoliquid with variable thermal conductivity,Brownian moment,and thermophoresis aspects

The improvement of heat transport is a very important phenomenon in nuclear reactors, solar collectors, heat exchangers, and coolers, which can be achieved by choosing the nanofluid as the functional fluid. Nanofluids improve thermophysical properties; as a result, they have made great progress in engineering, biomedical, and industrial applications. Therefore, a numerical study has been proposed to analyze the flow and heat transport of nanoliquids over an extendable surface near a stagnation point with variable thermal conductivity under the influence of the magnetic field, due to their importance in the engineering field. Nanoliquid attributes explain the Brownian motion and the diffusion of thermophoresis. The effects of the chemical reaction and the uniform internal heat source/heat sink are also considered. The Nachtsheim‐Swigert shooting procedure based on the Runge‐Kutta scheme is used for numerical calculation. The impact of effective parameters on velocity, temperature, and volume fraction of the nanoparticles is shown in the graphs and reported in detail. The surface criteria are also estimated with respect to the shear stress and the rate of heat and mass transfer. The aspects of the Brownian moment and Lorentz force are positively correlated to the thermal field of the nanoliquid. Also, the variable thermal conductivity aspect favors the growth of the thermal boundary layer.     

Unsteady natural convection magnetohydrodynamic stagnation point flow with assisting and opposing characteristics: Buongiorno model

In this investigation, the flow of an unsteady mixed convection boundary layer viscous nanofluid on a stretchable sheet is considered. The flow examination is affected by a magnetic field. The reason for the examination exhibited is to create models for nanomaterials that incorporate the Brownian movement and thermophoresis phenomena. The created nonlinear standard differential condition is illuminated numerically utilizing the Runge-Kutta-Gill technique and the start program. The different factors of speed, temperature, and concentration are reported and discussed. The examination shows that the speed, temperature, and concentration are lower in contrast with the consistent stream on account of an assisting flow, whereas the opposite situation is noticed in the opposing flow case. The effects of Brownian movement and thermophoresis in the concentration case are totally different.     

Impact of porosity and radiation on two-dimensional unsteady magnetohydrodynamics heat transfer stagnation point flow with viscous dissipation

The current study aims to investigate the influence of porosity in the presence of radiation, and viscous dissipation on two-dimensional unsteady magnetohydrodynamics mixed convection heat and mass transfer flow at the stagnation point. The governing time-dependent nonlinear partial differential equations are converted into a nonlinear ordinary differential equation by utilizing similarity transformations. The transformed equations are solved by employing the bvp4c technique, a well-known numerical approach. The influence of nondimensional factors on fluid velocity, temperature, and species concentration profiles is explored and graphically represented. For varied values of the Prandtl number, magnetic field, and Schmidt number, the friction factor, Nusselt, and Sherwood numbers are also explored and provided in tabular format. As increasing the porosity parameter, the temperature profile, concentration profile is growing, and velocity profile diminishes. The conclusions of this study are widely accepted by the scientific community.     

Stagnation‐point flow of Maxwell fluid with magnetic field and radiation effects

The effect of heat transfer on the steady two‐dimensional stagnation point flow of a Maxwell fluid over a stretching sheet is discussed. The fluid is electrically conducting in the presence of a uniform applied magnetic field. The radiation effect in the energy equation is taken into account. The arising nonlinear problem is solved by a homotopy analysis method (HAM). Convergence of the series solutions is checked. The values of skin friction coefficient and local Nusselt number have been computed and discussed. © 2011 Wiley Periodicals, Inc. Heat Trans Asian Res; Published online in Wiley Online Library ( wileyonlinelibrary.com/journal/htj ). DOI 10.1002/htj.20385     

Effect of thermophoresis and Brownian motion on the melting heat transfer of a Jeffrey fluid near a stagnation point towards a stretching surface: Buongiorno's model

This analysis intends to address the coupled effect of phase change heat transfer, thermal radiation, and viscous heating on the MHD flow of an incompressible chemically reactive nanofluid in the vicinity of the stagnation point toward the stretching surface, taking a Jeffrey fluid as the base fluid. Convergent analytical solutions for the nonlinear boundary layer equations are obtained by the successive application of scaling variables and the highly efficacious homotopy analysis method. Error analysis is implemented to endorse the convergence of the solutions. Through parametric examination, influence of various physical parameters occurring in analysis of the profiles of velocity, temperature, and nanoparticle concentration, coefficient of surface drag, rates of mass and heat transfer is explored pictorially. The Deborah number and the melting parameter are found to enhance velocity, and the associated momentum boundary layers are thicker, whereas the magnetic field depreciates the flow rate. Temperature is observed to enhance with the thermophoresis parameter, Prandtl number and Eckert number, whereas a reduction is seen with the thermal radiation parameter and Brownian motion parameter. Nanoparticle concentration is depleted by the chemical reaction parameter, the thermophoresis parameter, and the Lewis number.     

Magnetohydrodynamic flow of a micropolar nanofluid in association with Brownian motion and thermophoresis: Irreversibility analysis

This investigation was carried out with the purpose of presenting the flow of micropolar fluid flowing in the microchannel placed parallel to the ground. The prime aim of the work was to study the behavior of micropolar fluid and the response of the microrotation component when the two significant mechanisms namely Brownian movement and thermophoresis are accounted for, as these effects are mainly concerned with the motion of the particles of nano-dimensions. For the flow of micropolar, we account for the extra kinematics variables combined with the classical continuum mechanics namely microinertia moment tensor and gyration tensor. Magnetic effect and suction/injection of the fluid through the channel walls are also facilitated. The influence on the fluid concentration due to the presence of activation energy was accounted in the present examination. On considering all of these effects, equations are carefully modeled and the solution was attained with the aid of Runge–Kutta Fehlberg 4–5th order method using a shooting scheme. The results have deciphered that the presence of material parameter elevates the microrotation component on the upper half of the channel and depletes it at the lower half. The microinertia parameter shows the opposite behavior of the material parameter. Brownian motion parameter is found to enhance the thermal profile and concentration profile. Lesser entropy was generated when the material parameter was high.     

Impact of the Cattaneo‐Christov thermal and solutal diffusion models on the stagnation point slip flow of Walters' B nanofluid past an electromagnetic sheet

The present study investigates the effect of Cattaneo‐Christov thermal and solutal diffusion on the stagnation point flow of Walters' B nanofluid past an electromagnetic sheet subject to velocity, thermal, and solutal slips. The study analyzed the role of electromagnetic fields. In addition, the authors introduced the heat transfer aspect due to Brownian motion and thermophoretic force. The numerical solution of the transformed governing equations employed the spectral local linearization method. Comparisons showed an excellent agreement with the numerical data presented in previous notable works. The study reveals that the developed electromagnetic field due to the arrangement of the sheet causes accelerated fluid motion, and diminution of nondimensional temperature and concentration. In addition, augmented velocity, thermal, and solutal slips develop the corresponding descending boundary layers. The augmented short memory coefficient enhances the skin friction coefficient. The Cattaneo‐Christov thermal and solutal diffusion upsurge the heat and mass transfer rates from the electromagnetic sheet, respectively.     

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.      

Time-dependent hydromagnetic stagnation point flow of a Maxwell nanofluid with melting heat effect and amended Fourier and Fick's laws

An unsteady stagnation point flow of a Maxwell fluid over a unidirectional linearly stretching sheet is studied under the influence of a magnetic field. The parabolic energy equation, which is based on parabolic Fourier law is replaced with a hyperbolic energy equation incorporating the heat flux model of Cattaneo–Christov. The Buongiorno model is used to characterize the properties of nanofluids using thermophoresis and Brownian diffusion coefficients. The phenomenon of melting heat transfer and slip mechanism is also embodied in the present study. Coupled nonlinear differential equations have appeared when the specified similarity transformations are applied. The mathematical problem is tackled via the homotopy analysis method. The impact of important physical parameters on the velocity, concentration, and temperature are highlighted via graphs. To verify our present results, a comparison is given with a limiting case with an already published article. It is witnessed through the graphs that the higher unsteadiness parameter and melting heat coefficient both are responsible for the reduction in the velocity and temperature of the nanofluid. Also, the velocity slip parameter detracts the velocity profile and affiliated boundary layer thickness of the Maxwell nanofluid.     

Physical insight into thermal analysis of magnetohydrodynamic stagnation point flow of micropolar nanofluid across a flexible surface equipped with porous medium and Fourier and Fick's law

Nanofluids represent a novel heat transfer liquid, making them an efficient medium for enhancing energy transmission. Nevertheless, significant knowledge gaps still exist concerning current strategies for improving heat transfer in nanofluids, underscoring the necessity for comprehensive research on these fluid systems. Therefore, this study considered theoretical analysis retrieves the influence of radiative two-dimensional stagnation point flow of second-grade micropolar fluid flow about an elongated channel surface implanted in porous media with magnetic effect, and modified heat and mass flux is under consideration. The major novel effect of the current study is to analyze the activation energy and thermal aspect of the system in the presence of nonlinear radiation effects that are considered in the revised mathematical framework by utilizing the boundary layer theory. The resulting set of coupled partial differential equations is further reduced and transformed into a dimensionless system of ordinary differential equations through appropriate scaling invariants. We initiate the RKF-45 investigation scheme to numerically analyze the transformed dimensionless system, considering relevant parameters. The computational algorithm is implemented using MATLAB programming syntax. Plotted visuals are revealed for leading parameters against pertinent flow profiles graphically and with numerical data. Additionally, the convergence analysis of the numerical results for various flow profiles of the fluids were compared to establish the authenticity of the proposed flow problem. These research findings play a significant role in controlling heat transfer rates and fluid velocities in diverse manufacturing processes and industrial applications, ultimately aiding in achieving the desired product quality.     

Transport of thermal energy in the magnetohydrodynamic oblique stagnation point flow in a hybrid nanofluid with nanoparticle shape effect

In the present paper, the augmented heat characteristics of a hybrid nanofluid which is a blend of Al₂O₃ (alumina) and Ag (silver) in the host hybrid fluid (C₂H₆O₂-H₂O) (50%–50%) impinging obliquely on an elastic surface with magnetic lines of force are investigated. The properties of the nanofluid are assessed through the computational solutions established with the aid of the popular Runge–Kutta–Fehlberg fifth-order (RKF 5) numerical technique. Outputs of the analysis reveal that the rate of thermal energy transport in the hybrid (mono) nanofluid is enhanced by 11.5% (5.8%) by using blade-shaped nanoparticles in comparison to that of the spherical particles. Stream contours of both nanofluids are inclined to the left (right) of the stagnation-point for positive (negative) values of the stagnation flow parameter.     

Unsteady mixed convection flow of a micropolar fluid near the stagnation point on a vertical surface

The unsteady mixed convection boundary-layer flow of a micropolar fluid near the region of the stagnation point on a double-infinite vertical flat plate is studied. It is assumed that the unsteadiness is caused by the impulsive motion of the free stream velocity and by sudden increase or sudden decrease in the surface temperature from the uniform ambient temperature. The problem is reduced to a system of non-dimensional partial differential equations, which is solved numerically using the Keller-box method. This method may present well-behaved solutions for the transient (small time) solution and those of the steady-state flow (large time) solution. It was found that there is a smooth transition from the small-time solution (initial unsteady-state flow) to the large-time solution (final steady-state flow). Further, it is shown that for both assisting and opposing cases and a fixed value of the Prandtl number, the reduced steady-state skin friction and the steady-state heat transfer from the wall (or Nusselt number) decrease with the increase of the material parameter. On the other hand, it is shown that with the increase of the Prandtl number and a fixed value of the material parameter, the reduced steady-state skin friction decreases when the flow is assisting and it increases when the flow is opposing.     

Exploration of radiative heat energy on the peristaltic blood flow of a nanofluid in conjunction with a heat source

The peristaltic flow of a conducting nanofluid associated with the Buongiorno model observed within a wavy channel is proposed in this article. In a peristaltic flow, the process of pumping takes place from a lower pressure to a high-pressure region. It is treated as a vehicle through which the liquid passes in a channel due to its dynamic rush and expands in its length. Therefore, an analysis is carried out for the interaction of thermal radiation and heat source on the peristaltic flow of nanofluid past a tapered channel. The crux of this investigation is the interaction of Hall current due to the conjunction of conducting medium. An analytical technique is used to get the solution of the transformed governing equations, and furthermore, the pressure gradient is also evaluated. The flow phenomena characterized by certain parameters are obtained and presented via graphs. An important observation is seen in that the contribution of magnetic field and Hall current may favor the pumping process and the pressure gradient lowering in the conducting medium is one of the important characteristics.     

Features of Cattaneo‐Christov heat flux model for Stagnation point flow of a Jeffrey fluid impinging over a stretching sheet: A numerical study

The present work focuses on a two‐dimensional steady incompressible stagnation point flow of a Jeffery fluid over a stretching sheet. The Cattaneo‐Christov heat flux model is incorporated into this study. Simulation is conducted via the Runge‐Kutta fourth‐order cum shooting method for the transformed system of nonlinear equations. The influence of the governing parameters on the dimensionless velocity, temperature, skin friction, streamlines, and isotherms is incorporated. A significant outcome of the current investigation is that an increase in the relaxation time parameter uplifts temperature; however, a gradual decrease is observed in the velocity field. Another important outcome of the present analysis is that the momentum boundary layer augments due to an increase in the Deborah number; however, a decrease is observed in the temperature. Furthermore, it is also observed that the skin friction coefficient escalates with an increase in the relaxation time parameter for the assisting flow, but a reverse trend is observed for the opposing flow.     

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.     

Influence of heat transfer on the nonorthogonal stagnation point flow of a third‐order fluid towards a stretching surface

The present article looks at the theoretical analysis of a steady stagnation‐point flow with heat transfer of a third‐order fluid towards a stretching surface. The formulation of the problem has been carried out for a third order fluid and constructed partial differential equations are rehabilitated into ordinary differential equations. The consequential ordinary differential equations are solved analytically using the homotopy analysis method (HAM). Graphical illustrations are shown for various parameters involved in the flow equations. Numerical values of skin friction coefficients and heat flux are computed and presented through tables. © 2013 Wiley Periodicals, Inc. Heat Trans Asian Res; Published online in Wiley Online Library (wileyonlinelibrary.com/journal/htj). DOI 10.1002/htj.21042     

Radiation and slip effects on MHD point flow of nanofluid towards a stretching sheet with melting heat transfer

In the present paper, the melting heat transfer of a nanofluid over a stretching sheet is investigated. Magnetohydrodynamic stagnation point flow with thermal radiation and slip effects is considered for this study. The governing model of the flow is solved by Runge–Kutta fourth-order method using appropriate similarity transformations. Temperature and velocity fields are presented for various flow pertinent parameters. Nondimensional physical parameters such as Prandtl number, radiation parameter, Brownian motion parameter, Lewis number, thermophoresis parameter, magnetic parameter, and melting parameter on fluid velocity, heat, concentration, skin friction, Sherwood number, and Nusselt number are presented graphically and discussed numerically. Heat transfer rate can be increased by increasing slip, melting, or radiation parameter. Mass transfer increases for greater values of melting parameter or slip parameter while radiation parameter shows the opposite impact on mass transfer.