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.     

Numerical simulation of mixed convective 3D flow of a chemically reactive nanofluid subject to convective Nield's conditions with a nonuniform heat source/sink

The present investigation aims to explore the influence of a mixed convection and nonuniform heat source/sink on unsteady flow of a chemically reactive nanofluid driven by a bidirectionally expandable surface. Convective heat transport phenomenon is used to maintain the temperature of the surface. Moreover, zero mass flux is also accounted at the surface such that the fraction of nanomaterial maintains itself on strong retardation. The governing nonlinear set of partial differential equations is transformed into a set of ordinary differential equations via a suitable combination of variables. The Keller‐Box scheme has been incorporated to make a numerical inspection of the transformed problem. The spectacular impacts of the pertinent constraints on thermal and concentration distributions are elucidated through various plots. Graphical outcomes indicate that the thermal state of nanomaterial and nanoparticles concentration are escalated for elevated amounts of Biot number, porosity parameter and nonuniform heat source/sink constraints. Furthermore, it is also seen that escalating amounts of unsteady parameter, temperature controlling indices, Prandtl number, and expansion ratio parameter reduce the thermal and concentration distributions. Numerical results for the rate of heat transference have been reported in tabular form. The grid independence approach is used to verify the convergence of the numerical solution and the CPU run time is also obtained to check the efficiency of the numerical scheme adopted for finding the solution.     

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.     

Maxwell nanofluid heat and mass transfer analysis over a stretching sheet

The Buongiorno model Maxwell nanofluid flow, heat and mass transfer characteristics over a stretching sheet with a magnetic field, thermal radiation, and chemical reaction is numerically investigated in this analysis. This model incorporates the effects of Brownian motion and thermophoresis. The governing partial differential equations are transformed into a coupled nonlinear ordinary differential equation by using the similarity transformation technique. The resultant nonlinear differential equations are solved by using the Finite element method. The sketches of velocity, temperature and concentration with diverse values of magnetic field parameter (0.1 ≤ M ≤ 1.5), Deborah number (0.0 ≤ β ≤ 0.19), radiation parameter (0.1 ≤ R ≤ 0.7), Prandtl number (0.5 ≤ Pr ≤ 0.8), Brownian motion parameter (0.1 ≤ Nb ≤ 0.7), thermophoretic parameter (0.2 ≤ Nt ≤ 0.8), Chemical reaction parameter (1.0 ≤ Cr ≤ 2.5) and Lewis number (1.5 ≤ Le ≤ 3.0) have investigated and are depicted through plots. Moreover, the values of the Skin-friction coefficient, Nusselt number, and Sherwood numbers are also computed and are shown in tables. The sequels of this analysis reviewed that the values of Skin-friction coefficient and Sherwood number intensified with hiked values of Deborah number (β), whereas, the values of Nusselt number decelerate as values of (β) improves.     

Application of heat transfer on MHD shear thickening fluid flow past a stretched surface with variable heat source/sink

The purpose of this study is to explore the viscous dissipation stimulus on the steady convective magnetohydrodynamic shear thickening liquid stream across a vertically stretched sheet. The impact of thermic heat, first-order velocity slip, and variable heat generation/absorption are considered and also ignored the effect of magnetic Reynold's number. We converted flow controlling equations into the set of dimensionless nonlinear ordinary differential equations by employing similarity variables to solve these coupled equations by R–K and shooting technique. The effect of different dimensionless variables on velocity, heat, friction factor, and local Nusselt numbers are presented through graphs and tables. Depreciation in velocity and growth in temperature distribution is detected when the Casson fluid parameter is increased. Temperature is the increasing function of the Eckert number.     

Squeezing flow of Casson hybrid nanofluid between parallel plates with a heat source or sink and thermophoretic particle deposition

The investigations on the flow of non-Newtonian fluids are becoming one of the major topics in the research field. These liquids have substantial applications in industrial and engineering fields such as drilling rigs, food processing, paint and adhesives, nuclear reactors and cooling systems. On the other hand, hybrid nanofluids play a major role in the heat transfer process. Keeping this in mind, the motion of Casson hybrid nanofluid squeezing flow between two parallel plates with the effect of heat source and thermophoretic particle deposition is examined here. The partial differential equations that govern fluid flow are converted into ordinary differential equations using appropriate similarity variables and those equations are numerically solved using the Runge–Kutta–Fehlberg fourth–fifth-order method by implementing the shooting scheme. The graphs depict the effects of a number of key parameters on fluid profiles in the absence and presence of the Casson parameter. These graphs show that fluid velocity enhances with the augmentation of the local porosity parameter. Thermal dispersal upsurges for enhancement of heat source/sink parameter and the concentration profile escalates for an upsurge of the thermophoretic parameter. Skin friction enhances with enhancement in the local porosity parameter.     

Multiple slip impacts on unsteady MHD micropolar fluid past a stretching sheet with non-Darcy porous medium and uneven heat source/sink

The major scope of this research is to scrutinize the effects of multiple slips on unstable magnetohydrodynamic micropolar fluid past a stretched sheet with a non-Darcy porous medium. In the momentum equation, the non-Darcy porous medium effect is also taken into consideration. The effects of uneven heat source/sink and thermal radiation in the energy equation are also analyzed. By implementing the similarity transmission, the mathematical modeling of the set of managing partial differential equations is reframed into nonlinear ordinary differential equations. These equations are numerically solved by applying Matlab built-in solver bvp5c. The implications of foremost parameters such as micropolar parameter, magnetic parameter, permeability parameter, Prandtl, Eckert, and Schmidt numbers, Chemical reaction, slip parameters on velocity, microrotation, temperature as well as concentration profiles are displayed pictorially and explained. It is worthwhile to mention that the improving values of micropolar parameter $K$ escalate the velocity as well as microrotation profiles. However, the upsurge in non-Darcy porous medium $F_s$ will cause a declining nature in the velocity profile. Also, an enhancement in the unsteadiness parameter $A$ brings about a lessening in all the profiles. Increment in all the three usual slip parameters will bring a declining nature in the respective profiles. An increase in Schmidt number will give a deduction nature in velocity as well as concentration profile. Moreover, the physical quantities are defined and Nusselt numbers are formulated in the table, and it enlarges while boosting up $Pr$ and $R$, whilst a reverse nature is noticed for others. This present study compared with the earlier studies in special cases holds a better agreement.     

Study of chemically reactive and thermally radiative Casson nanofluid flow past a stretching sheet with a heat source

Nanoparticle (NP) delivery is an exciting and rapidly developing field that adequately takes care of thermal radiation in blood flow and is likely to have bearing on the therapeutic procedure of hyperthermia, blood flow, and heat transfer in capillaries. The NP parameters such as size, shape, and surface characteristics can be regulated to improve nano-drug delivery efficiency in biological systems. The NPs outperform traditional drug delivery processes in drug carrying capacity and controlled release. The current article investigates the boundary layer flow and heat transfer of thermally radiative Casson nanofluid (NF) over a stretching sheet with chemical reaction and internal heat source. In our study, Cu and Al₂O₃ are taken as NPs in a suitable base fluid. The problem is analyzed by using similarity transformations and is solved with MATLAB's built-in solver bvp4c. The effects of pertinent parameters characterizing the flow model are presented through graphs and tables. The important findings of the investigation are noted as: the use of metallic oxide is more beneficial to attain higher temperature within a few layers close to the bounding surface; the appearance of convexity and concavity in the concentration profile attributed to flow instability, and the constructive and destructive heterogeneous reactions at the bounding surface have distinct roles to modify the NF flow in the boundary layer.     

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.      

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.     

Impact of relative motion of a magnetic field on unsteady magnetohydrodynamic natural convection flow with a constant heat source/sink

This study analyzes time‐dependent magnetohydrodynamics natural convective flow of a viscous incompressible fluid in an annulus with ramped motion of the boundaries. The governing momentum and energy equations are solved analytically, in terms of the modified Bessel function of the first and second kinds. The influence of governing parameters such as the Hartman number, radius ratio, Grashof number, heat absorption parameter, and Prandtl number are discussed with the help of line graphs. It is found that the Hartmann number has a retarding effect on fluid velocity when K = 0.0 and K = 0.5, while the reverse effect is noticed when K = 1.0. The Hartman number also decreases the mass flow rate for all cases of $K$ while it enhances the skin friction at the inner surface of the outer cylinder. It increases the skin friction at the outer surface of the inner cylinder when K = 1.0 and K = 0.0, but decreases the skin friction at the outer surface of the inner cylinder when K = 0.5.     

Irreversibility analysis of micropolar nanofluid flow in a vertical channel with the impact of inclined magnetic field and heat source or sink

This article examines the inclined magnetic field effect on the flow of micropolar nanofluids in a vertical channel with convective boundary conditions and heat source or sink. Thermodynamics second law is employed to analyze the aspects of entropy generation. The governing differential equations are modified into dimensionless form by using suitable nondimensional variables. These transformed equations are solved by implementing the differential transform technique. The results are analyzed graphically. Skin friction and Nusselt number values are evaluated at the boundary walls of the channel. The major findings of the study are material parameter enhances the microrotation but suppresses both velocity and temperature. Magnetic parameter and angle of the implication of magnetic field decrease the velocity and microrotation. Material parameter and angle of imposed magnetic field minimize the entropy generation.     

Interpretation of infinite shear rate viscosity and a nonuniform heat sink/source on a 3D radiative cross nanofluid with buoyancy assisting/opposing flow

With respect to bionomical concerns and energy security, the performance of refrigeration systems should be enriched, which can be done by improving the characteristics of working liquids. Nanoliquids have attracted interest in the fields of engineering and industry due to their prominent thermophysical characteristics. Researchers have used nanoliquids as working liquids and noticed significant fluctuations in thermal execution. In this study,  our prime aim was to study the impact of thermal radiation and varying thermal conductivity on a cross-nanofuid with the addition of a nonuniform heat sink–source, chemical process, and activation energy (AE) together with effects of assisting and opposing buoyancy. Furthermore, the relationship of zero-mass flux together with the mechanism of thermophoresis and Brownian motion is considered. Traditionalistic transformations gave the ordinary differential equations (ODEs), which are further dealt with the approach of the Shooting Scheme to change the boundary value problem (BVP) into an initial value problem (IVP) and a numerical comparison is made with the Matlab solver package bvp4c. Bvp4c is based upon a collocation scheme, which yields numeric outcomes for nonlinear ODEs with IVP. Impacts of the involved parameters on mass transfer profile, heat, and momentum fields are shown through graphs. Mass transfer of the cross nanofluid increases with increasing values of AE parameter. Values of physical quantities like drag forces, rate of transport of heat and mass in the case of assisting/opposing flow are tabulated. The drag force magnitudes are greater for enhancing values of M, a, and n, while on the other hand, the opposing tendency is seen for We₁ and We₂. The magnitude of the rate of heat transport (Nusselt number) falls for greater values of m, σ, δ, and N_t, but in contrast, it accelerates for E, Pr, and n.     

Interaction of radiation and chemical reaction on Walters' B fluid over a medium porosity of a vertical stretching surface

The study of non-Newtonian fluid has gained more significant attention recently than ever due to its various applications in the associated discipline, among which is polymer processing. On the account of its application, this present work analytically investigates a steady-state boundary layer flow on Walters' B fluid over a vertical plate, embedded in a porous medium. The model equations for momentum, heat, and mass transfer are transformed to the associated ordinary differential equation by suitable similarity variables which are executed by means of the homotopy analysis method. The results of various parameters encountered are discussed accordingly. The novel results showcase among others that various values of radiation parameters amplify the radiative flux, which intensifies the polymeric flow and magnifies the rate of heat exchange to the liquid. This increases the thermal energy and accentuates the temperature distribution, while the interaction of Biot number pioneers strong convective heating which overshoots the temperature of which its vast application is rooted in industries as well as technology disciplines for the drying of substances/components.     

Multiple slip effects on MHD nanofluid flow over an inclined,radiative, and chemically reacting stretching sheet by means of FDM

In the present study, the magnetohydrodynamic characteristics of an electrically conducting nanofluid flowing past an inclined stretching sheet have been studied numerically. The governing partial differential equations were transformed to nonlinear ordinary differential equations (ODEs) via suitable similarity variables. The wall suction/injection as well as Navier's first‐order slip has been considered for velocity, temperature, and concentration at the wall. The ODEs were solved in a finite difference framework via a computer program written in Engineering Equation Solver platform. The effect of different parameters on the velocity, temperature, and concentration field has also been presented. Multiple slip flow finds its application in many practical fields such as microelectromechanical systems, nanoelectromechanical systems, flow of micro‐organisms, rarefied gas flow, to name a few.     

Multiple characteristics of three-dimensional radiative Cross fluid with velocity slip and inclined magnetic field over a stretching sheet

This article focuses on the three-dimensional Cross fluid flow of a radiative nanofluid over an expanding sheet with aligned magnetic field, chemical reaction, and heat generation phenomenon. The stretching sheet has convective heat and slip boundary conditions. The similarity variables are properly used for the conversion of a dimensional mathematical model into a nondimensional one. The transformed ordinary differential equations are handled for the numerical outcomes of the suggested fluidic model by incorporating the shooting scheme. Furthermore, the numeric investigations are also compared by bvp4c MATLAB built-in package. In a limited case, both the techniques are checked with already published articles, thereby revealing good agreement. Furthermore, the effects of few parameters like Prandtl number, Weissenberg number, heat generation, stretching rate parameter, magnetic parameter, thermal radiation, Brownian and thermophoresis parameters, and Lewis number on concentration, temperature, and velocity profiles have been presented using figures and numerical tables. The strong intensity of the magnetic field across the fluid and increment in the inclination angle (ϑ) result in a lower velocity profile. Temperature is more prominent for the higher slip mechanism. Furthermore, there in an increase in thermophoretic force, which pushes the nanoparticles, and this mixing of nanoparticles helps to increase the concentration profile. A higher Cross fluid index responds to a larger velocity.     

Heat and mass transfer characteristics of radiative hybrid nanofluid flow over a stretching sheet with chemical reaction

Unsteady magnetohydrodynamic heat and mass transfer analysis of hybrid nanoliquid flow over a stretching surface with chemical reaction, suction, slip effects, and thermal radiation is analyzed in this study. A combination of alumina (Al₂O₃) and titanium oxide (TiO₂) nanoparticles are taken as hybrid nanoparticles and water is considered as the basefluid. Using the similarity transformation method, the governing equations are changed into a system of ordinary differential equations. These equations together with boundary conditions are numerically evaluated by using the Finite element method. The influence of various pertinent parameters on the profiles of fluids concentration, temperature, and velocity is calculated and the outcomes are plotted through graphs. The values of nondimensional rates of heat transfer, mass transfer, and velocity are also analyzed and the results are depicted in tables. Temperature sketches of hybrid nanoliquid intensified in both the steady and unsteady cases as the volume fraction of both nanoparticles rises.     

Flow and heat transfer in a porous medium saturated with a Sisko nanofluid over a nonlinearly stretching sheet with heat generation/absorption

This work is focused on steady flow and heat transfer in a porous medium saturated with a Sisko nanofluid (non‐Newtonian power‐law) over a nonlinearly stretching sheet in the presence of heat generation/absorption. Nonlinear PDEs are transformed into a system of coupled nonlinear ODEs with related boundary conditions using similarity transformation. The reduced equations are then solved numerically using the Runge–Kutta–Fehlberg fourth–fifth order method (RKF45) with Maple 14.0 software. The solutions depend on the power‐law index n and the effect of pertinent parameter such as the Brownian motion parameter, thermophoresis parameter, Lewis number, the permeability, and the heat generation/absorption on the dimensionless velocity, temperature, and nanoparticles volume fraction and also on the skin friction, local Nusselt, and Sherwood numbers are produced for values of the influence parameter. A rapprochement of the numerical results of the actual study with formerly published data detected an excellent agreement.