Exact solutions for magnetohydrodynamic nanofluids flow and heat transfer over a permeable axisymmetric radially stretching/shrinking sheet

We report on the magnetohydrodynamic impact on the axisymmetric flow of Al_(2)O_(3)/Cu nanoparticles suspended in H_(2)O past a stretched/shrinked sheet.With the use of partial differential equations and the corresponding thermophysical characteristics of nanoparticles,the physical flow process is illustrated.The resultant nonlinear system of partial differential equations is converted into a system of ordinary differential equations using the suitable similarity transformations.The transformed differential equations are solved analytically.Impacts of the magnetic parameter,solid volume fraction and stretching/shrinking parameter on momentum and temperature distribution have been analyzed and interpreted graphically.The skin friction and Nusselt number were also evaluated.In addition,existence of dual solution was deduced for the shrinking sheet and unique solution for the stretching one.Further,Al_(2)O_(3)/H_(2)O nanofluid flow has better thermal conductivity on comparing with Cu/H_(2)O nanofluid.Furthermore,it was found that the first solutions of the stream are stable and physically realizable,whereas those of the second ones are unstable.

Three-dimensional mixed convection stagnation-point fow past a vertical surface with second-order slip velocity

This study is concerned with the three-dimensional(3D)stagnation-point for the mixed convection flow past a vertical surface considering the first-order and secondorder velocity slips.To the authors’knowledge,this is the first study presenting this very interesting analysis.Nonlinear partial differential equations for the flow problem are transformed into nonlinear ordinary differential equations(ODEs)by using appropriate similarity transformation.These ODEs with the corresponding boundary conditions are numerically solved by utilizing the bvp4c solver in MATLAB programming language.The effects of the governing parameters on the non-dimensional velocity profiles,temperature profiles,skin friction coefficients,and the local Nusselt number are presented in detail through a series of graphs and tables.Interestingly,it is reported that the reduced skin friction coefficient decreases for the assisting flow situation and increases for the opposing flow situation.The numerical computations of the present work are compared with those from other research available in specific situations,and an excellent consensus is observed.Another exciting feature for this work is the existence of dual solutions.An important remark is that the dual solutions exist for both assisting and opposing flows.A linear stability analysis is performed showing that one solution is stable and the other solution is not stable.We notice that the mixed convection and velocity slip parameters have strong effects on the flow characteristics.These effects are depicted in graphs and discussed in this paper.The obtained results show that the first-order and second-order slip parameters have a considerable effect on the flow,as well as on the heat transfer characteristics.

MHD stagnation-point flow and heat transfer towards stretching sheet with induced magnetic field

The problem of the steady magnetohydrodynamic （MHD） stagnation-point flow of an incompressible viscous fluid over a stretching sheet is studied. The effect of an induced magnetic field is taken into account. The nonlinear partial differential equations are transformed into ordinary differential equations via the similarity transformation. The transformed boundary layer equations are solved numerically using the shooting method. Numerical results are obtained for various magnetic parameters and Prandtl numbers. The effects of the induced magnetic field on the skin friction coefficient, the local Nusselt number, the velocity, and the temperature profiles are presented graphically and discussed in detail.

MHD flow and heat transfer of a hybrid nanofluid past a permeable stretching/shrinking wedge

The steady flow and heat transfer of a hybrid nanofluid past a permeable stretching/shrinking wedge with magnetic field and radiation effects are studied. The governing equations of the hybrid nanofluid are converted to the similarity equations by techniques of the similarity transformation. The bvp4c function that is available in MATLAB software is utilized for solving the similarity equations numerically. The numerical results are obtained for selected different values of parameters. The results discover that two solutions exist, up to a certain value of the stretching/shrinking and suction strengths. The critical value in which the solution is in existence decreases as nanoparticle volume fractions for copper and wedge angle parameter increase. It is also found that the hybrid nanofluid enhances the heat transfer rate compared with the regular nanofluid. The reduction of the heat transfer rate is observed with the increase in radiation parameter. The temporal stability analysis is performed to analyze the stability of the dual solutions, and it is revealed that only one of them is stable and physically reliable.

Comparative numerical study of single and two-phase models of nanofluid heat transfer in wavy channel

The main purpose of this study is to survey numerically comparison of two- phase and single phase of heat transfer and flow field of copper-water nanofluid in a wavy channel. The computational fluid dynamics （CFD） prediction is used for heat transfer and flow prediction of the single phase and three different two-phase models （mixture, volume of fluid （VOF）, and Eulerian）. The heat transfer coefficient, temperature, and velocity distributions are investigated. The results show that the differences between the temperature fie].d in the single phase and two-phase models are greater than those in the hydrodynamic tleld. Also, it is found that the heat transfer coefficient predicted by the single phase model is enhanced by increasing the volume fraction of nanoparticles for all Reynolds numbers; while for the two-phase models, when the Reynolds number is low, increasing the volume fraction of nanoparticles will enhance the heat transfer coefficient in the front and the middle of the wavy channel, but gradually decrease along the wavy channel.

Mixed convection in gravity-driven nano-liquid film containing both nanoparticles and gyrotactic microorganisms

Analysis of a gravity-induced film flow of a fluid containing both nanoparticles and gyrotactic microorganisms along a convectively heated vertical surface is presented.The Buongiorno model is applied. Two kinds of boundary conditions, the passive and the active boundary conditions, are considered to investigate this film flow phenomenon.Through a set of similarity variables, the ordinary differential equations that describe the conservation of the momentum, the thermal energy, the nanoparticles, and the microorganisms are derived and then solved numerically by an efficient finite difference technique.The effects of various physical parameters on the profiles of momentum, thermal energy,nanoparticles, microorganisms, local skin friction, local Nusselt number, local wall mass flux, and local wall motile microorganisms flux are investigated. It is expected that the passively controlled nanofluid model can be much more easily achieved and applied in real circumstances than the actively controlled model.

Natural convection of an alumina-water nanofluid inside an inclined wavy-walled cavity with a non-uniform heating using Tiwari and Das' nanofluid model

The present study is devoted to numerical analysis of natural convective heat transfer and fluid flow of alumina-water nanofluid in an inclined wavy-walled cavity under the effect of non-uniform heating. A single-phase nanofluid model with experimental correlations for the nanofluid viscosity and thermal conductivity has been included in the mathematical model. The considered governing equations formulated in dimensionless stream function, vorticity, and temperature have been solved by the finite difference method. The cavity inclination angle and irregular walls(wavy and undulation numbers)are very good control parameters for the heat transfer and fluid flow. Nowadays, optimal parameters are necessary for the heat transfer enhancement in different practical applications. The effects of the involved parameters on the streamlines and isotherms as well as on the average Nusselt number and nanofluid flow rate have been analyzed. It has been found that the heat transfer rate and fluid flow rate are non-monotonic functions of the cavity inclination angle and undulation number.

Mixed convection boundary layer flow near stagnation-point on vertical surface with slip

This paper considers the steady mixed convection boundary layer flow of a viscous and incompressible fluid near the stagnation-point on a vertical surface with the slip effect at the boundary. The temperature of the sheet and the velocity of the external flow are assumed to vary linearly with the distance from the stagnation-point. The governing partial differential equations are first transformed into a system of ordinary differential equations, which are then solved numerically by a shooting method. The features of the flow and heat transfer characteristics for different values of the governing parameters are analyzed and discussed. Both assisting and opposing flows are considered. The results indicate that for the opposing flow, the dual solutions exist in a certain range of the buoyancy parameter, while for the assisting flow, the solution is unique. In general, the velocity slip increases the heat transfer rate at the surface, while the thermal slip decreases it.

Flow and heat transfer of nanofluid past stretching/shrinking sheet with partial slip boundary conditions

The boundary layer flow of a nanofluid past a stretching/shrinking sheet with hydrodynamic and thermal slip boundary conditions is studied. Numerical solutions to the governing equations are obtained using a shooting method. The results are found for the skin friction coefficient, the local Nusselt number, and the local Sherwood number as well as the velocity, temperature, and concentration profiles for some values of the velocity slip parameter, thermal slip parameter, stretching/shrinking parameter, thermophoresis parameter, and Brownian motion parameter. The results show that the local Nusselt number, which represents the heat transfer rate, is lower for higher values of thermal slip parameter, thermophoresis parameter, and Brownian motion parameter.

Impact of anisotropic slip on the stagnation-point flow past a stretching/shrinking surface of the Al2O3-Cu/H2O hybrid nanofluid

The characteristics of heat transfer in the three-dimensional stagnationpoint flow past a stretching/shrinking surface of the Al2O3-Cu/H2O hybrid nanofluid with anisotropic slip are investigated.The partial differential equations are converted into a system of ordinary differential equations by valid similarity transformations.The simplified mathematical model is solved computationally by the bvp4c approach in the MATLAB operating system.This solving method is capable of generating more than one solutions when suitable initial guesses are proposed.The results are proven to have dual solutions,which consequently lead to the application of a stability analysis that verifies the achievability of the first solution.The findings reveal infinite values of the dual solutions at several measured parameters causing the non-appearance of the turning points and the critical values.The skin friction increases with the addition of nanoparticles,while the escalation of the anisotropic slip effect causes a reduction in the heat transfer rate.

Numerical simulation of MHD natural convection flow in a wavy cavity filled by a hybrid Cu-Al2O3-water nanofluid with discrete heating

We consider the combined effect of the magnetic field and heat transfer inside a square cavity containing a hybrid nanofluid(Cu-Al2O3-water).The upper and bottom walls of the cavity have a wavy shape.The temperature of the vertical walls is lower,the third part in the middle of the bottom wall is kept at a constant higher temperature,and the remaining parts of the bottom wall and the upper wall are thermally insulated.The magnetic field is applied under the angleγ,an opposite clockwise direction.For the numerical simulation,the finite element technique is employed.The ranges of the characteristics are as follows:the Rayleigh number(10^3≤Ra≤10^5),the Hartmann number(0≤Ha≤100),the nanoparticle hybrid concentration(ϕAl2O3,ϕCu=0,0.025,0.05),the magnetic field orientation(0≤γ≤2π),and the Prandtl number Pr,the amplitude of wavy cavity A,and the number of waviness n are fixed at Pr=7,A=0.1,and n=3,respectively.The comparison with a reported finding in the open literature is done,and the data are observed to be in very good agreement.The effects of the governing parameters on the energy transport and fluid flow parameters are studied.The results prove that the increment of the magnetic influence determines the decrease of the energy transference because the conduction motion dominates the fluid movement.When the Rayleigh number is raised,the Nusselt number is increased,too.For moderate Rayleigh numbers,the maximum ratio of the heat transfer takes place for the hybrid nanofluid and then the Cu-nanofluid,followed by the Al2O3-nanofluid.The nature of motion and energy transport parameters has been scrutinized.

Mixed convection stagnation-point flow on vertical stretching sheet with external magnetic field

The problem of steady laminar magnetohydrodynamic （MHD） mixed con- vection stagnation-point flow of an incompressible viscous fluid over a vertical stretch- ing sheet is studied. The effect of an externally magnetic field is taken into account. The transformed boundary layer equations are solved numerically by using an implicit finite-difference scheme. Numerical results are obtained for various values of the mixed convection parameter, Hartmann number, and Prandtl number. The effects of an exter- nally magnetic field on the skin friction coefficient, local Nusselt number, velocity, and temperature profiles for both A 〉 1 and A ~ 1, where A is the velocity ratio parameter, are presented graphically and discussed in detail. Both assisting and opposing flows are considered, and it is found that dual solutions exist for the opposing flow.

Mixed convection boundary layer flow past vertical flat plate in nanofluid: case of prescribed wall heat flux

An analysis is carried out to investigate the steady mixed convection bound- ary layer flow of a water based nanofluid past a vertical semi-infinite flat plate. Using an appropriate similarity transformation, the governing partial differential equations are transformed into the coupled, nonlinear ordinary （similar） differential equations, which are then solved numerically for the Prandtl number Pr = 6.2. The skin friction coeffi- cient, the local Nusselt number, and the velocity and temperature profiles are presented graphically and discussed. Effects of the solid volume fraction φ and the mixed convection parameter λ on the fluid flow and heat transfer characteristics are thoroughly examined. Different from an assisting flow, it is found that the solutions for an opposing flow are non-unique. In order to establish which solution branch is stable and physically realizable in practice, a stability analysis is performed.

Magnetohydrodynamic flow past a shrinking vertical sheet in a dusty hybrid nanofluid with thermal radiation

The magnetohydrodynamic(MHD)mixed convection flow past a shrinking vertical sheet with thermal radiation is considered.Besides,the effects of Cu-Al_(2)O_(3) nanoparticles and dust particles are considered.The similarity variables reduce the governing equations to the similarity equations,which are then solved numerically.The outcome shows that,for the shrinking case,the solutions are not unique.The rate of heat transfer and the friction factor enlarge with increasing the values of the copper nanoparticle volume fraction as well as the magnetic parameter.Meanwhile,the assisting flow and the rise of the thermal radiation reduce these quantities.Two solutions are found,and the boundary layer separation is dependent on the mixed convection parameter.

Unsteady flow of a Maxwell hybrid nanofluid past a stretching/shrinking surface with thermal radiation effect

The non-Newtonian fluid model reflects the behavior of the fluid flow in global manufacturing progress and increases product performance.Therefore,the present work strives to analyze the unsteady Maxwell hybrid nanofluid toward a stretching/shrinking surface with thermal radiation effect and heat transfer.The partial derivatives of the multivariable differential equations are transformed into ordinary differential equations in a specified form by applying appropriate transformations.The resulting mathematical model is clarified by utilizing the bvp4c technique.Different control parameters are investigated to see how they affect the outcomes.The results reveal that the skin friction coefficient increases by adding nanoparticles and suction parameters.The inclusion of the Maxwell parameter and thermal radiation effect both show a declining tendency in the local Nusselt number,and as a result,the thermal flow efficacy is reduced.The reduction of the unsteadiness characteristic,on the other hand,considerably promotes the improvement of heat transfer performance.The existence of more than one solution is proven,and this invariably leads to an analysis of solution stability,which validates the first solution viability.

Unsteady three-dimensional boundary layer flow due to a permeable shrinking sheet

The unsteady viscous flow over a continuously permeable shrinking surface is studied. Similarity equations are obtained through the application of similar transformation techniques. Numerical techniques are used to solve the similarity equations for different values of the unsteadiness parameter, the mass suction parameter, the shrinking parameter and the Prandtl number on the velocity and temperature profiles as well as the skin friction coefficient and the Nusselt number. It is found that, different from an unsteady stretching sheet, dual solutions exist in a certain range of mass suction and unsteadiness parameters.

Slip effects on unsteady mixed convection of hybrid nanofluid flow near the stagnation point

The unsteady mixed convection of the Al_(2)O_(3)-Cu/H_(2)O hybrid nanofluid flow near the stagnation point past a vertical plate is analyzed.The bvp4c technique is used to solve the resulting ordinary differential equations.The combined effects of the velocity and thermal slip are addressed.The effects of different relevant physical parameters are studied numerically.The results show that the heat transfer rate is reduced when the volume fraction of the nanoparticles increases,while the unsteadiness parameter has an opposite effect in the opposing flow.The presence of the slip parameter is proven to increase the skin friction coefficient while reduce the local Nusselt number in the buoyancy opposing flow.A contradictory result is observed in the buoyancy assisting flow.Meanwhile,the heat transfer rate is reduced in the buoyancy of the assisting and opposing flows when the thermal slip effect is considered.

Fluid flow driven along microchannel by its upper stretching wall with electrokinetic effects

We develop a mathematical model to describe the flow in a microchannel driven by the upper stretching wall of the channel in the presence of electrokinetic effects. In this model, we avoid imposing any unphysical boundary condition, for instance, the zero electrostatic potential in the middle of the channel. Using the similarity transformation, we employ the homotopy analysis method （HAM） to get the analytical solution of the model. In our approach, the unknown pressure constant and the integral constant related to the electric potential are solved spontaneously by using the proper boundary conditions on the channel walls, which makes our model consistent with the commonly accepted models in the field of fluid mechanics. It is expected that our model can offer a general and proper way to study the flow phenomena in microchannels.

Stretching surface in rotating viscoelastic fluid

The boundary layer flow over a stretching surface in a rotating viscoelastic fluid is considered. By applying a similarity transformation, the governing partial differ- ential equations are converted into a system of nonlinear ordinary differential equations before being solved numerically by the Keller-box method. The effects of the viscoelastic and rotation parameters on the skin friction coefficients and the velocity profiles are thor- oughly examined. The analysis reveals that the skin friction coefficients and the velocity in the x-direction increase as the viscoelastic parameter and the rotation parameter in- crease. Moreover, the velocity in the y-direction decreases as the viscoelastic parameter and the rotation parameter increase.

Comment on the paper “Hydromagnetic thin film flow of Casson fluid in non-Darcy porous medium with Joule dissipation and Navier's partial slip”

The present comment concerns some doubtful results included in the abovepaper.