F.Mabood,T.A.Yusuf,A.M.Rashad,W.A.Khan and Hossam A.Nabwey

1Department of Information Technology,Fanshawe College London,ON,N5Y 5R6,Canada

2Department of Mathematics,University of Ilorin,Ilorin,Kwara,240003,Nigeria

3Department of Mathematics,Faculty of Science,Aswan University,Aswan,81528,Egypt

4Department of Mechanical Engineering,College of Engineering,Prince Mohammad Bin Fahd University,Al Khobar,31952,Saudi Arabia

5Department of Mathematics,College of Science and Humanities in Al-Kharj,Prince Sattam bin Abdulaziz University,Al-Kharj,11942,Saudi Arabia

6Department of Basic Engineering Science,Faculty of Engineering,Menoufia University,Shebin El-Kom,32511,Egypt

Abstract:The current investigation aims to explore the combined effects of heat and mass transfer on free convection of Sodium alginate-Fe3O4 based Brinkmann type nanofluid flow over a vertical rotating frame.The Tiwari and Das nanofluid model is employed to examine the effects of dimensionless numbers,including Grashof,Eckert,and Schmidt numbers and governing parameters like solid volume fraction of nanoparticles,Hall current,magnetic field,viscous dissipation,and the chemical reaction on the physical quantities.The dimensionless nonlinear partial differential equations are solved using a finite difference method known as Runge-Kutta Fehlberg(RKF-45)method.The variation of dimensionless velocity,temperature,concentration,skin friction,heat,and mass transfer rate,as well as for entropy generation and Bejan number with governing parameters,are presented graphically and are provided in tabular form.The results reveal that the Nusselt number increases with an increase in the solid volume fraction of nanoparticles.Furthermore,the rate of entropy generation and Bejan number depends upon the magnetic field and the Eckert number.

Keywords:Nanofluid flow;entropy generation;heat and mass transfer;viscous dissipation;chemical reaction

1 Introduction

Most of the conventional liquids such as saltwater,liquid metal,plasma,etc.are conducting fluids that have over the years captured immense attention of renowned researchers to the study of the dynamics of these fluids because of their significant engineering applications like MHD generators,flow meters,metal purification,metallurgy,geothermal energy extractions,and polymer technology.Some studies[1–3]involve fluid flow analysis of the electrically conducting fluids.However,Hall current effect is significant for a strong magnetic field and a low density[4].In the pioneering work of[5],the Hall current effect was taken into consideration to examine the magnetohydrodynamic flow of a viscous ionized gas passing through parallel plates.Further studies of Hall effects have been communicated by[6]who reviewed the peristaltic flow of a Jeffrey non-Newtonian fluid over vertical walls in the presence of porous medium and Hall influence.Muthucumaraswamy et al.[7]studied the unsteady flow of a viscous fluid over an exponential plate accelerating due to density difference with Hall effects and thermal radiation.

Hydromagnetic fluid flow problems are essential in the field of earth science,Meteorology.Interestingly,the Hall current induces both the primary and secondary flows in fluid governed of Coriolis force.Very recently,Krishna et al.[8]have been analyzed the mixed convection laminar flow of hydromagnetic viscous rotating fluid flow over a porous vertical sheet with Hall effects.Hall current influence on the unsteady flow of an oscillating fluid over an exponential slip plate with chemical reaction is investigated by[9].They concluded that the Coriolis force and Hall current tend to augment the fluid velocity in the secondary flow direction whereas,in the primary flow direction,Ion-slip current enhanced.Given these applications,Hall current effects on rotating magnetohydrodynamic have been studied in various flow geometries,for example[10,11].

Above mentioned literature was performed in the fluid flow models of a conventional fluid flow of an electrically conducting fluid.Still,fluids with the inclusion of nanometer-sized particles(nanofluid)behave quite differently from that of the traditional fluid in several vital aspects.A report from the current trend in research has shown that heat transfer is enhanced in the thermal system through the embedded nanoparticle into conventional liquids.The applications exist in a solar receiver,nuclear reactor,microbial fuel cell,thermal storage,biomedical applications,heat exchangers,industrial cooling medium.Authors have established several results about nanofluid flow in various geometries,for example,see Ali et al.[12,13].

In energy management,minimizing entropy production in a thermal system cannot be overemphasized because of its limited percentage of energy available as heat.It is,however,imperative to improve the amount of energy available for work through entropy generation.Some relevant articles that analyzed the flow and heat transfer using the 2nd law of thermodynamics are[14–17].Opanuga et al.[18]have examined the Hall current and ion-slip on a steady flow of micropolar fluid through an infinite vertical channel with entropy generation.

The objective of the current analysis is to examine the rate of entropy optimization on MHD Brinkmantype nanofluid flow over a vertical rotating plate with the influence of radiation and chemical reaction.It is,therefore,pertinent to examine the effect of this feature because entropy production occurs in moving fluid with high temperature.To the best of our knowledge,the present study has not remained investigated.By applying suitable transformations,the governing equations of the model are converted to non-dimensional form and then solved by employing the Runge–Kutta–Fehlberg scheme.Effects of all the pertinent parameters on velocity,temperature,nanoparticle concentration,skin friction coefficient,Nusselt number,Sherwood number,entropy generation,and the Bejan number profiles are shown through graphs and extensively discussed.

2 Problem Formulation

A magnetohydrodynamic convective flow of Sodium alginate-Fe3O4based Brinkmann type nanofluid is examined in a vertical rotating frame.The flow is assumed to be incompressible and time-dependent.

Fig.1 explains the coordinate system of the vertical rotating frame in a nanofluid.The system spins about the normal axis with an angular velocity Ω.Consider the physical quantities depend only on y.Also,a magnetic field of constant strengthB0is introduced in a direction parallel to the y-axis in direction to the fluid flow.Considering the effects of thermal radiation and chemical reaction,the governing equations are:

Figure 1:Schematic diagram

Subject to the initial and boundary conditions

Defining

The dimensionless variables are given by

Thus,the governing equations are:The boundary conditions are given by

where the parameters are defined by

Sc(Schmidt number);(Diffusive constant parameter);γ(Brinkmann parameter);R(Ideal gas constant);Tdiff(Temperature difference);Cdiff(Concentration difference);Gr(Thermal Grashof number);Gc(Solutal Grashof number);Preff(Effective Prandtl number);Pr(Prandtl number);Nr(Radiation parameter);Ec(Eckert number);M(Magnetic parameter);Cm(Chemical reaction parameter);m(Hall current parameter);g(Acceleration due to gravity);(Induced magnetic field);δ(Non dimensional rotation parameter).

3 Entropy Generation

Table 1:Thermophysical properties of base fluid(Sodium Alginate)and nanoparticle

4 Results and Discussion

The system of partial differential Eqs.(8)–(11)with associated initial and boundary conditions Eq.(12)are solved numerically using a finite difference method.In all cases,we have adopted the following default values of parameters γ=m=M=Nr=0.5,Gr=Gc=0.1,Ec=?=0.01,Pr=6,

Cm=Sc=ξ=δ=0.2 Unless individually shown in the appropriate separately.Variations of the dimensionless velocitiesu(y,t)(P/velocity)andw(y,t)(S/velocity)withM(magnetic field)depicted in Fig.2.Figs.2a and 2b display the influence of the primary and secondary velocities with increasing valuesM.Higher values ofMcause both velocities plots to decelerate.The physics of this trend is that as the magnetic field is applied,a resistance force opposing the fluid motion is produced,thereby causing a decrease in the velocity of the liquid.Comparatively,a gradual drop in the secondary velocity is noticed,as shown in Fig.2b.

Figure 2:Effects of magnetic parameter on(a)primary and(b)secondary velocities

The impacts ofM(magnetic parameter),Nr(thermal radiation parameter),?(volume fraction parameter),andEc(Eckert number)on the temperature profileT(y,t)are shown in the Figs.3 and 4.Fig.3a elucidates the impact ofMonT(y,t).Inside the thermal boundary layer,the dimensionless temperature increases with the magnetic field.In this plot,higher estimations ofMconnects presence of Lorentz heating in the flow,the force boost the fluid temperature,and a distinct trend is perceived within 0.4 ≤y≤2.0.The impact of the radiation parameter on the dimensionless temperatureT(y,t)is presented in Fig.3b.However,an enhancement in the temperature profile is observed at all points in the presence of thermal radiation.The reason for this trend is that bigger estimations of Nr produce more heat into the fluid,causing a rise in the temperature.

Figure 3:Variation of temperature with(a)magnetic(b)radiation parameters

Figure 4:Variation of temperature with(a)solid volume fraction(b)Eckert number

The effects of the solid volume fraction of nanoparticles and Eckert number on the dimensionless temperature are shown in the Figs.4a and 4b respectively.As shown in Fig.4a that enhancement in the solid volume fraction of nanoparticles ? leads to an increase in the temperature profile.Moreover,an increase in Ec corresponds to a significant rise in the temperature profile.Fig.4b demonstrates this behavior.Physically,frictional heating produces more heat with an increase in Ec.

Figs.5a and 5b illustrate the impacts of the chemical reaction(Cm)and Schmidt number(Sc)on the dimensionless concentration,respectively.The behavior of the dimensionless concentration for different values of destructive chemical reaction parameters(Cm＞0)is portrayed in Fig.5a.It is noticed that the dimensionless concentration is a decreasing function ofCm.In true sense,the amount of nanomaterials presence in the fluid becomes smaller as the destructive chemical reaction occurs.Meanwhile,Fig.5b displays the concentration profile decreases rapidly with an increase in the Schmidt number.Physically,Scis the ratio of the momentum to the mass diffusivity,so the relative effect of momentum diffusion to species diffusion is signified by Schmidt number.A drop in concentration profile gives the impression that the diffusion of species dominates the momentum diffusivity.

Figure 5:Variation of concentration(a)chemical reaction parameter(b)Schmidt number

Plots of physical quantities such asCf(Skin friction coefficient)Nu(rate of heat transfer)and theSh(rate of mass transfer)as a function of ? are shown in Figs.6a and 6b for various pertinent parameters.The behavior of the magnetic parameterM,Brinkman parameter γ,Hall current parameterm,and the thermal Grashof numberGron the skin friction coefficient is displayed in Figs.6a and 6b.The rise in the magnitude ofCfhas been noticed for higher values ofM.The physics behind this is an increase inMgenerates a drag like force which reduces the friction on the wall surface.As the Brinkmann parameter γ rises and for all values of ?,a high impact of skin factor at the wall is observed(see Tab.2).Similarly,from Fig.6b,the skin factor enhanced with increasing values of parametersmandGr.

Figure 6:Variation of skin friction with ?(a)M and γ(b)Gr and m

Table 2:Skin friction values when t =1,m=ξ=δ=Sc=Cm =0.2,Gr=Gc=Ec=Nr=0.1

The rate of heat transfer as a function of ? is exhibited in Fig.7.for different values ofM,Ec,ξ,and γ.These plots show that due to the temperature gradient,the heat flux is an increasing function ofM.Tab.3 reports that the Nusselt number increases with an increase in the volume fraction of nanoparticles and the radiation parameter.The higher rate of heat transfer from the moving fluid to the wall with larger values of the Brinkman parameter γ.However,as shown in Fig.7,an augmentedm,γ,Ecand slow down the heat transfer rateNu.Physically,enhancement inEccorresponds to upsurge in the thermal field via dissipation,hence,boosting the heat transfer rate.

Figure 7:Variation of Nusselt number with ?(a) M and Ec(b)γ and m

The influence of the Schmidt numberScand chemical reaction parameterCmon the Sherwood numberShis displayed in Fig.8 and Tab.4.It may be noted that with a rise in the concentration gradient,mass transport increases for increasing values of bothScandCm.

Table 3:Nusselt number values when t =1,m=ξ=δ=Sc=Cm =0.2,Gr=Gc=Ec=M =0.1

Figure 8:Variation of Sherwood number with ? for different values of Sc and Cm

The impacts of governing parameters,includingM,γ,Ec,andGron the entropy generation rate,NGare shown in Fig.9.It is noticed that the rate of disorderliness becomes low in the absence ofM.However,the magnetic field produces a Lorentz force,which boosts the rate of entropy generation.Also,higher values ofEcescalate the entropy production.It is observed from the same plot that improving the magnitude of γ marginally suppressed the rate of entropy generation.Moreover,throughout the fluid system,enhancedGrsuppressed the rate of entropy generation.This is an indication that there is more fluid-particle disorder via augmentation inM,Ec,γ,andGr.

Table 4:Sherwood number values when t =1,m=ξ=δ=0.2,Gr=Gc=γ=M =Ec=Nr=0.1

Figure 9:Variation of entropy generation rate with ?(a)M and γ(b)Gr and Ec

Fig.10 elucidates the Bejan numberBeagainst ? for different values ofM,γEc,andGr.The contribution of fluid friction is more dominant via enhancement ofM,andEc.Also,since an increase in γ andGrcorrespondingly reduce the Bejan number,the fluid friction irreversibility dominates throughout the mainstream.Generally,we observed that the governing parameters enhance the rate of entropy production and correspondingly decrease the Bejan number.

Figure 10:Variation of Bejan number with ?(a)M and γ(b)Gr and Ec

5 Conclusions

In this paper,the effects of viscous dissipation and chemical reaction on MHD flow with combined heat and mass transfer of incompressible sodium-alginate based Fe3O4in a rotating frame have been analyzed.The following are the main results of the present study:

●The dimensionless velocityu(y,t)increases with the augmentation ofGrand ? while it peters out via incrementedM.

●The dimensionless temperature θ(y,t)increases with the augmentation ofM,?,andEc.

●An increase in the chemical reaction parameter and Schmidt number has shown a declining trend for the dimensionless concentrationC(y,t).

●Viscous drag decreases due toM,?,and γ while shows the opposite fashion viaGrandm.

●The rate of heat transfer is decreasing due to the rise inEcandm.

●The mass transfer rate increases with an increase inScwhile it decreases with ?.

●Rate of Entropy generation and Bejan number shows the opposite trend forMandEc.

Funding Statement:The authors received no specific funding for this study.

Conflicts of Interest:The authors declare that they have no conflicts of interest to report regarding the present study.