M.Ijaz Khan ,Seifedine Kadry ,Yuming Chu ,M.Waqas

1 Department of Mathematics,Riphah International University,Faisalabad Campus,Faisalabad 38000,Pakistan

2 Department of Mathematics and Computer Science,Beirut Arab University,Beirut,Lebanon

3 Department of Mathematics,Huzhou University,Huzhou 313000,China

4 Hunan Provincial Key Laboratory of Mathematical Modeling and Analysis in Engineering,Changsha University of Science &Technology,Changsha 410114,China

5 NUTECH School of Applied Sciences and Humanities,National University of Technology,Islamabad 44000,Pakistan

Keywords:Darcy–Forchheimer porous medium Titanium dioxide and graphene oxide nanoparticles Second order velocity slip Convective boundary condition Activation energy Heat generation/absorption

ABSTRACT The prime objective of the present communication is to examine the entropy-optimized second order velocity slip Darcy–Forchheimer hybrid nanofluid flow of viscous material between two rotating disks.Electrical conducting flow is considered and saturated through Darcy–Forchheimer relation.Both the disks are rotating with different angular frequencies and stretches with different rates.Here graphene oxide and titanium dioxide are considered for hybrid nanoparticles and water as a continuous phase liquid.Joule heating,heat generation/absorption and viscous dissipation effects are incorporated in the mathematical modeling of energy expression.Furthermore,binary chemical reaction with activation energy is considered.The total entropy rate is calculated in the presence of heat transfer irreversibility,fluid friction irreversibility,Joule heating irreversibility,porosity irreversibility and chemical reaction irreversibility through thermodynamics second law.The nonlinear governing equations are first converted into ordinary differential equations through implementation of appropriate similarity transformations and then numerical solutions are calculated through Built-in-Shooting method.Characteristics of sundry flow variables on the entropy generation rate,velocity,concentration,Bejan number,temperature are discussed graphically for both graphene oxide and titanium dioxide hybrid nanoparticles.The engineering interest like skin friction coefficient and Nusselt number are computed numerically and presented through tables.It is noticed from the obtained results that entropy generation rate and Bejan number have similar effects versus diffusion parameter.Also entropy generation rate is more against the higher Brinkman number.

1.Introduction

Quality development and the amount of heat energy are two main constraints in thermal reactions.During such thermal processes the energy consumption and useful work are seen through a tool which is called the entropy.The entropy is determined through second law of thermodynamics.According to this law,there must be some energy losses which results in decreasing the efficiency of the system.Such losses of energy are more carefully observed through entropy generation.In thermal processes,the efficiency can be improved by decreasing the entropy.So the main purpose must be the reduction of entropy production.Second law characteristics of heat transport through forced convection are discussed by Bejan [1,2]in terms of four fundamental flow configurations:(i)Cross flow,(ii)pipe flow,(iii)entrance region,and(iv)flat plates.He also discussed in this research article the interplay between irreversibility subject to heat transport via finite temperature gradients.Entropy generation in the problem of fluid flow along a flat surface with suction/injection is examined by Reveillere and Baytas[3].The flow behavior is discussed by an isothermal porous flat plate.The governing flow expressions are first altered into ordinary ones which are tackled numerically by using the Shooting technique along with Fourth-Order-Kutta Method.They found that the entropy generation is dramatically enhanced on the wall subject to heat transfer irreversibilities and high fluid friction.Their obtained results also show that the total entropy generation number reduces via blowing fluid except for small dimensionless temperature parameter(γ)values and large Prandtl number.Muhammad et al.[4,5]studied the effects of magnetohydrodynamics (MHD),second order velocity slip,viscous dissipation,mixed convection,activation energy and Darcy–Forchheimer medium,thermal radiation on heat transport and entropy optimized fluid flow towards a curved stretched surface.Their obtained results demonstrate that both entropy generation rate and Bejan number enhance against higher values of heterogeneous reaction parameter.Besides this,numerous researchers and analyst have focused and discussed the entropy generation rate in fluid flow towards a stretched surface as given in Refs.[6–10].

In the recent era,the flow analysis due to stretching surfaces caught the attention of researchers for their widespread applications in technological and engineering processes.Nuclear reactors,rubber production,glass production,drawing of plastic thin films,textile industries,aerodynamics,plasma study,fiber coating and geothermal sources of energy are the different fields using stretching flows applications.The combinations of nanometer-sized particles in fluids are used to enhance their thermal characteristics like heat transfer rate,thermal conductivity and viscosity in the above processes.They have potential applications in various fields.Crane[11]initiated the work by studying stretching problems and their exact solutions.The boundary layer flow situation with uniform moving surface is examined by Sakiadis[12].Khan et al.[13]scrutinized MHD stagnation point flow of non-Newtonian material(Casson fluid) with homogeneous-heterogeneous reactions.They assumed the homogeneous process by first order kinetics in the ambient fluid and on the wall surface the heterogeneous process is isothermal cubic autocatalytic kinetics.The study of boundary layer flow caused by a vertically heated plate in a porous medium with suction/injection parameters is done by Ali[14].Abel et al.[15]scrutinized the impact of thermal radiation in MHD boundary layer flow generated by a moving sheet.Noreen and Nadeem[16]and Nandeppanavar et al.[17]deliberated the effect of heat transfer on stretching sheet with variable thermal conductivity,partial slip and heat source.Chaim[18]and Cortell[19]computationally examined the heat transfer phenomenon with linear and nonlinear stretching respectively.Vyas and Ranjan [20]computed the behaviors of viscous dissipation and thermal radiation on the MHD boundary layer flow through a non-linear stretching surface.Salem and Fathy[21]described the impacts of thermal conductivity and variable viscosity in an incompressible stagnation point flow towards a stretching surface.The flow of non-Newtonian liquid over a stretching plate surface with viscous dissipation is analyzed by Chen[22].Further,the behavior of various fluids caused by stretching surfaces with various geometries is shown through Refs[23–25].

Recently the nanotechnology has presented new advancements in many technological and industrial processes.Classical methods of heat transfer in various fluid models are insufficient to fulfill the challenges faced in modern times due to the lower thermal properties.The mixtures of metals or metal oxides with the base fluids are used to enhance the thermal capacities of coolants [26].For these novel characteristics of such fluids,they are beneficial in quick heat transfer rate in industries such as power generation,thermal heating and chemical processes.The enhancement of thermal properties of nanofluids is quite advantageous in nuclear reactors [27].The use of such particles in nuclear systems is economical and provides safety as well,which is seen in Refs [28–30].More heat produces inside the electronic systems which reduces the efficiency as well as their reliability.These devices are cooled down by the help of nanofluids.Nanofluids are also used in lubrication processes of machine parts,refrigerators,cooling processes and power engines etc.Buongiorno[31]designed a model consisting of inertia,Brownian diffusion,thermo-phoresis,Magnus impact,diffusion-phoresis,fluid drainage,and gravity.He proposed that Brownian motion and thermophoresis are having a more dominating influence.Kuznetsov and Nield [32]computationally analyzed the above effects in a boundary layer flow produced by a vertical sheet.They noted that cooling process decreases for decreasing both Brownian motion as well as thermophoresis effects.Nield and Kuznetsov[33]studied the thermal stability of different nanofluids through porous medium.Some recent articles on this topic can be found in Refs.[34–40].

The prime interest of the present research work is to investigate the electrical conducting entropy optimized Darcy–Forchheimer hybrid nanofluid flow between two rotating disks.The flow is electrically conducting through applied magnetic field and saturated via Darcy–Forchheimer relation.Here,Darcy–Forchheimer hybrid nanofluid flow between two rotating disks is discussed first time in the presence of titanium oxide and graphene oxide.The hybrid nanoparticles i.e.,(titanium oxide and graphene oxide)are utilized for increasing the thermal conductivity of base fluid (water),because these hybrid nanoparticles gave better results than the traditional nanoparticles.That is why we have used these hybrid nanoparticles.Both the disks stretch with different rates and rotate with different angular frequencies.The activation energy is considered.The nonlinear governing flow expressions is first altered into ordinary ones and solved via Built-in-Shooting method.The impact of sundry flow parameters on the entropy generation rate,concentration,Bejan number,temperature,velocity,Nusselt number and skin friction coefficient are discussed graphically.

2.Coordinate System and Modeling

Fig.1.Schematic flow diagram.

Table 1 Mathematical representation of transport properties of base fluid and nanomaterials

3.Modeling of Entropy

4.Engineering Curiosity

4.1.Skin friction coefficient

4.2.Nusselt number

5.Results and Discussion

Here mathematical modeling is presented for the entropy optimized electrical conducting Darcy–Forchheimer hybrid nanomaterial flow between two stretchable and rotating disks,where the disks rotate with different angular frequencies and stretching rates.Second order velocity slip is imposed at the boundary of both disks.The governing flow expressions are converted into ordinary differential equations through implementation of appropriate similarity transformations.Numerical results are found out for the velocity,skin friction coefficient,temperature,Nusselt number,concentration,Bejan number and entropy generation via built-in-Shooting technique.Impacts of sundry flow variables are discussed graphically subject to both hybrid nanoparticles i.e.,Graphene Oxide (GO) and Titanium dioxide (TiO2).Table 1 is portrayed for the transport characteristics of hybrid nanomaterials i.e.,thermal conductivity,heat capacity,electrical conductivity,viscosity and density.In Table 1,the subscripts f stands for the continuous phase liquid or base fluid,nf for the nanomaterials,φ*stands for the nanoparticles volume fraction and s nano-solid particles.Table 2 is revealed the numerical values of specific heat(cp),electrical conductivity(σ),thermal conductivity(k)and density(ρ)of the hybrid nanoparticles i.e.,titanium oxide (TiO2),graphene oxide (GO) and base fluid water.Table 3 scrutinize the computational analysis ofskin friction coefficients i.e.,where Cf1for the lower disk and Cf2for the upper disk subject to both hybrid nanoparticles(TiO2and GO).It is remarked from Table 3,that magnitude of skin friction coefficients increases for higher estimations of Reynolds number,nanoparticle volume fraction and magnetic parameter.It is noticed that the results for titanium oxide is more prominent than the graphene oxide.Table 4 shows the computational representation of Nusselt number or heat transfer rates i.e.,where Nux1for the lower disk and Nux2for the upper disk subject to both hybrid nanoparticles (TiO2and GO).The magnitude of heat transfer rate at lower disk decreases for higher values of Reynolds number and magnetic parameter while boosts versus larger nanoparticles volume frac-tion.It is also noticed from Table 4,that the heat transfer rate at upper disk upsurges for both hybrid nanoparticles subject to rising magnetic parameter,nanoparticles volume fraction and Reynolds number.The results for titanium oxide at upper disk are more than the graphene oxide.In graphs the solid line represents the impact of titanium oxide and dotted line highlights the behavior of graphene oxide.Table 5 is plotted for the comparative analysis of present results with Stewartson [42]and Imtiaz et al.[43]and found very good agreement.

Table 2 Numerical values of specific heat,electrical conductivity,thermal conductivity and density

Table 3 Analysis of skin friction coefficient versus magnetic parameter,Reynolds number and nanoparticles volume fraction for both titanium oxide and graphene oxide hybrid nanoparticles

Table 4 Analysis of Nusselt number versus magnetic parameter,Reynolds number and nanoparticles volume fraction for both titanium oxide and graphene oxide hybrid nanoparticles

The numerical solutions of ordinary differential Eqs.(9)–(13)with boundary conditions (Eq.(14)) are obtained through Builtin-Shooting method.Our main purpose here is to investigate the contributions of various pertinent flow variables i.e.,first order slip parameter,Darcy–Forchheimer number,second order slip parameter,magnetic parameter,thermal Biot number,activation parameter,Brinkman number and chemical reaction parameter on the temperature,Nusselt number,velocity field,skin friction coefficient,entropy generation rate,concentration and Bejan number.For this purpose,Figs.2–18 are portrayed to investigate the behaviors of sundry variables.Figs.2 and 3 depict the impact of first order slip parameter (L1=0.3,0.5,1.0,1.5) on (f′(ξ)) and (f(ξ))respectively.Here dual behavior of (f′(ξ)) is remarked for higher values of first order slip parameter for both titanium oxide (TiO2)and graphene oxide hybrid nanoparticles (see Fig.2).Near the lower disk the magnitude of (f′(ξ)) decreases and boosts near the upper disk surface.Fig.3 portrays how the first order slip variable affects (f(ξ)).It can be examined from Fig.3 that an increment in first order slip variable results the magnitude of fluid velocity diminishes in the presence of both (TiO2) and (GO).Figs.4 and 5 display the salient characteristics of second order slip parameter(L2=-0.3,-0.5,-0.7,-0.9) on (f′(ξ)) and (f(ξ)) respectively in radial direction.One can examine that magnitude of (f′(ξ))decreases versus larger second order slip parameter for both(TiO2) and (GO) near the lower disk surface and increases closed to the surface of upper disk (see Fig.4).Also the magnitude of (f(ξ)) diminishes against larger second order slip variable (Fig.5).Fig.6 elucidates the influence of velocity field under the variation of Darcy-Forchheimer number (Fr=1.0,1.2,1.4,1.6).Here velocity field shows dual behavior against higher Darcy-Forchheimer number for both titanium oxide and graphene oxide.Near the lower disk surface the magnitude of velocity field declines and then upsurges closed to the upper disk surface.Figs.7 and 8 elucidates the impact of first order slip parameter (L3=0.7,1.0,1.3,1.6) and second order slip parameter (L4=-0.5,-0.8,-1.1,-1.4) on (g(ξ))in tangential direction respectively.Here(g(ξ))is an increasing function of (L3).Also magnitude of (g(ξ)) displays dual impact against larger second order slip parameter in tangential direction.Near the lower disk the magnitude of (g(ξ)) increases and decreases closed to the surface of upper disk for larger estimations of (L4).Fig.9 highlights the influence of rotation parameter(Ω=0.0,0.2,0.4,0.6) on tangential component of velocity.Clearly,(g(ξ)) is an increasing function of rotation parameter.One can justify this phenomenon with the fact that due to the higher angular frequency of lower disk,more rotation occurs in the working fluid and consequently there is a rise in magnitude of tangential component of velocity.

Table 5 Comparative analysis of present results with Stewartson [42]and Imtiaz et al.[43]when M=β=Fr and Re=1

Fig.2.f′(ξ)versus ξ for L1 with 0.3(blue line),0.5(brown line),1.0(purple line)and 1.5 (green line).

Fig.3.f(ξ)versus ξ for L1 with 0.3(blue line),0.5(brown line),1.0(purple line)and 1.5 (green line).

Fig.4.f′(ξ) versus ξ for L2 with -0.3 (blue line),-0.5 (brown line),-0.7 (purple line) and-0.9 (green line).

Fig.5.f(ξ)versus ξ for L2 with-0.3(blue line),-0.5(brown line),-0.7(purple line)and-0.9 (green line).

Fig.6.f′(ξ)versus ξ for Fr with 1.0(blue line),1.2(brown line),1.4(purple line)and 1.6 (green line).

Fig.7.g(ξ)versus ξ for L3 with 0.7(blue line),1.0(brown line),1.3(purple line)and 1.6 (green line).

Fig.8.g(ξ) versus ξ for L4 with -0.5 (blue line),-0.8 (brown line),-1.1 (purple line) and-1.4 (green line).

Fig.9.g(ξ)versus ξ for Ω with 0.0(blue line),0.2(brown line),0.4(purple line)and 0.6 (green line).

Fig.10.θ(ξ) versus ξ for M with 0.1 (blue line),0.5 (brown line),1.0 (purple line)and 1.5 (green line).

Fig.11.θ(ξ) versus ξ for B1 with 0.0 (blue line),0.4 (brown line),0.8 (purple line)and 1.2 (green line).

Fig.12.θ(ξ) versus ξ for Br with 0.0 (blue line),0.5 (brown line),1.0 (purple line)and 1.5 (green line).

Fig.13.φ(ξ) versus ξ for k1 with 0.0 (blue line),0.4 (brown line),0.8 (purple line)and 1.2 (green line).

Fig.14.φ(ξ)versus ξ for E with 0.0(blue line),0.5(brown line),1.0(purple line)and 1.5 (green line).

Fig.15.NG (ξ)versus ξ for L1 with 0.2(blue line),0.25(brown line),0.3(purple line)and 0.35 (green line).

Fig.16.Be versus ξ for L1 with 0.7(blue line),0.75(brown line),0.8(purple line)and 0.85 (green line).

Fig.17.NG (ξ) versus ξ for Br with 0.0 (blue line),0.4 (brown line),0.8 (purple line)and 1.2 (green line).

Fig.18.Be versus ξ for Br with 0.0(blue line),0.4(brown line),0.8(purple line)and 1.2 (green line).

Figs.10–12 elucidate the variation of temperature distribution for various pertinent flow variables like magnetic parameter,thermal Biot number and Brinkman number.Fig.10 is sketched for rising estimations of magnetic parameter (M=0.0,0.5,1.0,1.5).Mathematically,it is the ratio of electromagnetic force and viscous force.It is remarked from Fig.10 that for rising values of magnetic parameter,the magnitude of temperature distribution is significantly increases,because Lorentz force inside the working fluid increases which provides more resistance to the motion of material particles.Fig.11 describes the feature of thermal Biot number(B1=0.0,0.4,0.8,1.2)on temperature distribution.Here Fig.11,give the increasing behavior subject to the rising estimations of thermal Biot number.In addition,the behavior of graphene oxide is more than the titanium oxide.Fig.12 depict the salient aspects of Brinkman number (Br=0.0,0.5,1.0,1.5) on temperature distribution.Same behavior is noticed for Brinkman number (like behavior of thermal Biot number).

Features of chemical reaction,activation energy parameter,first order slip parameter and Brinkman number on the concentration field,entropy generation rate and Bejan number are sketched in Figs.13 to 18.Fig.13 depicts the effects of chemical reaction parameter (k1=0.0,0.4,0.8,1.2) on concentration field.An increment in chemical reaction parameter,the magnitude of concentration field increases.Fig.14 is scrutinized to investigate the influence of activation energy parameter (E=0.0,0.5,1.0,1.5) on the concentration field.Clearly,the concentration field is an increasing function of activation energy variable.Figs.15 and 16 exhibit the attributes of first order slip parameter (L1=0.2,0.25,0.3,0.35) on NGand Be.Here entropy generation rate boosts up when an enhancement occurs in first order slip parameter.In fact for increasing estimations (L1),the deformation is partially transferred to the fluid and thus consequently the entropy generation rate boosts.It is also noticed that impact of graphene oxide is more than the titanium oxide.Fig.16 depicts the variation in Bejan number for varying first order slip parameter (L1=0.7,0.75,0.8,0.85).An enhancement in (L1) correspond to higher Bejan number.It is also noticed that impact of graphene oxide is more than the titanium oxide.Figs.17 and 18 depict the curves of entropy generation and (Be) for rising estimations of Brinkman number.The curves of entropy generation rate increases versus higher Brinkman number(Br=0.0,0.4,0.8,1.2).Fig.18 is plotted to explore the effect of Brinkman number(Br=0.0,0.4,0.8,1.2)on Bejan number.Here,Bejan number is a decreasing function of Brinkman number.Physically,for higher Brinkman number the thermal conductivity declines and consequently the graph of Bejan number decreases.

6.Conclusions

The prime interest of the present communication is to examine the magnetohydrodynamics (MHD) entropy optimized Darcy–Forchheimer hybrid nanoliquid flow between two stretchable and rotating disk surfaces.The flow is saturated through Darcy–Forchheimer porous medium and electrically conducting via applied magnetic field of strength B0.Here,two different type of hybrid nanoparticles are considered i.e.,titanium oxide and graphene oxide and water as a continuous phase liquid or base liquid.Heat generation/absorption,Joule heating and viscous dissipation effects are utilized in the modeling of energy equation.Furthermore,binary chemical reaction with Arrhenius activation energy is accounted.Total entropy generation rate which depends on five different type of irreversibilities i.e.,fluid friction irreversibility,heat transfer irreversibility,porosity irreversibility,chemical reaction irreversibility and Joule heating irreversibility is calculated via second law of thermodynamics.The system of ordinary differential equations is solved numerically with the help Built-in-Shooting method.Influences of pertinent flow parameters on the entropy generation rate,concentration,velocity,Bejan number,temperature are discussed for both titanium oxide and graphene oxide.the engineering quantities like surface drag force and Nusselt number are discussed numerically in the presence of magnetic parameter,Reynolds number and nanoparticles volume fraction.Some fruitful results are summarized as:

· Dual behavior(f′(ξ)) is remarked for higher values of first order slip parameter for both titanium oxide (TiO2) and graphene oxide hybrid nanoparticles.

· The behavior of graphene oxide is more dominant on the velocity of fluid particles than the titanium oxide.

· The magnitude of velocity field declines near the surface of lower disk for higher values of Darcy-Forchheimer number while increases closed to the surface of upper disk.

· (g(ξ)) is an increasing function of rotation parameter.

· Temperature distribution is more against rising values of magnetic parameter,thermal Biot number and Brinkman number.

· Concentration field boosts versus larger activation parameter.

· Entropy generation rate and Bejan number show contrast impact versus Brinkman number.

· The magnitude of surface drag force at both the disk surfaces increases versus Reynolds number and magnetic parameter.

· Magnitude of heat transfer rate at both disk surfaces boosts against nanoparticles volume fraction.

Acknowledgements

The research was supported by the National Natural Science Foundation of China (Grant Nos.11971142,11871202,61673169,11701176,11626101,and 11601485).