# Entropy Analysis in Bidirectional Hybrid Nanofluid Containing Nanospheres with Variable Thermal Activity

Authors: Iftikhar Ahmad; Qazi Zan-Ul-Abadin; Muhammad Faisal; K. Loganathan; Tariq Javed; Sonam Gyeltshen
Journal: Journal of Nanomaterials
 (2022)

---

## Abstract

An investigation of the thermal performance of water-conveying nanospheres (magnetite (Fe3O4) and silver (Ag)) subject to variable thermal controls, namely, variable surface temperature and variable surface normal heat flux, has been made. A bidirectionally elongating surface is used to generate an unsteady flow mechanism with the action of the Lorentz force. Derived equations of basic laws are firstly nondimensionalized and then numerically solved by applying the Keller-Box method. The local Nusselt number for both the thermal cases is calculated and discussed. Percent-wise enhancement in the rate of heat transport has also been included in the analysis. It was concluded through the present exploration that at lower volume fractions of magnetite and silver, the rate of heat transport is observed to be dominant. The rate of heat transference has attained identical values for both the provided thermal conditions at the surface. Moreover, intensities of velocity and thermal profiles diminish with the appreciation of the choice of unsteadiness. The temperature-controlling indices also affect the thermal profile, and it is reduced with the intensification in the considerations of these indices. The values of thermal conductivity, density, and electrical conductivity have been improved with the inclusion of nanospheres (magnetite (Fe3O4) and silver (Ag)), whereas the value of specific heat is reduced with the mixture of these nanospheres. The Nusselt number is increased up to 5% with the involvement of magnetite nanospheres, and it is enhanced up to 4% with the involvement of silver nanospheres.


---

## Body

## 1. Introduction

A staggering and unprecedented advancement in the field of microelectronics, microfluidics, chemical synthesis, optical devices, high-power engines, shipping, microsystems, and electrical and mechanical components has entirely transformed the foundations of human life. These developments additionally demand effective cooling procedures in order to explore the performance of thermal conductivity and reliability of operational devices. The basic idea of controlling the thermal efficiency via cooling through fluids seems to be deficient, and hence, this issue has been resolved by submerging the tiny particles into the host fluid that certainly affects its thermomechanical possessions. This colloidal mixture of tiny particles into the host fluid is famous for the name nanofluid. Famous theoretical models regarding the study of thermal conductivity have been proposed by Choi et al. [1], Maxwell [2], and Bruggeman [3]. The pioneer nanofluid model for the study of tiny particle shapes/sizes is presented by Hamilton and Crosser [4], and it shows that the shapes of the dispersed tiny particles sufficiently affect the thermophysical behavior of heterogeneous substances. Later on, Masoumi et al. [5] presented the model to compute the operative viscosity of nanomaterials. The development of knowledge in the field of nanofluid has also led us to the understanding of tiny particle shapes/sizes as an important factor causing changes in thermal conductivity and viscosity. Some more recent developments in the existing nanofluid models regarding the shapes/sizes of nanoparticles are disclosed by Mooney [6] and Ohshima [7]. In these nanofluid models, it is inferred that the thermal conductivity performance of water-conveying nanomaterial is significantly improved by submerging the nanospheres. Some more extraordinary explorations regarding the control of the nanoparticle shape factor on the thermal executions of various substances are made by the researchers via Refs. [8–12].Recently, an improved class of nanofluids with the name hybrid nanofluid has come into existence to bear the high performance of thermal conductivity associated with the mono-nanofluid. With the utilization of hybrid nanofluids, revolutionary applications in the field of heat transfer processes are found in generator and nuclear cooling systems, automobile radiators, electronic cooling with coolants in the machinery, solar heating, welding, lubrication, drug reduction, biomedicine, cooling and heating in buildings, and defense. The main issue for the regular nanofluid network is that they either have a good thermal conductive performance or have a better rheological setup. Single-handedly, they do not possess all the necessary properties which are mandatory for a certain heat exchanger application. Therefore, with the suitable choice of two or more nanoparticles, hybrid nanofluid can provide us with a well-suited mixture, which owns all physicochemical features of different substances that can scarcely be found in a separate substance. These distinctive properties of hybrid nanofluids have increased the attention of worldwide investigators, and therefore, many research articles have been published in this fascinating field over the last decade. Numerical exploration regarding the 3D flow of hybrid nanofluidCu−Al2O3/H2O towards an expanding obstacle with the significance of the Newtonian heating and Lorentz force is made by Devi and Devi [13] and obtained that the frequency of heat transference for the hybrid nanofluid is much better than that for the ordinary nanofluid even with the effect of the magnetic environment. Hayat et al. [14] conferred the 3D radiative dynamics of hybrid nanofluid Ag−CuO/H2O impinging over a bidirectionally expanding slippery surface in a rotating frame and concluded that the rate of heat transference of the hybrid nanofluid is improved as compared to that of the traditional nanofluid. The detailed analysis of the thermal conductivity and viscosity of hybrid nanofluid Al2O3−CuO with ethylene glycol and propylene glycol binary host fluid with different shapes of nanoparticles (i.e., spherical, cylindrical, platelet, blade, and brick) is disclosed by Kumar and Sahoo [15] and concluded that the nanoparticles having spherical shapes have two percent higher thermal conductivity to that of platelet-shaped nanoparticles. Moreover, the impact of volume fraction on nanoparticles is detected incredibly in the thermal conductivity of nanofluids. Nabil et al. [16] comprehensively discussed the thermophysical features of hybrid nanofluids and hybrid nanolubricants with potential applications in the engineering sector with the predictions of their future utilization. Rashid and Liang [17] presented the numerical solution for the dynamics of MHD nanofluid with the significance of different shapes of nanoparticles over a rotating stretchable obstacle through a porous domain. A theoretical investigation regarding the transitory steady flow of water-conveying hybrid nanofluid containing magnetite and silver nanospheres is conducted by Nabwey and Mahdy [18]. Zainal et al. [19] numerically explored the impact of MHD on the steady bidirectional flow of hybrid nanofluid containing copper and aluminum oxide nanoparticles with water as host fluid. Besides, many researchers [20–25] have contributed to the field of hybrid nanofluid with various thermal engineering aspects.Passive/variable thermal aspects play a significant role in the development of aircraft cooling frames, heat exchanger devices, fuselage condensation processes, air conditioning condensers, refrigeration systems, electric heaters, power converters, motor controllers, and many other industrial and mechanical applications. Oliveira et al. [26] presented a design for a passive heat exchanger system to control the cooling system of aircraft for variable thermal conditions. In this investigation, water is considered a host liquid with the features of natural, forced, and mixed convection. The aspects of passive heating on the dynamics of fluid in the boundary layer zone are initially discussed by Liu and Andersson [27] by considering the variable surface temperature (VST) and variable surface heat flux (VHF) mechanisms. The rate of heat transference is expressively enhanced with the presence of these heating mechanisms. Ahmad et al. [28] numerically disclosed the unsteady 3D flow of copper-water nanofluid with variable thermal aspects. Ahmad et al. [29] also reported the unsteady 3D rotating flow of nanofluid with the aspects of the prescribed thermal environment. The hybrid nanofluid flow over an expanding cylinder with the characteristic of prescribed surface heat flux is examined by Waini et al. [30]. Waini et al. [31] also disclosed the significance of prescribed surface heat flux on the dynamics of hybrid nanofluid past over a thin needle. Khashi’ie et al. [32] discussed the flow of hybrid nanofluid over a shrinking domain with prescribed thermal conditions. The effect of suspended particles on the flow of nanofluid with the effect of VST and VHF mechanisms is also analyzed by Gireesha et al. [33]. Some leading explorations regarding the outcomes of prescribed thermal conditions on the flow of nanofluids as well as hybrid nanofluids are found via Refs. [34–37].Entropy is a physical property of the system that obeys the thermodynamic second law. It measures the level of disorderliness and unproductive energy within the environment. The entropy of a reversible system is constant, whereas the entropy of an irreversible system varies with the factors involved in transport equations. In general, real processes are reversible (i.e., the system responds according to the environmental scenario). Entropy transport in hybrid nanofluid is a new area of research and got considerable importance nowadays. Entropy transport in nanofluids as well as hybrid nanofluids within the thermal environment is inquired by Huminic and Huminic [38]. Khan et al. [39] scrutinized the entropy transport in the rotatory flow of hybrid nanofluid influenced by magnetization. Heat transport and entropy transport in hybrid nanofluids packed in a flattened tube are examined by Huminic and Huminic [40]. Entropy transport with temperature-controlled viscosity in hybrid nanofluid comprising single-walled as well as multiwalled carbon nanotubes is elaborated by Ahmad et al. [41]. Entropy transport in the peristaltic-type flow of hybrid nanofluid is reported by Zahid et al. [42]. Hussien et al. [43] discussed the entropy and thermal transport in hybrid nanofluids by considering microtubes. Sheikholeslami et al. [44] considered the entropy transport in hybrid nanofluid contained in a porous tank with the impact of the Lorentz force. Heat and entropy transport assessment in the bidirectional flow of hybrid nanofluid by the movement of convectively heating devices is disclosed by Upreti et al. [45]. Saleh and Sundar [46] focused on entropy and exergy transport in hybrid nanofluid containing nanodiamond in a circular tube. Recently, Eswaramoorthi et al. [47] investigated the entropy and heat transport in magnetized water-driven nanofluid over a bidirectional moving device with a radiation phenomenon.The above literature survey reveals the statement that no attention has been given in order to investigate the unsteady 3D entropy optimized flow of water-based hybrid nanofluid containing nanospheres, i.e., magnetite and silver, towards an elongating surface with passive thermal controls. The effect of the Lorentz force is also considered in the modeling for physical relevancy. A passive heat controlling system consists of both variable surface temperature and variable surface heat flux mechanisms. Similarity transformations have been used to accomplish the mathematical model for solvable situations. The Keller-Box method [48–52] with Newton’s linearization scheme has been endorsed to find the numerical solution of the modeled problem. The validity of the acquired results is discussed through the grid-independent tactic. The graphical aids have been provided to discuss the influence of various emerging entities on thermal as well as entropy setups. The relations based on the coefficient of skin friction, entropy formation, Bejan number, and local Nusslet number have also been derived and discussed through various scientific patterns.


## 2. Mathematical Formulation

The mathematical formulation of the problem has been completed in view of the undermentioned assumptions:(i)
Incompressible, laminar, and unsteady flow(ii)
Bidirectional stretching(iii)
Thermal equilibrium situation of nanospheres (i.e., magnetite and silver) with water(iv)
No-slip condition at the surface within the boundary layer region(v)
Utilization of variable thermal conditions(vi)
Magnetohydrodynamic three-dimensional flow(vii)
Entropy and heat transport


### 2.1. Flow Description

The Cartesian configuration is used to frame an unsteady mathematical model regarding the dynamics of hybrid nanofluid containing nanospheres (Fe3O4andAg) suspended in water towards a bidirectionally elongating surface. Magnetite nanospheres can be used in the process of water purification, drug delivery, and steel manufacturing, whereas silver is the best electrical as well as thermal conductor among all metals and hence ideal for electrical applications. The concept of variable magnetic field Bt=B0/1−ct having strength B0 is externally applied through the Lorentz force in the absence of an electric field. The nanospheres are in thermal equilibrium with the base liquid. The concept of the no-slip condition at the surface within the boundary stream is also utilized for laminar and incompressible flow. The hybrid nanofluid is restricted in the domain 0<z<∞, whereas the disturbance in the hybrid nanofluid is created through expansion velocity uw=ax/1−cta>0,c>0 along the x-direction and expansion velocity vw=by/1−ctb≥0 along the y-direction of the system configuration (as shown in Figure 1). The thermophysical properties of water H2O, magnetite Fe3O4, and silver Ag are included in the mathematical modeling through Table 1.Figure 1
Graphical abstract of the mathematical model.Table 1
Thermophysical features of water, magnetite nanoparticles, and silver nanoparticles (Ref. [20]).
Thermophysical featuresBase fluidNanoparticlesH2OFe3O4ψ1Agψ2Density:ρ (kgm−3)997.1518010500Thermal conductivity:k (Wm−1K−1)0.6139.7429Specific heat:Cp (Jkg−1K−1)4179670235Electrical conductivity:σ (Ω−1m−1)0.052.5×1046.3×107Prandtl number:Pr6.20——In view of the aforementioned assumptions, component forms of governing equations are composed as (Ref. [12])
(1)∂u∂x+∂v∂y+∂w∂z=0,(2)∂u∂t+u∂u∂x+v∂u∂y+w∂u∂z=μhnfρhnf∂2u∂z2−σhnfρhnfBt2u,(3)∂v∂t+u∂v∂x+v∂v∂y+w∂v∂z=μhnfρhnf∂2v∂z2−σhnfρhnfBt2v,with (Ref. [9])
(4)ρhnf=ψ1ρp1+ψ2ρp2+1−ψ1−ψ2ρf,σhnfσbf=1+3σp2/σbf−1ψ2σp2/σbf+2−σp2/σbf−1ψ2,σbfσf=1+3σp1/σf−1ψ1σp1/σf+2−σp1/σf−1ψ1,μhnfμbf=1+2.5ψ2+6.2ψ22,μbfμf=1+2.5ψ1+6.2ψ12.Supported BCs are (Ref. [34])
(5)z=0:u=uwx,t=ax1−ct,v=vwy,t=by1−ct,w=0,(6)z⟶∞:u⟶0,v⟶0.Similarity variables comprise (Ref. [35])
(7)u=ax1−ctf′η,(8)v=ay1−ctg′η,(9)w=−aϑf1−ct1/2fη+gη,(10)η=aϑf1−ct1/2z.In the application of equation (7), equation (1) is identically balanced and equations (2)–(5) are of the following forms:
(11)ε1f′′′−f′2+f+gf′′−Sf′+η2f′′−ε2M2f′=0,ε1g′′′−g′2+f+gg′′−Sg′+η2g′′−ε2M2g′=0,with
(12)f0+g0=0,(13)f′0=1,(14)g′0=α,(15)f′∞⟶0,(16)g′∞⟶0.In the bidirectional flow, authors are generally concentrating on choosing the range ofα as 0<α<1 because for α>1, the axes x and y are interchanged, and for α=1, the stretching rate becomes alike in both directions; thus, the flow profile turns out to be axisymmetric. Also, considering the limiting situation (i.e., α⟶0), the flow becomes unidirectional. Wang [53] reported that without any loss of generality, the condition f0+g0=0 can be restored by f0=0 and g0=0. Further, in equations (1)–(12), velocity measures corresponding to the x,y,z-directions are u,v,w, respectively. The presence of time is denoted by t, whereas the temperature is symbolized by T, and hnf stands for the hybrid nanofluid, bf symbolizes the base fluid, nf discusses the nanofluid, μ expresses the viscosity, ρ reports the density, σ evokes the electrical conductivity, p1 depicts the magnetite nanoparticles having concentration ψ1, p2 illustrates the silver nanoparticles having concentration ψ2, and f,g,η posts the similarity variables with the involvement of ϑf (i.e., kinematic viscosity). Moreover, S=c/a, M=σf/aρf1/2B0, and α=b/a diagnose the unsteady, magnetic, and expansion ratio parameters, respectively. The coefficients ε1 and ε2 are communicated as (Refs. [9, 12])
(17)ε1=1+2.5ψ2+6.2ψ221+2.5ψ1+6.2ψ12ψ1ρp1/ρf+ψ2ρp2/ρf+1−ψ1−ψ2,ε2=1+3σp2/σbf−1ψ2/σp2/σbf+2−σp2/σbf−1ψ21+3σp1/σf−1ψ1/σp1/σf+2−σp1/σf−1ψ1ψ1ρp1/ρf+ψ2ρp2/ρf+1−ψ1−ψ2.


### 2.2. Heat Transport

In the current model, the heat transport phenomenon can be described by the following equation:(18)∂T∂t+u∂T∂x+v∂T∂y+w∂T∂z=khnfρCphnf∂2T∂z2,with (Ref. [9])
(19)ρCphnf=ψ1ρCpp1+ψ2ρCpp2+1−ψ1−ψ2ρCpf,khnfkbf=kp2+2kbf+2ψ2kp2−kbfkp2+2kbf−ψ2kp2−kbf,kbfkf=kp1+2kf+2ψ1kp1−kfkp1+2kf−ψ1kp1−kf.Here, two types of thermal BCs are used to discuss the variable thermal activity. These are variable surface temperature (VST) and variable surface heat flux (VHF). Mathematically, we have(20)VSTcase:z=0:T=Twx,y,t=T∞+T0xrys1−ct,z⟶∞:T⟶T∞,VHFcase:z=0:−kf∂T∂zw=qwx,y,t=T1xrys1−ct,z⟶∞:T⟶T∞.Similarity transformations for both the thermal activities are (Ref. [12])
(21)VSTcase:θη=Tx,y,z,t−T∞Twx,y,t−T∞,(22)VHFcase:T−T∞=T1kfϑfa1−ct1/2xrysϕη.As an application of equation (21), we have
(23)VSTcase:ε3θ′′+Prf+gθ′−rf′+sg′θ−Sθ+η2θ′=0,VHFcase:ε3ϕ′′+Prf+gϕ′−rf′+sg′ϕ−Sϕ+η2ϕ′=0,with
(24)θ0=1,(25)ϕ′0=−1,(26)θ∞⟶0,(27)ϕ∞⟶0.In equations (18)–(24), the free stream temperature is specified by T∞, whereas T0,T1 are dimensional constants. The variation in the surface temperature is controlled by the indices r,s along the x,y-directions, respectively. Thermal conductivity and specific heat capacitance are conveyed by k and ρCp, respectively. The Prandtl number is expressed by Pr=ρCpfυf/kf, and the coefficient ε3 may be expressed as
(28)ε3=kp2+2kbf+2ψ2kp2−kbf/kp2+2kbf−ψ2kp2−kbfkp1+2kf+2ψ1kp1−kf/kp1+2kf−ψ1kp1−kfψ1ρCpp1/ρCpf+ψ2ρCpp2/ρCpf+1−ψ1−ψ2.


### 2.3. Entropy Transport

The equation that represents the entropy transport (ET) in the present situation may be expressed as
(29)ET=khnfT∞2∂T∂x2+∂T∂y2+∂T∂z2+σhnfBt2T∞u2+v2+μhnfT∞∂u∂z2+∂v∂z2.In equation (29), the first term is due to the heat transport and the second term is due to the magnetic field. It is appropriate to formulate the entropy transport as a factor form of the entropy transport (ET) rate and characteristic entropy (Ec) rate. Characteristic entropy for both the thermal regulations is defined as
(30)Ec1=khnfTw−T∞2T∞2x2VSTcase,Ec2=khnfT−T∞2T∞2x2VHFcase.


#### 2.3.1. Local Entropy Transport for the VST Case

The local entropy transport in dimensionless form for the VST case can be calculated as(31)ET1=ETEc1=Rexθ′2+r2θ2+s2θ2+RexBrα1Mε4f′2+ε5f′′2+ReyBrα1Mε4g′2+ε5g′′2.Here,Br=μfuw2/kfTw−T∞ constitutes the Brinkman number, α1=Tw−T∞/T∞ organizes the temperature difference parameter, and ε4,ε5 are due to the hybrid mixture of nanoparticles, and these may be expressed as
(32)ε4=1+3σp2/σbf−1ψ2/σp2/σbf+2−σp2/σbf−1ψ21+3σp1/σf−1ψ1/σp1/σf+2−σp1/σf−1ψ1kp2+2kbf+2ψ2kp2−kbf/kp2+2kbf−ψ2kp2−kbfkp1+2kf+2ψ1kp1−kf/kp1+2kf−ψ1kp1−kf,(33)ε5=1+2.5ψ1+6.2ψ121+2.5ψ2+6.2ψ22.


#### 2.3.2. Local Entropy Transport for the VHF Case

The local entropy transport in dimensionless form for the VHF case can be calculated as(34)ET2=ETEc2=Rexϕ′2+r2ϕ2+s2ϕ2+RexBrα1Mε4f′2+ε5f′′2+ReyBrα1Mε4g′2+ε5g′′2.


### 2.4. Physical Quantities

The key expressions for thermal engineering importance and heat exchanger devices are the skin friction coefficients (i.e.,Cfx and Cfy) for the measurement of applied stresses at the wall, local Nusslet number (i.e., Nux) for the measurement of the rate of heat transference through the surface, and Bejan number Be for the measurement of irreversibility distribution. These engineering expressions are calculated through the following relations:
(35)Cfx=τwxρfuw2,Cfy=τwyρfvw2,Nux=xqhkfTw−T∞,Be=EGduetoheattransferTotalEG,τwx=μhnf∂u∂zz=0,τwy=μhnf∂v∂zz=0,qh=−khnf∂T∂zz=0.The aforementioned quantities in equations (33) and (34) utilizing Reynold’s numbers Rex=xuw/ϑf and Rey=yvw/ϑf are expressed as
(36)Rex1/2Cfx=ε5f′′0,Rey1/2Cfy=α−3/2ε5g′′0,Rex−1/2Nux=−khnfkfθ′0VSTcase,khnfkf1ϕ0VHFcase,Be1VSTcase=Rexθ′2+r2θ2+s2θ2Rexθ′2+r2θ2+s2θ2+RexBr/α1Mε4f′2+ε5f′′2+ReyBr/α1Mε4g′2+ε5g′′2,Be2VHFcase=Rexϕ′2+r2ϕ2+s2ϕ2Rexϕ′2+r2ϕ2+s2ϕ2+RexBr/α1Mε4f′2+ε5f′′2+ReyBr/α1Mε4g′2+ε5g′′2.


## 2.1. Flow Description

The Cartesian configuration is used to frame an unsteady mathematical model regarding the dynamics of hybrid nanofluid containing nanospheres (Fe3O4andAg) suspended in water towards a bidirectionally elongating surface. Magnetite nanospheres can be used in the process of water purification, drug delivery, and steel manufacturing, whereas silver is the best electrical as well as thermal conductor among all metals and hence ideal for electrical applications. The concept of variable magnetic field Bt=B0/1−ct having strength B0 is externally applied through the Lorentz force in the absence of an electric field. The nanospheres are in thermal equilibrium with the base liquid. The concept of the no-slip condition at the surface within the boundary stream is also utilized for laminar and incompressible flow. The hybrid nanofluid is restricted in the domain 0<z<∞, whereas the disturbance in the hybrid nanofluid is created through expansion velocity uw=ax/1−cta>0,c>0 along the x-direction and expansion velocity vw=by/1−ctb≥0 along the y-direction of the system configuration (as shown in Figure 1). The thermophysical properties of water H2O, magnetite Fe3O4, and silver Ag are included in the mathematical modeling through Table 1.Figure 1
Graphical abstract of the mathematical model.Table 1
Thermophysical features of water, magnetite nanoparticles, and silver nanoparticles (Ref. [20]).
Thermophysical featuresBase fluidNanoparticlesH2OFe3O4ψ1Agψ2Density:ρ (kgm−3)997.1518010500Thermal conductivity:k (Wm−1K−1)0.6139.7429Specific heat:Cp (Jkg−1K−1)4179670235Electrical conductivity:σ (Ω−1m−1)0.052.5×1046.3×107Prandtl number:Pr6.20——In view of the aforementioned assumptions, component forms of governing equations are composed as (Ref. [12])
(1)∂u∂x+∂v∂y+∂w∂z=0,(2)∂u∂t+u∂u∂x+v∂u∂y+w∂u∂z=μhnfρhnf∂2u∂z2−σhnfρhnfBt2u,(3)∂v∂t+u∂v∂x+v∂v∂y+w∂v∂z=μhnfρhnf∂2v∂z2−σhnfρhnfBt2v,with (Ref. [9])
(4)ρhnf=ψ1ρp1+ψ2ρp2+1−ψ1−ψ2ρf,σhnfσbf=1+3σp2/σbf−1ψ2σp2/σbf+2−σp2/σbf−1ψ2,σbfσf=1+3σp1/σf−1ψ1σp1/σf+2−σp1/σf−1ψ1,μhnfμbf=1+2.5ψ2+6.2ψ22,μbfμf=1+2.5ψ1+6.2ψ12.Supported BCs are (Ref. [34])
(5)z=0:u=uwx,t=ax1−ct,v=vwy,t=by1−ct,w=0,(6)z⟶∞:u⟶0,v⟶0.Similarity variables comprise (Ref. [35])
(7)u=ax1−ctf′η,(8)v=ay1−ctg′η,(9)w=−aϑf1−ct1/2fη+gη,(10)η=aϑf1−ct1/2z.In the application of equation (7), equation (1) is identically balanced and equations (2)–(5) are of the following forms:
(11)ε1f′′′−f′2+f+gf′′−Sf′+η2f′′−ε2M2f′=0,ε1g′′′−g′2+f+gg′′−Sg′+η2g′′−ε2M2g′=0,with
(12)f0+g0=0,(13)f′0=1,(14)g′0=α,(15)f′∞⟶0,(16)g′∞⟶0.In the bidirectional flow, authors are generally concentrating on choosing the range ofα as 0<α<1 because for α>1, the axes x and y are interchanged, and for α=1, the stretching rate becomes alike in both directions; thus, the flow profile turns out to be axisymmetric. Also, considering the limiting situation (i.e., α⟶0), the flow becomes unidirectional. Wang [53] reported that without any loss of generality, the condition f0+g0=0 can be restored by f0=0 and g0=0. Further, in equations (1)–(12), velocity measures corresponding to the x,y,z-directions are u,v,w, respectively. The presence of time is denoted by t, whereas the temperature is symbolized by T, and hnf stands for the hybrid nanofluid, bf symbolizes the base fluid, nf discusses the nanofluid, μ expresses the viscosity, ρ reports the density, σ evokes the electrical conductivity, p1 depicts the magnetite nanoparticles having concentration ψ1, p2 illustrates the silver nanoparticles having concentration ψ2, and f,g,η posts the similarity variables with the involvement of ϑf (i.e., kinematic viscosity). Moreover, S=c/a, M=σf/aρf1/2B0, and α=b/a diagnose the unsteady, magnetic, and expansion ratio parameters, respectively. The coefficients ε1 and ε2 are communicated as (Refs. [9, 12])
(17)ε1=1+2.5ψ2+6.2ψ221+2.5ψ1+6.2ψ12ψ1ρp1/ρf+ψ2ρp2/ρf+1−ψ1−ψ2,ε2=1+3σp2/σbf−1ψ2/σp2/σbf+2−σp2/σbf−1ψ21+3σp1/σf−1ψ1/σp1/σf+2−σp1/σf−1ψ1ψ1ρp1/ρf+ψ2ρp2/ρf+1−ψ1−ψ2.


## 2.2. Heat Transport

In the current model, the heat transport phenomenon can be described by the following equation:(18)∂T∂t+u∂T∂x+v∂T∂y+w∂T∂z=khnfρCphnf∂2T∂z2,with (Ref. [9])
(19)ρCphnf=ψ1ρCpp1+ψ2ρCpp2+1−ψ1−ψ2ρCpf,khnfkbf=kp2+2kbf+2ψ2kp2−kbfkp2+2kbf−ψ2kp2−kbf,kbfkf=kp1+2kf+2ψ1kp1−kfkp1+2kf−ψ1kp1−kf.Here, two types of thermal BCs are used to discuss the variable thermal activity. These are variable surface temperature (VST) and variable surface heat flux (VHF). Mathematically, we have(20)VSTcase:z=0:T=Twx,y,t=T∞+T0xrys1−ct,z⟶∞:T⟶T∞,VHFcase:z=0:−kf∂T∂zw=qwx,y,t=T1xrys1−ct,z⟶∞:T⟶T∞.Similarity transformations for both the thermal activities are (Ref. [12])
(21)VSTcase:θη=Tx,y,z,t−T∞Twx,y,t−T∞,(22)VHFcase:T−T∞=T1kfϑfa1−ct1/2xrysϕη.As an application of equation (21), we have
(23)VSTcase:ε3θ′′+Prf+gθ′−rf′+sg′θ−Sθ+η2θ′=0,VHFcase:ε3ϕ′′+Prf+gϕ′−rf′+sg′ϕ−Sϕ+η2ϕ′=0,with
(24)θ0=1,(25)ϕ′0=−1,(26)θ∞⟶0,(27)ϕ∞⟶0.In equations (18)–(24), the free stream temperature is specified by T∞, whereas T0,T1 are dimensional constants. The variation in the surface temperature is controlled by the indices r,s along the x,y-directions, respectively. Thermal conductivity and specific heat capacitance are conveyed by k and ρCp, respectively. The Prandtl number is expressed by Pr=ρCpfυf/kf, and the coefficient ε3 may be expressed as
(28)ε3=kp2+2kbf+2ψ2kp2−kbf/kp2+2kbf−ψ2kp2−kbfkp1+2kf+2ψ1kp1−kf/kp1+2kf−ψ1kp1−kfψ1ρCpp1/ρCpf+ψ2ρCpp2/ρCpf+1−ψ1−ψ2.


## 2.3. Entropy Transport

The equation that represents the entropy transport (ET) in the present situation may be expressed as
(29)ET=khnfT∞2∂T∂x2+∂T∂y2+∂T∂z2+σhnfBt2T∞u2+v2+μhnfT∞∂u∂z2+∂v∂z2.In equation (29), the first term is due to the heat transport and the second term is due to the magnetic field. It is appropriate to formulate the entropy transport as a factor form of the entropy transport (ET) rate and characteristic entropy (Ec) rate. Characteristic entropy for both the thermal regulations is defined as
(30)Ec1=khnfTw−T∞2T∞2x2VSTcase,Ec2=khnfT−T∞2T∞2x2VHFcase.


### 2.3.1. Local Entropy Transport for the VST Case

The local entropy transport in dimensionless form for the VST case can be calculated as(31)ET1=ETEc1=Rexθ′2+r2θ2+s2θ2+RexBrα1Mε4f′2+ε5f′′2+ReyBrα1Mε4g′2+ε5g′′2.Here,Br=μfuw2/kfTw−T∞ constitutes the Brinkman number, α1=Tw−T∞/T∞ organizes the temperature difference parameter, and ε4,ε5 are due to the hybrid mixture of nanoparticles, and these may be expressed as
(32)ε4=1+3σp2/σbf−1ψ2/σp2/σbf+2−σp2/σbf−1ψ21+3σp1/σf−1ψ1/σp1/σf+2−σp1/σf−1ψ1kp2+2kbf+2ψ2kp2−kbf/kp2+2kbf−ψ2kp2−kbfkp1+2kf+2ψ1kp1−kf/kp1+2kf−ψ1kp1−kf,(33)ε5=1+2.5ψ1+6.2ψ121+2.5ψ2+6.2ψ22.


### 2.3.2. Local Entropy Transport for the VHF Case

The local entropy transport in dimensionless form for the VHF case can be calculated as(34)ET2=ETEc2=Rexϕ′2+r2ϕ2+s2ϕ2+RexBrα1Mε4f′2+ε5f′′2+ReyBrα1Mε4g′2+ε5g′′2.


## 2.3.1. Local Entropy Transport for the VST Case

The local entropy transport in dimensionless form for the VST case can be calculated as(31)ET1=ETEc1=Rexθ′2+r2θ2+s2θ2+RexBrα1Mε4f′2+ε5f′′2+ReyBrα1Mε4g′2+ε5g′′2.Here,Br=μfuw2/kfTw−T∞ constitutes the Brinkman number, α1=Tw−T∞/T∞ organizes the temperature difference parameter, and ε4,ε5 are due to the hybrid mixture of nanoparticles, and these may be expressed as
(32)ε4=1+3σp2/σbf−1ψ2/σp2/σbf+2−σp2/σbf−1ψ21+3σp1/σf−1ψ1/σp1/σf+2−σp1/σf−1ψ1kp2+2kbf+2ψ2kp2−kbf/kp2+2kbf−ψ2kp2−kbfkp1+2kf+2ψ1kp1−kf/kp1+2kf−ψ1kp1−kf,(33)ε5=1+2.5ψ1+6.2ψ121+2.5ψ2+6.2ψ22.


## 2.3.2. Local Entropy Transport for the VHF Case

The local entropy transport in dimensionless form for the VHF case can be calculated as(34)ET2=ETEc2=Rexϕ′2+r2ϕ2+s2ϕ2+RexBrα1Mε4f′2+ε5f′′2+ReyBrα1Mε4g′2+ε5g′′2.


## 2.4. Physical Quantities

The key expressions for thermal engineering importance and heat exchanger devices are the skin friction coefficients (i.e.,Cfx and Cfy) for the measurement of applied stresses at the wall, local Nusslet number (i.e., Nux) for the measurement of the rate of heat transference through the surface, and Bejan number Be for the measurement of irreversibility distribution. These engineering expressions are calculated through the following relations:
(35)Cfx=τwxρfuw2,Cfy=τwyρfvw2,Nux=xqhkfTw−T∞,Be=EGduetoheattransferTotalEG,τwx=μhnf∂u∂zz=0,τwy=μhnf∂v∂zz=0,qh=−khnf∂T∂zz=0.The aforementioned quantities in equations (33) and (34) utilizing Reynold’s numbers Rex=xuw/ϑf and Rey=yvw/ϑf are expressed as
(36)Rex1/2Cfx=ε5f′′0,Rey1/2Cfy=α−3/2ε5g′′0,Rex−1/2Nux=−khnfkfθ′0VSTcase,khnfkf1ϕ0VHFcase,Be1VSTcase=Rexθ′2+r2θ2+s2θ2Rexθ′2+r2θ2+s2θ2+RexBr/α1Mε4f′2+ε5f′′2+ReyBr/α1Mε4g′2+ε5g′′2,Be2VHFcase=Rexϕ′2+r2ϕ2+s2ϕ2Rexϕ′2+r2ϕ2+s2ϕ2+RexBr/α1Mε4f′2+ε5f′′2+ReyBr/α1Mε4g′2+ε5g′′2.


## 3. Keller-Box Simulation

The next task after the mathematical modeling of the physical model is to construct the solution of the modeled equations. Here, we preferred the implicit finite difference technique, namely, the Keller-Box method, for the numerical solution of the modeled equations because it has second-degree accuracy with the flexibility of step size adoption. This method is most suitable for the solution of boundary layer flow problems because of its faster convergence rate as compared to ordinary numerical approaches (i.e., shooting method, BVP4c, and RK method). By using this method, the higher-order differential equations are converted to first-order differential equations, and then the first-order differential equations are transformed into difference equations by using central difference formulae. The difference equations are converted into linearized forms by means of Newton’s linearization approach, and then these equations are arranged into the matrix-vector form. The solution of the matrix-vector form of linearized equations is obtained by using the LU decomposition method. In the whole process of the numerical solution, the physical domain0∞ is shortened to the finite domain η0η∞ by adjusting η0=0, η∞=20, np=500, and h=η∞−η0/np to obtain the first approximation of the numerical solution, and then the numbers of grid points np are increased by reducing the step size h to achieve the desired accuracy, i.e., ε=10−6. The complete description of the Keller-Box simulation/numerical code is presented via a flow chart (Figure 2).Figure 2
Flow chart for the Keller-Box simulation.The grid-independent solution of the modeled problem via the Keller-Box method is presented in Table2 for suitable choices of involved physical parameters with the help of thermophysical features of water, magnetite, and silver. Table 2 discloses that with the upgradation of grid points, the solution approaches its convergence criterion. 1000 grid points are enough for the convergence of f′′0, 2000 grid points are enough for the convergence of g′′0, 5000 grid points are sufficient for the convergence of θ′0, and 3000 grid points are necessary for the convergence of ϕ0. However, the Keller-Box solution of the mathematical model is presented with up to 10,000 grid points for the declaration of its stability. This grid-independent solution is further used to manipulate the outcomes for physical quantities of thermal importance and for graphical illustrations against various choices of pertinent involved entities.Table 2
Keller-Box solution of the present mathematical model using the grid-independent approach whenα=S=M=0.5, r=s=1.0, and ψ1=ψ2=0.02.
Grid points:np−f′′0−g′′0−θ′0ϕ05001.4234140.6399613.5027190.28549310001.4234130.6399533.5022530.2855315001.4234130.6399523.5021670.28553820001.4234130.6399513.5021370.2855425001.4234130.6399513.5021230.28554130001.4234130.6399513.5021150.28554235001.4234130.6399513.502110.28554240001.4234130.6399513.5021070.28554245001.4234130.6399513.5021050.28554250001.4234130.6399513.5021050.28554210,0001.4234130.6399513.5021050.285542


### 3.1. Robust Validation

In order to the validation of the numerical technique using the grid-independent approach, a restricted comparison is incorporated with the existing literature by using 10,000 grid points in Table3. A convincing pact is found among current scrutiny and previous investigations.Table 3
Robust validation of the present outcomes with previous investigations whenS=M=ψ1=ψ2=0.0.
f∞g∞−f′′0−g′′0α=1.0Current outcomes0.7514940.7514941.1737211.173721Liu and Andersson [27]0.7514940.7514941.1737211.173721Wang [53]0.7515270.7515271.1737201.173720α=0.5Current outcomes0.8423600.4516631.0930960.465206Liu and Andersson [27]0.8423600.4516631.0930960.465206Wang [53]0.8423600.4516711.0930970.465205α=0.0Current outcomes1.00.0-1.00.0Liu and Andersson [27]1.00.0-1.00.0Wang [53]1.00.0-1.00.0


## 3.1. Robust Validation

In order to the validation of the numerical technique using the grid-independent approach, a restricted comparison is incorporated with the existing literature by using 10,000 grid points in Table3. A convincing pact is found among current scrutiny and previous investigations.Table 3
Robust validation of the present outcomes with previous investigations whenS=M=ψ1=ψ2=0.0.
f∞g∞−f′′0−g′′0α=1.0Current outcomes0.7514940.7514941.1737211.173721Liu and Andersson [27]0.7514940.7514941.1737211.173721Wang [53]0.7515270.7515271.1737201.173720α=0.5Current outcomes0.8423600.4516631.0930960.465206Liu and Andersson [27]0.8423600.4516631.0930960.465206Wang [53]0.8423600.4516711.0930970.465205α=0.0Current outcomes1.00.0-1.00.0Liu and Andersson [27]1.00.0-1.00.0Wang [53]1.00.0-1.00.0


## 4. Results and Discussion

The major and foremost outcomes of the present investigation are discussed in this section of the report in the form of tables and various graphical representations. The impact of unsteady factorS on hybrid nanofluid velocity component f′η, velocity component g′η, thermal setup θη for the VST environment, and thermal setup ϕη for the VHF environment is discussed through Figures 3(a)–3(d), respectively. It is observed through Figures 3(a)–3(d) that the progression in the choice of the unsteady factor from S=0.0−2.0 reduces the impact of velocities f′ and g′, impact of thermal setup θη for the VST approach, and impact of thermal setup ϕη for the VHF approach nearer to the stretching device. Opposite trends are noticed in the far-field environment. Physically, the unsteady factor is the ratio of the dimensional constant c to the expansion rate a, and thus, the escalation in the choice of S produces a lower expansion rate and enhances the impact of c (i.e., positive expansion). Mathematically, the unsteady factor is also involved in the modeled equations in the product form with the quantities f′η, g′η, θη, and ϕη, respectively. As a result, these quantities are reduced with the escalation in the choice of the unsteady factor nearer to the stretching device. Moreover, the thickness of the boundary stream is also reduced with the improvement in the choice of the unsteady factor.Figure 3
(a–d) The impact of the unsteady factor onf′η via panel (a), on g′η via panel (b), on θη via panel (c), and on ϕη via panel (d).
(a)(b)(c)(d)The significance of temperature-controlling indicesr and s on the thermal setup is discussed in Figures 4(a) and 4(b) for the VST environment and Figures 4(c) and 4(d) for the VHF environment. It is noticed through these figures that intensifying choices of r and s diminish the thermal setup for both the prescribed heating mechanisms. Physically, r and s are involved in the energy equation as the product of θη and ϕη, that is, the main cause of temperature reduction. Moreover, negative choices of r and s produce more temperature distribution than positive choices of r and s. Furthermore, the temperature distribution is activated higher for the VHF environment than the VST environment for r=−2, whereas it is triggered higher for VST aspects than VHF aspects for all the adopted choices of the index s. The augmentation in the choice of r reduces the thickness of the thermal layer, and the thickness of the thermal layer dominates for the diverse considerations of r than the considerations of s (i.e., thermal index along the y-direction). Figure 5(a) accompanies the significance of M and implication of α on f′η. The first velocity component f′η is abbreviated with the enlargement of M from M=0.0−1.6 and amplification of α for α=0.4 and α=0.7 (i.e., dashed lines). Physically, the stretching rate a lies in the denominator of both factors M and α, and therefore, the reduction to f′η can be observed.Figure 4
(a–d) Impacts of indicesr on θη via panel (a), s on θη via panel (b), r on ϕη via panel (c), and s on ϕη via panel (d).
(a)(b)(c)(d)Figure 5
(a–d) Impacts ofM and α on f′η via panel (a), on g′η via panel (b), on θη via panel (c), and on ϕη via panel (d).
(a)(b)(c)(d)Figure5(b) conveys the importance of M and inference of α on g′η. The second velocity component g′η is abridged with the development of M from M=0.0−1.6, and it is enhanced with the magnification of α for α=0.4 and α=0.7 (i.e., dashed lines). Physically, the stretching rate b lies in the numerator of α, whereas the stretching rate a deceits in the denominator of M (i.e., magnetic parameter). Therefore, the proliferation in g′η is noticed for α and reduction in g′η is observed for the magnetic factor. The thickness of the momentum layer is also improved due to the resistivity impact of the Lorentz force. Figure 5(c) illustrates the consequence of M and significance of α on θη. Temperature θη is condensed with the progression of α, whereas it is intensified with the evolution of the magnetic factor. Physically, the Lorentz force slows down the velocity of the hybrid nanofluid and gave the domination chance to θη (i.e., temperature for the VST process). Figure 5(d) elucidates the importance of M and worth of α on ϕη. Temperature ϕη is shortened with the evolution of α, whereas it is deepened with the advancement of the magnetic feature. Actually, the Lorentz force acts as a resistive agent for the velocity of the hybrid mixture, and hence, ϕη is increased with its positive approach.Table4 provides the significance of magnetite tiny particle volume fraction ψ1 and silver tiny particle volume fraction ψ2 on the skin friction coefficients and local Nusslet number for fixed choices of other involved entities. It is elaborated in Table 4 that higher choices of ψ1 and ψ2 enhance the worth of skin friction coefficients. Mathematically, the skin friction coefficients consist of the factor of tiny particle volume fractions, and therefore, strictly increasing behavior in the amount of skin friction is achieved with the development of volume fractions of tiny particles. Moreover, the skin friction coefficient is detected higher for the y-direction as compared to the x-direction as its mathematical expression involves the term of expansion ratio α with negative fraction power. The rate of heat transference is increased for both 0.01≤ψ1≤0.07 and 0.01≤ψ2≤0.07. Physically, the rate of heat transference depends on the value of thermal conductivity for magnetite and silver nanoparticles. As the thermal conductivity is higher for silver, silver is higher than magnetite, and therefore, silver nanoparticles have produced a faster rate of heat transference as compared to magnetite nanoparticles. Table 5 provides the impact of volume fraction ψ2 on the thermophysical features of hybrid nanofluid with the present mixture of the substances, whereas Table 6 discloses the effect of volume fraction ψ1 on the thermophysical features of hybrid nanofluid with the mixture of water, magnetite, and silver. It is noticed through these tables that higher choices of ψ1 and ψ2 reduce the effective heat capacity of hybrid nanofluid, whereas the escalating amounts of these substances (magnetite and silver) boost the worth of thermal conductivity, density, and electrical conductivity of the hybrid nanofluid. These outcomes are favorable for a lot of industrial and engineering applications. Table 7 is included to discuss the percentage increase in the rate of heat transport for the suspensions of magnetite and silver in the host liquid, i.e., water. According to the calculated results, silver is more reliable and efficient in order to enhance the rate of heat transport than magnetite. Physically, the thermal conductivity of silver is much higher than that of magnetite, and so the percent-wise enhancement in the rate of heat transport is observed to be higher for silver nanoparticles.Table 4
Significance of nanoparticle volume fractionsψ1 and ψ2 on the physical quantities having thermal engineering interests when α=S=M=0.5 and r=s=1.0.
Nanoparticle volume fractions−Rey1/2Cfx−Rey1/2CfyRex−1/2Nuxψ1ψ20.010.011.4565361.8537453.8551860.030.011.547971.9697333.9037030.050.011.6431192.0905273.9513040.070.011.7420642.2162313.9979920.070.031.8985862.4128164.0469190.070.052.0580132.6133094.0944550.070.072.2210412.8185524.140588Table 5
Variation in the thermophysical features for the present mixture of substances withψ1=0.01.
ψ2ρhnfρCphnf×103khnf×10−1σhnf×10−20.001038.9294.143916.2842297425.1515142330.011133.9584.104476.4738181375.3076207250.031324.0164.025596.8646690775.6294897790.051514.0743.946717.2719010125.9649112150.071704.1323.867837.6965658026.314759380.091894.193.788958.1398073346.679985485Table 6
Variation in the thermophysical features for the present mixture of substances withψ2=0.01.
ψ1ρhnfρCphnf×103khnf×10−1σhnf×10−20.001092.1294.139566.314955575.1515151510.011133.9584.104476.4738181375.3076207250.031217.6164.034296.7996703935.6294878270.051301.2743.964117.1368287955.9649071430.071384.9323.893937.4858921686.1347530030.091468.593.823757.4875023916.679976602Table 7
Percent-wise heat transport calculation for the various possessions of nanoparticles,α=S=M=0.5 and r=s=1.0, by using the following formulae, i.e., %Increase=Withnanoparticles−Withoutnanoparticles/Withoutnanoparticles×100=Result and %Enhancement=Result−100%.
ψ1,ψ2Rex−1/2NuxFe3O4ψ2=0.0Agψ1=0.00.003.7895313.7895310.013.813768 (0.6395778% increase)3.814161 (0.6499485% increase)0.023.837764 (1.272796% increase)3.838539 (1.2932471% increase)0.043.885047 (2.520523% increase)3.886515 (2.5592613% increase)0.063.931391 (3.7434712% increase)3.933425 (3.7971453% increase)0.083.976797 (4.9416669% increase)3.979245 (5.0062659% increase)Figure6(a) is included to discuss the control of Brinkman number Br on entropy transport for the VST case, whereas Figure 6(b) illustrates its impact on entropy transport for the VHF case. Entropy is augmented with the upgradation of Br from Br=0.2−1.1, and the worth of this augmentation is almost doubled for the VST case as compared to the VHF case. Physically, the velocity of the fluid enhances with the progression of Br and declines with the augmentation of the temperature difference between the wall and the far-field. So, these spectacular trends are achieved with the variation of the Brinkman number. Figure 7(a) is incorporated to confer the influence of temperature difference parameter α1 on the Bejan number for the VST case, whereas Figure 7(b) elucidates its impression on the Bejan number for the VHF case. The Bejan number is amplified with the evolution of α1 from α1=0.1−0.9, and the net impact of this intensification is almost doubled for the VST case as compared to the VHF case. Physically, the far-field temperature contracts with the advancement of α1 and amplifies with the growth of the temperature difference between the wall and the far-field. So, these enormous tendencies are attained with the variant of the temperature difference parameter.Figure 6
(a, b) Entropy transport for the variation of the Brinkman number.
(a)(b)Figure 7
(a, b) Bejan number for the variation of the temperature difference parameter.
(a)(b)


## 5. Conclusions

Heat transport and entropy transport in bidirectional hybrid nanofluid containing nanospheres (magnetite and silver) with VST (variable surface temperature) and VHF (variable surface heat flux) mechanisms are scrutinized. The Hamilton-Crosser nanofluid model is used to inspect the impact of nanospheres. Water is considered the base fluid with the amalgamations of nanospheres. The Keller-Box method has been used to solve the modeled problem, and outcomes have been discussed graphically as well as in tabular arrangements. The pertinent observations are as follows:(i)
The velocity setup for the present mixture of nanospheres into the host liquid remains on the inferior side(ii)
The reduction in the thermal setup has been chronicled for the amplification in the choices of heat distribution indices(iii)
The Nusselt number for the present combination of nanospheres shows its proficiency for escalating nanosphere volume fractions(iv)
The nanospheres play a vital role in the improvement of the thermal conductivity of hybrid nanofluid(v)
Percentage enhancement in the rate of heat transport is observed to be higher for silver nanospheres as compared to magnetite nanospheres(vi)
Entropy is amplified with the positive estimation of the Brinkman number, and the Bejan number is also amplified with the positive tendency of the temperature difference parameterThe outcomes of the present investigation are helpful in the development of heat exchangers, coating a sheet with nanoparticles, the purification process of water, steel manufacturing, drug delivery, etc.



---

*Source: 1915185-2022-07-18.xml*