# Entropy Generation in MHD Radiative Flow of CNTs Casson Nanofluid in Rotating Channels with Heat Source/Sink

Authors: Poom Kumam; Zahir Shah; Abdullah Dawar; Haroon Ur Rasheed; Saeed Islam
Journal: Mathematical Problems in Engineering
 (2019)

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## Abstract

We presented the applications of entropy generation for SWCNTs and MWCNTs based on kerosene oil for Casson nanofluid flow by rotating channels. Kerosene oil has advanced thermal conductivity and exclusive features and has a lot of practical uses due to its unique behavior. That is why we have used kerosene oil as a based fluid. For the entropy generation second law of thermodynamics is applied and implemented for the nanofluid transport mechanism. In the presence of magnetic field, the effects of thermal radiations and heat source/sink on the temperature profiles are studied. The fluid flow is supposed in steady state. With the help of suitable similitude parameters, the leading equations have been transformed to a set of differential equations. The solution of the modeled problem has been carried out with the homotopic approach. The physical properties of carbon nanotubes are shown through tables. The effects of the imbedded physical parameters on the velocities, temperature, entropy generation rate, and Bejan number profiles are investigated and presented through graphs. Moreover, the impact of significant parameters on surface drag force and heat transfer rate is tabulated.


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## Body

## 1. Introduction

Nanofluid is the potential heat transfer fluid which is one of the fundamental fragments of nanotechnology. Engine transmission oil, boiler exhaust flue gas recovery, cooling of electronics, nuclear systems cooling, nanofluids in drilling, defense, high-power lasers, space, biomedical applications, microwave tubes, drilling, thermal storage, lubrications, drag reductions, solar water heating, in diesel electric generator as jacket water coolant, heating and cooling of buildings, engine cooling, nanofluids in transformer cooling oil, cooling of welding, refrigeration (chillers, domestic refrigerator), etc. are the applications of carbon nanotubes (CNTs). These applications got the researchers attention to work on CNTs. Choi [1] has presented the idea of nanofluids for the first time by dipping the nanometer sized particle into the base fluid. The single-walled carbon nanotubes (SWCNTs) have higher heat transfer rate and surface drag force than multiwalled carbon nanotubes (MWCNTs) reported by Haq et al. [2]. Liu et al. [3] studied the synthetic engine oil and ethylene glycol in the presence of MWCNTs. They claimed that ethylene glycol with CNTs has higher thermal conductivity. Nadeem and Lee [4] presented the concept of nanofluid past over an exponentially stretching sheet. The boundary layer flow of nanoparticles over a stretching/shrinking surface is studied by Nadeem and Lee [5–8]. The mathematical model for the examination of nanofluids flow was established by Boungiorno [9]. Sheikholeslami [10–15] experimentally and theoretically studied the properties, applications, and impacts of nanofluids. Using mesoscopic approach, Sheikholeslami and Rokni [16–19] examined simulation of nanofluids and CuOeH2O nanofluid in a curved porous enclosure. Elias et al. [20] have inspected the impact of shape on heat transmission.As the time passes the geometry of the problem translated to more complex nature. One factor of the complexity of the problem geometry is the magnetic effect. Fluids coming under this umbrella are called magneto fluids, for example, liquid metals, plasmas, salt water, and electrolytes. Crystal growth, cooling of devices, magnetic drug targeting, and electromagnetic casting are the real-world uses of magnetic hydrodynamics (MHD) in the field of engineering and technology. Magnetic hydrodynamics (MHD) was proposed by Hannes Alfven [21]. Hayat et al. [22] studied the Magneto hydrodynamic flow of Casson nanofluid effects past over a stretched surface. Ali et al. [23] examined the blood flow of Casson fluid under the impact of Magneto hydrodynamics (MHD) in axisymmetric cylindrical tube. Shah et al. [24–26] examined the MHD nanofluid in rotating systems. Dawar et al. [27] examined the MHD carbon nanotube Casson nanofluid and radiative heat transfer in rotating channels. Kumar et al. [28] examined the MHD Williamson fluid over a curved sheet under the impact of nonthermal heat source/sink. Kumaran and Sandeep [29] studied the parabolic flow of MHD Casson and Williamson fluids under the impact of thermophoresis and Brownian motion parameters. The impact of Brownian motion and thermophoresis on bioconvective flow of nanoliquids has been examined by Kumar et al. [30]. The other related studies of Kumar et al. can be seen in [31–36].Casson fluid model characterizes a shear retreating fluid which is presumed to obligate infinite viscosity at zero rate of shear stress. This famous model after the initiation becomes the main point of interest for researchers. Development has been made in model by adding extra agents to the geometry of the modeled problem. Recently, Mehmood et al. [37, 38] numerically inspected Casson micropolar fluid over a stretched sheet with internal heat transmission. Singh Megahe et al. [39] examined the film flow of Casson fluid in the existence of varied heat flux using slip velocity. Abolbashari et al. [40] have studied the Casson nanofluid with entropy generation. The study about the fundamental models of CNTs can be read in [41–45].The goal of this research work is to examine the nanoparticles of SWCNTs and MWCNTs Casson fluid between two rotating parallel plate. The kerosene oil is taken as base fluid. The solution of the modeled problem has been carried out with the homotopic approach. Homotopy analysis method (HAM) is used due to better results. Many researchers [46–55] used HAM. The influence of thermal radiations, magnetic parameter, and heat source/sink is taken into account. The effect of all embedded parameters on velocities and temperature profiles has been studied graphically and discussed in detail.


## 2. Some Basic Thermal Conductivities and Fluid Models

Model of thermal conductivity for CNTs proposed by Maxwell [41] defined as(1)knfkf=1+3knf-kf/kfφknf-kf/kf+2-knf-kf/kf-1φ.where φ is known as nanoparticle volume fraction.Model of thermal conductivity for CNTs proposed by Jeffery [42] is the form:(2)knfkf=1+3knf-kfknf+2kfφ+3knf-kfknf+2kf2+34knf-kfknf+2kf3+⋯φ2.Davis [43] modified Jeffery’s model as(3)knfkf=1+3knf-kf/kfφknf+2kf/kf-knf-kf/kfφφ+φknf-kfkfφ2+knf-kfkfφ3.Hamilton and Crosser [44] proposed a new model defined as(4)knfkf=knf/kf+Φ-1-knf-1/kfΦ-1φknf/kf+Φ-1-knf-1/kfφ.Models (1)-(3) did not depend on shape of particles and deliver the approximations of thermal conductivity for small φ. On the other hand, model (4) depends on the shape of particles. The symbol Φ in (4) represents shape factor of the particles. Recently, Xue [45] proposed a model of a very large axial ratio recompensing the CNTs space distribution.(5)knfkf=1-φ+2knf/knf-kfln⁡knf+kf/2kfφ1-φ+2kf/knf-kfln⁡knf+kf/2kfφ.The rheological model for Casson fluid is defined as(6)Τij=2μΒ+Ρy2κmij,κ>κc,μΒ+Ρy2κcmij,κ>κc.In (1)-(6), φ,f,nf,Τij,μB,κ=mij. mij, Ρy, κc represent the nanoparticle volume fraction, base fluid, the nanofluid, Cauchy stress tensor, dynamic viscosity, components of strain rate, the yield stress of fluid and the critical value of κ.


## 3. Mathematical Modeling

Assume the flows of CNTs nanofluid based on kerosene liquid between two parallel (lower and upper) plates which are kept at the distanceh. The nanoscale materials (SWCNTs and MWCNTs) are used in this model. The mechanism is examined through the thermal radiation and heat transportation. The constant angular velocity γ rotates around y-axis. The three possible cases γ>0 which specify that both upper and lower plates rotated in the same direction γ<0 specify that both the plates rotated in opposite directions and γ=0 for the static case. The lower plate is moving quickly with a velocity Uw=cxc>0 than the upper plate. A three dimensional coordinate system x,y,z is chosen such that x is parallel to the plate, y is normal to the plate, and z is normal to xy-plane. The upper and lower plates are kept at y=0 and y=h respectively. With the help of two opposite forces with same magnitude, the lower plate is being kept stretchable so the position 0,0,0 cannot change. The fluid flow and heat transfer is assumed in steady state which is incompressible, laminar, and stable. The magnetic field B0 is applied in y-direction with which the fluid is rotating as shown in Figure 1 [46].Figure 1
Physical sketch of the problem.The governing equations for the state of problem are as follows [46]:(7)∂u∂x+∂v∂y=0,(8)ρnfu∂u∂x+v∂u∂y+2γw=-∂P∗∂x+1+1βμnf∂2u∂x2+∂2u∂y2-σnfB02u,(9)ρnfu∂v∂x+v∂v∂y=-∂P∗∂y+μnf1+1β∂2v∂x2+∂2v∂y2,(10)ρnfu∂w∂x+v∂w∂y-2γu=μnf1+1β∂2w∂x2+∂2w∂y2+σnfB02w,where P∗=P-γ2x2/2, σnf and μnf represent the modified pressure, electrical conductivity, and dynamic viscosity, respectively. The absenteeism of Pz∗ signifies the mesh cross flow beside z-axis and β is the Casson fluid parameter.The heat transfer phenomenon is specified as [46]:(11)u∂Τ∂x+v∂Τ∂y=knfρcnf∂2Τ∂x2+∂2Τ∂y2-∂Qr∂y+Q0T-T0ρcpf.Here Τ is the fluid temperature, αnf=knf/(ρc)nf is the thermal diffusivity, and Qr is the radiative heat flux, respectively.The radiative heat fluxQr is defined as(12)Qr=-4σ∗3k∂Τ4∂y,where σ∗ and k indicate the absorption coefficient and Stefan Boltzmann constant, respectively.SinceΤ4=4Τh3Τ-3Τh4, (11) becomes(13)u∂Τ∂x+v∂Τ∂y=knfρcnf1+16σ∗Τh33k∂2Τ∂y2+Q0T-T0ρcpf.The density, heat capacity, dynamic viscosity, and thermal conductivity of the nanofluid are mathematically represented as follows [47–50]:(14)ρnf=1-φρcf+φρcCNT,ρcnf=1-φρf+φρCNT,μnf=μf1-φ2.5,knf=kf1-φ+2φkCNT/kCNT-kfln⁡kCNT+kf/2kf1-φ+2φkf/kCNT-kfln⁡kCNT+kf/2kf.where CNT signifies the carbon nanotubes.Τ h is the temperature at lower wall and Τ0 is the temperature at upper wall which is taken as Τh>Τ0.The above system of equations has the following boundary conditions.(15)u→=Uw=cx,v→=0,w→=0,Τ=Τhaty=0,u→=0,v→=-A0,w→=0,Τ=Τ0aty=h.In (15), at the upper wall of the channel, A0 is the uniform suction/injection velocity. If A0>0, it is called uniform suction velocity and if A0<0, it is called uniform injection velocity.For the proposed problem we have defined the following similarities variables:(16)u→=cxf′ξ,v→=-chfξ,w→=cxgξ,θξ=Τ-Τ0Τh-Τ0,ξ=yh.In view of (8)-(10) and (13), the nondimensional system of equations is(17)1+1βfiv-1-φ2.51-φ+φρCNTρfRef′f′′-ff′′′-1-φ2.51-φ+φρCNTρfΩg′-σnfσfΜf′′=0,(18)1+1βg′′-1-φ2.51-φ+φρCNTρfRegf′-fg′+21-φ2.51-φ+φρCNTρfΩf′+σnfσfΜg=0,(19)ddξknfkf+Rd1+θ-1θ3θ′+1-φ2.51-φ+φρCNTρfPrRefθ′-γθ=0,where the relevant boundary conditions are(20)f0=0,f′0=1,g0=0,θ0=1,f1=ε,f′1=0,g1=0,θ1=0.In (17)-(20), ε=A0/ch is the suction and injection parameter. ε>0 is for the case suction parameter and if ε<0 is for the case of injection. Re=ch2/νf signifies the Reynolds number, Ω=γh2/νf represents the rotation parameter, Pr=μfcp/kf signifies the Prandtl number, Rd=16σ∗Τh3/3kk∗ indicates the thermal radiation parameter, Μ=σfB02h2/ρfνf represents the magnetic parameter, and γ=Q0/cρcpf signifies the heat generation or absorption coefficient.The equations for the surface drag force and heat transfer rate are defined as follows [51]:(21)C~f=1+1β1-φ2.51-φ+φρCNT/ρff′′0(22)Nux=-knfkf1+Rdθ′0.


## 4. Expression for Entropy Analysis and Bejan Number

For the stated problem of nanofluid, the local entropy rateSg.t per unit volume is [52](23)Sg.t′′′=knfT02∇T2+16σ∗Th33kk∗∇T2+μnfT0Φ+σnfB02T0u→2,(24)Sg.t′′′=knfT021+16σ∗Th33k∇T2+μnfT0Φ+σnfB02T0u→2,where ∇T=Τx+Τy and Φ is related to viscous dissipation.In our case(25)Sg.t′′′=knfT021+16σ∗Th33kk∗∇T2+μnfT0u→y2+σnfB02T0u→2.The characteristic entropy generation rate Sg.c is defined as(26)Sg.c′′′=knf∇Τ2Th2L2.The entropy generation Ns is defined as(27)Ns=Sg.t′′′Sg.c′′′.To calculate the entropy generation Ns, from (25), (26) with (16), implementing in (27), we obtain(28)Ns=Re1+Rdθ′2+11-φ2.5Brωf′′2+M2f′2,where Re,Br,ω, and M are the Reynolds number, Brinkman number, dimensionless temperature, and Hartmann number, respectively, whose expressions are given by(29)Re=Τh2L2,ω=Τh-Τ0Τ0,Br=μfUw2knfΤh-Τ0.The Bejan number Be is defined as(30)Be=knf/Τ021+16σ∗Th3/3kk∗Τy2μnf/Τ0u→y2+σnfB02/Τ0u→2,or(31)Be=Re1+Rdθ′2Re1/1-φ2.5Br/ωf′′2+M2f′2.


## 5. Solution by HAM

We use homotopy analysis method (HAM) to solve (17)-(19) with (20) by the succeeding process.The primary assumptions are as follows:(32)f0ξ=ξ+12ε-1ξ2,g0ξ=0,θ0ξ=1-ξ.The linear operators Lf,Lg, and Lθ are chosen such that(33)Lff=f′′′,Lgg=g′′,Lθθ=θ′′,which have the properties(34)Lfb1+b2ξ+b3ξ2=0,Lgb4+b5ξ=0,Lθb6+b7ξ=0,where bi(i=1→7).The consequences of nonlinear operatorsNf,Ng, and Nθ are indicated as follows:(35)Nffξ;Ύ,gξ;Ύ=1+1β∂4fξ;Ύ∂ξ4-1-φ2.51-φ+φρCNTρfRe∂fξ;Ύ∂ξ∂2fξ;Ύ∂ξ2-fξ;Ύ∂3fξ;Ύ∂ξ3-1-φ2.51-φ+φρCNTρfΩ∂gξ;Ύ∂ξ-σnfσfM∂2fξ;Ύ∂ξ2,(36)Ngfξ;Ύ,gξ;Ύ=1+1β∂2gξ;Ύ∂ξ2-1-φ2.51-φ+φρCNTρfRegξ;Ύ∂fξ;Ύ∂ξ-fξ;Ύ∂gξ;Ύ∂ξ+21-φ2.51-φ+φρCNTρfΩ∂fξ;Ύ∂ξ+σnfσfMgξ;Ύ,(37)Nθfξ;Ύ,θξ;Ύ=ddξknfkf+43Rd1+∂θξ;Ύ∂ξ-1θξ;Ύ3∂2θξ;Ύ∂ξ2+1-φ+φρcpCNTρcpfPrRefξ,Ύ∂θξ;Ύ∂ξ-γθξ;Ύ.Zeroth-order problems are(38)1-ΎLffξ;Ύ-f0ξ=ΎħfNffξ;Ύ,gξ;Ύ,(39)1-ΎLggξ;Ύ-g0ξ=ΎħgNgfξ;Ύ,gξ;Ύ,(40)1-ΎLθθξ;Ύ-θ0ξ=ΎħθNθfξ;Ύ,θξ;Ύ.The equivalent boundary conditions are(41)fξ;Ύξ=0=0,∂fξ;Ύ∂ξξ=0=1,fξ;Ύξ=1=ε,∂fξ;Ύ∂ξξ=1=0,gξ;Ύξ=0=0,gξ;Ύξ=1=0,θξ;Ύξ=0=1,θξ;Ύξ=1=0.where Ύ∈[0,1] is the emerging parameter. In case of Ύ=0 and Ύ=1, we have(42)fξ;1=fξ,gξ;1=gξandθξ;1=θξ.Expanding f(ξ;Ύ),g(ξ;Ύ) and θ(ξ;Ύ) by Taylor’s series(43)fξ;Ύ=f0ξ+∑q=1∞fqξΎq,gξ;Ύ=g0ξ+∑q=1∞gqξΎq,θξ;Ύ=θ0ξ+∑q=1∞θqξΎq,where(44)fqξ=1q!∂fξ;Ύ∂ξΎ=0,gqξ=1q!∂gξ;Ύ∂ξΎ=0andθqξ=1q!∂θξ;Ύ∂ξΎ=0.As the series (44) converges at Ύ=1, using Ύ=1 in (44), we get(45)fξ=f0ξ+∑q=1∞fqξ,gξ=g0ξ+∑q=1∞gqξ,θξ=θ0ξ+∑q=1∞θqξ.The qth-order problem gratifies the following:(46)Lffqξ-χqfq-1ξ=ħfUqfξ,Lggqξ-χqgq-1ξ=ħgUqgξ,Lθθqξ-χqθq-1ξ=ħθUqθξ.The equivalent boundary conditions therefore become(47)fq0=fq′0=fq1=fq′1=0,gq0=gq1=0,θq0=θq1=0.Here(48)Uqfξ=1+1βfq-1iv-Re1-φ2.51-φ+φρCNTρf∑k=0q-1fq-1-k′fk′′-∑k=0q-1fq-1-kfk′′′-Ω1-φ2.51-φ+φρCNTρfgq-1-σnfσfMfq-1′′,(49)Uqgξ=1+1βgq-1′′-1-φ2.51-φ+φρCNTρfRe∑k=0q-1gq-1-kf′k-∑k=0q-1fq-1-kg′k+21-φ2.51-φ+φρCNTρfΩfq-1′+σnfσfMgq-1,(50)Uqθξ=ddξknfkf+43Rd1+θq-1-1θq-13θq-1′+1-φ+φρcpCNTρcpfPrRe∑k=0q-1fq-1-kθk′′-γθq-1,where(51)χq=0,ifΎ≤11,ifΎ>1.


### 5.1. HAM Solution Convergence

The assisting parametersħf,ħg, and ħθ appear when we calculate the series solution of velocities (f′(ξ),g(ξ)) and temperature (θ(ξ)) functions by using HAM. These parameters lead the convergence of the modeled problem. Figure 2 is schemed to see the convergence of the problem. The velocity function f′(ξ) converges in the region -4.0≤ħf≤2.0, the velocity function g(ξ) converges in the region -3.0≤ħg≤1.0, and the temperature function θ(ξ) converges in the region -2.0≤ħθ≤1.0, respectively.Figure 2
ħ - c u r v e s for f′′0,g′0,θ′0.


## 5.1. HAM Solution Convergence

The assisting parametersħf,ħg, and ħθ appear when we calculate the series solution of velocities (f′(ξ),g(ξ)) and temperature (θ(ξ)) functions by using HAM. These parameters lead the convergence of the modeled problem. Figure 2 is schemed to see the convergence of the problem. The velocity function f′(ξ) converges in the region -4.0≤ħf≤2.0, the velocity function g(ξ) converges in the region -3.0≤ħg≤1.0, and the temperature function θ(ξ) converges in the region -2.0≤ħθ≤1.0, respectively.Figure 2
ħ - c u r v e s for f′′0,g′0,θ′0.


## 6. Results and Discussion

To discuss the impact of embedded parameters on velocitiesf′(ξ),g(ξ), temperature θ(ξ), entropy generation rate (Ns), and Bejan numbers (Be), Figures 3–21 are plotted. These parameters are Reynolds number (Re), Rotation parameter (Ω), magnetic parameter (M), Casson parameter (β), Prandtl number (Pr), thermal radiation parameter (Rd), and heat source/sink (γ). Figure 1 shows physical shape of the nanofluid flow. Figure 2 is plotted to see the convergence region of modeled problem.Figure 3
The impact ofRe on f′(ξ), when φ=0.04,A2=0.6,ε=0.8,β=0.7,M=1.0.Figure 4
The impact ofRe on g(ξ), when φ=0.04,Ω=0.6,β=0.7,M=1.0.Figure 5
The impact ofRe on θ(ξ), when φ=0.04,Pr=7.1,γ=0.8,Rd=0.9,M=1.0.Figure 6
The impact ofΩ on f′(ξ), when φ=0.04,Re=0.5,ε=0.8,β=0.7,M=1.0.Figure 7
The impact ofΩ on g(ξ), when φ=0.04,Re=0.5,M=1.0,β=0.7.Figure 8
The impact ofM on f′(ξ), when φ=0.04,Re=0.5,Ω=0.6,β=0.7,ε=0.8.Figure 9
The impact ofM on g(ξ), when φ=0.04,Re=0.5,Ω=0.6,β=0.7.Figure 10
The impact ofβ on f′(ξ), when φ=0.04,Re=0.5,Ω=0.6,M=1.0.Figure 11
The impact ofβ on g(ξ), when φ=0.04,Re=0.5,Ω=0.6.Figure 12
The impact of Pr onθξ, when φ=0.04,Re=0.5,γ=0.8,Rd=0.9.Figure 13
The impact ofRd on θ(ξ), when φ=0.04, Re = 0.5, γ=0.8, Pr = 0.9.Figure 14
The impact ofγ on θ(ξ), when φ=0.04,Re=0.5,Rd=0.8,Pr=0.9.Figure 15
The impact ofM on Ns, when φ=0.04,Rd=0.9,ω=0.5,Br=0.25,Re=1.0.Figure 16
The impact of Re onNs, when φ=0.04,Rd=0.9,ω=0.5,Br=0.25,M=1.0.Figure 17
The impact ofBr on Ns, when φ=0.04,Rd=0.9,ω=0.5,M=0.25,Re=1.0.Figure 18
The impact ofRd on Ns, when φ=0.04,Re=1.0,ω=0.5,Br=0.5,M=0.25.Figure 19
The impact ofM on Be, when φ=0.04,Rd=0.9,ω=0.5,Br=0.25.Figure 20
The impact ofBr on Be, when φ=0.04,Rd=0.9,ω=0.5,M=0.25.Figure 21
The impact ofRd on Be, when φ=0.04,Br=0.25,ω=0.5,M=0.25.


### 6.1. Velocities and Temperature Profiles

Figures3–5 are plotted to observe the effect of (Re) on (f′(ξ)andg(ξ)) and θ(ξ). Reynolds is a dimensionless number used to calculate the flow pattern in different situation of the fluid flow. At low Reynolds numbers, the fluid flow tends to laminar while at low Reynolds numbers, the fluid flow tends to turbulent flow. From here, we observed that with the escalating values of (Re) the velocities f′ξandgξ and temperature θ(ξ) profiles decrease. This is due to the stretching of the lower plate. The effect of (Ω) on velocities (f′(ξ)andg(ξ)) profiles is shown in Figures 6-7. From Figure 6, we observed the increasing behavior with the increasing rotation parameter while the opposite result was observed in Figure 7. It is interesting to mention that in the absence of rotation parameter the problem reduces to 2-dimensional flow in a channel. Figures 8-9 are plotted to observe the impact of (M) parameter on velocities (f′(ξ)andg(ξ)) profiles. According to Lorentz force theory, the magnetic field parameter has an inverse effect on (f′(ξ)andg(ξ)) profiles, that is, the upsurges in magnetic field parameter the (f′(ξ)andg(ξ)) profiles reduce. Figures 10-11 depict the impact of (β) on f′ξandgξ profiles. In practice, the increasing (β) results in the rise in dynamic viscosity that creates a resistance in the flow and a decline in fluid velocity thereof. Therefore, the increasing in Casson fluid parameter reduces the velocities f′ξandgξ profiles. Figure 12 is schemed to observe the influence of (Pr) on temperature profile θ(ξ). Physically, nanofluids have large thermal diffusivity with small (Pr) and vice versa. Therefore, the liquid temperature decreases. From here we observed that the growing (Pr) shows decline in θ(ξ). Figure 13 is schemed to observe the impact of (Rd) on θ(ξ). Thermal radiations have leading rules in heat transmission, when the coefficient of convection heat transmission is small. From here we observed that the increasing (Rd) shows a rapid increase in θ(ξ). Figure 14 specifies the impact of (γ) on θ(ξ). The porosity parameter has commanding character in fluid motion. Physically, the higher values of porosity parameter improve the porous surface which resists to the flow motion and as a result the motion of the fluid reduces. From here we see that θ(ξ) decreases with increasing (γ).


### 6.2. Entropy Generation and Bejan Number

Figures15–21 are plotted to see the influences of emerging parameters for both SWCNTs and MWCNTs based on kerosene nanoliquids on entropy generation Ns and Bejan number (Be). Generally, the entropy generation rate is more for SWCNTs as compared to MWCNTs. Figure 15 shows the impact of (M) on (Ns). Physically, due to magnetic parameter Lorentz force generates and decreases velocity of the flow and thus entropy generation declines. Figure 16 expresses the impact of (Re) on (Ns). With high Reynolds number, hectic motion arises for the reason that as (Re) escalates, the fluid moves more upsettingly and hence contribution of the fluid fraction and heat transfer on entropy result tends to escalation in entropy generation rate. Figure 17 shows the influence of (Br) on (Ns). The entropy generation rate escalates with the escalation in (Br). In addition, an increase in (Ns) created by fluid friction and joule dissipation happens with the escalating values of (Br). Figure 18 shows the influence of (Rd) on (Ns). From the figure we see that at 0.0≤ξ<0.5 and 0.5<ξ≤1.0 the escalating values of (Rd) show increasing behavior in (Ns). But at the mean position (i-e ξ=0.5), the radiation parameter has no impact on (Ns). It is due to the stretching of the lower plate. Figure 19 shows the influence of (Μ) on (Be). It is clear from the figures that the accelerating values of (Μ) shows decline in (Be). Figure 20 depicts the influence of (Br) on (Be). The increasing values of (Br) escalate the (Be). Figure 21 indicates the influence of (Rd) on (Be). From the figure we see that at 0.0≤ξ<0.5 and 0.5<ξ≤1.0 the increasing values of (Rd) show increasing behavior in (Be), while at ξ=0.5 the radiation parameter does not show increasing or decreasing behavior. It is due to the stretching of the lower plate.


### 6.3. Tables Discussion

Table is schemed to observe the impact of(Re),(Ω),(M), and (β) on skin fraction coefficient C~f. From Table 1, we observed that the surface drag force increases with the escalation in (Re) and (β) while it reduces with the escalation in (Ω) and (M). Table 2 is schemed to observe the impact of (Re),(Rd),(Pr), and (γ) on local Nusselt number Nux. From here it is observed that the heat transfer rate increases with the escalation in (Re) and (γ) while it reduces with the escalation in (Rd) and (Pr). In Tables 3–5, some physical and thermal properties of CNTs with thermal conductivity are presented.Table 1
The arithmetical values of surface drag forceC~f=μnf/μff′′(0), when σ=0.04.
R e Ω Μ β C ~ f 0.1 0.2 0.3 1.0 -0.524696  0.2    -0.521175  0.3 0.3   -0.520833  0.4   -0.497588  0.5 0.3  -0.507138  0.5  -0.515129  1.0 1.0 -0.526225  1.1 -0.516946  1.2 -0.509188Table 2
The arithmetical values of heat transfer rateNux=-knf/kf1+Rdθ′0, when σ=0.04.
R e P r R d γ N u x 0.1 7.1 0.1 0.1 -0.001095  0.2    -0.001094  0.3 7.1   -0.001091  7.2   -0.001095  7.3 0.1  -0.001099  0.2  -0.001103  0.3 0.1 -0.001349  0.2 0.209913  0.3 0.214909Table 3
Physical properties of CNTs [45].
Materials SWCNT MWCNT Thermal Conductivity 1 × 1 0 3 1 × 1 0 3 knf W / m K  Electrical Conductivity 1 × 1 0 6 - 1 × 1 0 7 1 × 1 0 6 - 1 × 1 0 7 φ n f S / m  Tensile StrengthGPa 1.5 × 1 0 2 1.5 × 1 0 2  Young’s ModulusGPa 10.54 × 1 0 2 1.2 × 1 0 3Table 4
The thermophysical properties of CNTs and nanofluids of some base fluids [53].
Physical Properties Specific Heatcp(J/kgK) Densityρ(kg/m3) Thermal Conductivityk(W/mk) Base fluid Water 41.97 × 1 0 2 0.997 × 1 0 3 61.3 × 1 0 - 2 Kerosene (lamp) oil 20.90 × 1 0 2 0.783 × 1 0 3 14.5 × 1 0 - 2 Engine oil 19.10 × 1 0 2 0.884 × 1 0 3 14.4 × 1 0 - 2  Nanofluids SWCNT 0.425 × 1 0 3 2.6 × 1 0 3 6.6 × 1 0 3 MWCNT 0.796 × 1 0 3 1.6 × 1 0 3 30 × 1 0 2Table 5
Thermal conductivity (knf) of CNTs with different volume fraction (φ).
φ k n f for SWCNT k n f for MWCNT 0 14.5 × 1 0 - 2 14.5 × 1 0 - 2  0.01 17.4 × 1 0 - 2 17.2 × 1 0 - 2  0.02 20.4 × 1 0 - 2 2 × 1 0 - 1  0.03 23.5 × 1 0 - 2 22.8 × 1 0 - 2  0.04 26.6 × 1 0 - 2 25.7 × 1 0 - 2


## 6.1. Velocities and Temperature Profiles

Figures3–5 are plotted to observe the effect of (Re) on (f′(ξ)andg(ξ)) and θ(ξ). Reynolds is a dimensionless number used to calculate the flow pattern in different situation of the fluid flow. At low Reynolds numbers, the fluid flow tends to laminar while at low Reynolds numbers, the fluid flow tends to turbulent flow. From here, we observed that with the escalating values of (Re) the velocities f′ξandgξ and temperature θ(ξ) profiles decrease. This is due to the stretching of the lower plate. The effect of (Ω) on velocities (f′(ξ)andg(ξ)) profiles is shown in Figures 6-7. From Figure 6, we observed the increasing behavior with the increasing rotation parameter while the opposite result was observed in Figure 7. It is interesting to mention that in the absence of rotation parameter the problem reduces to 2-dimensional flow in a channel. Figures 8-9 are plotted to observe the impact of (M) parameter on velocities (f′(ξ)andg(ξ)) profiles. According to Lorentz force theory, the magnetic field parameter has an inverse effect on (f′(ξ)andg(ξ)) profiles, that is, the upsurges in magnetic field parameter the (f′(ξ)andg(ξ)) profiles reduce. Figures 10-11 depict the impact of (β) on f′ξandgξ profiles. In practice, the increasing (β) results in the rise in dynamic viscosity that creates a resistance in the flow and a decline in fluid velocity thereof. Therefore, the increasing in Casson fluid parameter reduces the velocities f′ξandgξ profiles. Figure 12 is schemed to observe the influence of (Pr) on temperature profile θ(ξ). Physically, nanofluids have large thermal diffusivity with small (Pr) and vice versa. Therefore, the liquid temperature decreases. From here we observed that the growing (Pr) shows decline in θ(ξ). Figure 13 is schemed to observe the impact of (Rd) on θ(ξ). Thermal radiations have leading rules in heat transmission, when the coefficient of convection heat transmission is small. From here we observed that the increasing (Rd) shows a rapid increase in θ(ξ). Figure 14 specifies the impact of (γ) on θ(ξ). The porosity parameter has commanding character in fluid motion. Physically, the higher values of porosity parameter improve the porous surface which resists to the flow motion and as a result the motion of the fluid reduces. From here we see that θ(ξ) decreases with increasing (γ).


## 6.2. Entropy Generation and Bejan Number

Figures15–21 are plotted to see the influences of emerging parameters for both SWCNTs and MWCNTs based on kerosene nanoliquids on entropy generation Ns and Bejan number (Be). Generally, the entropy generation rate is more for SWCNTs as compared to MWCNTs. Figure 15 shows the impact of (M) on (Ns). Physically, due to magnetic parameter Lorentz force generates and decreases velocity of the flow and thus entropy generation declines. Figure 16 expresses the impact of (Re) on (Ns). With high Reynolds number, hectic motion arises for the reason that as (Re) escalates, the fluid moves more upsettingly and hence contribution of the fluid fraction and heat transfer on entropy result tends to escalation in entropy generation rate. Figure 17 shows the influence of (Br) on (Ns). The entropy generation rate escalates with the escalation in (Br). In addition, an increase in (Ns) created by fluid friction and joule dissipation happens with the escalating values of (Br). Figure 18 shows the influence of (Rd) on (Ns). From the figure we see that at 0.0≤ξ<0.5 and 0.5<ξ≤1.0 the escalating values of (Rd) show increasing behavior in (Ns). But at the mean position (i-e ξ=0.5), the radiation parameter has no impact on (Ns). It is due to the stretching of the lower plate. Figure 19 shows the influence of (Μ) on (Be). It is clear from the figures that the accelerating values of (Μ) shows decline in (Be). Figure 20 depicts the influence of (Br) on (Be). The increasing values of (Br) escalate the (Be). Figure 21 indicates the influence of (Rd) on (Be). From the figure we see that at 0.0≤ξ<0.5 and 0.5<ξ≤1.0 the increasing values of (Rd) show increasing behavior in (Be), while at ξ=0.5 the radiation parameter does not show increasing or decreasing behavior. It is due to the stretching of the lower plate.


## 6.3. Tables Discussion

Table is schemed to observe the impact of(Re),(Ω),(M), and (β) on skin fraction coefficient C~f. From Table 1, we observed that the surface drag force increases with the escalation in (Re) and (β) while it reduces with the escalation in (Ω) and (M). Table 2 is schemed to observe the impact of (Re),(Rd),(Pr), and (γ) on local Nusselt number Nux. From here it is observed that the heat transfer rate increases with the escalation in (Re) and (γ) while it reduces with the escalation in (Rd) and (Pr). In Tables 3–5, some physical and thermal properties of CNTs with thermal conductivity are presented.Table 1
The arithmetical values of surface drag forceC~f=μnf/μff′′(0), when σ=0.04.
R e Ω Μ β C ~ f 0.1 0.2 0.3 1.0 -0.524696  0.2    -0.521175  0.3 0.3   -0.520833  0.4   -0.497588  0.5 0.3  -0.507138  0.5  -0.515129  1.0 1.0 -0.526225  1.1 -0.516946  1.2 -0.509188Table 2
The arithmetical values of heat transfer rateNux=-knf/kf1+Rdθ′0, when σ=0.04.
R e P r R d γ N u x 0.1 7.1 0.1 0.1 -0.001095  0.2    -0.001094  0.3 7.1   -0.001091  7.2   -0.001095  7.3 0.1  -0.001099  0.2  -0.001103  0.3 0.1 -0.001349  0.2 0.209913  0.3 0.214909Table 3
Physical properties of CNTs [45].
Materials SWCNT MWCNT Thermal Conductivity 1 × 1 0 3 1 × 1 0 3 knf W / m K  Electrical Conductivity 1 × 1 0 6 - 1 × 1 0 7 1 × 1 0 6 - 1 × 1 0 7 φ n f S / m  Tensile StrengthGPa 1.5 × 1 0 2 1.5 × 1 0 2  Young’s ModulusGPa 10.54 × 1 0 2 1.2 × 1 0 3Table 4
The thermophysical properties of CNTs and nanofluids of some base fluids [53].
Physical Properties Specific Heatcp(J/kgK) Densityρ(kg/m3) Thermal Conductivityk(W/mk) Base fluid Water 41.97 × 1 0 2 0.997 × 1 0 3 61.3 × 1 0 - 2 Kerosene (lamp) oil 20.90 × 1 0 2 0.783 × 1 0 3 14.5 × 1 0 - 2 Engine oil 19.10 × 1 0 2 0.884 × 1 0 3 14.4 × 1 0 - 2  Nanofluids SWCNT 0.425 × 1 0 3 2.6 × 1 0 3 6.6 × 1 0 3 MWCNT 0.796 × 1 0 3 1.6 × 1 0 3 30 × 1 0 2Table 5
Thermal conductivity (knf) of CNTs with different volume fraction (φ).
φ k n f for SWCNT k n f for MWCNT 0 14.5 × 1 0 - 2 14.5 × 1 0 - 2  0.01 17.4 × 1 0 - 2 17.2 × 1 0 - 2  0.02 20.4 × 1 0 - 2 2 × 1 0 - 1  0.03 23.5 × 1 0 - 2 22.8 × 1 0 - 2  0.04 26.6 × 1 0 - 2 25.7 × 1 0 - 2


## 7. Conclusion

Entropy generation examination in two-dimensional mixed convection flow of nanofluid over an inclined stretching sheet has been analyzed. The kerosene liquid is taken as base fluid. The impact of magnetohydrodynamics, thermal radiation, and heat/source has been examined. The transformed system of equations has been solved by homotopy analysis method. On the achieved study, the key remarks are listed below.(i)
The velocity functionf′(ξ) escalates with the escalation in rotation parameter and reduces with rise in Reynolds number, magnetic field parameter, and Casson fluid parameter.(ii)
The velocity functiong(ξ) is reduced with escalation in Reynolds number, rotation parameter, magnetic field, and Casson fluid parameter.(iii)
The temperature functionθ(ξ) is increased with the escalation in thermal radiation parameter while reduced with the escalation in Reynolds number, Prandtl number, and heat source/sink.(iv)
The entropy generationNs is increased with the escalation in Reynolds number, Brinkman number, and thermal radiation parameter while reduced with the escalation in magnetic field parameter.(v)
The Bejan number is increased with the increase in Brinkman number and thermal radiation parameter while reduced with the upsurge in magnetic field parameter.(vi)
The surface drag force increases with the escalation in Reynolds number and Casson fluid parameter while reduces with the escalation in rotation and magnetic parameter.(vii)
The heat transfer rate increases with the escalation in Reynolds number and heat source/sink while reduces with the escalation in thermal radiation parameter and Prandtl number.



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