Thermodynamic entropy of a magnetized nanofluid flow over an inclined stretching cylindrical surface

Abstract

Keywords: Bejan Number; Entropy Generation; Inclined Stretching Cylinder; Nanofluid; Variable Thermal Conductivity

Introduction

Nanofluid technology has emerged in recent years as a promising methodology to improve the thermal conductivity of the original fluid by mixing nanomaterials with the base fluid. Nanofluids have been implemented in electronics cooling, air conditioning, cancer therapy, and renewable energy systems. Many authors developed such innovative aspects and contributed massively to nanofluids. The term “nanofluid” was first proposed by Choi [1]. Buongiorno [2] contributed conservation equations in nanofluid that are based on these two mechanisms of random motion of air bubbles in still air and a kind of force generated by heat air and cool wall of particles. Nield and Kuznetsov [3] applied the Buongiorno model to investigate the patterns of nanofluid flow that is porous in nature and the flow is passing through a vertical plate. The used nanofluid contains sub-microsonic solid particles that create porosity in the medium. Meantime, distinct boundary layer conditions were incorporated in the studies done by Mansur and Ishak [4], Kuznetsov and Nield [5]. Rana and Thakur [6] used the Buongiorno model and noticed critical values for the existence of the dual solution along an inclined stretching cylinder. Recently, V.K. Narla et al. [7] investigated the transport of nanofluid flow using electroosmosis in a curved channel. Magnetohydrodynamic (MHD) nanofluid flow has numerous applications like those of the earth generated heat energy eradication, nuclear reactors, and the petroleum industry. The MHD nanofluid flow with different geometries has been focused on by several authors (Mahatha et al. [8], Nagaraju et al. [9], Mondal et al. [10] and Kandwal et al. [11]). The transport of heat from one solid medium or place to another is studied as thermal conductivity. The phenomenon of thermal transport is very common in every fluid transfer process. Elbashbeshy et al. [12] studied the impact of heat treatment of materials with time-dependent nanofluid flows in moving cylinders. Bisht et al. [13] studied mixed convection to investigate the effects of chemical reactions and heat transport of variable nature on fluid flows through long cones of circular geometry. Some theoretical and experimental studies by Ravi Kanth and Kumar [14] have provided close correction with variation in temperature with thermal conductivity. The most significant properties of variable thermal conductivity for stretching cylinder is working with different nanofluids are reported in (Salahuddin et al. [15], Ramzan et al. [16], and Gajjela and Garvandha [17]). Boundary layer theory plays a significant role in fluid flow systems. Sakiadis [18] initiated to examine the behavior of boundary layer flow along rigid surfaces. The author suggested this boundary layer flow for both laminar and turbulent flow. Crane [19] developed a model for steady, quiescent, two-dimensional flow along a stretching sheet moving with linearly varying speed followed by Sakiadis [18]. After that many researchers examined

various aspects of the boundary layer flow along continuous and finite rigid surfaces and came up with some interesting results. Wang [20] explored the steady, continuous 2-dimensional fluid flow in stretching cylinder in outer stretched surfaces. Gorla et al. [21] studied phase change(melting) condition at cylindrical wall for nanoparticle heat transfer flow. Dhanai et al. [22] used the Buongiorno model to analyze the slip effects over an inclined stretching cylinder. The impacts of Thermodiffusion and Diffuso-thermal in mixed systems, which are frequently encountered in chemical engineering applications, are noteworthy for coupled heat, and mass transport would be much more substantial if the fluid medium is kept at lower density compared to the surrounding medium by suspending chemical species. In numerous chemical processes, there is indeed a chemical species that involves latent mass transport. These processes can be found in a variety of industrial challenges, such as ceramics production, glassware industry, food processing, polymer production, and many more. EL-Kabeir [23] examined the effects of Cross-diffusions within existence of a chemical reaction over a stretching cylinder which is embedded in the porous medium. Dzulkifli et al. [24] studied nanofluid transport to identify Soret and Dufour effect over stretching/shrinking sheet. Some of the interesting results on Cross-diffusions with different flow geometry have been reported in previous studies (Gajjela et al. [25], Jain and Bohra [26], Gajjela and Garvandha [27], Idowu and Falodun [28]). Entropy-generation minimization is required from the engineering point of view and sustains the long life of the system. Bejan and Kestin [29] invented the entropy generation minimization approach and suggested its wide use in research, engineering, and industrial applications. Butt et al. [30] developed a theoretical model for entropy in the transport of Newtonian fluid over a stretching cylinder. Srinivasacharya and Shafeeurrahman [31] investigated the entropy generation in a nanofluid flow in concentric cylinders. Some of the impressive papers in entropy generation analysis for various fluids with diverse geometries can even be found in previous studies (Taghizadeh and Asaditaheri [32], Tufail et al. [33], Zheng et al. [34], Jha and Yusuf [35]). Recently Sahoo and Nandkeolyar [36] investigated the mixed convective flow of Casson nanofluid. The authors studied entropy generation and loss of heat energy. Thermal radiations and hall current effects are seen in entropy generation. Specific quasi-linearization and chebyshev collocation method is used to observe the effects of various parameters. Zahor et al. [37] suggested a review of the previously established models for irreversibility in magnetohydrodynamics (MHD) nanofluid transport systems. The authors used numerical methods to solve the equations in their modeling. Ankush Raje et al. [38] analyzed the entropy generation for viscoelastic Jeffery fluid transport through an inclined circular pipe. They considered the porous fluid medium

under constant pressure and applied a uniform magnetic field to the system. The finite difference approach is used in the mathematical solution. Copper nanoparticles are studied with blood due to low cost and other factors. Teh and Asghar [39] carried out investigations on MHD hybrid nanofluid transport in three dimensions through a shrinking/rotating sheet that is rotating. In the application of Joule heating effects authors reported that there is a significant proportion with the transport profiles, hybrid nanofluid, heat, and rotating parameters. Ferroudj et al. [40] examined the square cavity filled with fluid for mixed convection for finding facts for entropy generation due to irreversibility distribution ratio with Prandtl numbers. The authors have significantly quoted the irreversibility distribution ratio increases the entropy generation for all Prandtl numbers like the fluids under certain thermal conditions, air (mixture of gases), and water. Most recently, Sharma et al. [41] investigated entropy optimization for micropolar ternary hybrid nanofluid flow of blood. The authors have studied Bayesian regularization networks with homogeneous as well as heterogeneous chemical reactions. Mishra et al. [42] investigated the nanofluid flow behaviour in arteries having stenosis. The flow patterns and their impacts are seen with MHD applications. Paul et al. [43] investigated the flow rates for hybrid nano fluids over the nanofluids. The hybrid nanofluid is flowing through a vertically stretching cylinder. The authors found that thermal behaviors of hybrid nanofluid is significantly better over nanofluids as the thermal transport enhances up to 7.5% more under the impact of thermal stratification. The previously reported literature encourages validating the existing entropy optimization and understanding the variations in transport phenomenon due to various parameters reported. The findings listed above reveal that

not much more has been reported on the second law analysis over a stretching cylinder with heat and mass transfer. Such attempts at varying a fluid’s thermal conductivity have not been demonstrated. Motivated by this fact, we seek to quantitatively investigate the impacts of entropy, Soret, and Dufour across an inclined stretching cylinder in nanofluid flow with chemical reaction, heat generation.The non-dimensional equations are solved by an efficient algorithm NDSolve in MATHEMATICA and validated the solutions with Bvph2.0 which is based on Homotopy analysis method developed by Zhao and Liao [44].

Mathematical Formulation

Let us consider a 2-dimensional, time independent, boundary layer flow of incompressible nanofluid due to an inclined stretching cylinder of radius a. A magnetic field B0 normally gets applied to the stretching surface. The physical model and coordinates system are depicted in Figure 1. The cylindrical coordinates are employed, with the z -axis aligned with the axial direction and the r -axis aligned with the cylinder’s normal. Furthermore, the energy equation is taken into account with the combined effects of heat source and viscous dissipation. The thermal conductivity varies with temperature. The Cross-diffusions (Thermo-diffusion and Diffusothermal) effects are put into consideration, and the flow region is governing with a homogeneous chemical reaction of first-order. The continuity, momentum, energy, and concentration equations of nanofluid flow after using boundary layer approximation with the stated assumptions are written as [27, 30, 33]: (1)

Thermal energy Equation: (11) (3) (12) Concentration Equation: (4)

The physical quantity of interests Cf (Skin-friction coefficient), Nu (Nusselt number), and Sh(Sherwood number) are, in terms of similarity variables, given by: (13)

Below are the boundary conditions defined for the above model:

Equation (14) clearly depicts the entropy effects generated by heat transfer (HT), fluid friction (FF), diffusion (DF), and magnetic field (MF). In dimension-less form, the entropy generation number is

temperature, and concentrations respectively, T∞, C∞ is the ambient temperature and concentrations respectively. Let us consider the set of similarity transformation and non-dimensional variables [27, 30, 33] as

The Bejan number (Be) [29] is expressed as (16) The Equation (1) is verified by the aforementioned similarity transformations, and the corresponding governing equations of (2)-(6) are transformed by using the similarity variables in Equation (7). The obtained equations with boundary conditions are as follows: (8)

Results And Discussion

The numerical solutions of highly nonlinear, coupled, boundary value problems (8)-(12) are attained by Shooting technique with fourth order Runge-Kutta-Gill method. Further the obtained numerical results are validated by Liao’s Homotopy Analysis method using Bvph2.0

in MATHEMATICA. The asymptotic boundary conditions in Equation (12) as η → ∞ are substituted for those at such a large but finite value of η = 6. Figures 2 to 4 show the axial velocity variation for variant values of γ, M, and α. It is clear that the axial velocity rises with an increase of γ. Away from the axis of the cylinder,

the rate of transfer declines until it eventually disappears. In Figure 3(a), as M increases, the axial velocity decreases. This is due to Lorentz force which causes the reduction in the fluid velocity, and it increases the temperature. Figure 4(a) illustrates the axial velocity declining as α changes from π⁄ to π⁄ . This is due to the maximum gravitational force 6 3

Figure 2. Axial velocity variation with (a) distinct γ values (b) contour plot, when Pr = 0.7; M = 1; Grt = 0.3; Grc = 0.3; α = π/4; Q = 0.3; ε = 0.1; Nb = 0.1; Nt = 0.1; Br = 0.35; Df = 0.5; Sc = 0.5; Sr = 0.5; Cr= 0.3.

Figure 3. Axial velocity variation with (a) distinct M values (b) contour plot.

Figure 4. Axial velocity variation with (a) distinct α values (b) contour plot.

(i) (j) Figure 5. Temperature profile for (a) distinct ε values (b) contour plot (c) distinct Sr values (d) contour plot (e) distinct Df values (f) contour plot (g) distinct M values (h) contour plot (i) distinct γ values (j) contour plot , when γ = 0.2; Pr = 0.7; M = 1; Grt = 0.3; Grc = 0.3; α = π/4; Q = 0.3; ε = 0.1; Nb = 0.1; Nt = 0.1; Br = 0.35; Df = 0.5; Sc = 0.5; Sr = 0.5; Cr = 0.3

Figure 6. Concentration profile for (a) distinct Sr values (b) contour plot (c) distinct Df values (d) contour plot (e) distinct M values (f) contour plot (g) distinct γ values (h) contour plot , when γ = 0.2; Pr = 0.7; M = 1; Grt = 0.3; Grc = 0.3; α = π/4; Q = 0.3; ε = 0.1; Nb = 0.1; Nt = 0.1; Br = 0.35; Df = 0.5; Sc = 0.5; Sr = 0.5; Cr = 0.3.

experienced by the fluid and consequently, the velocity and the thickness of the boundary layer dropped significantly. Figure 5 depicts the temperature profile variation for different values of ε, Sr, Df, M, γ. From Figures 5(a) and 5(b), we can see that temperature increases with variable thermal conductivity (ε). This is due to our assumption that the thermal conductivity varies linearly with temperature. The transverse velocity is reduced, and the temperature profile rises as the variable thermal conductivity parameter increases. The temperature decelerates with increasing values of Soret (Sr) and Dufour number (Df) is shown in Figures 5(c)-5(f). Physically, Sr and Df provide a thermal and concentration gradient, respectively, which results in a decreased convective flow and a corresponding drop in temperature. In Figure 5(g), as M increases, the temperature increases. This is due to Lorentz force which causes the reduction in the fluid velocity and it increases the temperature. Temperature shows the rising trends with higher values of γ in Figure 5(i). As there will be an inverse proposition between curvature and radius, higher curvature leads to a reduction of the cylinder surface area. Figure 6 illustrates the concentration profile variation for different Sr, Df, M, and γ. We can see that from Figures 6(a)-6(d) concentration field enhances when Soret number (Sr), and Dufour number (Df) increases. The considered

higher numerical values of Sr and Df, results in a stronger convective flow that increases in concentration profile but in case of magnetic parameter and curvature parameter, the concentration profile decelerates as shown in Figures 6(e)-6(h). The reduced surface area enhances fluid flow over the surface, temperature, and decelerating concentration. To analyze the thermodynamic second law for this problem, the graphs (contour plots) are portrayed for entropy generation number (Ns) and Bejan number (Be) in Figures 7, 8. Figure 7(a) exhibits that the group parameter (Br/Ω1) increases, the Ns increases. If the value of the group parameter rises, fluid friction, and magnetic field influence the entropy effects of heat transfer on the stretching pipe. Yet, in the case of expanding pipe, such results are not in absolute control. It is evident from Figures 7(b)-7(d) that entropy generation number increases for ε, M and γ. Due to the magnetic force’s resistive nature, Ns increases as M increases. We can minimize entropy generation by weakening the magnetic field. The impacts of entropy begin to depreciate as one moves away from the surface, as shown in Figure 7(c). Figure 8 reveals that the maximum value of Bejan number is 1. Bejan number decreases with group parameter (Br/Ω1 ) and increases for ε, M and γ. The graph (8(c)-8(d)) shows that as M increases, heat transfer irreversibility at the surface of

Figure 7. Contour plot for Average entropy generation rate for (a) distinct modified Brinkman number (b) distinct ε (c) distinct M (d) distinct γ, when Grt = 0.3; Grc = 0.3; α = π/4; Q = 0.3; ε = 0.1; Nb = 0.1; Nt = 0.1; Br = 0.35; Df = 0.5; Sc = 0.5; Sr = 0.5; Cr = 0.3; γ = 1; Pr = 1; M = 1; Re = 2, ϕ1 = 1; ϕ2 = 1; Z = 1.

Figure 8. Contour plot for Bejan with (a) distinct modified Brinkman number (b) distinct ε (c) distinct M (d) distinct γ, when Grt = 0.3; Grc = 0.3; α = π/4; Q = 0.3; ε = 0.1; Nb = 0.1; Nt = 0.1; Br = 0.35; Df = 0.5; Sc = 0.5; Sr = 0.5; Cr = 0.3; γ = 1; Pr = 1; M = 1; Re = 2, ϕ1= 1; ϕ2 = 1; Z = 1.

Table 1. Comparison of the numerical solution with Bvph 2.0 when γ = 0.1; Pr = 0.7; M = 1; Grt = 0.7; Grc = 0.2; α = π⁄4; Q = 0.1; ϵ = 0.1; Nb = 0.1; Nt = 0.1; Br = 0.35; Df = 0.5; Sc = 0.5; Sr = 0.5; Cr = 0.3 f ' (η)

Error

the stretching pipe is dominated by viscosity and magnetic irreversibility. Table 1 shows that both the techniques, numerical solution by shooting technique with Gill’s fourth order method, and Bvph2.0 suits each other. From Table 2, we observed that the drag force at the walls/ surfaces, heat transfer rate

increases, and mass transfer rate decreases for increasing values of γ, Pr, Nt, Df. For increasing values of α, the skin friction, Sherwood number increases and Nusselt number decreases. The drag force at the walls/ surfaces, Sherwood number decreases, and heat transfer rate increases for increasing values of ε, Nb, Sr.

Table 2. Numerical values of -f ''(0), -θ'(0), -ϕ'(0) at M = 1; Grt = 0.7; Grc = 0.2; Q = 0.1; Cr = 0.3; Sc = 0.5 γ

Conclusion

The impacts of variable thermal conductivity, entropy generation rate, heat transfer rate, Soret and Dufour parameters in nanofluid flow over inclined stretching cylinder are studied under the influence of viscous dissipation and internal heat source using Buongiorno model. The major findings are concluded as; (i) The higher intensity of the magnetic field (M = 0, 1, 2) decreases axial velocity and concentration, but a reverse trend appears for the temperature field. This is happening due to the force generated in the opposite direction to the flow. (ii) The temperature profile rises with increasing values of ϵ, γ (0 ≤ γ ≤ 1) and decelerates with α, Sr, and Df. Rise of temperature produces more mass transfer that disturbs the energy of the system. This effect needs to administer the entropy. (iii) The axial velocity decelerates with inclination angle (π⁄6 ≤ α ≤ π⁄3) increases. The inclination angle is affecting the flow transport rate that will create the major change in the energy of the system that will be leading cause of administering the entropy of the system.

(iv) The concentration distribution increases for a given increase in Sr, and decrease in M, γ. (v) Entropy increases with magnetic parameter, curvature parameter, group parameter, and variable thermal conductivity. (vi) Bejan number increases with increase of γ, ϵ, and M and decreases with group parameter. It is observed that some parameters like magnetic field, curvature, group, and variable thermal conductivity increase entropy that will increase the energy of the system. Higher energy in the system will create the disorder. This disorder causes more turbulence in physical systems, especially in biological processes. From the administrative/clinical point of view entropy must be optimized that will be good for the industrial, medical, and clinical applications. Future Scope The mathematical techniques employed to solve the current physical model exhibit great agreement with the past research that are now available, which supports the veracity of the current physical findings. Engineers can benefit from this paper’s extension to hybrid nanofluid or two-phase fluids based on these findings. Engineers have

good options for comparing the amount of heat released by the system due to nanoparticle movements in thermal processes experimentally and theoretically.

Nomenclature

Be Br Bo C C∞ Cp C *p Cs Cr Cw D DB DT Df Ec g Grc Grt k k1 kT l M Nb Nt Pr Ns Q Qo Re Rez Sc SG T Tm T0 Tw T∞ (r, z) U0 (u, w)

Bejan number Brinkman number constant magnetic field fluid concentration (kg/m3 ) ambient nanoparticle concentration specific heat (J/kg-K) specific heat of nanofluid concentration susceptibility Chemical reaction parameter Concentration at the surface molecular diffusivity (m2 /s) Brownian diffusion coefficient (m2 /s) thermophoresis diffusion coefficient (m2 /s) Dufour number Eckert number gravitational acceleration (m/s2 ) Local concentration Grashof number Local temperature Grashof number thermal conductivity (Wb/m2 ) Rate of reaction Thermal diffusion ratio the characteristic length(m) Magnetic field number(W/mK) Brownian motion parameter Thermophoretic parameter Prandtl number Non dimensional Entropy generation number heat source/sink parameter heat coefficient Reynolds number local Reynolds number Schmidt number Entropy generation fluid temperature(K) Mean temperature(K) reference temperature(K) Temperature at the surface(K) ambient temperature(K) Radial and axial coordinates in cylindrical system reference velocity velocity components in the (r, z) direction(m/s)

Greek Symbols α Angle of inclination the thermal expansion coefficient βT concentration expansion coefficient βC ϵ Thermal conductivity parameter μ dynamic viscosity of fluid(Pa.s) ρ density of fluid(Kg/m3 ) ρ* density of nanofluid

Similarity variable Stefan–Boltzmann constant cylinder curvature parameter kinematic viscosity electrical conductivity (or specific conductance) (S/m) dimensionless temperature dimensionless concentration Temperature and concentration difference coefficient irreversibility distribution ratios

Acknowledgement

The conduct of this research study is not currently supported by any funding source.

Data Availability Statement

The authors confirm that the data that supports the findings of this study are available within the article. Raw data that support the finding of this study are available from the corresponding author, upon reasonable request.

Conflict Of Interest

The author declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Ethics

There are no ethical issues with the publication of this manuscript.

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