Irreversibility Analysis of Hybrid Nanoﬂuid Flow over a Thin Needle with E ﬀ ects of Energy Dissipation

: The ﬂow and heat transfer analysis in the conventional nanoﬂuid Al 2 O 3 − H 2 O and hybrid nanoﬂuid Cu − Al 2 O 3 − H 2 O was carried out in the present study. The present work also focused on the comparative analysis of entropy generation in conventional and hybrid nanoﬂuid ﬂow. The ﬂows of both types of nanoﬂuid were assumed to be over a thin needle in the presence of thermal dissipation. The temperature at the surface of the thin needle and the ﬂuid in the free stream region were supposed to be constant. Modiﬁed Maxwell Garnet (MMG) and the Brinkman model were utilized for e ﬀ ective thermal conductivity and dynamic viscosity. The numerical solutions of the self-similar equations were obtained by using the Runge-Kutta Fehlberg scheme (RKFS). The Matlab in-built solver bvp4c was also used to solve the nonlinear dimensionless system of di ﬀ erential equations. The present numerical results were compared to the existing limiting outcomes in the literature and were found to be in excellent agreement. The analysis demonstrated that the rate of entropy generation reduced with the decreasing velocity of the thin needle as compared to the free stream velocity. The hybrid nanoﬂuid ﬂow with less velocity was compared to the regular nanoﬂuid under the same circumstances. Furthermore, the enhancement in the temperature proﬁle of the hybrid nanoﬂuid was high as compared to the regular nanoﬂuid. The inﬂuences of relevant physical parameters on ﬂow, temperature distribution, and entropy generation are depicted graphically and discussed herein.


Introduction
The first law of thermodynamics deals with conservation and conversion of energy. It stipulates that when a thermodynamic process is carried out, energy is neither gained nor lost. Energy only transforms from one form into another, and the energy balance is maintained. The law presumes that any change of a thermodynamic state can take place in either direction. However, this is not true, particularly in the inter-conversion of heat and work. Processes proceed spontaneously in certain directions-but not in opposite directions-even though the reversal of processes does not violate

Mathematical Formulation
Consider the flow of hybrid nanofluid Cu − Al 2 O 3 − H 2 O over a thin needle with frictional heating (energy dissipation). The coordinate system and the geometry of the physical flow model are shown in Figure 1. The free stream velocity (u * ∞ ) is in the direction of the x-axis, i.e., along the surface of the thin needle. The velocity of the thin needle (u * w ) and the temperature at the surface of the thin needle (T * w ) are supposed to be constant. Furthermore, the temperature in the free stream region (T * ∞ ) is considered to be constant such that T * w > T * ∞ . Following the Tiwari and Dass Model (TDM) [36] with boundary layer approximations, the governing equations for the hybrid nanofluid are: with the following boundary conditions: mathematical formulation, a numerical solution, and a graphical representation of the numerical solution, qualitative discussions, and concluding remarks.

Mathematical Formulation
Consider the flow of hybrid nanofluid Model (TDM) [36] with boundary layer approximations, the governing equations for the hybrid nanofluid are: with the following boundary conditions:  Here, (u * , v * ) represents the velocity components along the x * and r * -axes, respectively, T * shows the local temperature within the boundary layer, R(x * ) = aν b f x * U * [37] describes the shape of the surface of the thin needle, a shows the size of the needle, x * is the axial coordinate, ν b f represents the kinematic viscosity of the base fluid, and U * shows the composite velocity. On the other hand, the effective thermal conductivity k hn f , viscosity µ hn f , specific heat at constant pressure ρC p hn f , density ρ hn f , and thermal diffusivity α hn f of the hybrid nanofluid are defined as [33]: We introduce the dimensionless variables as [37]: where ψ denotes stream function defined such that u * = 1 r * ∂ψ ∂r * and v * = − 1 r * ∂ψ ∂x * , g(ξ) stands for dimensionless stream function, ξ is the similarity variable, θ presents the dimensionless temperature distribution, the subscripts hn f and b f denote the hybrid nanofluid and the base fluid, respectively, and U * = u * w + u * ∞ 0 indicates composite velocity. Equation (1) is identically satisfied by Equation (9), whereas Equations (2)-(4) yield: Here, ε stands for velocity ratio parameter, φ Cu and φ Al 2 O 3 for the solid volume fraction of copper and aluminum oxide nanoparticles, respectively, Ec for Eckert number, and Pr for Prandtl number. The above-mentioned parameters are defined as follows:

Irreversibility Analysis
The entropy generation due to frictional and thermal irreversibilities in the two-dimensional flow of the hybrid nanofluid over a thin needle is given by [38]: Symmetry 2019, 11, 663 5 of 14 The first term on the right side of Equation (14) indicates the heat transfer irreversibility (H.T.I) (entropy generation due to heat transfer), and the last term represents the frictional irreversibility (entropy generation due to frictional heating). To non-dimentionalized Equation (14), the characteristic entropy generation . E Gen c is defined as given below: .
With the utilization of dimensionless variables defined in Equations (9) and (15), Equation (14) takes the following useful form: Here, N HT = is the fluid friction irreversibility.

Numerical Solution
The non-linear self-similar Equations (10) and (11) with the dimensionless boundary conditions were solved by applying the Fehlberg fourth order Runge-Kutta method and the shooting technique. The Fehlberg fourth order Runge-Kutta method works for first order initial value problems. Since our problem was boundary value and higher order, we first reduced our problem to a first order initial value problem. The following are the three basic steps: i.
Convert Equations (10) and (11) to a set of first order initial value problems. ii.
The shooting technique is used to determine the missing initial conditions such that the conditions at ξ → ∞ are satisfied. iii.
Finally, the Fehlberg fourth order Runge-Kutta method (initial value problem method) is utilized to get the required numerical solutions.
The fourth order accurate solution is defined as follows: Each iteration consists of five steps that are given below: Equations (10) and (11) with the corresponding boundary conditions (12) are transformed to a system of initial value problems by taking: By substituting Equation (19) in the Equations (10)- (12), we obtain a system of first order differential equations with initial conditions, as given below: The missing initial conditions s 1 and s 2 are calculated by the utilization of an iterative scheme known as the shooting technique such that the boundary conditions g (ξ → ∞) → 1−ε 2 and θ(ξ → ∞) → 0 are satisfied. The iterative procedure continues until the solution converges to the desired accuracy of 10 −5 . Furthermore, the step size used in the simulation is ∆ξ = 0.001.

Results and Discussion
The dimensionless Equations (10) and (11) were highly nonlinear and had variable coefficients. Therefore, it was not possible to obtain the closed form solution of these equations. This fact forced us to solve these equations numerically. The Runge-Kutta Fehlberg schemes (RKFS) and the shooting method (SM) were utilized to get the solution of Equations (10) and (11) numerically. To analyze the impacts of different parameters on g (ξ), θ(ξ) and Ns(ξ), the obtained numerical results were plotted against the similarity variable ξ for various values of emerging parameters. The solid volume fractions of aluminum oxide and copper nanoparticles are represented by φ 1 and φ 2 , respectively, whereas φ = φ 1 + φ 2 . The thermophysical properties of nanoparticles and base fluid (water) are tabulated in Table 1. Table 2 shows the comparison of the present numerical values and the existing values in the literature when φ 1 = φ 2 = 0 and ε = 0. It was found that the numerical values were close enough and validated our numerical simulation.  Figure 2 demonstrates the variation of the velocity profile g (ξ) with the thickness of the thin needle a. The plot shows that g (ξ) enhanced with the decreasing of a. Physically, the drag force decreased as the size of the needle diminished; consequently, the velocity enhanced. The hybrid nanofluid flowed with less velocity compared to the regular nanofluid. This was because the density rose with the hybridity and consequently slowed down the fluid motion.  enhanced with the decreasing of a . Physically, the drag force decreased as the size of the needle diminished; consequently, the velocity enhanced. The hybrid nanofluid flowed with less velocity compared to the regular nanofluid. This was because the density rose with the hybridity and consequently slowed down the fluid motion.   Figure 3 also reveals that, under the same circumstances, the temperature of the hybrid nanofluid was higher than the regular nanofluid. High thermal conductivity of the hybrid nanofluid was responsible for these high values of temperature.  Figure 3 also reveals that, under the same circumstances, the temperature of the hybrid nanofluid was higher than the regular nanofluid. High thermal conductivity of the hybrid nanofluid was responsible for these high values of temperature.  The increase in the thermal dissipation parameter (Eckert number Ec) enhanced the temperature of both types of nanofluids, as demonstrated in Figure 4. This rise in temperature was due to the increment in friction between the nanofluid layers. The kinetic energy of the nanofluid converted into the thermal energy because of the frictional heating, which lead to rise in the temperature. The increase in the thermal dissipation parameter (Eckert number Ec ) enhanced the temperature of both types of nanofluids, as demonstrated in Figure 4. This rise in temperature was due to the increment in friction between the nanofluid layers. The kinetic energy of the nanofluid converted into the thermal energy because of the frictional heating, which lead to rise in the temperature.  The temperature had direct relation with the solid volume fraction of the nanoparticles, as shown in Figure 5. Furthermore, it was also observed that the hybrid nanofluid had a thick thermal boundary layer as compared to the regular nanofluid, and this was because of the high thermal conductivity of the hybrid nanofluid. The temperature had direct relation with the solid volume fraction of the nanoparticles, as shown in Figure 5. Furthermore, it was also observed that the hybrid nanofluid had a thick thermal boundary layer as compared to the regular nanofluid, and this was because of the high thermal conductivity of the hybrid nanofluid. The effect of the thickness of the thin needle on entropy generation is represented in Figure   6. The plot reveals that Ns decreased with increasing for both types of nanofluids, but for the fixed value of , less entropy was generated in the regular nanofluid as compared to the hybrid nanofluid. Furthermore, the surface of the thin needle was the region where maximum entropy was generated.  The effect of the thickness of the thin needle on entropy generation Ns is represented in Figure 6. The plot reveals that Ns decreased with increasing a for both types of nanofluids, but for the fixed value of a, less entropy was generated in the regular nanofluid as compared to the hybrid nanofluid. Furthermore, the surface of the thin needle was the region where maximum entropy was generated.
The effect of the thickness of the thin needle on entropy generation is represented in Figure   6. The plot reveals that Ns decreased with increasing for both types of nanofluids, but for the fixed value of , less entropy was generated in the regular nanofluid as compared to the hybrid nanofluid. Furthermore, the surface of the thin needle was the region where maximum entropy was generated.     The reduction in entropy generation was observed with the rising of ω , as shown in Figure 9. The reduction in entropy generation was observed with the rising of ω, as shown in Figure 9. The impacts of the velocity ratio parameter ε on entropy generation Ns are shown in Figures 10 and 11. It was found that entropy generation enhanced with the velocity ratio parameter ε subjected to the condition that the free stream velocity was less than the velocity of the thin needle. The decrement in Ns was observed with the rising of the velocity ratio parameter subjected to the condition that the velocity of the thin needle was less than that of the free stream velocity.  The impacts of the velocity ratio parameter ε on entropy generation Ns are shown in Figures 10  and 11. It was found that entropy generation enhanced with the velocity ratio parameter ε subjected to the condition that the free stream velocity was less than the velocity of the thin needle. The decrement in Ns was observed with the rising of the velocity ratio parameter subjected to the condition that the velocity of the thin needle was less than that of the free stream velocity.
The impacts of the velocity ratio parameter ε on entropy generation Ns are shown in Figures 10 and 11. It was found that entropy generation enhanced with the velocity ratio parameter ε subjected to the condition that the free stream velocity was less than the velocity of the thin needle. The decrement in Ns was observed with the rising of the velocity ratio parameter subjected to the condition that the velocity of the thin needle was less than that of the free stream velocity.

Conclusions
A theoretical study of heat transfer and entropy generation in the flow of dissipative hybrid nanofluid over a needle was conducted. The results of this study led to the following prime conclusions: • The temperature and the entropy generation were found to decrease with needle size decrement.

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The velocity profile was reduced with the increment in the size of the needle.

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The velocity of the hybrid nanofluid  . Effects of ε on Ns(ξ) when u * w < u * ∞ .

Conclusions
A theoretical study of heat transfer and entropy generation in the flow of dissipative hybrid nanofluid over a needle was conducted. The results of this study led to the following prime conclusions: • The temperature and the entropy generation were found to decrease with needle size decrement.

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The velocity profile was reduced with the increment in the size of the needle.

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The velocity of the hybrid nanofluid Cu − Al 2 O 3 − H 2 O was observed to be lower than the regular nanofluid Al 2 O 3 − H 2 O, whereas the rate of heat transfer was greater in the hybrid nanofluid as compared to the regular nanofluid. • A reduction in entropy generation Ns was found by raising the values of ω.

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It was perceived that Ns and temperature distribution were directly proportional to Eckert number Ec and φ.

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High entropy generation was found in the hybrid nanofluid as compared to the regular one.