Statistical Analysis of the Mathematical Model of Entropy Generation of Magnetized Nanofluid

Abstract

This investigation introduces a mathematical model of entropy generation for Magnetohydrodynamic (MHD) peristaltic wave of nanofluid. The governing equations have been created by the supposition of low Reynolds number and long wavelength estimation. The scientific arrangement has been procured with the help of perturbation technique. The concentration profile, temperature profile, pressure distribution and friction forces are shown graphically for some important parameters. Further, the eventual outcomes of connection between the entropy generation and some various parameters have been plotted by means of correlation and regression. It is fundamental to find the affectability of each parameter on entropy generation.

1. Introduction

In fluid mechanics, we think about the conduct of particles at each point inside a space under different physical conditions. Numerical models have been utilized for various sorts of fluids, such as Newtonian fluid [1,2] and non-Newtonian fluid [3,4] to depict the physical marvels in fluid mechanics. There are numerous special instances of non-Newtonian fluid (e.g., nanofluids, micropolar fluids) and numerous studies [5,6,7,8,9,10,11,12] have utilized a certain logical technique to illuminate distinctive kinds of fluids by creating models. Nanotechnology is considered to be an ideal innovative answer for tackling the worldwide energy crisis. Indeed, nanofluid is the fluid suspension of nanostructures, which predominates and guarantees the fundamental amplification of heat transfer properties of the fluid. This helps us to encounter the prospective complexities of fluids in various fluid configuration.

The use of heat transfer fluids is one of the technological applications of nanoparticles, and it possesses an enormous capacity to suspend nanoparticles and confront cooling problems in thermal systems. Due to the great demands placed upon heat transfer fluids in terms of decreasing or increasing energy release to systems, a significant research work was undertaken by Choi and Eastman [13,14] in which a mixture of nanoparticles and base fluid were designated as “nanofluid”. They defined a liquid of ultra-fine particles with sizes less than 100 nm. In the field of thermal engineering and heat transfer, nanofluid has always been an engrossing term. Peristalsis has various applications in connection with nanofluid, such as in engineering, bio-sciences and industries. Several theoretical and experimental attempts in this area have been conducted in the past. Specifically, the works of Latham [15] play a very important role in this connection. Similarly, because of its multiple advantages, research findings on peristaltic flow have received wider applications in industries, and numerous attempts have been made in literature to explore this direction of research. Abbas et al. [16] discussed the application of drug delivery systems under the influence of MHD with peristaltic motion. Bhatti et al. [17] examined the combined effects of MHD and partial slip on the peristaltic flow of nanofluid. Similarly, Salleh et al [18] studied nanofluid under the influence of magnetic fields in a moving vertical thin needle. A few more magnificent researches can be viewed in the available reference [19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42].

Use of second law investigation in heat design provides a likelihood of advancing a given framework or procedure on the premise of vitality quality, which is altogether different from first law examination. Sustainable power source is the type of vitality determined/gathered from different regular procedures and, as the name proposes, renewable energy sources always recharge inside nature. Specifically, entropy generation investigation amid natural convection heat movement in encased cavities (inside characteristic convection) with different setups has been an area of concentrated examination throughout the previous two decades. Entropy generation determines the dimension of irreversible heat in a procedure. Thus, entropy creation can be utilized as a standard for the evaluating the exhibition of building gadgets. Lately, a huge mass of research has been conducted to check the rate of the entropy generation amid normal convection in different design and ecological applications, for the productive utilization of the accessible vitality. At present, entropy generation advancement is the subject of interests in multiple areas, including heat exchange devices, combustion, electric cooling and permeable media. Rashidi et al. [43] explored the entropy of peristaltic flow in nanofluids with magnetic effects. Mohesen Turabi et al. [44] reviewed entropy generation in a thermal engineering system with solid structures. A comparison table between entropy generation and energy efficiency in natural convection has been analyzed by Pratibha and Tanway [45]. Some later works on entropy generation are [46,47,48,49,50].

Keeping in mind the aforementioned discussion, correlation and regression have not been investigated in any of the aforementioned studies. Therefore, the aim of the present study is to investigate the correlation and regression of entropy generation of MHD peristaltic flow of nanofluid with a porous medium. For this purpose, the study applies the situation of a low Reynolds number and a long wavelength using an analytical technique named the homotopy perturbation method (HPM), which is used to solve the simplified partial differential equations. Expression of temperature, concentration pressure and entropy generation have been obtained graphically. Based on the results of the entropy generation, correlation and regression were derived and explained the role of some pertinent parameters on entropy generation. Due to the vast importance of entropy generation in engineering, exchanging heat devices and electric cooling, this kind of investigation can be much beneficial to finding the sensitivity of each parameter on objective function that is considered to be entropy generation in this model.

2. Mathematical Formulation

We present a demonstration of the peristaltic movement with thick, electrically leading and incompressible nanofluid properties through a two-dimensional, non-uniform channel with sinusoidal wave engendering towards down its wall. As shown in Figure 1, a Cartesian coordinate framework is used so that the x axis is considered alongside the middle line, toward the wave propagation, with y axis traversing to it. The $B_0$, a uniform, outer attractive field, is forced along the y pivot and the initiated attractive field is thought to be irrelevant. The geometry of the divider surface is characterized as

$$H\left(\tilde{x},\tilde{t}\right)=\check{a}\sin \frac{2\pi }{\lambda } \left(\tilde{x}-C\tilde{t}\right)+b\left(\check{x}\right)$$

(1)

where $b\left(\tilde{x}\right)= b_0+ \mathsf{{\rm K}}\tilde{x}$.

The governing equation of motion, continuity, thermal energy and nanoparticle fraction for peristaltic nanofluid can be written as [16]

$$\frac{\partial \tilde{u}}{\partial \tilde{x}}+\frac{\partial \tilde{v}}{\partial \tilde{y}}=0,$$

(2)

$${\rho }_f\left(\frac{\partial \tilde{u}}{\partial \tilde{t}}+\tilde{u}\frac{\partial \tilde{u}}{\partial \tilde{x}}+\tilde{v}\frac{\partial \tilde{u}}{\partial \tilde{y}}\right)=-\frac{\partial \tilde{p}}{\partial \tilde{x}}+\frac{\partial }{\partial \tilde{x}}{\mathrm{S}}_{\tilde{x}\tilde{x}}+\frac{\partial }{\partial \tilde{y}}{\mathrm{S}}_{\tilde{x}\tilde{y}}-\sigma B_0\tilde{u}-\frac{\mu }{\tilde{k}}\tilde{u}+g\left[\begin{array}{c}\left(1-F\right){\rho }_{f_0}\zeta \left(T-T_0\right)-\\ \left({\rho }_p-{\rho }_{f_0}\right)\left(F-F_0\right)\end{array}\right]$$

(3)

$${\rho }_f\left(\frac{\partial \tilde{v}}{\partial \tilde{t}}+\tilde{v}\frac{\partial \tilde{u}}{\partial \tilde{x}}+\tilde{v}\frac{\partial \tilde{v}}{\partial \tilde{y}}\right)=-\frac{\partial \tilde{p}}{\partial \tilde{y}}+\frac{\partial }{\partial \tilde{x}}{\mathrm{S}}_{\tilde{y}\tilde{x}}+\frac{\partial }{\partial \tilde{x}}{\mathrm{S}}_{\tilde{y}\tilde{y}}-\sigma B_0\tilde{v}-\frac{\mu }{k}\tilde{v}+g\left[\begin{array}{c}\left(1-F\right){\rho }_{f_0}\zeta \left(T-T_0\right)-\\ \left({\rho }_p-{\rho }_{f_0}\right)\left(F-F_0\right)\end{array}\right]$$

(4)

$${\left(\rho c\right)}_f\left(\frac{\partial T}{\partial \tilde{t}}+\tilde{u}\frac{\partial T}{\partial \tilde{x}}+\tilde{v}\frac{\partial T}{\partial \tilde{y}}\right)=\kappa \left(\frac{{\partial }^{2}T}{\partial {\tilde{x}}^2}+\frac{{\partial }^{2}T}{\partial {\tilde{y}}^2}\right)+{\left(\rho c\right)}_p{\mathrm{D}}_B\left(\frac{\partial T}{\partial \tilde{x}}\frac{\partial F}{\partial \tilde{x}}+\frac{\partial F}{\partial \tilde{y}}\frac{\partial T}{\partial \tilde{y}}\right)+\frac{{\mathrm{D}}_T}{T_0}\left({\left(\frac{\partial T}{\partial \tilde{x}}\right)}^2+{\left(\frac{\partial T}{\partial \tilde{y}}\right)}^2\right)-\frac{\partial q_r}{\partial \tilde{y}}+Q_0,$$

(5)

$$\left(\frac{\partial F}{\partial \tilde{t}}+\tilde{u}\frac{\partial F}{\partial \tilde{x}}+\tilde{v}\frac{\partial F}{\partial \tilde{y}}\right)={\mathrm{D}}_B\left(\frac{{\partial }^{2}F}{\partial {\tilde{x}}^2}+\frac{{\partial }^{2}F}{\partial {\tilde{y}}^2}\right)+\frac{{\mathrm{D}}_T}{T_0}\left(\frac{{\partial }^{2}T}{\partial {\tilde{x}}^2}+\frac{{\partial }^{2}T}{\partial {\tilde{y}}^2}\right)-k_1\left(F-F_0\right),$$

(6)

Now let us consider the assumptions of a long wavelength number and low Reynolds approximations in the sense of creeping flow. By using the dimensionless quantities in Equations (2)–(6), we get the resulting equations in a simplified form as

$$\frac{{\partial }^{2}u}{\partial y^2}+We\frac{\partial }{\partial y}{\left(\frac{\partial u}{\partial y}\right)}^2-\frac{1}{k}u-M^{2}u-G{r}_F\Phi +G{r}_T\theta -\frac{\partial p}{\partial x}=0,$$

(7)

$$\left(\frac{1+{\mathrm{R}}_{\mathrm{n}}}{\mathrm{Pr}}\right)\frac{{\partial }^2\theta }{\partial y^2}+N_t{\left(\frac{\partial \theta }{\partial y}\right)}^2+\beta +N_b\frac{\partial \theta }{\partial y}\frac{\partial \mathsf{\Phi }}{\partial y}=0,$$

(8)

$$\frac{{\partial }^2\Phi }{\partial y^2} -\gamma \Phi +\frac{N_t}{N_b}\left(\frac{{\partial }^2\theta }{\partial y^2}\right)=0.$$

(9)

subject to the respective boundary conditions:

$$\Phi \left(0\right)=0, \frac{\partial u\left(0\right)}{\partial y}=0, \theta \left(0\right)=0,$$

(10)

$$\Phi \left(h\right)=1, \theta \left(h\right)=1, u\left(h\right)=0$$

(11)

In the presence of a magnetic field, the entropy generation can be derived from energy and entropy balance for the case of heat and mass transfer as [43]

$$\begin{array}{c}{\mathrm{S}}_{\mathrm{gen}}=\frac{{\mathcal{K}}_{nf}}{T_0^2}{\left(\nabla \mathrm{T}\right)}^2+\frac{{\mu }_{nf}}{\tilde{k}{T}_0}\left[2{\left(\frac{\partial \tilde{u}}{\partial \tilde{x}}\right)}^2+2{\left(\frac{\partial \tilde{v}}{\partial \tilde{y}}\right)}^2+{\left(\frac{\partial \tilde{u}}{\partial \tilde{y}}+\frac{\partial \tilde{v}}{\partial \tilde{x}}\right)}^2\right]+\frac{\sigma B_0^2}{T_0}{\left(\frac{\partial \tilde{u}}{\partial \tilde{y}}\right)}^2+\\ \frac{R{\mathrm{D}}_B}{F_0}{\left(\nabla F\right)}^2+\frac{R{\mathrm{D}}_B}{T_0}\left(\nabla F·\nabla T\right)\end{array}$$

(12)

The dimensionless form of the entropy generation number can be expressed as follows:

$$N_s=\frac{{\mathrm{S}}_{\mathrm{gen}}}{{\mathrm{S}}_{\mathrm{g}}}=\left(\frac{{\mathcal{K}}_{nf}}{{\mathcal{K}}_f}\right)\left({\left(\frac{\partial \theta }{\partial y}\right)}^2\right)+\left(1+M^2\right)B_r\frac{1}{\mathsf{\Omega }}\left(\frac{{\mu }_{nf}}{{\mu }_f}\right){\left(\frac{\partial u}{\partial y}\right)}^2+\mathsf{\Gamma }{\left(\frac{\mathsf{\Lambda }}{\mathsf{\Omega }}\right)}^2{\left(\frac{\partial \Phi }{\partial y}\right)}^2+ \zeta \left(\frac{\partial \theta }{\partial y}\right)\left(\frac{\partial \mathsf{\Phi }}{\partial y}\right),$$

(13)

where $\mathsf{\Omega }, B_r, \mathsf{\Lambda } ,\mathsf{\Gamma },\zeta$ are the dimensionless temperature difference, Brinkman number, concentration difference, diffusive coefficient and constant parameter, represented as

$$\mathsf{\Omega }=\frac{\left(T_1-T_0\right)}{T_0},B_r=\frac{{\tilde{c}}^2{\mu }_f}{\tilde{k}{\mathcal{K}}_f\left(T_1-T_0\right)}, \zeta =\frac{R{D}_B{T}_0}{{\mathcal{K}}_f}\left(\frac{F_1-F_0}{T_1-T_0}\right),\mathsf{\Gamma }=\frac{R{\mathrm{D}}_B{F}_0}{{\mathcal{K}}_f},\mathsf{\Lambda }=\frac{F_1-F_0}{F_0} .$$

(14)

For nanofluid, the viscosity model and thermal conductivity can be defined as [50]

$${\mu }_{nf}=\frac{{\mu }_f}{{\left(1-\overline{\varphi }\right)}^{2.5}},{\mathcal{K}}_{\mathrm{nf}}=\frac{{\mathsf{\kappa }}_{\mathrm{p}}+2{\mathsf{\kappa }}_{\mathrm{f}}+2\overline{\mathsf{\varphi }}\left({\mathsf{\kappa }}_{\mathrm{p}}-{\mathsf{\kappa }}_{\mathrm{f}}\right)}{{\mathsf{\kappa }}_{\mathrm{p}}+2{\mathsf{\kappa }}_{\mathrm{f}}-\overline{\mathsf{\varphi }}\left({\mathsf{\kappa }}_{\mathrm{p}}-{\mathsf{\kappa }}_{\mathrm{f}}\right)}{\mathsf{\kappa }}_{\mathrm{f}}$$

(15)

where ${\mu }_f$ ${\kappa }_f$ and ${\kappa }_p$, are the viscosity f base fluid, thermal conductivities of the nanofluid and nanoparticle, respectively.

3. Solution of Problem

With the help of HPM [16], Equations (7)–(9) can be written as:

$$\mathcal{H}\left(w,\tilde{q}\right)=\left(1-\acute{q}\right)\left(L_1\left(w\right)-L_1\left({\overline{w}}_0\right)\right)+\acute{q}\left(L_1\left(w\right)+ We\frac{\partial }{\partial y}{\left(\frac{\partial w}{\partial y}\right)}^2+G{r}_T\mathsf{\Theta }-G{r}_F\vartheta -\frac{\partial p}{\partial x}\right),$$

(16)

$$\mathcal{H}\left(\mathsf{\Theta },\tilde{q}\right)=\left(1-\acute{q}\right)\left(L_2\left(\mathsf{\Theta }\right)-L_2\left({\mathsf{\Theta }¯}_0\right)\right)+\acute{q}\left(L_2\left(\mathsf{\Theta }\right)+\frac{\mathrm{Pr}}{1+{\mathrm{R}}_{\mathrm{n}}\mathrm{Pr}}\left(N_b\frac{\partial \vartheta }{\partial y}\frac{\partial \mathsf{\Theta }}{\partial y}+N_t{\left(\frac{\partial \vartheta }{\partial y}\right)}^2\right)+\frac{\mathrm{Pr}\beta }{1+{\mathrm{R}}_{\mathrm{n}}\mathrm{Pr}}\right)$$

(17)

$$\mathcal{H}\left(\vartheta ,\tilde{q}\right)=\left(1-\acute{q}\right)\left(L_2\left(\vartheta \right)-L_2\left({\overline{\vartheta }}_0\right)\right)+\acute{q}\left(L_2\left(\vartheta \right)+ \frac{N_t}{N_b}\left(\frac{{\partial }^2\mathsf{\Theta }}{\partial y^2}\right)-\gamma \vartheta \right),$$

(18)

and the initial guess and linear operators for Equations (16)–(18) are defined as

$${\overline{w}}_0=\frac{\cosh {\mathrm{N}}^{2}y-\cosh {\mathrm{N}}^{2}h }{\cosh {\mathrm{N}}^{2}h},$$

(19)

$${\overline{\vartheta }}_0={\overline{\mathsf{\Theta }}}_0=\frac{y}{h}.$$

(20)

$$L_1=\frac{{\partial }^2}{\partial y^2}-M^2-\frac{1}{k}$$

(21)

$$L_2=\frac{{\partial }^2}{\partial y^2},$$

(22)

which defines the following expansion:

$$\vartheta \left(x,y\right)={\vartheta }_0\left(x,y\right)+\acute{q}{\vartheta }_1\left(x,y\right)+{\acute{q}}^2{\vartheta }_2\left(x,y\right)+\dots ,$$

(23)

$$\mathsf{\Theta }\left(x,y\right)={\mathsf{\Psi }}_0\left(x,y\right)+\acute{q}{\mathsf{\Psi }}_1\left(x,y\right)+{\acute{q}}^2{\mathsf{\Psi }}_2\left(x,y\right)+\dots ,$$

(24)

$$w\left(x,y\right)=w_0\left(x,y\right)+\acute{q}{w}_1\left(x,y\right)+{\acute{q}}^2{w}_2\left(x,y\right)+\dots ,$$

(25)

Using the expansion series defined in the terms mentioned in Equations (23)–(25) and incorporating them into the Equations (16)–(18) we get a system of linear differential equations and their relevant boundary conditions. Applying the scheme of HPM and comparing the powers of $\acute{q}$, we obtain the solution as $\acute{q} \to 1$. We obtained the temperature distribution, velocity profile, and concentration profile.

Utilizing the expensing arrangement characterized in terms of ($\vartheta \left(x,y\right)$,($\mathsf{\Theta }\left(x,y\right)$ and ($w\left(x,y\right)$) as referenced in Equations (23)–(25) and using into the Equations (16)–(18), we get an arrangement of direct differential equations with their significant limit conditions. By contrasting the forces of $\acute{q} \to 1$, we apply the scheme of HPM to determine the arrangement as q′ → 1, and obtain the required arrangements of temperature circulation, speed profile, and fixation profile.

4. Results and Discussion

In this section the obtained results are discussed. As shown in Figure 2, for higher values of $N_b$ and $N_t$ the temperature profile increases. This is because the Brownian motion creates micro- mixing, which increases thermal conductivity. Figure 3 shows that concentration force has the opposite behavior for various values of $N_b$ and $N_t$. It is observed from Figure 4 that pressure distribution has opposite effect for the various values of thermal Grashof parameter $G{r}_T$. and basic density Grashof number $G{r}_F$. The thermal Grashof parameter proclaims the general impact of thick hydrodynamic power and thermal buoyancy forces. For $G{r}_T<1$, the peristaltic routine is ruled by viscous powers, which is the other way around for $G{r}_T>1$. The $G{r}_F$ parameter is basically the proportion of species buoyancy forces to the thick hydrodynamic forces. For a situation where the two forces are equivalent (e.g., $G{r}_F=1$), velocity will be minimized. We can conclude from Figure 5a that pressure rise reduces for the larger values of magnetic parameter $M$, which shows the fact that pressure can be controlled by using the suitable magnetic field. Also, it is concluded from this figure that flow can pass easily without imposing higher pressure inside the channel. From Figure 5b and Figure 6, it is observed that friction force has completely the opposite behavior for the different values of the same physical parameters as compared with pressure rise distribution. Figure 7 shows a comparison of the graphical results of pressure rise and velocity profile. In Equation (7), by taking $W_e=0, M=0, \kappa \to \infty , G{r}_F=0, G{r}_T=0,$ the current results can be reduced to the results obtained by Shapiro et al. [39] for the Newtonian case. Further results can also be reduced to the similar results obtained by Gupta and Seshadri [41] and Mekheimer [42] by taking $W_e=0, \kappa \to \infty , G{r}_F=0, G{r}_T=0$.

From

Table 1. Model summary.

Model	R	R Square	Adjusted R Square	Standard Error of the Estimate
1	0.900 a	0.809	0.799	0.7558884
2	0.999 a	0.998	0.998	0.0550427
3	0.635 a	0.403	0.370	4.6675041
4	1.000 a	1.000	1.000	0.19437519

^a Predictors: (Constant), M.

, the R-square entropy generation for various values of magnetic parameter M is determined to be 0.809, meaning that approximately 81% of the variability of entropy generation is explained by parameter M in the model. On the other hand, the adjusted R-square 0.799 indicates that about 80% of the variability of entropy generation is accounted for Magnetic parameter M by the model. The entropy generation value for Brownian motion parameter $N_b$ is 0.998, which indicates that approximately 99% of the variability of entropy is due to parameter $N_b$ in the model, while the adjusted R-square 0.999 indicates that about 99% of the variability of entropy is accounted for $N_b$ by the model. In the R-square, the value of entropy generation for thermophoresis parameter $N_t$ is 0.403, which reveals that approximately 40% of the variability of entropy is explained by parameter $N_t$ in the model, while the adjusted R-square 0.370 indicates that about 37% of the variability of entropy is accounted for $N_t$ by the model. The entropy value for different values of $B_r$ is 1.00 with 100% of the variability of entropy accounted for parameter $B_r$ in the model, while the adjusted R-square 1.00 indicates that about 100% of the variability of entropy is accounted for $B_r$ by the Model.

From

Table 2. Coefficients.

Model		Unstandardized Coefficients		Standardized Coefficients	T	Significant
		B	Standard Error	Beta
1	(Constant)	74.223	0.351		211.381	0.000
	M	−2.562	0.293	−0.900	−8.739	0.000
2	(Constant)	29.097	0.026		1137.975	0.000
	M	2.049	0.021	0.999	95.977	0.000
3	(Constant)	65.565	2.168		30.239	0.000
	M	6.307	1.810	0.635	3.485	0.003
4	(Constant)	1.359	0.090		15.056	0.000
	M	68.492	0.075	1.000	908.676	0.000

, a decrease of −2.562 in entropy for independent variable M and an increase of 2.029 in Entropy for $N_b$ can be concluded. Similarly, an increase of 6.307 in entropy for parameter $N_t$ and an increase of 68.492 in entropy for $B_r$ scores can be concluded for every single-unit increase in Iv, assuming all other variables in the model are constant.

Table 3. Correlation table between entropy generation and parameters. **.

Entropy and Parameters	${\mathit{N}}_{\mathit{S}}{\mathit{V}}_{\mathit{S}}{\mathbf{B}}_{\mathbf{r}}$	${\mathit{N}}_{\mathit{S}}{\mathit{V}}_{\mathit{S}}{\mathit{N}}_{\mathit{t}}$	${\mathit{N}}_{\mathit{S}}{\mathit{V}}_{\mathit{S}}{\mathit{N}}_{\mathit{b}}$	${\mathit{N}}_{\mathit{S}}{\mathit{V}}_{\mathit{S}}\mathit{M}$
Values range	0.1 to 2.0	0.1 to 2.0	0.1 to 2.0	0.1 to 2.0
N	20	20	20	20
Pearson Correlation Siganificant (2-tailed)	1.000 ** 0.000	0.635 ** 0.003	0.999 ** 0.000	0.900 ** 0.000
Remarks	Perfect Relation	Strong Positive Relation	Strong Positive Relation	Very Strong Negative Relations

** Correlation is significant at the 0.01 level (2-tailed).

is plotted to break down the relationship of entropy generation for some delicate parameters. It is determined from these outcomes that a huge impeccable significant positive connection exists between Brinkman number $B_r$ and its entropy generation at 0.01 level. A solid positive relationship has been seen from the connection results between entropy and the parameters $N_t$ and $N_b$. An extremely negative relationship was seen between M and its entropy.

5. Conclusions

The following outcomes demonstrated through this study are:

• Temperature profile increases with higher values of $N_b$ and $N_t$.
• Pressure distribution and friction have an inverse conduct for bigger estimations of the magnetic parameter, Brownian movement parameter and thermophoresis parameter.
• The variability of entropy generation is 81% for the values of M, while 99% variability for the parameter $N_b$.
• The variability of entropy generation is 40% for the values of $N_t,$ while 100% variability for the parameter $B_r$.