Entropy Analysis for Hydromagnetic Darcy–Forchheimer Flow Subject to Soret and Dufour Effects

Abstract

Here, our main aim is to examine the impacts of Dufour and Soret in a radiative Darcy–Forchheimer flow. Ohmic heating and the dissipative features are outlined. The characteristics of the thermo-diffusion and diffusion-thermo effects are addressed. A binary chemical reaction is deliberated. To examine the thermodynamical system performance, we discuss entropy generation. A non-linear differential system is computed by the finite difference technique. Variations in the velocity, concentration, thermal field and entropy rate for the emerging parameters are scrutinized. A decay in velocity is observed for the Forchheimer number. Higher estimation of the magnetic number has the opposite influence for the velocity and temperature. The velocity, concentration and thermal field have a similar effect on the suction variable. The temperature against the Dufour number is augmented. A decay in the concentration is found against the Soret number. A similar trend holds for the entropy rate through the radiation and diffusion variables. An augmentation in the entropy rate is observed for the diffusion variable.

1. Introduction

Henry Darcy established the framework of homogeneous liquid flow through a permeable medium throughout his work on the progression of water over saturated sand [1]. At higher flow rates, when inertial and boundary impacts arise, then the Darcy law cannot work appropriately. To overcome such an issue, Forchheimer gave the concept of the non-Darcy model through the insertion of a quadratic velocity term in a momentum expression [2]. Later on, the Forchheimer term was so named by Muskat [3]. The moment of fluid flowing through a permeable surface is of keen interest due to its significance in technical, biological and scientific fields such as artificial dialysis, gas turbines, atherosclerosis, catalytic converters, geo energy production and many others. Hayat et al. [4] explored the 2-D Darcy–Forchheimer flow of non-Newtonian liquid with variable properties. Pal and Mondal [5] addressed the convective flow of Darcy–Forchheimer liquid subject to a variable heat sink or source and viscosity. Non-uniform heat conductivity analysis in the reactive flow of Darcy–Forchheimer Carreau nanomaterial subject to a magnetic dipole was presented by Mallawi and Ullah [6]. Alshomrani and Ullah [7] studied the convective flow of Darcy–Forchheimer hybrid nanomaterial subject to a cubic autocatalysis chemical reaction. Seth and Mandal [8] discussed the hydromagnetic effect in rotating the flow of Darcy–Forchheimer Casson liquid toward a permeable space. Little analysis concerning a porous medium is discussed in [9,10,11,12,13,14,15,16,17].

Thermal and solutal transportation in a permeable surface has been a significant consideration of researchers during the last two decades. This is due to its usefulness in geothermal systems, catalytic reactors, nuclear waste repositories, areas of geosciences, chemical engineering, energy storage units, drying technology, heat insulation, heat exchangers for packed beds and many others. Initially, the Dufour effect in liquid was explored by Rastogi and Madan [18]. After that, the diffusion-thermo impact in a homogeneous mixture was investigated in [19,20]. Moorthy and Senthilvalivu [21] studied the Dufour and Soret outcomes in the convective flow of liquid of a non-uniform viscosity subject to a permeable medium. Non-uniform temperature in a convective fluid flow subject to thermal-diffusion and diffusion-thermo effects was illustrated by El-Arabawy [22]. Few reviews with reference to relevant titles have been discussed through certain studies [23,24,25,26,27]. The important reason for entropy production is the conversion of thermal energy in the occurrence of numerous examples of processes, such as fluid friction, kinetic energy, rotational moments, molecular resistance, the Joule Thomson effect, mass transport rate, and molecular vibration. The concept of entropy optimization in liquid flow was initially given by Bejan [28,29]. Entropy analysis in a water-based hybrid nanoliquid subject to mixed convection was discussed by Buonomo [30]. Irreversibility exploration in the dissipative flow of nanomaterial with melting and radiation over a stretching sheet was addressed by Khan et al. [31]. Some investigations of the entropy rate are mentioned in [32,33,34,35,36,37,38,39].

To our knowledge, no study has reported about entropy-optimized radiative Darcy–Forchheimer flows with Soret and Dufour features yet. A porous medium through a Darcy–Forchheimer relation is discussed. Dissipation, Ohmic heating and radiation are scrutinized in energy equations. The physical characteristics for the Soret and Dufour impacts are addressed. A first-order chemical reaction is deliberated. To examine the thermodynamical system performance, we discuss entropy optimization. Nonlinear differential systems are obtained through appropriate transformations. Non-dimensional differential systems are solved through the finite difference method. The performance of appropriate variables concerning the velocity, entropy generation, concentration and thermal field have been scrutinized.

2. Formulation

An unsteady radiative hydromagnetic Darcy–Forchheimer flow saturating a porous medium is discussed. Joule heating, viscous dissipation and thermal radiation in energy expression have been scrutinized. The Soret and Dufour effects are inspected. The impact of the entropy rate is addressed. Additionally, the flow is subject to a chemical reaction of the first order. A constant magnetic field with a strength ($B_0$) is applied. Consider $u=u_w=ax$ as the stretching velocity, with $a>0$. The chosen magnetic Reynolds number is small. Figure 1 shows a flow sketch [31].

By taking into account the infinite plate, the term $\frac{\partial u}{\partial x}$ becomes zero. Here, the continuity equation becomes $\frac{\partial v}{\partial y}=0$.

Under the above discussion, the related expression for constant suction becomes

$$v=-v_0=\mathrm{constant}$$

(1)

$$\frac{\partial u}{\partial t}-v_0\frac{\partial u}{\partial y}=\nu \frac{{\partial }^{2}u}{\partial y^2}-\frac{\sigma B_0^2}{\rho }u-\frac{\nu }{k_p}u-F{u}^2,$$

(2)

$$\left.\begin{array}{c}\frac{\partial T}{\partial t}-v_0\frac{\partial T}{\partial y}=\alpha \frac{{\partial }^{2}T}{\partial y^2}+\frac{16}{3}\frac{{\sigma }^{\ast }{T}_{\infty }^3}{k^{\ast }\left(\rho c_p\right)}\frac{{\partial }^{2}T}{\partial y^2}+\frac{\mu }{\left(\rho c_p\right)}{\left(\frac{\partial u}{\partial y}\right)}^2\\ +\frac{\sigma B_0^2}{\left(\rho c_p\right)}{u}^2+\frac{D_B{K}_T}{C_s{c}_p}\frac{{\partial }^{2}C}{\partial y^2}\end{array}\right\},$$

(3)

$$\frac{\partial C}{\partial t}-v_0\frac{\partial C}{\partial y}=D_B\frac{{\partial }^{2}C}{\partial y^2}+\frac{D_B{K}_T}{T_m}\frac{{\partial }^{2}T}{\partial y^2}-k_r(C-C_{\infty }),$$

(4)

with

$$\left.\begin{array}{c}u=0,T=T_{\infty },C=C_{\infty },\mathrm{at}t=0\\ u=ax,T=T_w,C=C_w,\mathrm{at}y=0\\ u\to 0,T\to T_{\infty },C\to C_{\infty },\mathrm{as}y\to \infty \end{array}\right\}.$$

(5)

Consider the following formula:

$$\left.\begin{array}{c}\tau =\frac{v}{L_1^2}t,\xi =\frac{x}{L_1},\eta =\frac{y}{L_1},U(\tau ,\eta )=\frac{L_1}{\upsilon }u,\\ \theta \left(\tau ,\eta \right)=\frac{\left(T-T_{\infty }\right)}{\left(T_w-T_{\infty }\right)},\varphi \left(\tau ,\eta \right)=\frac{C-C_{\infty }}{C_w-C_{\infty }},\end{array}\right\},$$

(6)

Then, we have

$$\frac{\partial U}{\partial \tau }-S\frac{\partial U}{\partial \eta }=\frac{{\partial }^{2}U}{\partial {\eta }^2}-MU-\lambda U-Fr{U}^2,$$

(7)

$$\frac{\partial \theta }{\partial \tau }-S\frac{\partial \theta }{\partial \eta }=\frac{1}{Pr}\left(1+Rd\right)\frac{{\partial }^2\theta }{\partial {\eta }^2}+Ec{\left(\frac{\partial U}{\partial \eta }\right)}^2+MEc{U}^2+Du\frac{{\partial }^2\varphi }{\partial {\eta }^2},$$

(8)

$$\frac{\partial \varphi }{\partial \tau }-S\frac{\partial \varphi }{\partial \eta }=\frac{1}{Sc}\frac{{\partial }^2\varphi }{\partial {\eta }^2}+Sr\frac{{\partial }^2\theta }{\partial {\eta }^2}-\gamma \varphi ,$$

(9)

with

$$\left.\begin{array}{c}U=0,\theta =0,\varphi =0,\mathrm{at}\tau =0\\ U=\xi Re,\theta =1,\varphi =1,\mathrm{at}\eta =0\\ U\to 0,\theta \to 0,\varphi \to 0,\mathrm{as}\eta \to \infty \end{array}\right\}$$

(10)

Here the non-dimensional variables are $S\left(=\frac{v_0}{\nu }{L}_1\right)$, $M\left(=\frac{\sigma B_o^2{L}_1^2}{\nu \rho }\right)$, $Re\left(=\frac{a{L}_1^2}{\nu }\right)$, $Sr\left(=\frac{D{K}_T(T_w-T_{\infty })}{\nu T_m(C_w-C_{\infty })}\right)$, $F_r\left(=\frac{C_b}{\sqrt{k_p}}{L}_1\right)$, $Pr\left(=\frac{\nu }{\alpha }\right)$, $\lambda \left(=\frac{L_1^2}{k_p}\right)$, $E_C\left(=\frac{{\nu }^2}{c_p{L}_1^2(T_w-T_{\infty })}\right)$, $\gamma \left(=\frac{k_r{L}_1^2}{\nu }\right)$, $Du\left(=\frac{D_B{K}_T(C_w-C_{\infty })}{\nu C_s{c}_p(T_w-T_{\infty })}\right)$, $Rd\left(=\frac{16{\sigma }^{\ast }{T}_{\infty }^3}{3k^{\ast }k}\right)$, $Sc\left(=\frac{\nu }{D_B}\right)$ and $Br\left(=PrEc\right)$.

3. Engineering Contents of Interest

3.1. Nusselt Number

Here, we have

$$N{u}_x=\frac{x{q}_w}{k\left(T_w-T_{\infty }\right)},$$

(11)

with the heat flux $q_w$ given by

$$q_w=-{\left.\left(k+\frac{16{\sigma }^{\ast }{T}_{\infty }^3}{3k^{\ast }}\right)\left(\frac{\partial T}{\partial y}\right)\right|}_{y=0}$$

(12)

We finally have

$$N{u}_x=-\xi \left(1+Rd\right){\left(\frac{\partial \theta }{\partial \eta }\right)}_{\eta =0}.$$

(13)

3.2. Sherwood Number

This is given as

$$S{h}_x=\frac{x{j}_w}{D_B\left(C_w-C_{\infty }\right)},$$

(14)

in which the mass flux $j_w$ is defined as

$$j_w={\left.-D_B\left(\frac{\partial C}{\partial y}\right)\right|}_{y=0},$$

(15)

We can write

$$S{h}_x=-\xi {\left(\frac{\partial \varphi }{\partial \eta }\right)}_{\eta =0}.$$

(16)

4. Entropy

The important reason for entropy production is the conversion of thermal energy in the occurrence of numerous processes, such as fluid friction, kinetic energy, rotational moment, molecular resistance, the Joule–Thomson effect, mass transport rate and molecular vibration. We have the following [30,31,32,33]:

$$\left.\begin{array}{c}{S}_G=\frac{k}{T_{\infty }^2}\left(1+\frac{16{\sigma }^{\ast }{T}_{\infty }^3}{3k^{\ast }k}\right){\left(\frac{\partial T}{\partial y}\right)}^2+\frac{{\mu }_f}{T_{\infty }}{\left(\frac{\partial u}{\partial y}\right)}^2+\frac{\mu }{k_p{T}_{\infty }}{u}^2\\ +\frac{\sigma B_o^2}{T_{\infty }}{u}^2+\frac{R{D}_B}{T_{\infty }}\left(\frac{\partial T}{\partial y}\frac{\partial C}{\partial y}\right)+\frac{R{D}_B}{C_{\infty }}{\left(\frac{\partial C}{\partial y}\right)}^2\end{array}\right\},$$

(17)

The dimensionless expression is

$$\left.\begin{array}{c}{N}_G\left(\tau ,\eta \right)={\alpha }_1\left(1+Rd\right){\left(\frac{\partial \theta }{\partial \eta }\right)}^2+Br{\left(\frac{\partial U}{\partial \eta }\right)}^2+Br\lambda U^2\\ +MBr{U}^2+L\left(\frac{\partial \theta }{\partial \eta }\frac{\partial \varphi }{\partial \eta }\right)+L\frac{{\alpha }_2}{{\alpha }_1}{\left(\frac{\partial \varphi }{\partial \eta }\right)}^2\end{array}\right\}$$

(18)

In the above expression, the dimensionless parameters are ${\alpha }_1\left(=\frac{\left(T_w-T_{\infty }\right)}{T_{\infty }}\right)$, $N_G\left(=\frac{S_G{T}_{\infty }{L}_1^2}{k\left(T_w-T_{\infty }\right)}\right)$, $L\left(=\frac{R{D}_B(C_w-C_{\infty })}{k}\right)$ and ${\alpha }_2\left(=\frac{\left(C_w-C_{\infty }\right)}{C_{\infty }}\right)$.

5. Solution Methodology

Using the finite difference method, we can solve the nonlinear differential system [40,41,42,43] by writing

$$\left.\begin{array}{c}\frac{\partial U}{\partial \tau }=\frac{U_a^{n+1}-U_a^n}{\Delta \tau },\frac{\partial U}{\partial \eta }=\frac{U_{a+1}^n-U_a^n}{\Delta \eta }\\ \frac{\partial \theta }{\partial \tau }=\frac{{\theta }_a^{n+1}-{\theta }_s^k}{\Delta \tau },\frac{\partial \theta }{\partial \eta }=\frac{{\theta }_{a+1}^n-{\theta }_a^n}{\Delta \eta }\\ \frac{\partial \varphi }{\partial \tau }=\frac{{\varphi }_a^{n+1}-{\varphi }_a^n}{\Delta \tau },\frac{\partial \varphi }{\partial \eta }=\frac{{\varphi }_{a+1}^n-{\varphi }_a^n}{\Delta \eta }\\ \frac{{\partial }^{2}U}{\partial {\eta }^2}=\frac{U_{a+1}^n-2U_a^n+U_{a-1}^n}{{(\Delta \eta )}^2},\frac{{\partial }^2\theta }{\partial {\eta }^2}=\frac{{\theta }_{a+1}^n-2{\theta }_a^n+{\theta }_{a-1}^n}{{(\Delta \eta )}^2}\\ \frac{{\partial }^2\varphi }{\partial {\eta }^2}=\frac{{\varphi }_{a+1}^n-2{\varphi }_a^n+{\varphi }_{a-1}^n}{{(\Delta \eta )}^2}\end{array}\right\},$$

(19)

By employing Equation (23) in Equations (8)–(10), we obtain

$$\frac{U_a^{n+1}-U_a^n}{\Delta \tau }-S\frac{U_{a+1}^n-U_a^n}{\Delta \eta }=\frac{U_{a+1}^n-2U_a^n+U_{a-1}^n}{{(\Delta \eta )}^2}-M{U}_a^n-\lambda U_a^n-Fr{\left(U_a^n\right)}^2,$$

(20)

$$\left.\begin{array}{c}\frac{{\theta }_a^{n+1}-{\theta }_a^n}{\Delta \tau }-S\frac{{\theta }_{a+1}^n-{\theta }_a^n}{\Delta \eta }=\frac{\left(1+Rd\right)}{P_r}\frac{{\theta }_{a+1}^n-2{\theta }_a^n+{\theta }_{a-1}^n}{{(\Delta \eta )}^2}+Ec{\left(\frac{U_{a+1}^n-U_a^n}{\Delta \eta }\right)}^2\\ +MEc{\left(U_a^n\right)}^2+Du\left(\frac{{\varphi }_{a+1}^n-2{\varphi }_a^n+{\varphi }_{a-1}^n}{{(\Delta \eta )}^2}\right)\end{array}\right\}$$

(21)

$$\frac{{\varphi }_a^{n+1}-{\varphi }_a^n}{\Delta \tau }-S\frac{{\varphi }_{a+1}^n-{\varphi }_a^n}{\Delta \eta }=\frac{1}{Sc}\frac{{\varphi }_{a+1}^n-2{\varphi }_a^n+{\varphi }_{a-1}^n}{{(\Delta \eta )}^2}+Sr\frac{{\theta }_{a+1}^n-2{\theta }_a^n+{\theta }_{a-1}^n}{{(\Delta \eta )}^2}-\gamma {\varphi }_a^n,$$

(22)

with

$$\left.\begin{array}{c}{U}_a^0=0,{\theta }_a^0=0,{\varphi }_a^0=0,\\ U_0^n=1,{\theta }_0^n=1,{\varphi }_0^n=1,\\ U_{\infty }^n\to 0,{\theta }_{\infty }^n\to 0,{\varphi }_{\infty }^n\to 0\end{array}\right\}$$

(23)

The entropy generation expression yields

$$\left.\begin{array}{c}{N}_G\left(\tau ,\xi ,\eta \right)={\alpha }_1\left(1+Rd\right){\left(\frac{{\theta }_{a+1}^n-{\theta }_a^n}{\Delta \eta }\right)}^2+Br{\left(\frac{U_{a+1}^n-U_a^n}{\Delta \eta }\right)}^2+Br\lambda {\left(U_a^n\right)}^2\\ +MBr{\left(U_a^n\right)}^2+L(\frac{{\theta }_{a+1}^n-{\theta }_a^n}{\Delta \eta }.\frac{{\varphi }_{a+1}^n-{\varphi }_a^n}{\Delta \eta })+L\frac{{\alpha }_2}{{\alpha }_1}{\left(\frac{{\varphi }_{a+1}^n-{\varphi }_a^n}{\Delta \eta }\right)}^2\end{array}\right\}$$

(24)

6. Graphical Results and Review

The physical interpretation of the parameters of the concentration, entropy rate, velocity and temperature have been investigated. The present observations are compared with previous published results in

Table 1. Comparison of Nusselt numbers with [44].

Pr	Bidin and Nazar [44]	Recent Outcomes
1.0	0.9547	0.954710
2.0	1.4714	1.471409
3.0	1.8961	1.896115

, and excellent agreement is noticed.

6.1. Velocity

The influence of suction $\left(S\right)$ upon the velocity $\left(U\right(\tau ,\eta \left)\right)$ is sketched in Figure 2. Obviously, a higher estimation of the suction parameter $\left(S\right)$ decays the velocity. As expected, this is in accordance to the physical facts. Figure 3 displays the velocity against a magnetic field. Actually, reduction occurs in the velocity for $\left(M\right)$. A physically higher $\left(M\right)$ value corresponds to amplifying the Lorentz force which the flow opposes. Hence, velocity decay is guaranteed. Figure 4 was developed in order to recognize the velocity $\left(U\right(\tau ,\eta \left)\right)$ design with variation in the Forchheimer number $\left(Fr\right)$. A larger estimation for the Forchheimer number decays the velocity $\left(U\right(\tau ,\eta \left)\right)$. The influence of ($\lambda$) on $\left(U\right(\tau ,\eta \left)\right)$ is illustrated in Figure 5. Clearly, $U(\tau ,\eta )$ decays against higher $\lambda$ values.

6.2. Temperature

Figure 6 and Figure 7 are for the thermal field against the suction and magnetic variables $(S$ and $M)$. A similar scenario holds for the thermal field ($\theta (\tau ,\eta )$) through the suction and magnetic variables. Figure 8 portrays the performance of the radiation against the temperature ($\theta (\tau ,\eta )$). Larger radiation values lead to the temperature ($\theta (\tau ,\eta )$) increasing. Figure 9 displays the performance of the thermal field against the Prandtl number. A larger approximation of (Pr) corresponds to the decay of the thermal diffusivity, and consequently, the temperature ($\theta (\tau ,\eta )$) decreases. Figure 10 displays the impact of the thermal field $\theta (\tau ,\eta )$ on the Eckert number $\left(Ec\right)$. A higher estimation for ($Ec$) corresponds to the temperature being higher.

6.3. Concentration

Figure 11 exhibits the concentration performance against the suction variable $\left(S\right)$. Clearly, the concentration ($\varphi \left(\tau ,\eta \right)$) was reduced against larger (S) values. Figure 12 shows the performance of the concentration ($\varphi \left(\tau ,\eta \right)$) versus ($Sc$). An increment ($Sc$) decayed the mass diffusivity, and thus the concentration ($\varphi \left(\tau ,\eta \right)$) diminished. An amplification of the Soret number ($Sr$) led to a decaying value for $\varphi \left(\tau ,\eta \right)$ (see Figure 13). Figure 14 comprises the impact of $\varphi \left(\tau ,\eta \right)$ on $\gamma$. Here, $\varphi \left(\tau ,\eta \right)$ decreased against $\gamma$.

6.4. Entropy Generation Rate

The influence of the entropy rate ($N_G(\tau ,\eta )$) via the radiation variable is disclosed in Figure 15. Clearly, a greater $Rd$ value improved the radiation emission, which boosted the collision between the fluid particles, and so $N_G(\tau ,\eta )$ was enhanced. Figure 16 discloses the impact of L on $N_G(\tau ,\eta )$). As predicted, the entropy generation ($S_G\left(\eta \right)$) was greater via the higher approximation of L. A larger approximation of the Brinkman ($Br$) number enhanced the entropy generation ($S_G\left(\eta \right)$) (see Figure 17). This is because of augmentation through a higher $Br$ value causing the viscous features to improve. As a result, the entropy rate rose. Figure 18 demonstrates the entropy rate for the magnetic parameter. A larger approximation of the magnetic variable led to an increase in the entropy rate.

7. Closing Points

• The theermal field and velocity for the magnetic field had opposing trends.
• A decrease in velocity was noted for the Forchheimer number and suction variable.
• The velocity versus the porosity parameter was decreased.
• Similar behavior for the concentration and temperature against suction was noticed.
• The temperatures for the Eckert and Prandtl numbers were dissimilar.
• Radiation for the entropy and temperature had a similar role.
• The concentration decayed via larger approximation of the Soret number and reaction parameter.
• A decay in concentration against the Schmidt number held.
• Entropy generation enhancement against the Brinkman number and diffusion variable was noticed.
• The entropy rate was boosted with variation in the diffusion variable.