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SN Applied Sciences

, 1:1672 | Cite as

Entropy generation in MHD nanofluid flow with heat source/sink

  • Lalrinpuia Tlau
  • Surender OntelaEmail author
Research Article
  • 72 Downloads
Part of the following topical collections:
  1. Engineering: Fluid Mechanics, Computational Fluid Dynamics and Fluid Interaction

Abstract

This work investigates the generation of entropy in the presence of a heat source/sink in a sloping channel filled with porous medium in magnetohydrodynamic nanofluid flow. The regulating equations are nonlinear and coupled thermal and hydrodynamic equations. Homotopy analysis method is used in the handling of equations. Comparisons with existing literature have been produced and were discovered to be in excellent accord, which are a particular situation of the present issue. The impact on entropy generation, Bejan number, Nusselt number and skin friction of pertinent fluid parameters is addressed, developed and displayed graphically. Entropy generation was found to be minimum just above the center of the channel throughout the study. Skin friction and Nusselt number were found to be higher for the case of heat generation than heat absorption.

Keywords

Entropy Inclined channel Mixed convection Nanofluid Heat source MHD Homotopy analysis method 

1 Introduction

High performing supercomputers compute at extremely high speeds and generate high amount of heat. As a result, ultra-fast heat transportation is required to cool the supercomputers. This is one of the many problems the current generation of academicians and industrial technologies are facing. However, the current method of cooling machines need to be upgraded along with the advancement in technologies. Nanofluids were pioneered by Choi and Eastman [1] to aid in the cooling of high heat generating machines and consist of base fluid (either water, alcohol or oil) with suspended nanoparticles in them. It is a current niche research topic and has been tested for various applications. Application of nanofluid in real-life processes is still in its infancy though many applications have been tested including solar collectors, boiling processes, machining processes, heat exchangers, microelectromechanical systems (MEMS), refrigerators, etc. [2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15]

The magnetohydrodynamic (MHD) effect is an important effect that acts on nanofluid flow. Fakour et al. [16] found that with the increment of magnetic field effects, the maximum velocity and temperature decreased in the horizontal porous canal flow. In their inquiry of MHD nanofluid stream in a tube packed with a porous medium, Aaiza et al. [17] accounted for the form of nanoparticles. They noted that there was reduced velocity in elongated particle form relative to other particle shapes. Sheikholeslami et al. [18] regarded a flow in a semi-porous channel with an MHD impact. It was found that the velocity boundary layer thickness decreased with the increase in the Reynolds number. In their analysis for the MHD nanofluid flow in a rotating channel, Raza et al. [19] considered slip effects. When Azimi and Riazi [20] increased the Reynolds number, skin frictions and Nusselt numbers were found to be enhanced. Dogonchi et al. [21] analysis of MHD nanofluid flow in a porous canal took the thermal radiation effect into account and found an inverse relationship between the radiation and the temperature profile. Noreen et al. [22] found that velocity increases in the center of the channel where electroosmotic parameters have been reduced during a microchannel MHD nanofluid flow study. Ma et al. [23] studied the hybrid nanofluid flow on a channel and found that the Reynolds number significantly affected the velocity distribution. In the study performed by Selimefendigil and Öztop [24] for a pulsating MHD nanofluid flow in a channel with blocks, it was found that Nusselt number increases with an increased magnetic field angle. Chamkha et al. [25] conducted a further, but not exhaustive, literature review on the MHD nanofluid flow which could be referred for further insight by the reader.

A numerical study of flow of a nanofluid along a vertical plate with heat source and sink was carried out by Rana and Bhargava [26], observing that both the momentum and the thermal boundary layer decreased for the heat sink. Alsaedi et al. [27] noted that nanofluid flow suction and injection had opposite effects over a heat generated/absorbed surface. Hassan [28] showed the average Nusselt number to be increased when the amount of Cu nanoparticles in an enclosure with a heat source and sink was increased. In their analysis of the fluid flow in tapered channels with thermal source, radiation and magnetic field, Kothandapani and Prakash [29] noted that the mass transfer of nanoparticles and temperature distribution had opposite behavior. Umavathi and Sheremet [30] noted that heat generation has increased flow field velocity, while heat absorption has reduced flow rate for nanofluid flow in a vertical channel with heat source and sink. Reddy et al. [31] has investigated the impact of radiation and heat generation/absorption on the flow of nanofluids over a tilted porous plate and found that increasing the tilt angles has enhanced the distribution of temperature and concentration. Ali et al. [32] found the heat source to have a positive effect on the overall efficiency for a nanofluid flow in a grooved channel.

Recently, the importance of entropy analysis has picked up researcher’s attention for heat transfer phenomena. Bejan [33, 34] has brought it to the spotlight as a matter of fundamental importance. Li and Kleinstreuer [35] concluded it was recommended to use microchannels in nanofluid flows to reduce entropy. López et al. [36] demonstrated that entropy generation has intensified in an upright microchannel with radiation, slip and convective–radiative conditions at its boundaries with an increased nanofluid flow buoyancy forces. The study by Chamkha [37] found that an increase in nanoparticles led to a decline in entropy generation. Decrease in the temperature gradients in the wall, in the analysis carried out by Abbaszadeh et al. [38], was observed to cause decrease in the entropy generation. In their study of entropy generation in a porous square enclosure, Chamkha et al. [39] considered partial slip. The following papers can provide further reviews and discussions about entropy generation [40, 41, 42, 43].

With regard to the above-mentioned literature, the present research analyzes a MHD nanofluid flow in saturated porous medium in an angled channel with heat source/sink while considering hydrodynamic slip and convection at the boundary. The governing equations are solved using the homotopy analysis method (HAM) [44]. For clarity, the impacts of relevant flow parameters are graphically displayed and explained. To our best knowledge, no researchers have tried the current work.

2 Mathematical formulation

A laminar, fully developed incompressible MHD nanofluid flow in an inclined parallel plate channel filled with a saturated porous medium and a heat source/sink is taken into consideration. The x-axis is taken along the walls of the channel in the ascending direction. The y-axis is taken perpendicular to the walls of the channel. The walls of the channel are parallel to each other and at distance of L apart. The walls of the channel are assumed to be under same heat convection, as depicted in Fig. 1. The Boussinesq approximation [45] and viscous dissipation for the flow are taken into consideration, and the channel is kept inclined at an angle \(\psi\). Under these assumptions, the nonlinear coupled equations for momentum and temperature distribution in the channel are as follows:
Fig. 1

Problem configuration

$$\begin{aligned} \mu _{nf}\frac{\partial ^2 u}{\partial y^2}+g(\wp \beta )_{nf}\sin \psi (T-T_{0})-\frac{\mu _{nf}}{K}u-\sigma _{nf} B_{0}^2u= \frac{{\mathrm{d}}p}{{\mathrm{d}}x}, \end{aligned}$$
(1)
$$\begin{aligned} \kappa _{nf}\frac{\partial ^2 T}{\partial y^2}+\mu _{nf}\left( \frac{\partial u}{\partial y}\right) ^2+\bar{Q}(T-T_{0})+\sigma _{nf} B_{0}^2u^2= 0, \end{aligned}$$
(2)
with the associated boundary equations:
$$\begin{aligned} {\left\{ \begin{array}{ll} u=-\alpha \frac{\partial u}{\partial y}& \\ \kappa _{nf}\frac{\partial T}{\partial y} =-h_{f}(T_{1}-T)& {\text{at}}\quad y= 0 \end{array}\right. }, \end{aligned}$$
(3a)
$$\begin{aligned} {\left\{ \begin{array}{ll} u=-\alpha \frac{\partial u}{\partial y}& \\ \kappa _{nf}\frac{\partial T}{\partial y}=h_{f}(T_{2}-T) & {\text{at}}\quad y= L. \end{array}\right. } \end{aligned}$$
(3b)
In this study, we use the following relations for the physical properties of nanofluids and fluids [46]:
$$\begin{aligned} \mu _{nf}= \frac{\mu _{f}}{(1-\phi )^{2.5}}, \end{aligned}$$
(4)
$$\begin{aligned} \wp _{nf}= (1-\phi )\wp _{f}+\phi \wp _{s}, \end{aligned}$$
(5)
$$\begin{aligned} (\wp \beta )_{nf}= (1-\phi )(\wp \beta )_{f}+\phi (\wp \beta )_{s}, \end{aligned}$$
(6)
$$\begin{aligned} \frac{\kappa _{nf}}{\kappa _{f}}= \frac{(\kappa _{s}+2\kappa _{f})+2\phi (\kappa _{f}-\kappa _{s})}{(\kappa _{s}+2\kappa _{f})+\phi (\kappa _{f}-\kappa _{s})}. \end{aligned}$$
(7)
$$\begin{aligned} \frac{\sigma _{nf}}{\sigma _{f}}= 1+\frac{3\left( \frac{\sigma _{s}}{\sigma _{f}}-1\right) \phi }{\left( \frac{\sigma _{s}}{\sigma _{f}}+2\right) -\left( \frac{\sigma _{s}}{\sigma _{f}}-1\right) \phi } \end{aligned}$$
(8)
Introducing the following dimensionless variables in Eqs. (1) and (2):
$$\begin{aligned} \begin{aligned}&Y=\frac{y}{L} ; U=\frac{u}{U_{0}} ; \theta =\frac{T-T_{0}}{\Delta T};\epsilon =\frac{{\mathrm{Gr}}}{{\mathrm{Re}}};Q=\frac{4\bar{Q}L^2}{\kappa _{f}};\\&{\mathrm{Gr}}=\frac{8\wp _{f}^2g\beta _{f}\Delta TL^3}{\mu _{f}^2}; {\mathrm{Re}}=\frac{2U_{0}\wp _{f}L}{\mu _{f}}; {\mathrm{Br}}=\frac{\mu _{f}U_{0}^2}{\kappa _{f}\Delta T};\\&T_{0}=\frac{T_{1}+T_{2}}{2}; \Delta T=T_{2}-T_{1};~ r_{{\mathrm{T}}}=\frac{T_{2}-T_{1}}{\Delta T};~P=-\frac{L^2}{U_{0}\mu _{f}}\frac{{\mathrm{d}}p}{{\mathrm{d}}x};\\&U_{0}=-\frac{L^2}{12\mu _{f}}\frac{{\mathrm{d}}p}{{\mathrm{d}}x}; {\mathrm{Ha}}^2=\frac{4\sigma B_{0}^2L^2}{\mu _{f}}; \wedge _{1}=\frac{1}{(1-\phi )^{2.5}};\\&\wedge _{2}=(1-\phi )+\phi \frac{(\wp \beta )_{s}}{(\wp \beta )_{f}}; \wedge _{3}=\frac{\kappa _{nf}}{\kappa _{f}}, \wedge _{4}=\frac{\sigma _{nf}}{\sigma _{f}} \end{aligned} \end{aligned}$$
(9)
we have:
$$\begin{aligned} \wedge _{1}\frac{{\mathrm{d}}^2 U}{{\mathrm{d}}Y^2}+\frac{\wedge _{2}}{4}\epsilon \theta \sin \psi + P-\wedge _{1}\frac{U}{{\mathrm{Da}}}-\frac{\wedge _{4}}{4}{\mathrm{Ha}}^2U= 0, \end{aligned}$$
(10)
$$\begin{aligned} \frac{{\mathrm{d}}^2\theta }{{\mathrm{d}}Y^2}+{\mathrm{Br}}\frac{\wedge _{1}}{\wedge _{3}}\left( \frac{{\mathrm{d}}U}{{\mathrm{d}}Y}\right) ^2+\frac{Q\theta }{4\wedge _{3}}+\frac{\wedge _{4}}{\wedge _{3}}\frac{{\mathrm{Ha}}^2{\mathrm{Br}}}{4}U^2= 0, \end{aligned}$$
(11)
with the dimensionless boundary conditions:
$$\begin{aligned} {\left\{ \begin{array}{ll} U=-\gamma \frac{{\mathrm{d}}U}{{\mathrm{d}}Y} & \\ \frac{d\theta }{{\mathrm{d}}Y}=\frac{{\mathrm{Bi}}}{2}\left(\frac{r_{{\mathrm{T}}}}{2}+\theta \right)&\quad {\text{at}}\quad Y=0 \end{array}\right. } , \end{aligned}$$
(12a)
$$\begin{aligned} {\left\{ \begin{array}{ll} U=-\gamma \frac{{\mathrm{d}}U}{{\mathrm{d}}Y}& \\ \frac{d\theta }{{\mathrm{d}}Y}=\frac{{\mathrm{Bi}}}{2}\left(\frac{r_{{\mathrm{T}}}}{2}-\theta \right)& \quad {\text{at}}\quad Y=1\end{array}\right.}. \end{aligned}$$
(12b)
where Gr is the Grashof number, Re is the Reynolds number, Br is the Brinkman number, Da is the Darcy number, Ha is the Hartmann number, \(\varepsilon\) is the mixed convection parameter, Q is the heat source/sink parameter, P is the dimensionless pressure gradient, and \(U_{0}\) is the mean fluid velocity. \(r_{{\mathrm{T}}}\) is the dimensionless temperature difference ratio where \(r_{{\mathrm{T}}} =0\) denotes symmetric wall temperature; \(T_{1}=T_{2}\) and \(r_{{\mathrm{T}}}=1\) denote asymmetric wall temperature, \(T_{1}< T_{2}\) .

2.1 Skin friction and Nusselt number

Quantities which are of great interest to engineering applications are the skin friction and Nusselt numbers of the flow. On a rough basis, the skin friction is to the velocity boundary layer what the Nusselt number is to the thermal boundary layer. The skin friction quantifies the resistance offered by the surface over which the fluid is flowing. It is mainly caused due to the viscosity of the fluid. On the other hand, the Nusselt number is basically a ratio of the heat transfer due to convection to the heat transfer due to conduction. Hence, Nusselt number tells us the amount of enhancement in heat transfer due to movement of fluid. Thus, skin friction and Nusselt number are defined, respectively, as:
$$\begin{aligned} Cf=\frac{\tau _{w}}{\rho _{f}U_{0}^2} , {\mathrm{Nu}}=\frac{2Lq_{w}}{\kappa _{f}\Delta T} \end{aligned}$$
(13)
where the shear stress \(\tau\) and heat transfer coefficient \(q_{w}\) are defined, respectively, as:
$$\begin{aligned} \tau _{w}=\pm \mu _{nf}\left( \frac{\partial u}{\partial y}\right) _{y=0,L} , q_{w}=\pm \kappa _{nf}\left( \frac{\partial T}{\partial y}\right) _{y=0,L} \end{aligned}$$
(14)
Using the dimensionless transformation Eq. (9), the dimensionless skin friction and Nusselt number can be expressed as:
$$\begin{aligned} ReCf_{1}= \wedge _{1}\left( \frac{{\mathrm{d}}U}{{\mathrm{d}}Y}\right) _{Y=0} , ReCf_{2}=-\wedge _{1}\left( \frac{{\mathrm{d}}U}{{\mathrm{d}}Y}\right) _{Y=1} \end{aligned}$$
(15)
$$\begin{aligned} {\mathrm{Nu}}_{1}= 2\wedge _{3}\left( \frac{d \theta }{{\mathrm{d}}Y}\right) _{Y=0} , {\mathrm{Nu}}_{2}=-2\wedge _{3}\left( \frac{d \theta }{{\mathrm{d}}Y}\right) _{Y=1} \end{aligned}$$
(16)

3 Homotopy analysis method

We solve the above coupled Eqs. (10) and (11) with the associated boundary conditions Eq. (12) using HAM. We define a set of base functions to express the solution of the velocity and temperature as
$$\begin{aligned} \{Y^n|n=0,1,2,3,\ldots \} \end{aligned}$$
(17)
Therefore, the final expression for velocity and temperature can be defined as
$$\begin{aligned} U(Y)=\sum _{n=0}^{\infty }\zeta _{n}Y^n \text{ and } \theta (Y)=\sum _{n=0}^{\infty }\xi _{n}Y^n \end{aligned}$$
(18)
where \(\zeta _{n}\) and \(\xi _{n}\) are unknown coefficients to be determined. We take the linear auxiliary operators as:
$$\begin{aligned} \mathfrak {L}_{i}=\frac{{\mathrm{d}}^2}{{\mathrm{d}} Y^2} \quad {\text{where}}\quad i=1,2 \end{aligned}$$
(19)
Initial guess approximations of U(Y) and \(\theta (Y)\) are now chosen using boundary conditions (Eq. 12) as:
$$\begin{aligned} U_{0}=0, ~~~ \theta _{0}=\frac{{\mathrm{Bi}}r_{{\mathrm{T}}}}{{\mathrm{Bi}}+4}\left( Y-\frac{1}{2}\right) , \end{aligned}$$
(20)
such that
$$\begin{aligned} \mathfrak {L}_{1}(\Xi _{1}+\Xi _{2}Y)=0, ~~ \mathfrak {L}_{2}(\Xi _{3}+\Xi _{4}Y)=0 , \end{aligned}$$
(21)
where \(\Xi _{i}~~(i=1,2,3,4)\) are constants.
With nonzero auxiliary parameters \(h_{1}, h_{2}\), we develop the deformation equations of zeroth order as:
$$\begin{aligned} (1-\delta )\mathfrak {L}_{1}[U(Y;\delta )-U_{0}(Y)]=\delta h_{1}\aleph _{1}[U(Y;\delta )], \end{aligned}$$
(22)
$$\begin{aligned} (1-\delta )\mathfrak {L}_{2}[\theta (Y;\delta )-\theta _{0}(Y)]=\delta h_{2}\aleph _{2}[\theta (Y;\delta )], \end{aligned}$$
(23)
subject to the boundary conditions
$$\begin{aligned} U(0; \delta )&=-\gamma \left. \frac{{\mathrm{d}}U}{{\mathrm{d}}Y}\right| _{Y=0}\\ A_{3}\left. \frac{d\theta }{{\mathrm{d}}Y}\right| _{Y=0}&=\frac{{\mathrm{Bi}}}{2}\left( \frac{r_{{\mathrm{T}}}}{2}+\theta (0;p)\right) \end{aligned}$$
(24a)
$$\begin{aligned} U(1;\delta )&=-\gamma \left. \frac{{\mathrm{d}}U}{{\mathrm{d}}Y}\right| _{Y=1} \\ \left. \frac{d\theta }{{\mathrm{d}}Y}\right| _{Y=1}&=\frac{{\mathrm{Bi}}}{2}\left( \frac{r_{{\mathrm{T}}}}{2}-\theta (1;p)\right) \end{aligned}$$
(24b)
where \(\delta \in [0,1]\) is the embedding parameter and the nonlinear operators \(\aleph _{1}\) and \(\aleph _{2}\) are defined as
$$\begin{aligned}&\aleph _{1}[U(Y,\delta ),\theta (Y,\delta )]\nonumber \\&\quad =\wedge _{1}U''+\wedge _{2}\epsilon \theta \sin \psi +P-\wedge _{1}\frac{U}{{\mathrm{Da}}}-\frac{\wedge _{4}}{4}{\mathrm{Ha}}^2U, \end{aligned}$$
(25)
$$\begin{aligned}&\aleph _{2}[U(Y,\delta ),\theta (Y,\delta )]\nonumber \\&\quad =\theta ''+\frac{Q}{4\wedge _{3}}\theta +{\mathrm{Br}} \frac{\wedge _{1}}{\wedge _{3}}U'^2+ \frac{\wedge _{4}}{\wedge _{3}}\frac{{\mathrm{Ha}}^2{\mathrm{Br}}}{4}U^2. \end{aligned}$$
(26)
For \(\delta =0\) we have the initial guess approximations
$$\begin{aligned} U(Y;0)=U_{0}(Y),~~ \theta (Y;0)=\theta _{0}(Y), \end{aligned}$$
(27)
and for \(\delta =1\) we get the final solutions
$$\begin{aligned} U(Y;1)=U(Y),~~ \theta (Y;1)=\theta (Y). \end{aligned}$$
(28)
Hence, as \(\delta\) varies from 0 to 1, \(U(Y;\delta )\) and \(\theta (Y;\delta )\) vary continuously from the initial guesses \(U_{0}(Y)\) and \(\theta _{0}(Y)\) to the final solutions U(Y) and \(\theta (Y)\), respectively.
The mth-order deformation equations are written as
$$\begin{aligned} \mathfrak {L}_{1}[U_{m}(Y)-\daleth _{m}U_{m-1}(Y)]=h_{1}O_{m}^U(Y), \end{aligned}$$
(29)
$$\begin{aligned} \mathfrak {L}_{2}[\theta _{m}(Y)-\daleth _{m}\theta _{m-1}(Y)]=h_{2}O_{m}^\theta (Y), \end{aligned}$$
(30)
with the boundary conditions
$$\begin{aligned}& U(0;\delta )_{m-1}=-\gamma \left. \frac{{\mathrm{d}}U_{m-1}}{{\mathrm{d}}Y}\right| _{Y=0},\\ & \left. \frac{d\theta _{m-1}}{{\mathrm{d}}Y}\right| _{Y=0}-\frac{{\mathrm{Bi}}}{2}\theta (0;p)_{m-1}=0, \end{aligned}$$
(31a)
$$\begin{aligned}& U(1;\delta )_{m-1}=-\gamma \left. \frac{{\mathrm{d}}U_{m-1}}{{\mathrm{d}}Y}\right| _{Y=1} ,\\ &\left. \frac{d\theta _{m-1}}{{\mathrm{d}}Y}\right| _{Y=1}+\frac{{\mathrm{Bi}}}{2}\theta (0;p)_{m-1}=0, \end{aligned}$$
(31b)
where
$$\begin{aligned} O_{m}^U(Y)= \wedge _{1}U_{m-1}''+\wedge _{2}\epsilon \theta _{m-1} \sin \psi -\wedge _{1}\frac{U_{m-1}}{{\mathrm{Da}}}-\frac{\wedge _{4}}{4}{\mathrm{Ha}}^2U_{m-1}, \end{aligned}$$
(32)
$$\begin{aligned} O_{m}^N(Y)&= \theta _{m-1}''+ {\mathrm{Br}} \frac{\wedge _{1}}{\wedge _{3}}\left( \sum _{n=0}^{m-1}U_{m-1-n}'U_{n}'\right) +\frac{Q}{4\wedge _{3}}\theta _{m-1} \\&\quad +\frac{\wedge _{4}}{\wedge _{3}}\frac{{\mathrm{Ha}}^2{\mathrm{Br}}}{4}\left( \sum _{n=0}^{m-1}U_{m-1-n}U_{n}\right) , \end{aligned}$$
(33)
where m is an integer and
$$\begin{aligned} \daleth _{m}= {\left\{ \begin{array}{ll} 0, &{} \text {for}\ m\leqslant 1 \\ 1, &{} \text {for}\ m>1. \end{array}\right. } \end{aligned}$$
(34)
The initial approximations, linear operators and auxiliary parameters are chosen such that Eqs. (22) and (23) have a solution at each point of \(\delta \in [0,1]\). Using Taylor’s series and Eq. (27), the solutions expression can be written as
$$\begin{aligned} U(Y;\delta )= U_{0}(Y)+\sum _{m=1}^{\infty }U_{m}(Y)\delta ^m, \end{aligned}$$
(35)
$$\begin{aligned} \theta (Y;\delta )= \theta _{0}(Y)+\sum _{m=1}^{\infty }\theta _{m}(Y)\delta ^m, \end{aligned}$$
(36)
where \(h_{1}\) and \(h_{2}\) are chosen such that the series Eqs. (35) and (36) are convergent at \(\delta =1\). Hence, using Eq. (28), we have
$$\begin{aligned} U(Y)= U_{0}(Y)+\sum _{m=1}^{\infty }U_{m}(Y), \end{aligned}$$
(37)
$$\begin{aligned} \theta (Y)= \theta _{0}(Y)+\sum _{m=1}^{\infty }\theta _{m}(Y), \end{aligned}$$
(38)
with
$$\begin{aligned} \begin{aligned} U_{m}(Y)=\left. \frac{1}{m!}\frac{\partial ^m U(Y;\delta )}{\partial \delta ^m}\right| _{\delta =1}, \\ \theta _{m}(Y)=\left. \frac{1}{m!}\frac{\partial ^m \theta (Y;\delta )}{\partial \delta ^m}\right| _{\delta =1}. \end{aligned} \end{aligned}$$
Fig. 2

h-curve for \(h_{1}\)

Fig. 3

h-curve for \(h_{2}\)

Table 1

\(h_{1}\) and \(h_{2}\) optimum values for different approximation orders

Order (m)

\(h_{1}\)

Minimum error

\(h_{2}\)

Minimum error

14

− 0.62

6.33 × 10−6

− 1.16

3.24 × 10−6

16

− 0.62

2.92 × 10−6

− 1.16

1.63 × 10−6

18

− 0.62

8.11 × 10−7

− 1.16

3.55 × 10−8

Table 2

Velocity at different points on the channel for \(P = 12 ;~\varepsilon = 100; ~ r_{{\mathrm{T}}}=1; ~ {\mathrm{Ha}}=2; \psi = \pi /2; ~{\mathrm{Da}} = \infty ; ~ {\mathrm{Bi}} = 10^{10};~Q=\phi =\gamma =0\)

Y

Liu and Lo (DTM) [47]

Umavathi and Malashetty (Perturbation) [48]

Present (HAM)

\(\epsilon = -100\)

\(\epsilon = 100\)

\(\epsilon = 100\)

\(\epsilon = -100\)

\(\epsilon = 100\)

\(\epsilon = 100\)

\(\epsilon = -100\)

\(\epsilon = 100\)

\(\epsilon = 100\)

\({\mathrm{Br}}=0.08\)

\({\mathrm{Br}}=0\)

\({\mathrm{Br}}=0.08\)

\({\mathrm{Br}}=0.08\)

\({\mathrm{Br}}=0\)

\({\mathrm{Br}}=0.08\)

\({\mathrm{Br}}=0.08\)

\({\mathrm{Br}}=0\)

\({\mathrm{Br}}=0.08\)

0.2

0.965

0.681

0.822

0.985

0.69

0.805

0.982

0.681

0.821

0.4

1.23

1.21

1.432

1.235

1.22

1.395

1.266

1.208

1.428

0.6

1.04

1.4

1.63

1.1

1.392

1.55

1.073

1.402

1.626

0.8

0.571

1.071

1.22

0.585

1.075

1.18

0.596

1.071

1.216

Table 3

Temperature at different points on the channel for\(P = 12 ;~\varepsilon = 100; ~ r_{{\mathrm{T}}}=1; ~ {\mathrm{Ha}}=2; \psi = \pi /2; ~{\mathrm{Da}} = \infty ; ~ {\mathrm{Bi}} = 10^{10};~Q=\phi =\gamma =0\)

Y

Liu and Lo (DTM) [47]

Umavathi and Malashetty (Perturbation) [48]

Present (HAM)

\(\epsilon = -100\)

\(\epsilon = 100\)

\(\epsilon = 100\)

\(\epsilon = -100\)

\(\epsilon = 100\)

\(\epsilon = 100\)

\(\epsilon = -100\)

\(\epsilon = 100\)

\(\epsilon = 100\)

\({\mathrm{Br}}=0.08\)

\({\mathrm{Br}}=0\)

\({\mathrm{Br}}=0.08\)

\({\mathrm{Br}}=0.08\)

\({\mathrm{Br}}=0\)

\({\mathrm{Br}}=0.08\)

\({\mathrm{Br}}=0.08\)

\({\mathrm{Br}}=0\)

\({\mathrm{Br}}=0.08\)

0.2

− 0.249

− 0.3

− 0.231

− 0.25

− 0.3

− 0.233

− 0.253

− 0.3

− 0.232

0.4

− 0.035

− 0.1

− 0.008

− 0.04

− 0.1

− 0.01

− 0.045

− 0.1

− 0.009

0.6

0.171

0.1

0.1908

0.158

0.1

0.178

0.155

0.1

0.191

0.8

0.36

0.3

0.376

0.34

0.3

0.355

0.34

0.3

0.377

Table 4

Comparison of Nusselt numbers for \(P = 12 ; ~\epsilon = 100;~ \psi = \pi /2;~ {\mathrm{Bi}} = 10^{10};~{\mathrm{Da}} = \infty ;~Q={\mathrm{Ha}}=r_{{\mathrm{T}}}=\phi =\gamma =0\)

Br

Barletta [49]

Zanchini [50]

Present

− 0.04

− 43.211

− 43.211

− 43.212

− 0.03

− 33.219

− 33.219

− 33.218

− 0.02

− 22.722

− 22.722

− 22.722

− 0.01

− 11.669

− 11.669

− 11.669

− 0.005

− 5.916

− 5.916

− 5.916

0

0

0

0

0.005

6.087

6.087

6.087

0.01

12.356

12.356

12.354

0.02

25.482

25.482

25.469

0.03

39.478

39.478

39.427

0.04

54.463

54.464

54.322

Table 5

Thermophysical properties of base fluid and nanoparticles [46]

Property

Base fluid (water)

Nanoparticle (copper)

\(C_{p}\) (J/kg K)

4179

383

\(\sigma\) (\(\Omega\)m)\(^{-1}\)

0.05

5.96 × 107

\(\wp\) (kg/m\(^3\))

997.1

8954

k (W/mK)

0.6

400

\(\beta\) (K\(^{-1}\))

2.1 × 10−4

1.67 × 10−5

4 Convergence and validation

As the HAM solution is strongly dependent on the convergence control parameter h , \(h_{1}\) and \(h_{2}\) curves are plotted for various order of approximation so as to assure convergence as shown in Figs. 2 and 3, respectively, for \(P=12, {\mathrm{Br}}=0.05, \psi =\pi /4, \phi =5\%, {\mathrm{Da}}=0.5, {\mathrm{Ha}}=2, \varepsilon =100\). It is clear from the figures that the admissible range for \(h_{1}\) is \(-0.75\le h_{1}\le -0.25\) and that for \(h_{2}\) is \(-2\le h_{2}\le -1\). To choose the optimal value of the convergence control parameter, the average residual errors are defined as [51]:
$$\begin{aligned} E_{U,m}= \frac{1}{\mathbf {K}}\sum _{i=0}^{\mathbf {K}}\left[ \aleph _{1}\left( \sum _{j=0}^{m}U_{j}(i\Delta t)\right) \right] ^2 , \end{aligned}$$
(39)
$$\begin{aligned} E_{\theta ,m}= \frac{1}{\mathbf {K}}\sum _{i=0}^{\mathbf {K}}\left[ \aleph _{2}\left( \sum _{j=0}^{m}\theta _{j}(i\Delta t)\right) \right] ^2 , \end{aligned}$$
(40)
where \(\Delta t=1/\mathbf {K}\) and \(\mathbf {K}=20\). The minimum average residual errors are shown in Table 1 for different order of approximations. Thus, the minimum average residual error for U is taken at \(h_{1}=-0.62\) and for \(\theta\) is taken at \(h_{2}=-1.16\).

Using the technique of homotopy analysis method, the velocity and temperature characteristics of the issue were determined. Tables 2 and 3 show the numerical values of velocity and temperature field, respectively, obtained by HAM and compared with the results obtained by Liu and Lo [47], who used differential transformation method (DTM) and Umavathi and Malashetty [48], who used perturbation method. It can be seen that the answers agree with each other exceptionally. Moreover, Nusselt numbers were calculated for different values of Br and compared with existing results presented by Barletta [49] and Zanchini [50]. They were found to be in excellent agreement and shown in Table 4. The thermophysical properties of water and copper can be seen in Table 5.

5 Second law analysis

The local rate of generation of entropy for MHD nanofluid flow in a porous medium is given by [52, 53]
$$\begin{aligned} \begin{aligned} S_{G}=\frac{\kappa _{nf}}{T_{0}^2}\left( \frac{\partial T}{\partial y}\right) ^2 +\frac{\mu _{nf}}{T_{0}}\left( \frac{\partial u}{\partial y}\right) ^2 +\frac{\mu _{nf}}{KT_{0}}u^2+\sigma _{nf} B^2\frac{u^2}{T_{0}}. \end{aligned} \end{aligned}$$
(41)
The entropy generation for the flow in consideration is due mainly to four components. The first term on the right-hand side is the entropy generation due to heat transfer, the second due to fluid friction/viscous dissipation, the third term due to Darcy viscous dissipation as a result of the presence of porous medium and the fourth term due to the presence of magnetic field.
We apply the non-dimensional variables defined in Eq. (9) to the above equation, and using the non -dimensional entropy generation formula
$$\begin{aligned} N_{s}=\frac{L^2T_{0}^2}{\kappa _{f}\Delta T^2}S_{G}, \end{aligned}$$
(42)
we get the following equation
$$\begin{aligned} \begin{aligned} N_{s}&=\wedge _{3}\left( \frac{\partial \theta }{\partial Y}\right) ^2 +\wedge _{1}\frac{{\mathrm{Br}}}{\Omega }\left[ \left( \frac{\partial U}{\partial Y}\right) ^2 + \frac{U^2}{{\mathrm{Da}}} \right] \\&\quad +\wedge _{4}{\mathrm{Ha}}^2 \frac{{\mathrm{Br}}}{\Omega }U^2. \end{aligned} \end{aligned}$$
(43)
where \(\Omega =\Delta T/T_{0}\). The entropy generation rate can be roughly separated into two categories, namely entropy due to heat transfer and entropy due to viscous dissipation, and we write it as
$$\begin{aligned} N_{s}=\left. N_{h}\right| _{{\mathrm{heat\,transfer}}}+\left. N_{v}\right| _{\mathrm{fluid\,friction}}. \end{aligned}$$
(44)
such that
$$\begin{aligned} \left. N_{h}\right| _{\mathrm{heat\,transfer}}&=\wedge _{3}\left( \frac{\partial \theta }{\partial Y}\right) ^2 \\ \left. N_{v}\right| _{\mathrm{fluid\,friction}}&=\wedge _{1}\frac{{\mathrm{Br}}}{\Omega }\left[ \left( \frac{\partial U}{\partial Y}\right) ^2 + \frac{U^2}{{\mathrm{Da}}} \right] \\&\quad +\wedge _{4}{\mathrm{Ha}}^2 \frac{{\mathrm{Br}}}{\Omega }U^2. \end{aligned}$$
Another important aspect of thermodynamic analysis of heat transfer problems is the Bejan number. The Bejan number is defined as the ratio of entropy generation due to heat transfer to the total entropy generation, written as:
$$\begin{aligned} {\mathrm{Be}}=\frac{{\mathrm{Generation\,of\,Entropy\,by\,heat\,transfer}}}{{\mathrm{Total\,Entropy\,generated}}}. \end{aligned}$$
(45)
The Bejan number indicates the dominance of irreversibility in the process and ranges from 0 to 1. For \({\mathrm{Be}}>0.5\), the irreversibility due to heat transfer dominates, while for \({\mathrm{Be}}<0.5\), the irreversibility due to fluid friction dominates. At \({\mathrm{Be}} = 0.5\), the contribution to irreversibility from both heat transfer and fluid friction is equal.
Fig. 4

Graph of entropy generation with the influence of Brinkman number. Dotted lines denote heat sink, \(Q=-1\), and solid lines denote heat source, \(Q=1\)

Fig. 5

Graph of Bejan number with the influence of Brinkman number. Dotted lines denote heat sink, \(Q=-1\), and solid lines denote heat source, \(Q=1\)

Fig. 6

Graph of entropy generation with the influence of Biot number. Dotted lines denote heat sink, \(Q=-1\), and solid lines denote heat source, \(Q=1\)

Fig. 7

Graph of Bejan number with the influence of Biot number. Dotted lines denote heat sink, \(Q=-1\), and solid lines denote heat source, \(Q=1\)

Fig. 8

Graph of entropy generation with the influence of Darcy number. Dotted lines denote heat sink, \(Q=-1\), and solid lines denote heat source, \(Q=1\)

Fig. 9

Graph of Bejan number with the influence of Darcy number. Dotted lines denote heat sink, \(Q=-1\), and solid lines denote heat source, \(Q=1\)

Fig. 10

Graph of entropy generation with the influence of Hartmann number. Dotted lines denote heat sink, \(Q=-1\), and solid lines denote heat source, \(Q=1\)

Fig. 11

Graph of Bejan number with the influence of Hartmann number. Dotted lines denote heat sink, \(Q=-1\), and solid lines denote heat source, \(Q=1\)

6 Results and discussion

So as to conduct a thorough analysis of the various parameters governing the flow, computations were conducted and graphs were plotted accordingly. Unless mentioned otherwise, we have taken \(P=12,~ {\mathrm{Br}}=0.05,~ \phi =5\%,~ {\mathrm{Da}}=0.5, ~\varepsilon =100, ~\psi =\pi /4,~ \Omega =0.5, ~{\mathrm{Bi}}=10, ~\gamma =0.1, ~{\mathrm{Ha}}=2, ~r_{{\mathrm{T}}}=1\) throughout the study. In Figs. 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, the entropy generation profile and Bejan number profiles are shown graphically. The dotted lines denote the case of heat absorption, while the solid lines denote heat generation case.

6.1 Entropy generation and Bejan number

6.1.1 Brinkman number

Figure 4 shows the Brinkman number influence on the entropy generation profile. Increased Brinkman numbers can be found to lead to reduced entropy generation in the scheme. It may be noted that the rise in the Brinkman number shows that the viscous dissipation increases, and this reduces the fluid velocity. Entropy in the scheme therefore decreases. Figure 5 also shows the effect of rising Brinkman number on Bejan number. With the Brinkman increase, it may be observed that the Bejan number is declining. This shows the increasing impact of viscous dissipation on irreversibility.

6.1.2 Biot number

An increase in Biot number and its impact on the generation of entropy is shown in Fig. 6. An increase in the Biot number can be seen as leading to enhanced entropy. The increase in the Biot number indicates a higher convection on the channel walls that causes a greater temperature at the channel walls. As a result, entropy generation is on the rise. Bejan numbers also rise with the rise of the Biot number, a result shown in Fig. 7. As already stated, the rise in Biot number results in greater wall temperatures, leading in turn to a greater effect of the irreversibility by heat transfer. Therefore, the Bejan number is increasing.

6.1.3 Darcy number

An increase in Darcy number leads to a complete increase in entropy generation in the system, as illustrated in Fig. 8. Increased permeability of the porous material increases the Darcy number. This contributes to a rise in the liquid temperature which in turn increases heat production in the scheme. Figure 9 shows the influence on the Bejan number of the increment of Darcy number. The increase in the Darcy number leads to a reduction in the Bejan number near the channel walls. But Bejan number just above the middle of the channel is slightly rising.
Fig. 12

Graph of entropy generation with the influence of slip parameter. Dotted lines denote heat sink, \(Q=-1\), and solid lines denote heat source, \(Q=1\)

Fig. 13

Graph of Bejan number with the influence of slip parameter. Dotted lines denote heat sink, \(Q=-1\), and solid lines denote heat source, \(Q=1\)

Fig. 14

Graph of entropy generation with the influence of nanoparticle volume fraction. Dotted lines denote heat sink, \(Q=-1\), and solid lines denote heat source, \(Q=1\)

Fig. 15

Graph of Bejan number with the influence of nanoparticle volume fraction. Dotted lines denote heat sink, \(Q=-1\), and solid lines denote heat source, \(Q=1\)

Fig. 16

Graph of entropy generation with the influence of Brinkman number. Dotted lines denote heat sink, \(Q=-1\), and solid lines denote heat source, \(Q=1\)

Fig. 17

Graph of Bejan number with the influence of channel inclination angle. Dotted lines denote heat sink, \(Q=-1\), and solid lines denote heat source, \(Q=1\)

6.1.4 Hartmann number

The rise in the Hartmann number causes a lower entropy generation near the walls of the canal, but around the center of the canal was seen a significant increase in entropy generation, as shown in Fig. 10. Hartmann number rise results in a rise in liquid temperature, which increases entropy generation. This rise in fluid temperature also results in an increase in Bejan number near the channel walls, as shown in Fig. 11. The boost in the Hartmann number results in a decrease in the fluid velocity that contributes to the decrease in the Bejan number.

6.1.5 Slip parameter

The rise in the slip parameter resulted to a higher entropy generation in the bottom of the channel, while the entropy generated at the top wall reduced as shown in Fig. 12. The increase in slip parameter decreased the Bejan number on the bottom of the channel, while the Bejan number increased on the top of the channel, as shown in Fig. 13.

6.1.6 Nanoparticle volume fraction

Figure 14 shows an increase in the amount of nanoparticles in the fluid and its consequence on entropy generation. The thermal conductivity of the liquid also improves with a rise in nanoparticles, and energy is transmitted more quickly through the fluid. Consequently, entropy produced increases. The irreversibility of the liquid also progressively depends on the heat transfer as the thermal conductivity of the liquid increases, an attribute that is demonstrated in Fig. 15.

6.1.7 Channel inclination angle

Figure 16 shows the increase in the channel’s tilt angle and its influence on entropy generation. The entropy generation at the bottom plate decreased with an increase in tilt angle but gradually increased from around the channel widths’ quarter point. On the other hand, as shown in Fig. 17, the increase in tilt angle initially increased the Bejan number. The Bejan number was found to be decreasing at around the channel’s quarter width with an increase in tilt angle.

It may be noted that in all the cases, entropy generated near the walls of the channel was found to be higher for heat absorption case than heat generation cases, but the entropy generated near the center of the channel was lower for heat absorption case than heat generation case. The same can be observed for Bejan number.
Fig. 18

Graph of Nusselt number with the influence of nanoparticle volume fraction. Solid lines denote lower channel wall, Nu\(_{1}\), and dotted lines denote upper channel wall, Nu\(_{2}\)

Fig. 19

Graph of local skin friction with the influence of nanoparticle volume fraction. Solid lines denote lower channel wall, \(ReCf_{1}\), and dotted lines denote upper channel wall, \(ReCf_{2}\)

Fig. 20

Graph of Nusselt number with the influence of Hartmann number. Solid lines denote lower channel wall, Nu\(_{1}\), and dotted lines denote upper channel wall, Nu\(_{2}\)

Fig. 21

Graph of local skin friction with the influence of Hartmann number. Solid lines denote lower channel wall, \(ReCf_{1}\), and dotted lines denote upper channel wall, \(ReCf_{2}\)

Fig. 22

Graph of Nusselt number with the influence of channel inclination angle. Solid lines denote lower channel wall, Nu\(_{1}\), and dotted lines denote upper channel wall, Nu\(_{2}\)

Fig. 23

Graph of local skin friction with the influence of channel inclination angle. Solid lines denote lower channel wall, \(ReCf_{1}\), and dotted lines denote upper channel wall, \(ReCf_{2}\)

6.2 Nusselt number and skin friction

Figures 18, 19, 20, 21, 22, 23 show the effect of various flow parameters on the Nusselt number and local skin friction, respectively. Figures 18 and 19 depict the influence of increasing nanoparticles in the fluid on Nusselt number and skin friction, respectively. It may be observed from Fig. 18 that increasing the amount of nanoparticles in the fluid increases the Nusselt number on the lower wall of the channel. This is due to the fact that nanoparticles in the fluid increase the thermal conductivity of the fluid. The local skin friction was observed to decrease with increasing the nanoparticles in the fluid. This is attributed to the fact that increasing the amount of nanoparticles in the fluid increases the density of the fluid and hence the velocity decreases. Thus, skin friction decreases. Increasing the Hartmann number and its impact on Nusselt number and skin friction are shown in Figs. 20 and 21, respectively. Increasing the Hartmann number leads to an increase in Lorentz forces and suppresses the heat transfer capability of the fluid, leading to declining Nusselt number as shown in Fig. 20. The same Lorentz force also retards the fluid flow velocity, and hence, the skin friction decreases, as depicted in Fig. 21. Nusselt number and skin friction as a function of channel tilt angle are shown in Figs. 22 and 23. Nusselt number was observed to increase with increasing the channel tilt angle, as demonstrated in Fig. 22. As the fluid flow increases, the heat transfer capability of the fluid also surges and hence, Nusselt number increases. The skin friction on the upper wall of the channel increased, while the skin friction on the lower wall of the channel decreased with increasing the channel angle of inclination, as depicted in Fig. 23. As the channel becomes increasingly perpendicular, the amount of fluid in contact with the lower wall of the channel depletes which leads to a decrease in the skin friction. On the other hand, the amount of fluid in contact with the upper wall of the channel increases which leads to more skin friction.

In the ongoing section, Nusselt number and skin friction were taken as a function of the heat source/sink parameter. It may be observed that both Nusselt number and skin friction are increasing functions of the heat source/sink parameter.

7 Conclusion

The current work investigated the entropy generation for a nanofluid flow in a tilted channel with heat source/sink filled with a saturated porous medium with asymmetric wall temperatures while also considering Navier slip and convection at the walls of the channel. The governing equations were reduced to non-dimensional form and solved using HAM. Special case of the problem was compared with existing literature and found to be in excellent agreement. Throughout the study, entropy generation was found to be highest at the lower wall of the channel and minimum just above the center of the channel width. At the center of the channel, irreversibility process was found to be dominated by heat transfer irreversibility. It may also be noted that Bejan number and entropy generation were lower for heat absorption case than heat generation case near the center of the channel for all cases in the study. However, both Nusselt number and skin friction were observed to be higher for heat generation than heat absorption for all cases.

Notes

Acknowledgements

The authors acknowledges TEQIP-III , NIT Mizoram, for the financial support granted to Mr. Lalrinpuia Tlau for his doctoral study.

Funding

Mr. Lalrinpuia Tlau was funded by TEQIP-III for his doctoral study.

Compliance with ethical standards

Conflict of interest

The authors declare that they have no conflict of interest.

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© Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.Department of MathematicsNational Institute of Technology MizoramAizawlIndia

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