Analysis of thermo-hydraulic characteristics for flow of MWCNT-Fe 3 O 4 /H 2 O hybrid nanofluid through a wavy channel under magnetic field

Abstract

We investigate computationally the transport characteristics for flow of MWCNT-Fe₃O₄/H₂O hybrid nanofluid through a wavy channel under the influence of an externally applied magnetic field. The flow and temperature fields are analyzed in terms of streamlines, isotherms, average Nusselt number ( $\overline{Nu}$ ), magnetic enhancement ratio (ER_m), and magnetic performance factor (PF_m) for the following range of parameters: 100 $\le \mathrm{Reynolds}\mathrm{number}($ Re $)\le$ 500, 0 $\le \mathrm{Hartmannnumber}($ Ha $)\le$ 10, and 0% $\le \mathrm{nano}-\mathrm{particle}\mathrm{volume}\mathrm{fraction}(\varphi )\le$ 0.3%. The strength of the reverse flow in the recirculatory zones decreases with Ha, and beyond a critical Ha, the flow becomes attached even in the diverging part of the channel. The hot spot intensity near the walls decreases and hence, the value of $\overline{Nu}$ increases with the increase in volume fraction of nano-particles and Ha. The rate of increment of $\overline{Nu}$ is steeper at lower values of Ha. It further reveals that PF_m monotonically decreases with Ha for lower Re ( = 100). For higher values of Re, PF_m decreases with Ha for its lower and higher values, while for the intermediate range of Ha, PF_m increases with Ha.

Introduction

In the last few decades, the design and development of efficient and compact thermal systems is one of the focus areas of research because of its potential applications in numerous fields, including solar collectors, heat exchangers, chemical process plants, geothermal processes and flow boiling.^1–3 The use of wavy surfaces instead of plane surfaces is one of the widely used passive techniques for heat transfer augmentation. For flow through wavy channels, recirculatory zones are formed and thereby augmenting the heat transfer rate by fluid mixing and alteration in thermal boundary layer.^4–7 Moreover, in several applications, two or more active and/or passive heat exchanging techniques, viz., wavy surface and porous media, ⁸ wavy surface and nanofluid, ⁹ wavy surface, nanofluid and porous media, ¹⁰ wavy surface, nanofluid and magnetic field ¹¹ have been used. The main focus of all these techniques is mainly to maximize the heat transfer enhancement with minimum pressure drop penalty.

In the last few decades, numerous research investigations have been reported analyzing the transport dynamics for flow through wavy channels.^12–22 Nandi and Chattopadhyay numerically explored the effect of inlet pulsation on the heat transfer for flow through a wavy channel.^14,21 They found that the heat transfer rate is augmented with the Strouhal number and wavy amplitude. Kumar et al.^15,16 numerically investigated the heat transfer in a plane circular wavy microchannel (PCWMC) and circular wavy microchannel (CWMC) with tangentially branched secondary channels (TBSC). Tiwari and Moharana ²⁰ studied the conjugate heat transfer characteristics for flow through three-dimensional raccoon and wavy microchannels and found that the thermal performance is better for raccoon microchannel. Recently, Tiwari and Moharana ¹² investigated the conjugate heat transfer characteristics for two-phase boiling of water inside a wavy microchannel and reported that the performance factor increases drastically with the increase in thermal conductivity ratio of solid to fluid domain.

The thermally conductive nano-particles have been extensively used in base fluid like water as a passive heat transfer technique.^23–36 In recent times, several researchers have employed hybrid nanofluid like MWCNT–Fe₃O₄/water,^25,33,36 water-magnetite-graphene, ³² Cobalt- CeO₂-kerosene, ³¹ Cu–Ag/water, ³⁰ water-C₂H₆O₂-GO-MoS₂ ²⁹ for various applications. The main advantage of hybrid nanofluid is that it has better rheological and thermal behaviours as compared to single component nanofluids.^37,38 The thermal transport characteristics for MWCNT–Fe₃O₄/water nanofluid are better among all the aforementioned hybrid nanofluids due to the presence of multi-walled carbon nanotubes and highly magnetic material Fe₃O₄ and moreover it is stable up to 0.3% volume fraction. ³³ Sundar et al. ³³ experimentally found that the use of MWCNT–Fe₃O₄/water nanofluid through copper tube enhances the heat transfer rate from 14.81% to 31.10% for the range of Reynolds number 3000 to 22,000 for nano-particle volume fraction of 0.3%. They also reported that the thermal performance is better for the hybrid nanofluid as compared to single component nanofluids like Al₂O₃, TiO₂, and Fe₃O₄. Based on experimental investigation, Alklaibi et al. ³⁹ reported that the use of Fe₃O₄ coated MWCNT/water nanofluid in a plate heat exchanger, the thermal performance has improved from 1.02 to 1.053 times with the change in nano-particle volume fraction from 0.05% to 0.3%. Also, the entropy generation reduces up to 32.87% for the change in nano-particle volume fraction 0% to 0.3%.

The flow field for an electrically conductive fluid can be modulated with the application of magnetic field due to the resistive Lorentz force and it is called magnetohydrodynamic (MHD) flow. Several research investigations have been reported in literature on MHD flow through wavy channels of different geometry with an objective to achieve an enhanced heat transfer.^40–48 Mayeli et al.^46,47 numerically studied the effect of applied magnetic field on the forced convective flow of Al₂O₃-water nanofluid through sinusoidal and ribbed wavy channels and reported that with the change in Hartmann number from 0 to 40 the value of the average Nusselt number enhances up to 34.4% and 41% for ribbed and sinusoidal wavy channels, respectively. Mehta and Pati ¹¹ explored the MHD heat transport characteristics for flow of Al₂O₃-water nanofluid through an asymmetric wavy channel for Reynolds number in the range of 10 to 500. They reported that the increase in nano-particle volume fraction enhances the magnetic performance factor for the lower and higher regimes of Reynolds number in the considered range.

It indicates from the extensive literature survey that heat transfer rate in any thermal system can be augmented using nanofluid as a working fluid, imposing magnetic field in case of electrically conductive fluids, and by making the walls of the channel as wavy. However, no one has investigated the effect of magnetic field on the thermo-hydraulic characteristics for flow of MWCNT–Fe₃O₄/water nanofluid through a wavy channel, which we set as the objective of the present work.

Theoretical formulation

The two-dimensional, steady, and incompressible flow of Newtonian MWCNT-Fe₃O₄/H₂O hybrid nanofluid through a wavy channel is considered under the influence of externally applied magnetic field strength, B in y*-direction as shown in Figure 1. The walls of the channel at the inlet and outlet are straight with a distance of separation between the walls as 2L. The profiles of top and bottom wavy walls in the intermediate part are the following: $s_T(x^{*})=L+0.4Lsin(\pi ((x^{*}-3L)/L)$ and $s_B(x^{*})=-L-0.4Lsin(\pi ((x^{*}-3L)/L)$ . The axial lengths of inlet flat part, intermediate wavy part and outlet flat part are taken as 3L, 12L, and 5L, respectively.^18,49,50 The thermo-physical properties of the nanofluid are temperature-independent and the effects of viscous heating and thermal radiation are neglected. Moreover, we assume that the nano-particles are uniformly distributed in the base fluid. ¹¹ Furthermore, the magnetic field is uniform inside the domain as the order of magnetic Reynolds number is very small (<<1). ¹¹

Figure 1.

Physical domain.

The dimensionless form of governing transport equations can be written as 1:1

$$\mathrm{\nabla }\cdot \mathbf{u}=0$$

(1)

$${\displaystyle \frac{{\rho }_{nf}}{{\rho }_{bf}}(\mathbf{u}\cdot \mathrm{\nabla })\mathbf{u}=-\mathrm{\nabla }p+{\displaystyle \frac{{\mu }_{nf}}{{\mu }_{bf}}{\displaystyle \frac{1}{\mathrm{Re}}{\mathrm{\nabla }}^2\mathbf{u}+{\mathbf{F}}_m}}}$$

(2)

$$(Pr\mathrm{Re}){\displaystyle \frac{{(\rho c_p)}_{nf}}{{(\rho c_p)}_{bf}}(\mathbf{u}\cdot \mathrm{\nabla }\theta )={\displaystyle \frac{k_{nf}}{k_{bf}}{\mathrm{\nabla }}^2\theta }}$$

(3)

Here, $\mathrm{\nabla }\equiv (\mathrm{\partial }/\mathrm{\partial }x,\mathrm{\partial }/\mathrm{\partial }y)$ , $x=x^{*}/L$ , $y=y^{*}/L$ , $\mathbf{u}=(=u\hat{i}+v\hat{j})={\mathbf{u}}^{*}/u_{avg}^{*}$ is the dimensionless velocity vector, and ${\mathbf{F}}_m=-({\sigma }_{nf}/{\sigma }_{bf})H{a}^2(u\hat{i})$ is the Lorentz force due to the externally applied magnetic field. The dimensionless pressure and temperature are defined as $p=p^{*}/({\rho }_{bf}{(u_{avg}^{*})}^2)$ and $\theta =(T-T_i)k_{bf}/qL$ , respectively, where, $T_i$ and q are the inlet fluid temperature and imposed constant heat flux at the wavy walls. The different dimensionless numbers, namely Reynolds number, Prandtl number, and Hartmann number are defined as: $\mathrm{Re}={\rho }_{bf}{u}_{avg}^{*}L/{\mu }_{bf}$ , $\mathrm{Pr}={\mu }_{bf}(c_p{)}_{bf}/k_{bf}$ and $Ha=LB\sqrt{{\sigma }_{bf}/{\mu }_{bf}}$ , where, ρ, σ, μ, c_p, and k are the density, electrical conductivity, dynamic viscosity, specific heat capacity and thermal conductivity of the fluid, respectively. The subscripts bf and nf indicate base fluid and nanofluid, respectively.

No slip boundary condition ⁵¹ is used at all the wall. The other boundary conditions employed are as follows:

At the inlet

$$u=1.5(1-y^2)$$

(4a)

$$v=0$$

(4b)

$$\theta =0$$

(4c)

At the wall

$$u=v=0$$

(4d)

$$\mathrm{\partial }\theta /\mathrm{\partial }n=1\mathrm{for}3\le x\le 15$$

(4e)

$$\mathrm{\partial }\theta /\mathrm{\partial }y=0\mathrm{for}x<3\mathrm{and}x>15$$

(4f)

At the outlet

$$p=p_{atm}$$

(4g)

$$\mathrm{\partial }\theta /\mathrm{\partial }x=0$$

(4h)

To the best knowledge of the authors, explicit expressions are not available for evaluating the thermo-physical properties of MWCNT-Fe₃O₄/H₂O hybrid nanofluid and accordingly, we use the values of different properties as available in the literature ³³ based on experimental findings for different values of nano-particle volume fraction (ϕ) as presented in Table 1. Moreover, the electrical conductivity of Fe₃O₄ ( = 25000Sm⁻¹) is much higher than the electrical conductivity of MWCNT ( = 10⁻⁷Sm⁻¹) and therefore, the effect of electrical conductivity of MWCNT on the electrical conductivity of nanofluid has been ignored. ³⁸ The following expression has been used to calculate the electrical conductivity of the nanofluid ³⁸ :

$${\sigma }_{nf}={\sigma }_{bf}\left(1+{\displaystyle \frac{3\left({\displaystyle \frac{{\sigma }_{np}}{{\sigma }_{bf}}-1}\right)\varphi }{\left({\displaystyle \frac{{\sigma }_{np}}{{\sigma }_{bf}}+2}\right)-\left({\displaystyle \frac{{\sigma }_{np}}{{\sigma }_{bf}}-1}\right)\varphi }}\right)$$

(5)

Table 1.

Thermo-physical properties of MWCNT-Fe₃O₄/H₂O hybrid nanofluid. ³³

$\varphi$ (%)	Density, $\rho (kg{m}^{-3})$	Thermal conductivity, $k(W{m}^{-1}{K}^{-1})$	Dynamic viscosity, Density, $\mu (Pas)$	Specific heat capacity, $c_p$ $(Jk{g}^{-1}{K}^{-1})$
0	998.5	0.602	0.79	4182
0.1	1002.34	0.6734	0.91	4182.66
0.3	1010.04	0.6856	1.01	4183.99

Here, the electrical conductivity of base fluid ( ${\sigma }_{bf}$ ) is 5.5 × 10⁻⁶ Sm⁻¹. ³⁸

The computed temperature field is analyzed in terms of the Nusselt number evaluated as $Nu=hL/k_{bf}$ , where, $h=(q/(T_{wall}-T_i))$ is the heat transfer coefficient. Using the dimensionless form of temperature, the Nusselt number is calculated as ¹¹ :

$$Nu=\left({\displaystyle \frac{1}{{\theta }_{wall}}}\right){\displaystyle \frac{k_{nf}}{k_{bf}}}$$

(6)

The overall heat transfer rate can be quantified in terms of average Nusselt number $(\overline{Nu})$ as ¹¹ :

$$\overline{Nu}={\displaystyle \frac{\underset{X=3}{\overset{15}{\int }}NudS}{\underset{X=3}{\overset{15}{\int }}dS}}$$

(7)

Here, S is the direction along the wavy wall.

The magnetic enhancement ratio ( $E{R}_m$ ) is used to express the relative enhancement in heat transfer rate due to the imposed magnetic field as ¹¹ :

$$E{R}_m={\overline{Nu}/\overline{Nu}}_{Ha=0}$$

(8)

The relative change in pressure-drop due to the imposed magnetic field is expressed as in terms of magnetic pressure ratio ( $P{R}_m$ ) as ¹¹ :

$$P{R}_m={\mathrm{\Delta }p/\mathrm{\Delta }p}_{Ha=0}$$

(9)

The magnetic performance factor ( $P{F}_m$ ) is used to assess the ratio of enhancement in heat transfer rate and the change in pressure drop due to imposition of an external magnetic field as ¹¹ :

$$P{F}_m={\displaystyle \frac{E{R}_m}{{(P{R}_m)}^{1/3}}}$$

(10)

Numerical methodology and validation of model

We use a finite element method based commercial solver COMSOL Multiphysics v 5.2 ⁵² to compute the flow and temperature fields. The physical domain is divided into large number of subdomains called as elements. We define shape functions for each of the transport variables, and gauss values of transport variables are substituted in the governing equations, resulting in the residual, R. The shape function is multiplied with R, and the corresponding integrals are equated to zero. Hence, a local matrix is created for each element. Using all the local matrices, a global matrix is generated for the entire domain, and using the boundary conditions, the transport variables are computed iteratively up to the relative residual criteria is achieved as 10⁻⁶. We also perform a grid independence test by computing the average Nusselt number ( $\overline{Nu}$ ) for different mesh type (M) as depicted in Table 2. The mesh type M4 is selected for the numerical simulation as the relative percentage difference of ( $\overline{Nu}$ ) is less than 0.1% compared to the next level of refined mesh type, M5.

We validate extensively the computational model with the reported works. First, a comparison of local Nusselt number has been made as shown in Figure 2(a) with those published by Wang and Chen ⁵ for pressure driven flow in a wavy channel for different parameters as given in the figure caption. The second validation has been done by comparing the magnetohydrodynamic (MHD) dimensionless flow velocity profile for flow through a plane channel at different Hartmann numbers with the results of Back ⁵³ as presented in Figure 2(b). Furthermore, a third validation is made to verify the correctness in predicting the heat transfer characteristics for nanofluids flow. The ratio of heat transfer coefficient ( $h/h_{\varphi =0})$ is compared in Figure 2(c) with those published by Heyhat et al. ⁵⁴ for different values of nano-particle volume fraction. The close matches of the present results with the published works^53,54 depicted in Figure 2 confirms the accuracy of the present model.

Figure 2.

(a) Comparison of local Nusselt number with the results of Wang and Chen ⁵ for the dimensionless amplitude 0.3, dimensionless wavelength 2 and Prandtl number 6.93, (b) Comparison of flow velocity with those published by Back ⁵³ for different values of Hartmann number for MHD flow through a plane channel (c) Comparison of the ratio of average heat transfer coefficient ( $h/h_{\varphi =0})$ with the experimental results of Heyhat et al. ⁵⁴ for different values of ϕ.

Table 2.

Grid independence test for the different mesh type (M) for Ha = 0 and Re = 500.

Mesh type (M)	Number of elements	$\overline{Nu}$	Magnitude of relative percentage difference (%)
M1	7738	3.6595	1.66
M2	12496	3.6462	1.294
M3	31404	3.585	0.405
M4	81190	3.5979	0.0472
M5	127864	3.5996	0

Results and discussion

We investigate the flow and heat transfer characteristics for flow of MWCNT-Fe₃O₄/H₂O hybrid nanofluid through a wavy channel under the effect of external magnetic field. We present the results in terms of flow and temperature fields, average Nusselt number ( $\overline{Nu}$ ), magnetic enhancement ratio ( $E{R}_m$ ), and magnetic performance factor ( $P{F}_m$ ) for 100 $\le \mathrm{Reynolds}\mathrm{number}($ Re $)\le$ 500, 0 $\le \mathrm{Hartmann}\mathrm{number}($ Ha $)\le$ 10, and 0% $\le \mathrm{nano}-\mathrm{particle}\mathrm{volume}\mathrm{fraction}(\varphi )\le$ 0.3%.^{11,17–19,33,55}

As the heat transfer and pressure drop both strongly depend on the velocity field, we first analyze the distribution of streamlines and also the variation of recirculation velocity. Figure 3(a) shows the streamlines for different values of Hartmann number for ϕ = 0.3% and Re = 500. It is seen that the shape of the streamlines follows the geometry of the channel along with the formation of zone of circulation in the diverging sections for Ha = 0 and lower values of Ha. The magnetic field has a strong influence on the development of the recirculation zones and accordingly the strength of flow reversal strongly varies with Ha. With the increase in Ha, the size of the recirculatory zones decreases and interestingly these zones are completely disappeared at the higher values of Ha (≥ 1.5). It is attributed to the increase in flow energy proximity to the walls as the core region velocity decreases with the increase in Ha due to the resistive Lorentz force following the principle of conservation of mass as depicted in Figure 3(b) for Re = 100 and 500. Accordingly, for higher values of Ha ( $\ge$ 1.5), the flow becomes fully attached to the wavy walls. The magnitude of the reverse flow velocity plays a crucial role to the transport characteristics and accordingly we present the variation of recirculation velocity ( $u_R$ ) with Ha for different values of Re with ϕ = 0.3% in Figure 3(c). Herein, the recirculation velocity is defined as the maxima magnitude of reverse flow velocity in the physical domain. ⁵⁶ It is observed that the value of $u_R$ decreases with the increase in Ha and becomes zero once the Ha reaches the critical value ( $H{a}_c$ ). Further, the value of $u_R$ increases with Re as the increase in inertial force promotes the momentum loss in diverging wavy section and enhances the recirculation strength. It is also noted that the value of $H{a}_c$ increases with Re. The explanation of this observation is based on the fact that the stronger recirculatory zones at higher Re need more flow energy near the walls to fully disappear at the higher Ha.

Figure 3.

(a) Streamlines for different values of Ha for ϕ = 0.3% and Re = 500, (b) dimensionless flow velocity profile for Re = 100 and 500, (c) Variation of recirculation velocity ( $u_R$ ) with Ha at the different values of Re for ϕ = 0.3%.

The isotherms for Ha = 0 and 10 are depicted in Figure 4(a) and (b), respectively at different ϕ ( = 0%, 0.3%) with Re = 500. It is observed that the presence of hot fluids near the diverging walls because of the reverse flow velocity generates hot spots (higher temperature zones) for Ha = 0. It is also noted that the intensity of hot spots is less for ϕ = 0.3% as compared to the case of ϕ = 0%. It is attributed to the increase in thermal conductivity of the nanofluid (ϕ = 0.3%) as compared to base fluid flow case (ϕ = 0%) causing higher heat transfer rate from the heated walls to the bulk fluid. Furthermore, it is observed that the fully attached flow at higher strength of magnetic field (Ha = 10) causes relatively smaller values of isotherms near the walls compared to the case of Ha = 0. It is also noted that the core cold fluid region is expanded towards to the walls at Ha = 10 due to the absence of recirculatory zone. Furthermore, the increase in ϕ from 0% to 0.3% allows smaller value of isotherms to shift towards the walls (see elliptical box in Figure 4(b)).

Figure 4.

Isotherms for different ϕ with Re = 500 (a) Ha = 0 and (b) Ha = 10.

Figure 5 depicts the variation of average Nusselt number ( $\overline{Nu}$ ) with Ha for ϕ = 0%, 0.1% and 0.3%, and Re = 500. It is found that $\overline{Nu}$ is augmented with the increase in Ha and based on the rate of increment with Ha, two regions of Ha, namely, I and II are identified. For lower Ha in region I, the rate of increase in $\overline{Nu}$ with Ha is higher, while it is gradual for higher values of Ha in region II. The explanation of these observations are as follows. For region I, the decrease in the size of recirculatory zones with Ha (see Figure 3(a)) decreases the size of hot spot intensity, which in turn increases the value of $\overline{Nu}$ significantly. Whereas, for higher values of Ha, the fully attached flow decreases the thermal boundary layer thickness gradually due to enhancement in flow velocity near the walls with Ha and hence, the gradual augmentation in $\overline{Nu}$ with Ha is seen at its higher values. The increase in ϕ enhances the value of $\overline{Nu}$ due to the decrease in hot spot intensity with ϕ (see Figure 4). The percentage enhancements in $\overline{Nu}$ for the change in Ha from 0 to 10 are obtained as 203.12%, 198.97%, and 197.88% for ϕ = 0%, 0.1%, and 0.3%, respectively for Re = 500. With the increase in $\varphi$ from 0% to 0.3%, the enhancements in the value of $\overline{Nu}$ are 23.84% and 21.7%, respectively, for Ha = 0 and 10 for Re = 500.

Figure 5.

Variation of $\overline{Nu}$ with Ha for different ϕ for Re = 500.

The relative heat transfer enhancement by the magnetic field is presented in terms of magnetic enhancement ratio ( $E{R}_m$ ) and corresponding variation with Ha is shown in Figure 6 at different Re values. The value of $E{R}_m$ sharply enhances with Ha for its lower values in region A because of the decrease in the area of hot spots inside the recirculatory zones. Moreover, the fully attached flow regimes at higher Ha causes gradual decrement in thermal boundary layer thickness near the wavy walls and accordingly there is a gradual enhancement in the value of $E{R}_m$ with Ha at its higher values in region B. Furthermore, the increase in Re enhances the value of $E{R}_m$ at higher values of Ha, while $E{R}_m$ is almost insensitive with Re for lower values of Ha. It can be explained as follows. For lower values of Ha, the weaker strength of recirculation zones for all Re causes higher intensity of hot spots and therefore, the augmentation in heat transfer rate due to increase in advection with Re is suppressed, thus causing almost insensitive variation of $E{R}_m$ with Re. On the other hand, the fully attached flow at higher values of Ha causes an increase in advection strength near the walls with Re.

Figure 6.

Variation of $E{R}_m$ with Ha for different Re with ϕ = 0.3%.

The relative increase in pressure drop by the magnetic field is presented in terms of magnetic pressure drop ratio ( $P{R}_m$ ) and the corresponding variation with $Ha$ is shown in Figure 7 for different values of $\mathrm{Re}$ . It is observed that the $P{R}_m$ increases with $Ha$ due to the augmentation in resistive Lorentz force with $Ha$ . Moreover, the value of $P{R}_m$ increases with $\mathrm{Re}$ , which is attributed to the steep gradient of the velocity in the near walls region.

Figure 7.

Variation of $P{R}_m$ with Ha at the different ϕ for Re = 500.

The variation of magnetic performance factor ( $P{F}_m$ ) with Ha is depicted in Figure 8(a). The value of $P{F}_m$ monotonically decreases with Ha for lower $\mathrm{Re}(=100)$ . While for higher values of Re, $P{F}_m$ decreases with Ha for its lower and higher values, while for the intermediate range, $P{F}_m$ increases with Ha. For lower Re( = 100), the pressure-drop is relatively less due to weaker strength of recirculation zones at smaller Ha values (See Figure 7), while for Re ≥ 200 with lower values of Ha, the strength of the recirculatory zones is relatively higher and it increases the pressure drop significantly (See Figure 7). As a result, the value of $P{F}_m$ decreases with Ha for its smaller values up to a critical value ( $H{a}_1^{*}$ ), when Re ≥ 200 and decrement is steeper at higher Re values. Beyond $Ha>H{a}_1^{*}$ , the augmentation in heat transfer rate due to fully attached flow increases the value of $P{F}_m$ and attains a local maxima at another critical ( $H{a}_2^{*}$ ). Moreover, beyond $Ha>H{a}_2^{*}$ , $P{F}_m$ decreases with Ha due the larger pressure drop caused by Lorentz resistive force (See Figure 7). Furthermore, it is noted that the value of $P{F}_m$ decreases with Re for lower values of Ha due to significant enhancement in pressure drop.

Figure 8.

Variation of $P{F}_m$ with $Ha$ for different $\mathrm{Re}$ with $\varphi$  = 0.3%.

An effort has been made to compare the findings of the current investigation with the available results in the literature. Mayeli et al. ⁴⁶ found the values of average Nusselt number for MHD flow of Al₂O₃-water nanofluid through a wavy channel in the range of 6.72 to 10.18 when the Hartmann number is changed from 0 to 40 with Re = 500 and ϕ = 4%. In contrast, the present study dealing with the flow of MWCNT-Fe₃O₄/H₂O hybrid nanofluid predicts a higher value of average Nusselt number even for a smaller Hartmann number ( $\overline{Nu}=12.24$ at Ha = 10, ϕ = 0.3% and Re = 500). In another work on MHD flow of Al₂O₃-water nanofluids through a ribbed-wavy channel, Mayeli et al. ⁴⁷ found the value of average Nusselt number as 9.26 at Ha = 20, Re = 200 and ϕ = 4.5%. In comparison, the present study predicts a higher value of average Nusselt number ( = 9.35) even for smaller Hartmann number (Ha = 10) and volume fraction (ϕ = 0.3%) for the same Re. For laminar MHD forced convective flow of Cu-water nanofluid in a partially filled porous wavy channel, Ashorynejad et al. ⁴² reported 5.28% to 6.23% enhancements in average Nusselt number for the change in ϕ from 0 to 4% for Ha = 10. While the present study predicts 21.7 to 23.8% enhancements in average Nusselt number with the change of ϕ from 0 to 0.3% with Ha = 10. Recently, for Al₂O₃-water nanofluid MHD flow inside an asymmetric wavy channel, Mehta et al. ¹¹ found that the magnetic enhancement ratio very close to unity for Ha = 10 for, 5 ≤ Re ≤ 500 and 0 ≤ ϕ ≤ 5%. In the present study, we found the range of magnetic enhancement ratio as 1 to 2.98 for the change in Ha from 0 to 10 at Re = 500 and ϕ = 0.3%.

Conclusions

We investigate computationally the heat transfer and pressure drop for flow of MWCNT-Fe₃O₄/H₂O hybrid nanofluid through a wavy channel under the effect of the externally applied magnetic field. The computed flow and temperature fields are analyzed in terms of streamlines, isotherms, average Nusselt number ( $\overline{Nu}$ ), magnetic enhancement ratio ( $E{R}_m$ ), and magnetic performance factor ( $P{F}_m$ ) by varying the Reynolds number (Re), Hartmann number (Ha), and nano-particle volume fraction (ϕ) in the following ranges: 100 ≤ Re ≤ 500, 0 ≤ Ha ≤ 10, and 0% ≤ ϕ ≤ 0.3%. We summarize the important findings as follows:

The strength of the reverse flow in the recirculatory zones decreases with Ha, and beyond a critical Ha, the flow becomes attached even in the diverging part of the channel.

The increase in nano-particle volume fraction decreases the hot spot intensity near the walls. Moreover, the area of hot spot intensity decreases with the increase in Ha.

The value of $\overline{Nu}$ is enhanced with the increase in nano-particle volume fraction. The value of $\overline{Nu}$ also increases with Ha and the rate of increment is steeper at its smaller values, while it is gradual at higher values of Ha. The percentage enhancements in $\overline{Nu}$ for the change in Ha from 0 to 10 are 203.12%, 198.97%, and 197.88% for ϕ = 0%, 0.1%, and 0.3%, respectively. With the increase in ϕ from 0% to 0.3%, the values of $\overline{Nu}$ are enhanced as 23.84% and 21.7% for Ha = 0 and 10, respectively for Re = 500.

The value of $E{R}_m$ sharply enhances with Ha for its lower values, while there is a gradual enhancement in the value of $E{R}_m$ with Ha at its higher values.

The value of that PF_m monotonically decreases with Ha for lower Re ( = 100), while for higher values of Re, PF_m decreases with Ha for its lower and higher values, while for the intermediate range, PF_m increases with Ha.