Numerical Investigation of a Rotating Magnetic Field Influence on Free Convective CNT/Water Nanofluid Flow within a Corrugated Enclosure

Abstract

This paper emphasizes the effect of applying a rotating magnetic field on the natural convective flow of CNT/Water nanofluid inside a corrugated square cavity differentially heated through its sidewalls, while the upper and lower boundaries are supposed to be perfectly insulated. The aim of this study is to highlight the impact of a large variety of parameters, namely Hartman number, frequency of rotation, Rayleigh number, nanoparticles volume fraction, and corrugation aspect ratio on the flow behaviour and thermal transport characteristics. The governing non-linear coupled differential equations are solved by using the finite element technique. Outcomes indicated that the thermal energy exchange is improved with the Rayleigh number increment and nanoparticles loading, while it is weakened with the rising of Ha, ascribed to the Lorentz force opposition to buoyancy. Moreover, enlarging the corrugation aspect ratio causes the apparition of stagnant fluid zones and the rate of heat transfer is reduced as a result.

1. Introduction

Magnetohydrodynamic (MHD) flow has become the focus of several researchers and engineers due to the diversity of its uses, such as nuclear engineering, electronic cooling, crystal growth, and control of transport phenomena [1,2]. Moreover, the key control parameters governing this type of flow are the boundary conditions, the geometry shape factor and the thermophysical properties of the working fluid. In the first studies related to MHD, the applied magnetic fields were time-independent, and by technological developments, some authors studied the effects of the time-dependent magnetic field. Dold and Benz [3] considered the flow of fluid and transfer of heat inside the Bridgman configuration subject to a rotating magnetic field. It was mentioned that the generated Lorentz force acts as a stirrer and causes an azimuthal-directed flow. Zou et al. [4] studied the 3D thermal and mass diffusion in a floating-zone crystal growth while considering the effects of the Lorentz force. They reported that the latter force causes considerable flow stability and uniformed concentration profiles and thus allows production of better quality crystals. Yao et al. [5] performed a numerical analysis of the 3D transient thermocapillary convection flow subject to Lorentz force. It was concluded that the transition to the oscillating flow depends on the Marangoni number and suggested extending the study by investigating the rotation frequency effect to optimize the flow control. Witkowski and Walker [6] considered the Marangoni convection in a floating zone under the effect of Lorentz force. The authors mentioned that crystal growth performance is improved at specific magnetic field intensities and specific Marangoni number values. Dold and Benz [7] studied the Lorentz force effect on crystal growth and flow structure and they noted a better homogenization of the concentration of the dopant and thus better crystallization performances. A similar study was performed by Feonychev and Bondareva [8] and similarly concluded there was a homogenization improvement. The influence of Lorentz force on the solidification of a metal alloy has been examined numerically by Nikrityuk et al. [9]. Because of the metal alloy’s strong electrical conductivity, the generated Lorentz force induced a forced convective flow. A rotating magnetic field is used to study Rayleigh–Benard convection by Friedrich et al. [10] via numerical and experimental studies. The authors mentioned that the flow is oscillating due to the existence of the Taylor vortices. Similar work has been presented by Volz and Mazuruk [11]. Lyubimov et al. [12] considered the magnetohydrodynamic convective thermal energy transport in a layer of conductive fluid and mentioned that, for strong Lorentz force intensity, the regime of thermal transport becomes mainly conductive. Kang and Hyun [13] inspected the magnetohydrodynamic buoyancy-driven convection in an air-filled enclosure, which is a paramagnetic fluid. The authors concluded that applying strong Lorentz force leads to an essential enhancement of thermal energy transport. Dold and Benz [14] experimentally studied the temporal variation during the transfer of heat by convection of gallium under static and rotating magnetic fields. It was revealed that the generated Lorentz force leads to the elimination of temperature fluctuations. Utilizing a cylindrical cavity exposed to a rotating magnetic field, Lyubimov et al. [15] analysed the stability of convective flow. The authors mentioned that a variety of flow structure appears by varying the intensity of the Lorentz force. Lyubimov et al. [16] considered the thermal energy exchange in a conductive fluid layer exposed to Lorentz force. The authors mentioned that the latter force opposes the convective thermal transport and reduces the instabilities.

With the discovery of nanofluids and their beneficial effect on heat and mass transfers, a sizable number of experimental investigations revealed that, for a given volume fraction, the suspensions of carbon nanotubes (CNT) is more prominent in terms of thermal properties when compared to other nanoparticles [17,18,19]. In fact, carbon nanotubes are chemically stable and have good electric and thermal conductivities. Moreover, the coupled effects of using nanoparticles and time-dependent magnetic fields have drawn significant attention. Fadaei et al. [20] examined the Lorentz force influence on nanofluid forced convection flow inside a 2D channel. As stated by the authors, the magnitude and rotation frequency have a crucial effect on flow structure and thermal energy transport. Boroun and Larachi [21] performed numerical and empirical investigations of ferrofluid mass transfer inside a T-mixer cavity exposed to a rotating magnetic field. The authors mentioned that their numerical model failed to give similar results to the experimental findings and proposed to improve this model. Yarahmadi et al. [22] analysed a ferrofluid’s oscillation and magnetic field arrangement effects on forced convection. Their experimental results revealed that, in comparison with the non-oscillating magnetic field, using an oscillating one could achieve 19.8% of heat transfer enhancement. Kolsi et al. [23] considered the effect of a time-dependent periodic field on the buoyancy-induced flow of MWCT-nanofluid. The authors concluded that the magnetic temporal periodicity induced the flow structure periodicity and the thermal behaviour, and the energy exchange is more intense for higher oscillation frequencies. Boroun and Larachi [24] performed experimental and numerical investigations of the Lorentz force effect on the nanofluid Poiseuille flow. The authors mentioned that the deviation of empirical results is due to the micro-convective flow that has not been considered numerically. Recently, Fan et al. [25] studied the melting process of NEPCM and the effects of a rotating magnetic field. It is concluded that adding nanoparticles and applying a rotating magnetic field reduces the melting-process time by about 23%. Rasaee et al. [26] conducted an empirical investigation of magnetohydrodynamic ferrofluid convection in a rifled pipe. They mentioned that the hydrothermal performances of the system are enhanced using the rotating magnetic field and adding nanoparticles. Hassan et al. [27] analysed the effects of an oscillating magnetic field on nanofluids flowing through a stretchable rotating disk. Nanoparticles addition and applying a highly oscillating magnetic field lead to essential changes in the flow structure and temperature field [28,29].

Based on the aforementioned literature survey, there is a lack of studies regarding the effect of rotating magnetic field application on carbon nanotube-based nanofluid within confined spaces. Accordingly, the aim of the current investigation is to analyse this effect on CNT/water nanofluid inside corrugated cavities for a wide range of governing parameters.

2. Problem Statement

The present problem is represented by a top and bottom corrugated square enclosure as displayed in Figure 1. Vertical walls are kept with variant temperatures, while the horizontal walls have corrugations with height (A) and are deemed as adiabatic. Furthermore, a CNT/water nanofluid fills the enclosure with Pr = 6.2, and an imposed rotative magnetic field ($\overrightarrow{B}$) that has a time-dependent inclination angle (γ) according to the x-axis.

Table 1. Thermophysical properties of water and CNT-nanoparticles [23]. Reprinted with permission from ref. [23]. Copyright 2019 MDPI.

Property	Water	CNT
ρ (kg m−3)	997.1	2600
Cp (J kg−1 K−1)	4179	425
k (W m−1 K−1)	0.613	6600
β.10−6 (K−1)	21 × 10−5	1.6 × 10−6
σ (W−1 m−1)	0.05	4.8 × 10−7

indicates the CNT nanoparticles and pure fluid (water) thermophysical properties. The flow is deemed as unsteady, two-dimensional, laminar, incompressible and governed by Boussinesq’s approximation, in which the nanofluid density will alter with fluid temperature.

2.1. Governing Equations

In accordance with Ohm’s law, the law of conservation of charge and the Lorentz force, the electric current density J and electromagnetic force F are expressed as [30].

$$\mathit{J}=\sigma \left(-\nabla \mathrm{\Phi }+\mathit{V}\times \mathit{B}\right)$$

(1)

$$\nabla \cdot \mathit{J}=0$$

(2)

$$\mathit{F}=\mathit{J}\times \mathit{B}$$

(3)

where V (u,v) is the velocity field. Supposing the electric potential F as constant at the electrically non-conducting boundaries, therefore Equations (1)–(3) lessen to:

$$\mathit{J}={\sigma }_l\left(\mathit{V}\times \mathit{B}\right)$$

(4)

$$\mathit{F}={\sigma }_l\left(\mathit{V}\times \mathit{B}\right)\times \mathit{B}$$

(5)

Based on the aforementioned assumptions, the continuity, momentum and energy equations can be expressed in the next dimensional form:

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

(6)

$$\frac{\partial u}{\partial t}+u\frac{\partial u}{\partial x}+v\frac{\partial u}{\partial y}=\frac{1}{{\rho }_{nf}}\left[-\frac{\partial p}{\partial x}+{\mu }_{nf}\left(\frac{{\partial }^{2}u}{\partial x^2}+\frac{{\partial }^{2}u}{\partial y^2}\right)++{\sigma }_{nf}{B}_o^2\left(v\sin \gamma \cos \gamma -u{\sin }^2\gamma \right)\right]$$

(7)

$$\frac{\partial v}{\partial t}+u\frac{\partial v}{\partial x}+v\frac{\partial v}{\partial y}=\frac{1}{{\rho }_{nf}}\left[\begin{array}{r}-\frac{\partial p}{\partial y}+{\mu }_{nf}\left(\frac{{\partial }^{2}v}{\partial x^2}+\frac{{\partial }^{2}v}{\partial y^2}\right)+g{\left(\rho \beta \right)}_{nf}\left(T-T_c\right)\\ +{\sigma }_{nf}{B}_o^2\left(u\sin \gamma (t).\cos \gamma (t)-v{\cos }^2\gamma (t)\right)\end{array}\right]$$

(8)

$$\frac{\partial T}{\partial t}+u\frac{\partial T}{\partial x}+v\frac{\partial T}{\partial y}={\alpha }_{nf}\left(\frac{{\partial }^{2}T}{\partial x^2}+\frac{{\partial }^{2}T}{\partial y^2}\right)$$

(9)

Moreover, ${\rho }_{nf}$ is the effective nanofluid density expressed as follows:

$${\rho }_{nf}=\left(1-\phi \right){\rho }_f+\phi {\rho }_{np}$$

(10)

and φ is the solid nanoparticles volume fraction. Moreover, α_nf is the nanofluid’s thermal diffusivity which is defined as:

$${\alpha }_{nf}=\frac{k_{nf}}{{\left(\rho C_p\right)}_{nf}}$$

(11)

where $C_p{}_{nf}$ is the nanofluid’s heat capacitance which is defined as:

$${\left(\rho C_p\right)}_{nf}=\left(1-\phi \right){\left(\rho C_p\right)}_f+\phi {\left(\rho C_p\right)}_{np}$$

(12)

The nanofluid’s thermal expansion coefficient ${\beta }_{nf}$ is defined as:

$${\left(\rho \beta \right)}_{nf}=\left(1-\phi \right){\left(\rho \beta \right)}_f+\phi {\left(\rho \beta \right)}_{np}$$

(13)

Furthermore, ${\mu }_{nf}$ is the effective dynamic viscosity defined by the Hamilton and Crosser model [31]:

$${\mu }_{nf}={\mu }_f\left(1+a\phi +b{\phi }^2\right)$$

(14)

Nanoparticles with tube shapes have a = 13.5 and b = 904.4, and the nanofluid thermal conductivity is approximated by Xue [26] as follows:

$$\frac{k_{nf}}{k_f}=\frac{1-\phi +2\phi \frac{k_s}{k_s-k_f}\ln \left(\frac{k_s+k_f}{2k_f}\right)}{1-\phi +2\phi \frac{k_f}{k_s-k_f}\ln \left(\frac{k_s+k_f}{2k_f}\right)}$$

(15)

The following dimensionless parameters are presented [31]:

$$\begin{gathered} \tau =\frac{{\alpha }_{f}t}{L^2}, X=\frac{x}{L}, Y=\frac{y}{L}, U=\frac{uL}{{\alpha }_f}, V=\frac{vL}{{\alpha }_f}, \theta =\frac{T-T_c}{T_h-T_c}, P=\frac{p{L}^2}{{\rho }_{nf}{\alpha }_f^2}, Ha=B_{o}L\sqrt{\frac{{\sigma }_f}{{\mu }_f}}, \\ Ra=\frac{g{\beta }_f\left(T_h-T_c\right)L^3}{{\nu }_f{\alpha }_f}, \mathrm{Pr}=\frac{v_f}{{\alpha }_f} \end{gathered}$$

(16)

By employing Equation (16), a dimensionless form for Equations (6)–(9) is produced as below:

$$\frac{\partial U}{\partial X}+\frac{\partial V}{\partial Y}=0$$

(17)

$$\begin{array}{l}\frac{\partial U}{\partial \tau }+U\frac{\partial U}{\partial X}+V\frac{\partial U}{\partial Y}=-\frac{\partial P}{\partial X}+\frac{{\mu }_{nf}}{{\rho }_{nf}{\alpha }_f}\left(\frac{{\partial }^{2}U}{\partial X^2}+\frac{{\partial }^{2}U}{\partial Y^2}\right)\\ +H{a}^2\mathrm{Pr}\frac{{\rho }_f}{{\rho }_{nf}}\left(V\sin \gamma \cos \gamma -U{\sin }^2\gamma \right)\left(1+\frac{3\left(\frac{{\sigma }_{np}}{{\sigma }_f}-1\right)\phi }{\left(\frac{{\sigma }_{np}}{{\sigma }_f}+2\right)-\left(\frac{{\sigma }_{np}}{{\sigma }_f}-1\right)\phi }\right)\end{array}\}$$

(18)

$$\begin{array}{l}\frac{\partial V}{\partial \tau }+U\frac{\partial V}{\partial X}+V\frac{\partial V}{\partial Y}=-\frac{\partial P}{\partial Y}+\frac{{\mu }_{nf}}{{\rho }_{nf}{\alpha }_f}\left(\frac{{\partial }^{2}V}{\partial X^2}+\frac{{\partial }^{2}V}{\partial Y^2}\right)+\frac{{\left(\rho \beta \right)}_{nf}}{{\rho }_{nf}{\beta }_f}Ra\mathrm{Pr}\theta \\ +H{a}^2\mathrm{Pr}\frac{{\rho }_f}{{\rho }_{nf}}\left(U\sin \gamma \cos \gamma -V{\cos }^2\gamma \right)\left(1+\frac{3\left(\frac{{\sigma }_{np}}{{\sigma }_f}-1\right)\phi }{\left(\frac{{\sigma }_{np}}{{\sigma }_f}+2\right)-\left(\frac{{\sigma }_{np}}{{\sigma }_f}-1\right)\phi }\right)\end{array}\}$$

(19)

$$\frac{\partial \theta }{\partial \tau }+U\frac{\partial \theta }{\partial X}+V\frac{\partial \theta }{\partial Y}=\frac{{\alpha }_{nf}}{{\alpha }_f}\left(\frac{{\partial }^2\theta }{\partial X^2}+\frac{{\partial }^2\theta }{\partial Y^2}\right)$$

(20)

The angle of the magnetic field (g) in Equations (18) and (19) is a function of dimensionless time ($\tau$), and frequency (f). It can be defined as:

$$\gamma =2\pi f\tau$$

(21)

The local Nusselt number (Nu) on the left hot wall is described as follows:

$$Nu=\frac{k_{nf}}{k_f}{\left(\frac{dT}{dx}\right)}_{x=0}=\frac{k_{nf}}{k_f}{\left(\frac{d\theta }{dX}\right)}_{X=0}$$

(22)

The average Nusselt number (Nu_av) is estimated by integrating the local Nu on the hot wall:

$$N{u}_{\mathrm{av}}={\displaystyle \underset{0}{\overset{L}{\mathrm{\int }}}Nudy}={\displaystyle \underset{0}{\overset{1}{\mathrm{\int }}}NudY}$$

(23)

2.2. Boundary Conditions

Based on the configuration given in Figure 1, boundary conditions can be expressed as:

• Vertical walls

$$u=v=0, T=T_h \mathrm{at} x=0$$

$$u=v=0, T=T_c \mathrm{at} x=L$$

• Horizontal walls

$$u=v=0, \frac{dT}{dy}=0 \mathrm{at} y=0 \mathrm{and} y=L$$

By employing the same parameters in Equation (16), boundary conditions can be transformed into dimensionless form in the following manner:

-

Vertical walls

$$U=V=0, \theta =1 \mathrm{at} X=0$$

$$U=V=0, \theta =0 \mathrm{at} X=1$$

-

Horizontal walls

$$U=V=0, \frac{d\theta }{dY}=0 \mathrm{at} Y=0 \mathrm{and} Y=1$$

2.3. Grid Independent and Validation

The grid type employed in the present study is non-uniform and unstructured. To examine the grid independency, Nu_av has been evaluated at five grid numbers characterized by Mesh 1–5 as exhibited in Figure 2. As indicated by the figure, when the grid number enlarges, Nu_av ascends. If the percentage error is estimated between each two successive mesh types from lowest to highest. The maximum error is 2.25% between Mesh 1 and Mesh 2, while the minimum error has happened at the higher mesh (i.e., Mesh 4 and Mesh 5), which is predicted as less than 1%. This indicates that these higher mesh numbers give more accurate results. In the present study Mesh 4 corresponds to grid number 12,412 and is selected for the calculations.

For the purpose of validation, the present model has been configured to the Hussam et al. [32] model. The streamlines and isotherms are evaluated at one period of time with the following parameters: A = 0.5, f = 5, φ = 0.2, Ra = 10⁶, Ha = 100, as revealed in Figure 3. It is clearly observed there is a precise agreement between the two studies, of the validity of the present model in studying the magnetohydrodynamic problems in square cavities.

3. Results and Discussion

Numerical simulations are conducted with numerous scopes of the studied parameters, namely Hartman number (0 ≤ Ha ≤ 100), volume fraction of CNT nanoparticles (0 ≤ φ ≤ 0.05), Rayleigh number (10⁴ ≤ Ra ≤ 10⁷), frequency (0 ≤ f ≤ 4), and aspect ratio (0.1 ≤ A/L ≤ 0.3). The impacts of these variables pertaining to the flow structure and average Nusselt number (Nu_av) have been studied in the next articles.

Figure 4, presents the streamlines for Ha = 0 and temporal snapshots of the flow structures for Ha = 100 for Ra = 10⁶, f = 4, and A/L = 0.1. It is noteworthy to mention that for Ha = 0 when the steady flow is achieved, its structure becomes time-independent. While for Ha = 100, the flow structure is periodic because of the magnetic field rotation. For Ha = 0, the flow structure is represented by two clockwise cells near the active walls. It is also mentioned that due flow is more intense close to the active walls, and this is due to the strong buoyancy forces generated by the density variations. For Ha = 100, the intensity of flow is lessened due to the generated Lorentz forces, and the flow structure becomes time-dependent because of the magnetic field rotation. The time needed for the magnetic field to perform a complete rotation (period) is denoted by “t_p”. The period is divided into four times where the flow structures are presented. For t = t_o and t = t_o + t_p/2, the flow structures are identical because, at these specific times, the magnetic fields have the same direction with opposite orientations (i.e., horizontal magnetic field). Similarly, for τ = t_o + t_p/4_o and τ = t_o + 3t_p/4 where the magnetic field is vertical. For τ = t_o and τ = t_o + t_p/2, flow is characterized by a single clockwise rotating cell at the core of the cavity and by a square-shaped structure close to the walls. The ascending and descending flows are reduced, and the damping effect is more pronounced close to the active walls. For τ = t_o + t_p/4 and τ = t_o + 3t_p/4 the flow becomes characterized by two co-rotative cells, indicating that the magnetic field impact is less important at these times compared to (τ = t_o and τ = t_o + t_p/2). This finding can also be concluded from the maximum values of the velocity magnitudes that are 50 and 90 for (τ = t_o and τ = t_o + t_p/2) and (τ = t_o + t_p/4 and τ = t_o + 3t_p/4), respectively.

To understand the impact of adding CNT nanoparticles on the flow behaviour we also plotted the case of CNT volume fraction φ = 0.05. The flow structure has mainly the same behaviour as for φ = 0, except for the intensification of the velocity magnitude, especially for Ha = 0. This intensification is due to the improvement of the thermophysical properties of the fluid, leading to more important buoyancy forces and thus more intensive convective flows especially close to the active walls.

Figure 5 is presented to investigate the effect of the corrugation heights. As mentioned above in the description of Figure 4, due to the periodicity of the flow structure, the streamlines are shown only at τ = t_o and τ = t_o + t_p/4. The increase of the A/L causes a pilling up of the streamlines at the cavity core and the apparition of low-velocity zones close to the adiabatic corrugations. It is also to be mentioned that the velocity magnitudes are lower for A/L = 0.3 compared to A/L = 0.1. For Ha = 0, the flow is represented by two co-rotative cells, which are more prominent when the CNT is added (φ = 0.05). For Ha = 100, at the beginning of the rotation period (t = t_o) a monocellular structure appears, indicating the reduction of the flow intensity because at this specific time, the Lorentz forces oppose the vertical ascending and descending flows that occur close to the active walls. In the mid-period, the Lorentz forces oppose mainly the horizontal flow causing the apparition of a multicellular structure. In addition, the stagnation zones disappear, and the flow in the vicinity of the upper and lower matches the corrugations patterns due to the interaction between the Lorentz force and the horizontal flow.

Figure 6, presents the isotherms for Ha = 0 and temporal snapshots of the temperature profiles for Ha = 100 for Ra = 10⁶, f = 4, φ = 0 and A/L = 0.1. When no magnetic field is applied, the temperature fields are denoted by a central vertical stratification of the isotherms, indicating the predominance of convection compared to conduction. This stratification becomes more significant with CNT-nanoparticles addition, indicating the intensification of the transfer of heat. The magnetic field application leads to lessening of the vertical stratification due to the generated Lorentz forces that oppose the buoyancy forces. For higher values of A/L, the adiabatic surface increases, causing an interruption of the isotherm distortions at the top and bottom regions, especially with magnetic field application, indicating the reduction of the convective effects.

To analyse the rotating magnetic field effect on the thermal energy exchange rate, the temporal variations of the Nu_av for different rotation frequencies at φ = 0.05, Ra = 10⁶, Ha = 100, A/L = 0.1 are presented (Figure 7a). As a result of the rotative aspect of the applied magnetic field, the temporal variation of Nu_av is periodic with a frequency identical to that of the magnetic field with a little lag (phase delay) caused by the imposed initial conditions.

Figure 7b depicts the effect of Ha and φ on the temporal variations of the Nu_av. Obviously, due to the generated Lorentz forces, the increase of Ha causes a lessening in the heat transfer amount, and the ascend of φ causes its intensification due to the enhancement of the thermophysical properties of the nanofluid. It is also to be mentioned that there are phase delays between all the variations.

For a greater understanding of the effect of the governing parameters on the rate of heat transfer, the average Nusselt number is averaged over a period and is noted Nu^t_av.

Plots in Figure 8 illustrate Nu^t_av against numerous factors such as Ra, φ, A/L and f. Figure 8a presents the effect of Ra on Nu^t_av at two specific values of Ha and φ with keeping A/L = 0.1. The figure reveals that Nu^t_av ascends as Ra rises with all studied scopes of Ha and φ. This reflected the convection effects become significant as Ra enlarges and then Nu^t_av enlarges as a result. It is obvious that the applied magnetic field and adding nanoparticles have opposing effects. Specifically, Nu^t_av lowers with Ha but ascends with φ. Figure 8b elucidates the influence of φ on Nu^t_av at two specific values of Ra and Ha with keeping A/L = 0.1. It is obvious that Nu^t_av ascends as φ augments with all studied scopes of Ra and Ha. This is owing to the augmentation of buoyancy force as φ augments. Moreover, the figure reveals that the impact of Ha and Ra is reversed on Nu^t_av. Because enlarging Ra enhances Nu^t_av, as stated previously, ascending Ha will reduce Nu^t_av due to the Lorentz force generated, and the fluid movement opposes the buoyancy force and thus reduces the rate of the thermal energy exchange. Figure 8c depicts the influence of A/L on Nu^t_av at two specific values of Ha and φ with keeping Ra = 10⁶. The figure shows that for the higher Ha, the Nu^t_av decreases as A/L enlarges; this refers to the reduction in cavity size, which leads to reduced fluid quantity, then buoyancy force reduced, leading to lessened heat transfer inside. Meanwhile, as Ha lowered, Nu^t_av remains approximately fixed with A/L until A/L = 0.25; then after this value, it decreases slightly. Figure 8d shows the variation of Nu^t_av with f for two values of A/L, Ha = 50 and Ra = 10⁶. It is noticed that for A/L = 0.3, the heat transfer is maximum at f = 0, and is a little bit higher for f = 2 compared to (f = 0.25 and f = 4). This behavior is reversed for low values of A/L (0.1) where the heat transfer increases by applying a rotating magnetic field and the maximum occurs for f = 2.

4. Conclusions

A computational analysis of MHD free convective CNT/water nanofluid flow in a 2D corrugated cavity subjected to an external rotative magnetic field is presented. The governing equations are solved using the FEM and the impacts of Rayleigh number, rotation frequency, corrugations aspect ratio and nanoparticles volume fractions on the flow structure, thermal field and thermal energy exchange rate are investigated. The main conclusions can be summarized as follows:

• Applying a magnetic field produces a Lorentz force that opposes the buoyancy force and reduces flow intensity and temperature gradient.
• The rotation of the magnetic field produces a time-dependent periodic behaviour of the flow.
• The increase of Ra values leads to the increase of the heat transfer for all the considered cases.
• The addition of CNT nanoparticles leads to a significant enhancement of thermal energy exchange.
• The increment of the corrugation aspect ratio causes the apparition of stagnant fluid zones and thus will cause a lessening in the rate of heat transfer.
• The effect of the frequency of the magnetic field rotation depends on the corrugations’ size. In fact, the heat transfer increases with the frequency for low values of A/L and the opposite occurs for lower values.