Natural convection heat transfer in a nanoﬂuid ﬁlled L -shaped enclosure with time-periodic temperature boundary and magnetic ﬁeld

The natural convective Cu-water nanoﬂuid ﬂow in L -shape cavity with an oscillating temperature proﬁle is studied numerically. The cavity’s lower horizontal and left vertical walls are heated sinusoidally with time about a high mean temperature ( T (cid:1) H ). In contrast, the cavity’s right vertical wall and its nearby horizontal lower wall are kept cold at a temperature ( T c ). The calculations have been performed over temperature oscillation amplitude (0 (cid:3) A (cid:3) 2 Þ , dimensionless temperature oscillation frequency 0 (cid:3) f (cid:3) 100 ð Þ , Rayleigh number (10 3 (cid:3) Ra (cid:3) 10 8 Þ , Hartmann number (0 (cid:3) Ha (cid:3) 100), the nanoparticles volume fraction 0 (cid:3) / (cid:3) 0 : 2 ð ), and enclosure aspect ratios (0 : 2 (cid:3) AR (cid:3) 0 : 8). Outcomes reveal that with AR = 0.2, heat transfer happens considerably through conduction at Ra = 10 3 –10 5 , while the time average Nusselt number ( Nu (cid:1) ) is independent of both Ha and Ra . Convection effects, on the other hand, become signiﬁcant at high Ra . Additionally, as Ha ascends from 0 to 50, Nu (cid:1) increases linearly with increasing / , while it remains steady at Ha = 75 and 100. (cid:1) 2022 THE AUTHORS. Published by Elsevier BV on behalf of Faculty of Engineering, Alexandria University. This is an open access article under the CC BY license (http://creativecommons.org/licenses/by/ 4.0/).


Introduction
In recent years, the steady/unsteady-state heat transport within various simple/complex shape cavities has been extensively studied.A new review of studies provides a complete overview of the flow field and heat transfer with/without nanofluid in various geometries of cavities [1][2][3].
Various solutions were considered to enhance convective heat transfer in many types of equipment [4,5], increasing, for example, the heating system efficiency and cost savings [6].Various modifications were considered, such as changing the geometry and the boundary conditions [7], utilizing fins [8] and vortex generators [9], and many others.The L-shaped cavity produces a circulation area and vortices [10] that enhance the heat transfer [11]; combined with the use of nanofluid [12], it optimizes the performance of many engineering systems.
In several investigations, nanofluid flows and heat transfer within different shaped cavities have been studied numerically [13][14][15][16][17][18].Using a nanofluid-equipped L-shaped enclosure, Mahmoodi [19] numerically studied free convection fluid flow phenomena.The governing equations were discretized using the finite volume method, with the velocity and pressure fields computed using the SIMPLER technique.Various Ra = 10 3 -10 6 , AR = 0.2-0.6, as well as / = 0-0.1,were used in this investigation.Their results showed that when raising the Ra and /, for all cavities' aspects ratios, Nu À increased.Sourtiji and Hosseinizadeh [20] employed nanofluids to augment heat transfer in a cavity with L-shaped subjected to magnetohydrodynamics (MHD) natural convection heat transmission effects in the horizontal direction.Their research revealed that AR determines a critical Ra value.Moreover, the existence of the magnetic field affects heat transport and the flow field significantly in cavities, where the Nu À decreases as the Ha increases.This was due to the Lorentz force, which acts to dampen fluid motion across magnetic field lines.
Utilizing the Lattice Boltzmann Method (LBM), Kalteh and Hasani [21] inspected free convective nanofluid flow in an L-shaped cavity.According to their findings, the nanofluid increases the rate of heat transfer, which is improved with lessening AR.Additionally, increasing the diameter of the nanoparticles has the opposite impact on Nu À .Mliki et al. [22,23] also utilized the LBM to investigate the heat transfer inside L-shaped chambers with nanofluids.Their research employed two different kinds of nanoparticles (Cu and Al 2 O 3 ).They investigated the effects of several relevant characteristics such as AR, Ha, and Ra on the Nu À .The findings indicate that nanofluid increases the quantity of heat transferred and that decreasing AR boosts this impact.
Mohebbi and Rashidi [24] have implemented a heating obstacle containing nanofluid in an L-shaped enclosure.They elucidated that, for the examined range of Ra, the greater Nu À is obtained for a high-volume fraction.Furthermore, it increased as AR and the nanoparticles' diameter were reduced.
The authors showed that the Nu À reached its maximum value when placing the heating element within the left wall, in a lower position.
The thermophoresis and the Brownian motion effects on fluid flow and natural convective heat transfer in an L- shaped cavity utilizing nanofluid were investigated by Sheikholeslami et al. [25].Based  decreased by increasing the AR and the concentration.Using the LBM, Rahimi et al. [26] numerically investigated the free convective flow of a hybrid nanofluid in a hollow L-shaped cavity.The thermophysical characteristics of a SiO 2 /TiO 2 / water-Ethylene-glycol-nanofluid were found experimentally, resulting in a decrease in the overall entropy production with the enlargement of nanoparticles loading.The fluid flow, heat transfer by natural convective, and irreversibility in a finned L-shaped enclosure filled with nanofluid were investigated numerically and experimentally by Zhang et al. [27].The thermal conductivity of TiO 2 -SiO 2 /W-EG and the dynamic viscosity were found experimentally, whereas the LBM was used for the numerical solution.The cavity's horizontal walls were insulated, while the other walls were maintained at different temperatures.The walls were mounted with four rectangular fins, two of which were hot and the other two were cold.They found that the thermal designs of the implanted fins substantially influenced both the fluid flow and temperature fields.
The AL 2 O 3 /water nanofluid convection thermogravitational within an L-shaped cavity containing a rectangular hot barrier was numerically considered using the LBM by Mohebbi et al. [28].The impact of various essential factors on the process of cooling and fluid circulation within the cavity were investigated; the nanoparticle volume fractions (0 / 5 %), the Rayleigh number (10 3  Ra 10 6 ), the aspect ratio (0.2 AR 0.6), as well as the height and the position of the heater.According to the findings, the Nu À increased as the Ra, and / increased, while the rates of heat transfer were discovered to depend strongly on the location of the heating part.
Another interesting numerical investigation of using an open inclined L-shaped cavity containing Ag/water was conducted by Armaghani et al. [29].They investigated at how heat transport and irreversibility were influenced by the position of the separate hot source.They found that the angle of inclination has a substantial impact on the transport of heat.
A numerical analysis of free convective flow in an L-shaped container using a flexible baffle was performed by Ghalambaz et al. [30].An Arbitrary Lagrangian-Eulerian moving mesh frame was used numerically to present the governing equations where FEM has been used.The results show that a stiffer baffle slows the heat convective heat by resisting fluid movement.Zhang et al. [31] looked at entropy distributions on the free convective flow with magnetohydrodynamics in an L-shaped enclosure containing Newtonian/non-Newtonian fluids using the finite-difference lattice Boltzmann model (FDLBM).Ahmed et al. [32] investigated MHD Ferro-convection heat transmission in an oblique L-shaped container heated on the corner with a double-lid drive.The flow domain has a heat generation/absorption source, whereas thermal radiation is ignored.The results show that the isothermal distribution increased sharply as the heated corner lengths increased.At constant heat generation parameter (Q = 2), when the lengths of the active sections are adjusted between 0.2 and 0.8, Nu À decreases by 4.7 percent.In addition, as the Hartmann number increases from 0 to 50, the heat transfer is decreased by 30.7 percent for ferrofluid condition with a concentration of 5 percent.
Seyyedi et al. [33] have provided the economic analyses and the entropy generation of convective Al 2 O 3 -water nanofluid flow within an L-shaped container under the impact of an oriented magnetic field.Their study elucidated that increasing the nanoparticles' loading from 0 to 0.1 decreases the entropy generation by 15.14 percent for Ha = 25, while for Ha = 75, it decreases by 8.15 percent.Nia et al. [34] utilized the LBM to evaluate the influence of different baffle designs, such as baffle length and location, on the free convection of Cu-water nanofluid fluid flow and transfer of heat within an L-shaped cavity with Ra (10 3 -10 5 ) and / (0-0.05).According to their finding, the addition of a baffle improves natural convection regardless of its length at lower Ra values, whereas for higher Ra, only the longer baffle, regardless of its location, can improve natural convection.
Recently several studies investigated convective heat transmission and fluid flow in L-shaped porous enclosures with/ without nanofluids [35][36][37].Using the Buongiorno two-phase model, Chamkha et al. [36] numerically explored the irreversibility and thermal analyses in a permeable L-shaped enclosure filled with nanofluids.A house code was written in C ++ and added to the Open Foam solver for achieving entropy generation and Buongiorno's model.The results showed that the optimum thermal performance occurred in a 0.5 aspect ratio cavity with Al 2 O 3 and CuO nanoparticles.Moria [37] performed a numerical investigation on the influence of heating and porous blocks on the heat transmission and flow of fluid features within an L-shaped enclosure for numerous relevant factors like Ra, Darcy number (Da), enclosure inclination angle, and porous block effective thermal conductivity.According to the heating position, blocks had a considerable influence on the heat transmission rates within the enclosure.Furthermore, the porous block was created to improve the transfer of heat within the cavity.
Almeshaal et al. [38] considered the effects of Ra, Ra, / on Rayleigh-Be´nard convective fluid flow with heat transmission and in a three-dimensional L-shaped cavity involving nanofluid (MWCNT-water).For Ra values, less than 10 5 and 3 percent of nanoparticle fractions, the AR of 0.4 gives optimum heat transfer.The AR and the percentage of nanoparticles were increased to improve the 3-D appearance.
Recently, LBM was used by Mohammadifar et al. [39] to investigate laminar free convective Cu/water nanofluid flow in an L-shape enclosure with an open end.The impact of Ra, AR, and / on heat transfer and fluid flow were examined using various boundary conditions at the east side of the open-end enclosure.The most significant heat transfer improvement of 186.92 percent was reached when the boundary conditions were changed from a fixed adiabatic wall to an open boundary condition.The oblique magnetic field effects on heat transport and irreversibility within an L-shaped cavity occupied with alumina/water were studied by Zhang et al. [40].The cavity's L- shape was modeled as a circle's quadrant.The results indicated that for high values of Ra, the perpendicular magnetic field attained the maximum rates of irreversibility and transfer of heat.
Armaghani et al. [41] considered the impact of the heat sink/source size and position on an L-shaped cavity having MHD mixed convection and filled with Al 2 O 3 -Cu/Water hybrid nanofluid.The findings show that using the highest sink power resulted in the optimum heat transfer.
As this review reveals, most studies supposed uniform temperatures at the walls of the L-shaped cavity.However, several applications in engineering are subject to a variable tempera-Natural convection heat transfer in a nanofluid filled L-shaped enclosure with time-periodic temperature ture, such as energy storage, thermal insulation in buildings, and solar thermal systems.System's walls with recirculating fluid under different conditions have grasped many researchers' attention.The importance is rising in many industrial sectors, such as optimizing the design of heat exchangers and energy recovery systems.To the best of our knowledge, the influence of time-periodic boundary temperature and magnetic field in an L-shaped cavity filled with nanofluid has not been studied in the literature.Therefore, the main objective of this study is to investigate the effect of different parameters (Rayleigh number, aspect ratio of the cavity, Hartmann number) on fluid flow and temperature fields and the corresponding heat transfer features within the L-shaped cavity subjected to time-varying thermal boundaries.

Problem statement and mathematical formulation
The configuration of the current work, elucidated in Fig. 1, involves a 2-dimensional L-shaped cavity of equal width and height H with a thickness L and aspect ratio AR = L/H.The cavity is filled with Cu-water nanofluid with Prandtl number Pr of 6.2.Cu-water nanofluid has been selected as the working fluid inside the cavity as they are widely used as coolants in industrial applications.For example, adding copper nanoparticles of / = 0.003 to ethylene glycol has increased thermal conductivity by 40 % [42].
The left vertical and the perpendicular lower walls are exposed to an unsteady temperature condition that varies sinusoidally with time (t) and has an average high temperature of T À h , angular frequency x, and temperature oscillation amplitude A, and expressed as: The outer walls of the L-shape are hot, and the inner walls are cold, with the ends being adiabatic.A uniform magnetic field of a strength B o is exerted in the positive direction of xaxis and permeates through the nanofluid.Those two conditions provide a combination of electromagnetic force and buoyancy forces that control the produced convective fluid flow.An electrically conducting fluid does not produce a magnetic field.The impact of the thermal Joule heating and viscous dissipation of the fluid are likewise neglected.Table 1 details the thermophysical properties of the carrier fluid (water) and the Cu-nanoparticle.For the fluid buoyancy, the Boussinesq approximation is used, where density differences within the fluid are ignored, with the exception of the gravitational force expression in the y-momentum equation.

Governing equations
According to the previously described assumptions, for an unsteady-two-dimensional Cartesian coordinate system, the conservation equations comprise a mass conservation equation, horizontal and vertical momentum equations and an energy equation that are respectively written as The thermophysical properties of the nanofluids are included in the governing Eqs.(2-5); density (q nf ), kinematic viscosity (m nf ), electrical conductivity (r nf ), coefficient of thermal expansion (b nf ) and thermal diffusivity (a nf ), are represented in section 2.3.The magnetic field (B) is discussed in section 2.2.

Electromagnetic forces
The Lorentz force J Â B, which is defined as the vector cross product of the density of the electric current (J) and magnetic field (B), describes the influence of electromagnetic fields in the momentum equation.This work proceeds under the assumption that flows herein satisfy the low magnetic Reynolds number assumption (R m ¼ ulrl 0 ( kl), where l 0 ¼ 1:2566Â 10 À6 kg m s À2 A À2 is the magnetic permeability of free space, u and l are characteristic velocity and length scales, and r is  the electrical conductivity [44].Under this assumption, the electric field may be represented by the gradient of an electric potential field (rU).Ohm's law may thus be written as The electric potential equastion is thus given by where U is the electric potential, V is the velocity vector, and B o is the strength of the magnetic field which leads to a Poisson equation for the electric potential field.For a 2D velocity field V(x,y) and B acting in the x-direction, V Â B is nonzero only in the z-direction, and its divergence must therefore be zero.This reduces the Poisson equation to According to Pirmohammadi et al. [45], for the flow in a cavity with electrically insulating boundaries the electric potential satisfies @U=@n ¼ 0, rendering the solution of Eq. ( 9) as U ¼ constant, the gradient of which is zero in the Lorentz force, V Â B. Thus, only the r V Â B ð Þpart of Ohm's law contributes to the Lorentz force, ultimately appearing as Àr nf B 2 v=q nf in the vertical momentum Eq. ( 4).
The water and nanoparticle electric conductivities are assumed as 0.05 and 5:96 Â 10 7 kg À1 m À3 s 3 A 2 , so that over the computed range of volume fractions, the product of characteristic velocity and length scales required to satisfy the R m ( 1 assumption range from ul ð Þ U¼0 k1:59 Â 10 7 $ m 2 s À1 to ul ð Þ U¼0:2 k6:68 Â 10 À2 $ m 2 s À1 .Thus, it is clear that at smaller volume fractions, the low-R m assumption is satisfied for any practical industrial speed and length scales, while at the upper end the assumption holds at small scales and low speeds.

Nanofluids thermophysical properties relations
The nanofluids' thermophysical properties, including the heat capacitance qC p À Á nf , are to be evaluated by the following relations [46][47][48]: Fig. 2 Schematic for the grid type employed on the study.Natural convection heat transfer in a nanofluid filled L-shaped enclosure with time-periodic temperature with the subscripts (f) and (nf) denoting the base-fluid and the nanofluid, respectively.
The kinematic viscosity of the nanofluids is expressed as with the effective dynamic viscosity l nf À Á of the nanofluid expressed by the result represented by Brinkman [49] for a fluid that contains suspension of small rigid spherical particles, as Also, the nanofluid effective thermal conductivity (k nf ) is to be evaluated using the result by Yu and Choi [50], which counts for the liquid nanolayer on the surface of a nanoparticle and is expressed as where g is a dimensionless number representing the ratio of the nanolayer thickness to particle's radius, taken as 0.1 in this work [43].

Normalized governing Equations, Boundary, and initial conditions
Defining the dimensionless parameters, The normalized forms of the Governing equations, Eqs. ( 2) -( 5), can be written as: The normalized initial and boundary conditions utilized in this study are expressed below: Initial condition: at s ¼ 0: U ¼ V ¼ 0 (no flow), h ¼ 0 (uniform T c though out).
Boundary conditions: at X ¼ 0 and Y ¼ 0: U ¼ V ¼ 0 (no-slip condition),  The local Nusselt number over the two hot walls is expressed as The average value along the two hot walls becomes

Grid independent and validation
Numerical simulations were performed using the COMSOL Multiphysics package, employing a finite-element method for spatial discretization and a backward differentiation method for time integration.A thorough grid resolution study was conducted to ensure adequate spatial and temporal resolution of the numerical simulations.The grid-type employed in the present study is a triangular grid that illustrated in Fig. 2, and the grid sensitivity is examined by evaluating Nu at the heated walls with six grid numbers and with AR = 0.2, Ra = 10 6 , Ha = 100, A = 2, and f = 1, as illustrated in Table 2.The relative error decreases considerably with increas- ing mesh resolution, which indicates that the reasonable accuracy in results is fulfilled with meshes 5 and 6, and mesh number 5 is selected for use hereafter due to reduced computational expense.The present formulation is validated against numerous cases.The first case is performed with reference to the results by Mahmoodi [19] and Saidi and Karimi [12] for an L- shaped cavity utilizing a nanofluid without a magnetic field and fixed hot temperature on the heated walls at AR = 0.4 and f = 0.06 as revealed in Table 3.The current results are in good agreement with the published results.
The second case of verification has been achieved by comparing with Hussam et al. [51] for a square cavity with a periodically varying temperature profile under a magnetic field at Ra = 10 6 , Ha = 25, A = 2, as given in Table 4.There is a good agreement between the two models with a maximum relative error of 3.19 percent.

Results and discussion
The numerical simulation results of the buoyancy-driven flow in L-shaped cavities filled with Cu-water nanofluid are discussed in this section.The effects of parameters including the temperature oscillation amplitude (0 A 2Þ, the oscillation frequency 0 f 100 ð Þ , the Rayleigh number (10 3 Ra 10 8 Þ, the Hartmann number (0 Ha 100), the nanoparticle volume fractions 0 / 0:2 ð ), and the aspect ratios of the enclosure (0:2 AR 0:8) are investigated.Between 2 and 20 oscillating periods were computed to ensure that the solutions reached a statistical steady state not contaminated by the initial conditions.A time step Ds ¼ 0:001 and integration over 200-time units are used throughout the computations.
Fig. 3 represents the transient response of the average Nusselt number (Nu

À
) with time during 6 periods of oscillation for selected values of forcing amplitude and with AR = 0.2, Ra = 10 6 , Ha = 100, and / = 0.2.This figure demonstrates that Nu À changes periodically, a similar pattern as the imposed temperature condition on the heated walls, with the maximum peak value at the largest amplitude of oscillation; this condition enhances the heat transfer significantly., which is also constant.The effect of the periodic temperature boundary conditions propagates through the cavity, causing an increase in the time-average Nu À , with the highest value corresponding to the larger amplitude for the same frequency.Interestingly, as the amplitude grows, the frequency for maximum Nu À remains nearly constant (f % 6) for all amplitudes considered in this study (A = 0.5, 1, 1.5, 2).The peak frequency response was different from that of a square cavity, where peak frequency remains constant for A > 0.5 while increasing for A 0.5 [51].
To gain further insight into the results represented in Fig. 4, the streamlines and isothermal contours of natural convection within the enclosure for different amplitudes are plotted in Fig. 5 (using the same parameters as in Fig. 4).The streamline contours show a slow flow near the boundaries, while in the arms of the cavity, the temperature varies almost linearly from the hot boundary to the adjacent cold boundary (a feature consistent with a conduction-dominated flow), and at low amplitudes, the outer corner of the L-bend features a relatively broad warm region, which becomes smaller as amplitude increases.Interestingly, in the vertical arm of the cavity, the streamlines demonstrate that the flow adopts a 1D profile that is invariant in y away from the bend and the end-wall, while in the horizontal arm, the flow circulates with a significant xdependence.
Also, the pattern of the streamlines changes slightly with the increase in the temperature amplitude.As the maximum amplitude of the oscillation increases, the temperature difference between the cold and hot walls increases.Therefore, the fluid touching the cold surface cools and migrates toward the hot region of the cavity, near the hot walls, due to the buoyancy effect, resulting in substantially improved heat transfer.Table 5 Critical Ra values for the departure from the conduction regime with reference to the selected Ha at the peak frequency (f = 6) for AR = / = 0.2 using Eq.(26).6 were cor-related using the power-law correlation with a coefficient r 2 = 0.9986 over the range of 10 Ha 100 and expressed below as: where C = À 28335.6,D = 17987.48and n = 0.8654.This is based on the buoyancy and Lorentz forces by Ra and Ha in Eq. ( 21), whose ratio is approximately Ra/Ha 2 .Natural convection and Lorentz force are balanced to determine the threshold between them.Table 5 lists the values of different Ha and the corresponding critical Ra, related to the minimum Nu À at which the departure from conduction to convection starts, obtained by using Eq. ( 26).smaller number of broader cells when compared to the Ha = 25 case.Secondly, observing the isotherm lines in both cases of low and high Ha, as shown in Figs.7 and 8, at Ra = 10 4 and 10 6 , the developed isotherm lines are primarily parallel to the nonadiabatic walls of the enclosure and characterize a diffusiondominated regime resulting in heat transfer mainly by conduction.With a further increase in the buoyancy force within the cavity at Ra = 10 8 , numerous thermal boundary layers are produced in the vicinity of the non-adiabatic walls, characterized by a convective-dominated regime resulting in heat transfer mainly by convection.For Ha = 25, it can be noted that after Ra = 10 5 , the flow transitions from diffusiondominated to convection-dominated regimes.For Ha = 100, however, the flow continues in a diffusion-dominated regime for Ra < 10 6 , followed by transitions to a convectivedominated regime beyond Ra = 10 6 owing to the ascend in buoyancy force.Furthermore, for both values of Ha and at Ra = 10 8 , the isotherm lines slope from the non-adiabatic walls in the vertical portion of the enclosure; however, it has an irregular shape in the horizontal portion due to the stronger convective effects replacing the smoother diffusion-dominated features.
The Nu À variation with the nanoparticle volume fraction (/) for selected Hartmann number, at the peak frequency, with AR = 0.2, Ra = 10 6 and A = 2, is depicted in Fig. 9.In general, Nu À ascends with U enlargement; due to the fact that using Cu-water nanofluid is thermally more conductive than pure fluid casing an increase in the transfer of heat.Additionally, it is realized that the gradient of the trends slope changes substantially as the Ha increases from 10 to 100.The growth is more identifiable at low Hartmann numbers; the Nu À augments linearly with enlarging nanoparticle loading as Ha increases from 0 to 50, while it remains nearly constant at Ha = 75 and 100.This result may be attributed to the Lorentz force damping the buoyant flows of nanofluid with high nanoparticle concentrations, which reduces the heat transfer by convection.
The Nu À variation with the aspect ratio (AR) for selected Hartmann number, at the peak frequency, with / = 0.2, Ra = 10 6 , and A = 2, is depicted in Fig. 10.It is observed that as AR varies from AR = 0.2 to 0.8, for a given Ha, there is a remarkable drop in the Nu À is negligible regardless of the aspect ratio.To furtherly explain these conditions, the streamlines and isotherms contours, at the peak frequency (f = 6), for low and high aspect ratios (AR = 0.2 and 0.8) and / = 0.2, Ra = 10 6 , A = 2, Ha = 100 are depicted in Fig. 11 (Ha = 25) and Fig. 12 (Ha = 100).
Observing the streamline contours in Fig. 11, at Ha = 25 and for a low aspect ratio, the streamlines show a large recirculation cell formed and expanding into the upper corners and the lower-left corner of the enclosure with a secondary counterrotating cell in the middle of the lower part of the enclosure.However, for a high aspect ratio, the recirculating cell expands further between the vertical walls, and the secondary counterrotating cell in the corner becomes more active, establishing a clockwise rotating vortex.In Fig. 12, at Ha = 100 and AR = 0.2, two clockwise rotating vortices are created within the enclosure, while only one large clockwise rotating vortex is formed at AR = 0.8.The increase Ha increases Lorentz force that suppresses the formation of flow vortexes, which in turn weakens the convection heat transfer, while the conduction governs thermal energy transport, consequently, lowering Nu À as was depicted in Fig. 10.It is found that the vortices in the streamlines become less dense with increasing

Conclusions
In this study, free convection heat transfer of copper-water nanofluid inside an L-shaped cavity has been investigated numerically for a sinusoidal oscillating hot wall temperature in the presence of a magnetic field.The effects of enclosure aspect ratio, Rayleigh number, the volume fraction of the copper nanoparticles, the amplitudes of temperature oscillation, forcing frequencies, and Hartmann number, on the flow dynamics and transfer of heat behaviors have been investigated at a Prandtl number Pr = 6.2.
The natural convection was consistently enhanced by the use of nanofluid, with a 47 % increase in Nu À from / = 0 to 0.2 at Ha = 0 and decreases to the smaller 8 % increase as Ha is ascended to Ha = 100.
The results showed that the higher the temperature oscillation amplitude applied to the hot walls, the higher the peak Nusselt number significantly improves heat transfer.In contrast to the square cavity case, the frequency at which the peak Nu À occurs stays nearly fixed as the oscillation amplitude grows.For AR = 0.2, heat transfer curried out mainly through conduction at Ra = 10 3 -10 5 , and Nu

Fig. 1
Fig. 1 Schematic for the studied configuration.

Fig. 4
Fig. 4 Nu À variation for / = AR = 0.2, Ra = 10 6 , and Ha = 100 for selected A and over a range of f.The horizontal dashed line depicts Nu À in the absence of oscillation.(d) represents the maximum NuÀ

Fig. 4
Fig. 4 depicts the Nu À variation for / = AR = 0.2, Ra = 10 6 , and Ha = 100 for selected amplitudes of oscillation and over a range of forcing frequencies.The case with A = 0 represents a constant surface temperature condition (T À h ) on the two heated walls; this case corresponds to the lowest value of the NuÀ

Fig. 6 Fig. 6
Fig. 6 portrays the alteration of the Nu À , at the peak oscillation frequency, with Ra and selected values of Hartmann numbers for AR = / = 0.2, and A = 2.The value of Nu À progresses with the increase of Ra, for a given value of Ha, a variety of zones can be distinguished from this; the first zone covers the small to moderate range of Rayleigh numbers

Â 10 5 (
Ra = 10 3 -10 5 ), and the second zone covers the high range of Ra > 10 5 .Through the first zone, Nu À is nearly constant with negligible effects due to the changes in Ra and Ha; the fluid flow is dominated by diffusion and the heat transfer is dominated by conduction.Also, the hot wall boundaries do not have a clearly visible boundary layer.In the second zone, the Nu À generally rises significantly as Ra increases and become sensitive to the changes in Ha; the transmission of heat and flow of fluid are dominated through convection.Observing the changes in the Nu À through the whole region, there is a critical Rayleigh number (Ra cr ) for every Ha value beyond which the Nu À rapidly increases.The Ra cr and Ha data in Fig.

À:
However, for a given AR value, smaller Hartmann number Ha = (0, 10, 25) produce the largest Nu À values.Additionally, the variance in Nu À between the cases of Ha = 0 (no magnetic field) versus Ha = 100 (largest effect of magnetic field) lessens with the rise in AR.The transfer of heat enhances as AR decreases, and this increase becomes more pronounced in the range of Ha 25 by almost 64 %.However, for low values of Ha 10, the effect of the Ha on Nu

À
is unrelated to both Hartmann and Rayleigh numbers.Convection effects, on the other hand, become significant at high Rayleigh numbers, and Nu À demonstrates Ha independence.The critical Rayleigh number for the transition from conduction to convection exhibits a power-law relationship.For low Hartmann number, at Ra = 10 4 , a single clockwise rotating vortex is evolved within the vertical segment of the enclosure, whereas the fluid in the horizontal segment is approximately stagnant.For Ra = 10 6 , Rayleigh-Be´nard like cell is induced in the horizontal enclosure segment.However, for Ra = 10 8 , the Rayleigh-
on the outcomes, Nu