1 Introduction

In the engineering and industrial context, a wide range of investigations has been performed for natural convection using nanofluid media with various techniques to improve the thermal conductivity and enhance the heat transfer performance of the fluid within the enclosure. To improve thermal behavior, numerous approaches have been proposed involving enclosure geometry, boundary conditions, filled media, etc. Nanofluid is defined as nanoscale particles (diameter less than 100 nm) or fibers diluted in a base fluid. In the literature, numerous studies have been conducted on the heat transfer and flow inside an enclosure filled with nanofluid medium [3, 34]. Heat transfer and Ag-water nanofluid flow inside a square enclosure with a center heater is analyzed by Mahalakshmi et al. [16]. Their outcomes indicate that an increase in heater length leads to an increase in heat transfer rate. Conjugate natural convection flow in a square cavity in presence of hybrid nanofluid is investigated by Ghalambaz et al. [9]. They stated that the heat transfer rate rises due to the rise in hybrid nanoparticles and thermal conductivity ratio. Natural convection of hybrid/water nanofluid inside a porous enclosure by applying Darcy model is studied by Mehryan et al. [23]. They concluded that the heat transport decreases due to the dispersion of hybrid nanoparticles in the base fluid. Balushi et al. [2] considered a nonhomogeneous dynamic model to investigate natural convection by using magnetic nanoparticles in a square enclosure. They found that the average Nusselt number augments as the nanoparticles volume fraction increase.

There are several cases of heat transfer under the impact of the magnetic field because of their various variety of application in engineering areas such as chemical industry, power and cooling industry for drying, chemical vapor deposition on surfaces, cooling of nuclear reactors, crystal growth in liquids, cooling of nuclear reactors, electronic packages, and petroleum industries. From the works of literature, the influence of the magnetic field on the natural convection inside the enclosure decreases the convection effect and reduces the heat transfer rate. Therefore, the interaction of nanoparticles and the addition of nanoparticles to the fluid can improve its thermal performance (since the thermal conductivity of solid is typically higher than that of liquids) and heat transfer mechanism in the enclosure. Ghasemi et al. [10] was the first to use nanofluid ( - water) as the working fluid in the presence of a magnetic field in a differentially heated square enclosure. Their result showed that the presence of a magnetic field adversely affects the fluid flow, and heat transfer rate increases and decreases with the increase of the Rayleigh number and Hartmann number, respectively. Extensive research has been performed considering the effect of magnetic fields on fluid flow inside a cavity [15, 30].

There are several studies in the literature based on the phenomena of mixed convection inside enclosures [4, 12, 17, 19]. Öztop and Salem [26] carried a review on entropy generation in natural and mixed convection heat transfer. Mahapatra et al. [18] analyzed the effect of buoyancy ratio on double-diffusive mixed convection with uniform and nonuniform heating of walls in a lid-driven square enclosure. Double-diffusive mixed convection under the influence of the magnetic field has been investigated by Uddin et al. [33] for both uniformly and nonuniformly heated and concentrated walls. Double- diffusive mixed convection in a shallow inclined cavities with different inclination angles, Lewis numbers, and buoyancy ratios is numerically performed by Teamah et al. [32]. They observed that the mass transfer rate increases by increasing Lewis number, but there is no change in heat transfer rate. Also, the average Nusselt number and Sherwood numberenhance with the rise in the buoyancy ratio.

Natural convection heat transfer in corrugated or wavy enclosures is gaining the attention of most researchers for enhancing the efficiency of heat and mass transfer. Over the last few years, natural convection in complex enclosures has been carried out by various researchers [20, 25, 31]. Rahimpour and Moraveji [28] examined the heat transfer of MHD free convection in a C-shaped chamber filled with /water nanofluids. They examined the impact of Hartmann, Rayleigh numbers, chamber tilted angle, nanoparticles concentration, and chamber geometrical parameter on thermal performance and fluid flow structures. Rashed et al. [29] carried out the buoyancy-driven flow of hybrid nanofluids within an inclined two-sided wavy enclosure considering magneto-convection flow. Bezi et al. [5] studied the entropy generation and heat transfer flow in a semiannular enclosure filled with nanofluid. Their results show that the entropy generation strengthened by increasing the Rayleigh number. Ahmed [1] performed natural convection of dusty hybrid nanofluids in diverging-converging cavities including volumetric heat sources. The heat transfer of MHD flow of Newtonian nanoliquid in a baffled U-shaped enclosure is investigated by Zaim et al. [35]. The nanofluid flow inside a Novel shape porous cavity is examined by Molana et al. [24] subjected to magnetic-field-dependent viscosity. They deduced that the Nusselt number has a direct relationship with the aspect ratio. Chamkha et al. [7] investigated the entropy generation of nanofluid flow inside a gamma-shaped porous cavity considering the magnetic field. They found that by raising the nanoparticle volume fraction the heat transfer rate and entropy generation enhance.

From our investigations there are limited attempts focused on mixed convection flow inside a complex enclosure filled with nanofluid. Thus, the aim of the present work is to investigate the steady MHD double-diffusive mixed convection in a curved enclosure filled with nanofluid. The reasons for selecting this model are:

• Importance of this curved fluid flow geometry (or curved duct) [8, 13, 21] in many engineering and technological systems such as compact heat exchangers, steam boilers, thermal power generation, aircraft engines, etc.

• Its originality since limited work has been done in such geometry considering double-diffusive mixed convection as the mode of heat transfer and nanofluid as the medium.

The influence of Richardson number, the solid volume fraction of nanoparticles, the type of nanofluids, and heat and mass source length and location on the average Nusselt number, Sherwood number, and total entropy generation have been studied. This paper is organized as follows. Section 2 presents the physical model. Section 3 includes mathemat- ical formulation. Section 4 discusses the numerical implementation and code validation. Section 5 presents the results and discussion. Finally, a conclusion is given in Section 6.

2 Physical model

The schematic diagaram of the system considered in this paper is depicted in Fig. 1. The enclousure is of lenght $\textcolor{red}{}L$, and a discrete heat and concentration source of dimesion less length $\textcolor{red}{}H$ are located at the left wall of the enclousure having constant hot temperature $T_h$ and concentration$C_h$. The remaining part of the left wall and the right wall are considered to be at constant temperature and concentration of and . The top and bottom walls are assumed to be adiabatic and impermeable. Furthermore, the right wall of the enclosure is allowed to move in upward direction with a constant velocity of , while all the remaining walls are at rest. An external magnetic field with uniform strength and the gravitational force are applied in the vertically downward direction. The fluid inside the enclosure is a water-based nanofluid containing different types of nanoparticles such as and . There is no slip between the particles of the base fluid and the nanoparticles, and they are assumed to be in thermal equilibrium. In dimensional form, the lower and upper wall of the enclosure is defined by the functions

where a is the amplitude of the sinusoidal walls. The nanofluid is considered to be New- tonian, and the flow is steady, incompressible, and viscous.The viscous dissipation and magnetic dissipation are expected to be negligible.The flow is directed by a combined buoyancy effect due to both temperature and concentration variations. The density varia- tion is described by the following formula:

where and are the dimensional temperature and solute concentration, respectively, and are the fluid density, volumetric coefficient of thermal expansion, and volumetric coefficient of solutal expansion, respectively.

3 Mathematical formulations

3.1 Thermo-physical properties of nanofluid

The thermo-physical properties of the base fluid and nano-sized particles are tabulated in Table 1, where, and indicate the specific heat, thermal conductivity, electrical conductivity, dynamic viscosity, and thermal diffusivity, respectively.

The effective density , specific heat , thermal expansion coefficient of the nanofluid, according to [3], are:

and

Effective dynamic viscosity and the effective thermal conductivity of the nano- fluid, which are obtained from Brinkman model [6] and Maxwell’s model [22], respec- tively, are introduced as

Finally, the thermal diffusivity and electrical conductivity of the nanofluid are defined, respectively, as

Here represents the volume fraction of the nanoparticle, and . The subscriptsf , and are used to refer base fluid, solid particle, and nanofluid, respectively.

3.2 Governing equations

The equations, which govern the double-diffusive flow, are given in dimensional form [4].

Continuity equation:

Momentum conservation equations:

Energy equation:

Here and are the dimensional velocities along the and directions, represents the dimensional fluid pressure, and is the mass diffusivity.

Following transformation of variables are used to convert system (1)–(5) into nondimensional form:

The resulting nondimensional continuity, momentum, energy, and concentration equations can be written as

where and are the dimensionless velocities along the and directions, ,, and represent the dimensionless pressure, temperature, and concentration, respectively.

The nondimensional parameters are given by

Where are Prandtl number, Lewis number, Grashof number, Reynolds number, Hartmann number, Richardson number, and buoyancy ratio, respectively.

The lower and upper sinusoidal wall in nondimensional form can be written as

Where is the dimensionless amplitude of the sinusoidal walls.

The imposed boundary conditions on the enclosure in nondimensional form on top and bottom wall:

On right wall:

On left wall:

Having heat and mass source:

without heat and mass source:

Here

3.3 Nusselt number and Sherwood number

The local Nusselt number and local Sherwood number at the left wall is defined as

The average Nusselt number and average Sherwood number are obtained by integrating local Nusselt number and local Sherwood number , respectively:

To evaluate the integrals, a Simpson’s rule of integration is implemented.

3.4 Governing equations for entropy generation

The boundary conditions imposes nonequilibrium flow in the flow field that causes en- tropy generation. The local entropy generation equations in dimensional form are

where are entropy generation due to heat transfer, fluid friction, magnetic field, and solutal concentration, respectively. The dimensionless forms of the local entropy generation expression using the dimensionless quantities in (6) are given by

where

are called irreversibility distribution ratios and taken as . Here is the gas constant, and are bulk temperature and bulk concentration, respectively, and could be calculated as and , respectively.

The average entropy generation due to heat transfer , fluid friction , magnetic field , and diffusionare obtained by integrating the local entropy generation by the system volume:

The total entropy generation is given by .

4 Numerical technique

4.1 Methodology

The stream function and vorticity in nondimensional form are given by

which gives a single equation

Using Eq. (10) and eliminating pressure term from Eqs. (7) and (8), we get

The presence of the curved walls make it difficult to impose the curve boundary on rectangular grids. Thus, transformation is required to convert the irregular physical domain into a regular (square) computational domain ; see Fig. 2. In this study, we have considered the following coordinate transformation:

Using the above transformation, the sinusoidal boundaries are transformed into the straight lines.

The equations can be evaluated in domain using the following relationship:

Here is the Jacobian of the transformation

Using the above relations, the governing equations (8), (9), (11), and (12) can be written in new transformed coordinates as

where

We get the following biharmonic equation in stream function-velocity formulation by substituting ω from Eq. (13) and writing in (14):

The calculations are performed in computational domain, and the results are then transformed back to the physical domain. The transformed equations are discretized using a second-order central difference scheme. The set of discretized equations for each vari- able are solved by using an outer-inner iteration procedure, biconjugate gradientstabilized method (BiCGStab) [14, 27]. The triagonal system

is solved by using Thomas algorithm to get and . After obtaining and , we compute and .

The numerical method was implemented in FORTRAN software. The convergence criterion is that the difference of values of all the variables between two consecutive iterations is less than .

4.2 Code validation and grid independence study

Accuracy tests were made for in Table 2 for grid independence using the four sets of grids:. A good agreement was found between and grids so the numerical computations were carries out using grid nodal points.

The present results have been authenticated successfully in three steps:

• With the work of Ghasemi et al. [10], for , using various Hartmann numbers in a differentially heated square cavity at; shown in Table 3.

• With the result of Mahapatra et al. [18], in a square enclosure maintained at different temperatures and concentrations as represented in Figs. 3(a) and 3(b). The average Nusselt number and average Sherwood number at the lower wall were compared when for various values of Rayleigh number.

• With the work of Gupta and Kalita [11], for , using various Reynolds numbers in a lid-driven square cavity; shown in Fig. 3(c).

It is clear from Table 3 and Fig. 3 that the obtained results of the present work illustrate good agreement with those from the earlier investigations, and therefore, we get the confidence of the present numerical code.

5 Results and discussion

The present work investigated numerically MHD double-diffusive mixed convective heat and mass transfer and entropy generation in a curved enclosure filled with different types of nanofluids and subjected to discrete heating. The right wall is moving in an upward direction. The influence of various physical parameters such as Richardson number, Hart- mann number, heat and mass source location and length, the volume fraction of nanoparticles are examined in terms of streamlines, isotherms, isoconcentrations, entropy generation, average Nusselt number, and average Sherwood number. We have carried out the simulation for fixed Prandtl number, Reynolds number , and Buoyancy ratio .

5.1 Effects of Richardson number

In the mixed convection problem, the Richardson number is used to show the importance of natural convection relative to the forced convection. It might be noticed that free convection phenomena are studied for , and mixed convection effects can be visualized for . Figure 4 describes the contour of streamlines, isotherms, and isoconcentrations for various values of Richardson number inside the enclosure filled with Cu-water nanofluid at .

At low Richardson number , the thermal buoyancy forces and inertia forces are in balance. The distribution of streamlines shows the mixed convection effect with two counter-rotating vortices, and the isotherm and isoconcentration patterns are ap- peared to be linear. As the Richardson number increases, the buoyancy forces predominate the inertia forces, and hence, the natural convection becomes more effective than mixed convection. From the figure it is observed that the secondary anticlockwise vortices adjacent to the right cold wall begin to vanish, and the enclosure is filled with primary clockwise vortices. The strength of fluid flow increases from for to . The isotherm and isoconcentration lines show a similar pattern due to similarity in energy and mass equation. They become irregular and spread in an upward direction signifying convection mode. The isoconcentration lines are found to be more compressed than isothermal lines resulting in a high mass transfer rate compared to heat transfer. From the observations we can conclude that the natural convection effect is dominating over the mixed convection. Figure 5(a) depicts the variation of total entropy generation with . It is observed that Stotal increases with and is found to be high for pure fluid than nanofluid. The heat and mass transfer rate are determined using the average Nusselt and Sherwood numbers, respectively. From Figs. 5(b) and 5(c) it can be seen that irrespective of other parameters, the increase in escalates and due to enhanced convection, and consequently, heat and mass transfer is increasing function of Richardson number. Also, the and decrease with the addition of nanoparticles.

5.2 Effects of Hartmann number

Figure 6 illustrates the effect of the magnetic field parameter in terms of streamlines, isotherms, and isoconcentrations.

As seen from the figure, the flow field, temperature and concentration distribution have an inverse relation with the magnetic field effect. In the absence of magnetic field, i.e., at , the intensity of circulation is strong and compact due to the dominant shear effect, which produces high convection mode. Further, the isotherm and isoconcentration contours are highly concentrated near heat and mass sources. But when the Hartmann number increases (i.e., and ), the strength of flow circulation weakens because of the presence of Lorentz force, which decelerates the flow velocity and results in an attenuation of convection mode and intensification of conduction mode. It is observed that value of for and drops to . The concentrated region of isotherms and isoconcentrations adjacent to the sources becomes less compressed and smooth. It is noted that the thermal boundary layer begins to fade with the ascending values of . Figure 7(a) shows the variation of total entropy generated versus . In our analysis, the entropy decreases when increases. Therefore, we may conclude that the presence of a high value of can be used to minimize entropy generation in engineering systems. Influence of Hartmann number on and is plotted in Figs. 7(b) and 7(c). and attenuate with the enhancement of as the presence of a strong magnetic field intensifies the conduction mechanism and consequently reduces the heat and mass transfer rate.

5.3 Effect of heat source location and length

The effects of heat and mass source locations are depicted in Fig. 8 at . The intensity of the fluid motion significantly rises when the source position moves up and also the size of the circulation cell changes. The locationof sources has a direct effect on the pattern of streamlines. It is found that the upper corner’s circulating zones grew up and the lower corner’s circulating zones dwindled by changing the source position from lower to upward. Also, the strength of isotherm and isoconcentration lines indicate that the conduction mode changes to convection mode with the increase in the source location.

Numerical simulations of heat and mass source length on the fluid flow, heat, and mass transfer are shown in Fig. 9. It is observed that when is small, the shear-driven force is dominant as compared to the buoyancy force. The streamlines show primary clockwise circulation in the major part of the enclosure and secondary anticlockwise circulation adjacent to the cold right wall. When increases, the buoyancy force increases considerably, which augments the circulation strength and decreases the effect of shear-driven force.

The size of the primary circulation grows bigger, while the secondary circulation reduces as increases and also the performance of the thermal process increases. The reason for all these behaviors is the total heat input that increases as the heat and mass source length increase.

The temperature and concen- tration contours become more skewed for due to an increase in temperature and concentration gradient with the increase in source length. Figure 10(a) presents the variation of total entropy generation with for different . As the heat transfer rises at , the total entropy is also expected to be maximum. It is also observed that for all R, the entropy generation drops with the increase in Figure 11 shows the variation of total entropy generated versus .

The total entropy generation is maximum for due to heat transfer irreversibility, solutal concentration irreversibility, and fluid friction irreversibility.Also, the addition of the nanoparticles to the pure fluid leads to a decrease in total entropy generation for all values of . This happens due to an increase in viscosity that increases the fluid friction and significantly affects the entropy generation. Figures 10(b), 10(c) and 11(b), 11(c) show the variation of average Nusselt number and Sherwood number with the volume fraction of nanoparticles for different heat and mass source size and location. Both Nuavg and Shavg number increase with rise in and . This happened because maximum temperature increases.

5.4 Effects of nanoparticle volume fraction

Figure 12 explains the effect of nanoparticles volume fraction on streamlines, isotherms, and isoconcentrations inside the enclosure at. It is seen that the effect of the volume fraction parameter is not prominent on streamlines.

The fluid motion is represented by one primary clockwise circulation and secondary counter-clockwise circulation for all cases. The increase in the nanoparticle volume fraction results in a decrease in strength of circulation. The value of . This behavior is due to an increase in the fluid viscosity caused by the increase in and hence results in a decrease in flow velocity. The convection effect decreases at a high solid volume fraction of nanoparticles, and hence, the isotherms and isoconcentrations change from irregular shape for pure fluid to regular shape for nanofluid . A rise in enhances the thermal conductivity that consequently enhances the temperature and concentration gradient and leads to a reduction in entropy generation.

Figure 13(a) shows the influence of various types of nanoparticles on total entropy generation. It is detected that the total entropy generation decreases with rising in the volume fraction of nanoparticles, and minimum entropy generation is obtained in case of nanoparticles. The variation of the heat and mass transfer rate is shown in Figs. 13(b) and 13(c) for various types of nanoparticles. Here we find that the highest rate of heat and mass transfer rate is obtained by adding compared to other nanoparticles. Due to the presence of nanoparticles, the fluid velocities decrease due to an increase in dynamic viscosity and induce resistance against the fluid motion, which in turn reduces the heat and mass transfer rate. Figures 14(a) and 14(b) show and for different Lewis number and volume fraction of nanoparticles. An increase in Le causes the solutal force to rise and tends to increase the mass transfer rate. A reverse trend is observed by the average Nusselt number, which decreases with a rise in.

6 Conclusion

In the present study, we have adopted a dome-shaped enclosure filled with nanofluid to study MHD double-diffusive mixed convection in the presence of discrete heating. We have considered water-based nanofluids using different types of nanoparticles. The dimensional governing equations are then converted to dimensionless forms using suitable transformations. The resulting nondimensional forms of the equations were written in stream function-velocity formulation and solved numerically using the BiCGStab method followed by the Thomas algorithm. The effect of governing parameters on the heat trans- fer as well as on the entropy generation is analyzed. It is found that the considered modelis convenient for entropy generation minimization. Several significant conclusions drawn from this study are listed in the following:

• An increase in Richardson number augments the strength of convection effect due to stronger buoyant force, and therefore, better enhances the heat and mass transfer rate

• At low Richardson number, the increase in the volume fraction of nanoparticles leads to an increase in average Nusselt number, while at high Richardson number, it decreases.

• An increase in Hartmann number reduces the convection effect.

• The increase in the volume fraction of nanoparticles drops the value of stream function strongly for all values of and and has an adverse effect on the average Nusselt number, Sherwood number, and entropy generation.

• The heat and mass transfer rate enhances when the source length and its location increase. The increase in source length leads to an increase in the temperature inside the enclosure.

• The total entropy generation is an increasing function of Richardson number and a decreasing function of Hartmann number, volume fraction of nanoparticles, size, and location of the source.

• The highest rate of heat and mass transfer can be obtained by adding compared to other nanoparticles. The heat transfer rate declines with the increase of Lewis number, while the mass transfer rate increases.

The current research work can be further extended by considering the porous media and heat generation effect inside the enclosure.

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