Magneto-convection of water near its maximum density in a cavity with partially thermally active walls

Abstract

The main objective of the present numerical investigation is to understand the effect of partially active thermal zones and density inversion on the buoyancy-driven convection of cold water near its maximum density under the influence of uniform magnetic field inside a square cavity. The partially heated or cooled zones are placed along the vertical sidewalls of the cavity while the other inactive portions of the walls are considered to be adiabatic and thermally insulated. The length of the partially active thermal zones is measured to be equal to half the length of the cavity’s height. When the locations of heating and cooling zones are placed in different positions, it provides a significant effect on flow pattern and the rate of heat transfer. To investigate this location effect, the thermally active zones are placed in a combination of five different relative positions along the vertical sidewalls. The modeled equations are solved using finite volume method with Semi-Implicit Method for Pressure Linked Equation algorithm. The numerical coding is done through the FORTRAN 95 programming for numerical simulation and the results attained are represented graphically in the form of flow field and temperature distribution for various values of density inversion parameter, Hartmann number, and for various positions of thermal zones. The results reveal that the heat transfer rate behaves nonlinearly on increasing the density inversion parameter values. The high transfer rate is perceived for the location middle–middle when $T_{m} < 0.5$ and for top–bottom when $T_{m} \geq 0.5$ for all values of the Hartmann number.

Keywords

Magneto hydrodynamics convection partially thermally active walls cavity maximum density

Introduction

Occurrence of natural convection in enclosures heated from its side walls is of rational interest to many researchers due to its versatile applications in many branches of science, engineering, and technology, specifically, in the study of energy conservation and built environment. Mostly in real-time applications like building insulations, home/industrial storage installations, cryogenics, electronic cooling, air-conditioning, etc. this mode of heat transfer remains common. Another important area focused by scientists, engineers, and researchers is magneto hydrodynamics (MHD), which involves the study of electrically conducting fluids in the presence of electromagnetic fields. The blend of natural convection and magnetic field includes widespread applications in various fields like liquid metal cooling of nuclear reactors, solar collectors, electromagnetic casting, material manufacturing technology.^1,2 In all these analyses, the study on temperature distribution and the flow pattern of the fluid in the presence of magnetic field becomes crucial. Thus, owed to its significance and multipurpose applications, the study on free convection in enclosures with partially heated or cooled walls in the presence of magnetic field has gained special interest among researchers.^3,4

Various studies regarding enclosures are extensively carried out both experimentally and numerically. Valencia and Frederick⁵ numerically analyzed the heat transfer rate for five different positions of half heated and half insulated vertical walls in square cavities. They established that the heat transfer rate can be controlled by varying the positions of the partial active walls along the cavity. Yucel and Turkoglu⁶ numerically analyzed the flow and heat transfer characteristics in partially heated and cooled rectangular cavities. They observed that the averaged Nusselt number declines as the heater size rises for a given cooler size and the averaged Nusselt number increases with increase in cooler size for a given heater size. Natural convection in a cubical enclosure with thermally active zones placed on the same wall was carried out by Frederick.⁷ Based on the study of flow pattern and temperature distribution, they proposed an expression for heat transfer coefficient. A numerical and experimental investigation was carried on free convection in a square cavity with partially active sidewalls by Paroncini et al.⁸ They noticed that there is an increase of Nusselt number with an increase of Rayleigh number. Cianfrini et al.⁹ numerically studied the laminar natural convection inside rectangular cavities bounded below by partially heated wall and cooled sidewall. They found that the heat transfer increases inside the cavity by increasing the heater size and Rayleigh number.

Venkatachalappa and Subbaraya¹⁰ numerically studied magneto-hydrodynamic convection in a two-dimensional rectangular enclosure filled with electrically conducting fluid, in which two side walls are maintained at uniform heat flux condition. They found that the magnetic field can be used effectively to control the convection inside the enclosure. Rudraiah et al.¹¹ evaluated a similar problem by differentially heated isothermal sidewalls. The numerical results showed that the effect of the magnetic field inside the cavity decreases the convective heat transfer. The application of magnetic field restrains the unavoidable convection current, which arises in an enclosure and thus upholding the controllability of the system. This effect has encouraged the study of laminar natural convection in tilted enclosure with transverse magnetic field by Al-Najeem et al.¹² It is established that the flow characteristics and heat transfer strongly depend on the magnetic field and the inclination angle of the cavity. Hossain et al.¹³ analyzed the effects of heat generation and the combined effect of magnetic field and surface tension on unsteady laminar natural convection flow. They observed that the increase of heat generation within the cavity increases the flow rate and temperature of the fluid. Ece and Buyuk¹⁴ explored the free convection in an inclined square differentially heated enclosure in the presence of magnetic field.

Study on effectiveness of steady laminar flow in the incidence of magnetic field on convection heat transfer in a tilted square enclosure was carried out by Pirmohammadi and Ghasemmi.¹⁵ They observed that the mechanisms of heat transfer and the characteristics of fluid flow inside the enclosure strongly depend on the strength of magnetic field, its inclination angle, and the Rayleigh number. The impact of thermally active zones on hydro-magnetic convection in an enclosure in different directions of magnetic field was studied by Sivasankaran and Bhuvaneswari.¹⁶ They observed that the heat transfer increases if the heater is located in the middle or bottom of the hot wall while the cooler is at the top or middle of the cold wall in the considered enclosure. Sivasankaran et al.¹⁷ analyzed the rate of heat transfer and fluid flow under the effect of magnetic field in a partitioned enclosure. In the study, they found that the heat transfer increases when the partition is affixed in the adiabatic wall and the same decreases when the Hartmann number is increased. A computational study of convection in a square cavity in the presence of magnetic field was studied by Bhuvaneswari et al.¹⁸ In the work, they had spatially varied sinusoidal temperature distributions in both the sidewalls and established the result as follows: the heat transfer rate increases first and then starts decreasing on increasing the phase deviation. Obayedullah and Chowdhury¹⁹ made a study on steady natural convection flow in a rectangular enclosure filled with electrically conducting fluid. They observed that the temperature distribution, fluid flow, and heat transfer rate robustly depend on the Rayleigh number and Hartmann number. Heat transfer enhancement of Cu–water nanofluid in a square cavity with a circular disk under the magnetic field was studied by Kobra et al.²⁰ It was noted that the overall Nusselt number significantly decreases on increasing the Hartmann number and Rayleigh number. To study the effect of magnetic field on flow field, heat transfer, and entropy generation, Aghaei et al.²¹ numerically investigated the effects of magnetic field on mixed convection and entropy generation of nanofluid in a trapezoidal enclosure.

Water has its maximum density at $3.98 ° C$ and the study on natural convection about this temperature is intricate. But, to understand the study concerning free convection in low-temperature water, the analysis on cold water near its density maximum has become very essential.²² Moreover, the study on density–temperature relationship is very much fundamental in the natural convection occurrence because it gives rise to abundant interesting facts which alter the fluid flow structure and thermal fluid properties. Inaba and Fukuda²³ studied the natural convection effect in the region of density inversion of water in an inclined square cavity. From the study, they noted that the density inversion of water influences the natural convection inside the cavity. Nansteel et al.²⁴ studied the convection of cold water about its density maximum in the limit of very small Rayleigh number in a rectangular enclosure. They found that the aspect ratio affects the flow structure and the heat transfer increases as aspect ratio increases. Then, Ishikawa and Hirata²⁵ numerically simulated the results of natural convection with density inversion in a square cavity filled with cold water. They proposed an expression for heat transfer correlation as a function of Galilei number, wall temperature ratio, and density function.

An investigation related to unsteady laminar natural convection flow of water near its maximum density in a rectangular cavity, bounded by isothermal walls with internal heat generation was carried out by Hussain and Rees.²⁶ It is found that the temperature and flow of water are strongly influenced by internal heat generation parameter. They also observed the circulation of flow getting reversed when internal heat generation parameter becomes strong. Sivasankaran and Ho^27–29 investigated the effect of temperature-dependent properties on natural convection of water near its maximum density. They observed that the fluid flow and heat transfer rate get strongly influenced by maximum density temperature of water due to the formation of bicellular structure. Sivasankaran et al.³⁰ explored the effect of variable fluid properties on magneto-convection of water near its maximum density in an enclosure. Numerical simulation on buoyancy-driven convection of water about its maximum density with partially active vertical walls along the sidewalls of the cavity was deliberated by Kandaswamy et al.³¹ It has been identified that the heat transfer rate becomes less near the density maximum region. The same model problem on buoyancy-driven convection was studied with time periodic partially active vertical walls by Nithyadevi et al.³² They found that density inversion of water has an effect on natural convection and hypothesized that the heat transfer rate enhances up to 80% when the heater is located in the middle of the wall.

Recently, due to the advancement of technology in industries, partially heated walls are widely used rather than enclosures with fully heated walls. This advancement of technology has attracted several researchers to learn more about the problems related to cavities with partially active thermal walls. This paper describes the behavior of cold water near its density maximum in the presence of magnetic field inside a square cavity with partially active thermal walls. The work illustrates the magnetic field’s effect on the nature of flow and subsequent temperature distribution inside the cavity while varying the location of the partially active thermal zones along the side walls of the cavity. All the related works cited above explain the buoyancy-driven convective flow of fluid for various geometry of cavity with partially heated/cooled walls either with or without magnetic field. However, the flow behavior and temperature distribution of buoyancy-driven flow of cold water near its density maximum in the influence of magnetic field in a cavity bounded with partial thermal walls are not dealt so far, and since this is a distinctive case of research, to observe the effect of magnetic field on cold water, the current study has been carried out.

Governing equations

A schematic representation of two-dimensional square cavity, taken for the present study is shown in Figure 1(a). The five different combinations of partially heating/cooling locations are depicted in Figure 1(b). The length of the square cavity is taken to be “L” and the two partially active thermal walls that are placed along the sidewalls measure “L/2.” The portion of heater along the left wall is maintained at temperature $T_{h}$ and the right wall with cooler is maintained at temperature $T_{c} (T_{h} > T_{c})$ . The remaining boundaries, namely the horizontal sides and the inactive portions of sidewalls of the cavity, are considered to be adiabatic and thermally insulated. The locations of the heater and cooler are moved only along the vertical side walls and their relative five different combinations of thermal wall locations are top–top (TT), middle–middle (MM), bottom–bottom (BB), bottom–top (BT), and top–bottom (TB). The gravitational force (g) is assumed to be acting in the downward direction along y-axis and the magnetic field applied acts along the X-direction. The magnetic field induced within the enclosure is very small and so it is assumed to be negligible when compared to the applied magnetic field. The electric current $J$ and the electromotive force $F$ are defined by $J = σ_{e} (V \times B)$ and $F = σ_{e} (V \times B) \times B$ , respectively. All the variations in fluid properties are ignored and the gravitational force is considered to be sufficiently strong to make the specific weight appreciably different within the fluid. So, the density variation is taken into consideration along with the gravitational force, thereby, satisfying Boussinesq approximation. The density variation of water is modeled by the relation $ρ = ρ_{m} 1 - β {| T - T_{m} |}^{b}$ , where $β$ being coefficient of thermal expansion and $β = 9.297173 \times 10^{- 6}$ . The subscript $m$ denotes the density inversion state and $ρ_{m} = 999.72$ is the density of the water at the inversion state or maximum density of water, $T_{m} = 3.98 ° C$ is the temperature of the fluid at the density inversion state and the constant $b = 1.894816$ .

Figure 1.

(a) Physical configuration and boundary conditions. (b) (i) TT, (ii) MM, (iii) BB, (iv) BT, (v) TB, and different thermally active locations.

The following assumptions are taken in the study:

The flow is assumed to be unsteady, laminar, and incompressible.

All the fluid properties are constant except the density in buoyancy term.

The Boussinesq approximation is valid.

The viscous dissipation is assumed to be negligible.

The induced magnetic field is neglected.

According to the above assumptions, the equations governing the conservation of mass, momentum, and energy for two-dimensional incompressible flow in Cartesian coordinate system are expressed as follows, (see appendix for nomenclature).

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

(1)

\frac{\partial u}{\partial t} + u \frac{\partial u}{\partial x} + v \frac{\partial u}{\partial y} = - \frac{1}{ρ_{0}} \frac{\partial p}{\partial x} + υ [\frac{\partial^{2} u}{\partial x^{2}} + \frac{\partial^{2} u}{\partial y^{2}}]

(2)

\frac{\partial v}{\partial t} + u \frac{\partial v}{\partial x} + v \frac{\partial v}{\partial y} = - \frac{1}{ρ_{0}} \frac{\partial p}{\partial y} + υ [\frac{\partial^{2} v}{\partial x^{2}} + \frac{\partial^{2} v}{\partial y^{2}}] + g β {| T - T_{m} |}^{b} - \frac{σ_{e} B_{0}^{2} υ}{ρ_{0}}

(3)

\frac{\partial T}{\partial t} + u \frac{\partial T}{\partial x} + v \frac{\partial T}{\partial y} = \frac{k}{ρ_{0} c} [\frac{\partial^{2} T}{\partial x^{2}} + \frac{\partial^{2} T}{\partial y^{2}}]

(4)

The appropriate initial and boundary conditions for the problem are defined as follows

For t = 0 : u = v = 0; T = 0, 0 \leq x \leq L, 0 \leq y \leq L

(5a)

For t > 0 : u = v = 0; \frac{\partial T}{\partial y} = 0, y = 0 and y = L

(5b)

u = v = 0; T = T_{h}, at x = 0, 0 \leq y \leq \frac{L}{2} for bottom heating

(5c)

$\frac{L}{4} \leq y \leq \frac{3 L}{4}$ for middle heating

$\frac{L}{2} \leq y \leq L$ for top heating

u = v = 0; T = T_{c}, at x = L, 0 \leq y \leq \frac{L}{2} for bottom cooling

(5d)

$\frac{L}{4} \leq y \leq \frac{3 L}{4} for middle cooling$

$\frac{L}{2} \leq y \leq L$ for top cooling

u = v = 0; \frac{\partial T}{\partial x} = 0, elsewhere on x = 0 and x = L

(5e)

Equations (1) to (4) are appropriately nondimensionalized for numerical study and are rewritten as

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

(6)

\frac{\partial U}{\partial τ} + U \frac{\partial U}{\partial X} + V \frac{\partial V}{\partial Y} = - \frac{\partial P}{\partial X} + \frac{\partial^{2} U}{\partial X^{2}} + \frac{\partial^{2} U}{\partial Y^{2}}

(7)

\frac{\partial V}{\partial τ} + U \frac{\partial V}{\partial X} + V \frac{\partial V}{\partial Y} = - \frac{\partial P}{\partial Y} + \frac{\partial^{2} V}{\partial X^{2}} + \frac{\partial^{2} V}{\partial Y^{2}} + G r {| θ - θ_{m} |}^{b} - H a^{2} V

(8)

\frac{\partial θ}{\partial τ} + U \frac{\partial θ}{\partial X} + V \frac{\partial θ}{\partial Y} = \frac{1}{P r} [\frac{\partial^{2} θ}{\partial X^{2}} + \frac{\partial^{2} θ}{\partial Y^{2}}]

(9)

where the nondimensionalized parameters are

G r = \frac{g β {| T - T_{m} |}^{b} L^{3}}{υ^{2}}

, the modified Grashof number;

H a = B_{0} L \sqrt{\frac{σ_{e}}{μ}}

, the Hartmann number;

P r = ν / α

, the Prandtl number; and

θ_{m} = \frac{T_{m} - T_{c}}{T_{h} - T_{c}}

is the density inversion parameter.

The dimensionless variables that appear in equations (6) to (9) are defined as follows

X = \frac{x}{L}, Y = \frac{y}{L}, U = \frac{u L}{υ}, V = \frac{v L}{υ}, τ = \frac{t}{\frac{L^{2}}{υ}}, θ = \frac{T - T_{c}}{T_{h} - T_{c}}, P = \frac{p L^{2}}{ρ}

(10)

Then, the initial and boundary conditions become

For τ = 0 : U = V = 0, 0 \leq X \leq 1 and 0 \leq Y \leq 1

(11a)

For τ > 0 : U = V = 0, \frac{\partial θ}{\partial Y} = 0, at Y = 0 and Y = 1

(11b)

U = V = 0, θ = 1 at X = 0, heating region

(11c)

U = V = 0, θ = 0 at X = 1, cooling region

(11d)

U = V = 0, \frac{\partial θ}{\partial X} = 0, at X = 0 & 1, elsewhere

(11e)

The most important characteristic of heat transfer from engineering perspective is the measure of heat transfer rate across the cavity. Hence, Nusselt number computations are carried out by the expression

N u = \frac{h L}{k} = \frac{q L}{k (T_{h} - T_{c})} = - \frac{L}{(T_{h} - T_{c})} \frac{\partial T}{\partial x}

(12)

Accordingly, the nondimensional form of local Nusselt number on the left side wall is defined by

N u = - {[\frac{\partial θ}{\partial X}]}_{X = 0}

(13)

The averaged Nusselt number which measures the overall heat transfer is obtained by integrating the local Nusselt number over the partially active thermal walls, which is given by the formula

\bar{N u} = \frac{2}{L} \int_{0 + h_{L}}^{\frac{L}{2} + h_{L}} N u d Y

(14)

where

h_{L} = 0

\frac{L}{4}, \frac{L}{2}

for bottom, middle, and top heating zones, respectively. Subsequent to the attainment of temperature field, the numerical integration of equation (14) is performed for each time step.

Numerical method

The governing equations (6) to (9) jointly with the boundary conditions (11) are discretized using finite volume method as explained by Patankar.³³ The discretization of convective terms is carried out by QUICK scheme and the diffusive terms are discretized through the central difference scheme as explained by Versteeg and Malalasekara.³⁴ An implicit scheme is employed for time marching with time step value chosen to be $10^{- 4}$ . The velocities and pressure terms are coupled with Semi-Implicit Method for Pressure Linked Equations algorithm.

The computational spatial domain is meshed with uniform grid on both directions. The grid size is calculated by $1 / (N - 1)$ , where N is the number of grid points in either X- or Y-directions, respectively. To establish a proper grid size and to study the problem, the grid independent examinations are carried out for different grid sizes from 41 × 41 to 121 × 121. While comparing the solutions of all the grid sizes, the results from the grid size of 101 × 101 showed significant accuracy in solutions with grid independency. Therefore, grid size of 101 × 101 is taken for the present study. The resulting set of simultaneous equations obtained from each scheme for every unknown variable is solved by Gauss–Seidel iterative method. The convergence criteria for all variables in each iteration process are taken as $| \frac{ϕ_{n + 1} (i, j) - ϕ_{n} (i, j)}{ϕ_{n + 1} (i, j)} | \leq 10^{- 6}$ , where the subscript $n$ denotes previous approximation value in the iteration and (n + 1) denotes the current approximation value. The trapezoidal rule is employed to calculate the averaged heat transfer rate along the heater.

The validation of the present in-house code is very important in numerical study. In order to find the accuracy and validity of the in-house computer code using Fortran 95 programming for the present problem, different numerical simulations obtained from the code are compared with natural convection in a cavity with partially active thermal walls reported by Sankar et al.³ and Valencia and Frederick.⁵ The results are shown in Table 1. It can be observed from the table that the results of present simulation agreed well with the earlier investigation. It provides the confidence on our numerical procedure to be used here.

Table 1.

Comparison of averaged Nusselt number for different thermal locations with T_m = 0, Ha = 0.

Ref.	$\bar{N u}$
	$R a = 10^{4}$			$R a = 10^{5}$			$R a = 10^{6}$
	TB	MM	BT	TB	MM	BT	TB	MM	BT
Valencia and Frederick⁵	2.142	3.399	2.997	3.295	6.383	5.713	4.505	12.028	11.659
Sankar et al.³	2.089	3.318	2.888	3.136	6.198	5.643	4.397	11.832	11.298
Present	2.107	3.327	2.989	3.196	6.298	5.679	4.458	11.984	11.573

BT: bottom–top; MM: middle–middle; TB: top–bottom.

Results and discussions

Numerical investigations are carried out to study the effect of the partially thermally active zones and external uniform magnetic field on the buoyancy-driven convective flow of cold water near its density maximum inside a square cavity. Based on the arrangement of the partially active thermal walls, this study has been carried out for five different locations of partial heating and cooling zones. The heating and cooling regions are, respectively, kept at various positions on the left and right walls and are referred to as follows: (i) TT, (ii) MM, (iii) BB, (iv) BT, and (v) TB. The analysis is performed for different density inversion parameter values and Hartmann numbers. Hartmann number generally varies from 0 to 400. But for a wide range of Hartmann numbers, the flow can be laminar or turbulent with laminar core in it. Since this study deals with laminar flow of cold water Hartmann number is varied between 0 and 50. The study is carried out for the density inversion parameter $(T_{m})$ varying between 0 and 1 while the Grashof number is fixed as $G r = 10^{6}$ for all the five different locations of partial heating/cooling which is taken for the study. An illustration for the flow pattern and temperature distribution in the cavity is demonstrated by plotting the contours of streamlines and isotherms.

Effect of partial heating and density inversion parameter on flow fields

Figure 2 exhibits the flow pattern inside the cavity for the five different locations of thermally active zones with density inversion parameters, $T_{m} = 0.5, H a = 25$ , and $G r = 10^{6}$ . In the beginning, when the thermal walls are placed in the BB location, formation of dual cell circulations is noticed inside the cavity. The pattern of circulation is symmetric with respect to the mid-section of the cavity. The occurrence of symmetric nature of cells happens near the density inversion plane when the value of density inversion parameter is 0.5. This symmetrical nature of the flow cell is due to the equal dominance of diffusion and advection within the fluid. When the locations of thermal zones are altered to BT position, temperature of the fluid near the hot wall raises resulting in the appearance of two contours within the enclosure: one, an elongated hot cell filling the major portion of the cavity and it suppresses the eddy that is formed near the cold wall and the other, density inversion taking place near the cold wall. In the MM case, a narrow vertically elongated eddy with single nucleus is formed near the hot wall location and a stretched cold cell, which rolls with elliptic pattern is observed inside the core region, is filling almost the entire cavity. This is due to the influence of bulk motion of fluid happening within the cavity near the hot wall. Likewise, when the heater and cooler zones are kept in the TB position, a very diminutive recirculating flow is again formed near the hot wall and major portion of the cavity is filled by the cold cell that rotates in anticlockwise sense. This rotating cell apparently indicates the change in density happening near the heater kept inside the cavity. Apparently, for both the MM and TB positions, the maximum change in density takes place near the hot wall reasonably representing the dominance of convection taking place within the fluid. Again, when the locations of the thermal zones are shifted to TT position, the flow pattern gets split into two equal parts with respect to the mid-section of the cavity, inherently creating two equally rotating cells, one happening along the hot wall in clockwise direction and the other about the cold wall in anticlockwise direction. From the analysis, it is generally noted that flow cells appear near the location of the thermally active portions and the bicellular formation is about the density inversion plane where the bulk movement of fluid takes place. Thus, from the results obtained, it is clearly noted that the bicellular formation which rotates as an elliptical contour takes place along the density inversion plane and this elongated nature of the flow pattern is due to the domination of convective heat transfer that takes place within the cavity. For the BB and TT cases, the density inversion plane is along the vertical mid-section of the cavity and so the circulations look symmetrical inside the cavity. On the other hand, for the MM location and the TB location, it is seen that the density inversion plane moves toward the hot wall from the cold wall. That is, the change in density happens near the hot wall thereby maintaining the convective mode of heat transfer as a dominant one. For the BT case, the inversion plane’s position is closer to the cold wall. So, we could not get the symmetrical flow circulations.

Figure 2.

Streamlines for different thermally active zones with Ha = 25, T_m=0.5, and Gr = 10⁶. (a) BB, (b) BT, (c) MM, (d) TB, and (e) TT.

Figure 3 illustrates the streamlines for the BB, MM, and TB thermal zones for various density inversion parameters under the existence of uniform magnetic field. For all the three positions of the partially active thermal walls, the density inversion parameter is varied from 0 to 1. When $T_{m} = 0$ , that is when there is no density inversion inside the cavity, for all three positions of partially active thermal walls, the fluid rises from segment of high temperature and flows down toward the portion of cold wall where it prevails as low temperature and so only a single flow cell occupies the entire cavity. Further, on increasing the density inversion parameter value, density inversion takes place within the cavity by forming two rotating cells for all the three considered locations of thermal zones. In particular, while analyzing the flow patterns of the taken three locations, the streamlines for density inversion parameter 0.4 clearly show a clockwise circulation flow filling the major portion of the cavity. Also, a small cold cell is formed near the cooler and this happens due to occurrence of density inversion taking place near the cold wall. It clearly emphasizes that the position of the maximum density plane falls near the cold wall. While increasing the density inversion parameter to 0.5 and for the location BB the maximum density plane moves away from the cold wall to the mid-portion of the cavity. This depicts that the density gradient takes place along the mid-section of the cavity due to which the formation of two flow cells occurs. It also indicates that the cells formed exhibit a symmetrical flow pattern about the maximum density plane which acts along the mid-section of the cavity. When the parameter value is increased to 0.6, the maximum density plane moves toward the hot wall and so, the secondary cell that rotates in anticlockwise sense occupies the major portion of the cavity by suppressing the primary cell near the hot wall. That is, the Boussinesq flow region is increased and it results in the flow pattern differently.

Figure 3.

Streamlines for different density inversion parameters with Ha = 25 and Gr = 10⁶. BB: bottom–bottom; MM: middle–middle; TB: top–bottom.

While analyzing the flow pattern for the MM location, symmetrical pattern is not exhibited here for $T_{m} = 0.5$ . In fact, the secondary cold cell grows in its size for this value. This is due to dominance of convective flow and so moving away of the density plane from cold wall to hot wall is noticed. Further, for $T_{m} = 1$ , the cold cell completely occupies the cavity and the formation of thermal boundary layers is observed near the thermally active hot wall. For the study of TB case, major portion of the cavity is occupied by the anticlockwise rotating secondary cold cell when the value of density inversion parameter is 0.5. Again, on increasing the parameter value to 0.6 and further to 1, it is evidently noted that the secondary cell grows in its size and it occupies the full cavity which reveals the high dominance of Boussinesq flow. Hence, it is remarked that the bicellular structure and the strength of the cell(s) depend on the value of density inversion parameter and the thermally active locations.

Effect of partial heating and density inversion parameter on temperature fields

Figure 4 displays the temperature distribution inside the cavity for various heating/cooling locations while $H a = 25, T_{m} = 0.5$ and $G r = 10^{6}$ . When the heating and cooling locations are placed in the BB location of the enclosure, a symmetrical pattern in the distribution of heat is observed with respect to the mid-section of the cavity. This implies that the change in density takes place along the mid-portion of the cavity thereby inducing temperature variations inside the cavity. When the locations are shifted to BT position, weak thermal boundary layers are formed along the heating and cooling regions. On the other hand, for the MM case, strong thermal boundary layers are observed near the heating zone and vertical thermal stratification is observed inside the cavity. Further, on placing the thermal zones to the TB position, again temperature distribution takes place in a stratified manner vertically thereby increasing the temperature gradients near the cold wall. In addition to the vertical stratum, strong thermal boundary layers are also formed along the heating zone. When thermal zones are placed in the TT position, thermal layers are observed only about the upper part of the cavity and poor heat transfer takes place about the bottom part of the cavity.

Figure 4.

Isotherms for different thermally active zones with Ha = 25, T_m=0.5, and Gr = 10⁶. (a) BB, (b) BT, (c) MM, (d) TB, and (e) TT.

Figure 5 displays the isotherms for BB, MM, and TB locations of partially thermally active walls and discussions are carried out for different density inversion parameters that vary from 0 to 1 with $H a = 25$ and $G r = 10^{6}$ . The Hartmann number is chosen as 25 which is the average of the range considered in the present study and the choice of Grashof number is $10^{6}$ which suggests the flow as laminar. Now, when $T_{m} = 0$ , it is observed that thermal boundary layers are formed near the heating and cooling zones for the BB and MM locations and vertical temperature stratification is found on heat transfer inside the cavity for all locations. When the density inversion parameter $T_{m}$ is increased to 0.5, vertical stratification of heat transfer takes place inside the cavity along with the formation of thermal boundary layer near the thermal zones for all considered locations. For the BB location, the stratification is observed as a symmetrical distribution about the mid-section of the cavity. This illustrates the domination of conduction mode of heat transfer inside the cavity. Further, an increase of density inversion parameter values as 0.6 and 1, the dominance of convection mode of heat transfer is observed inside the cavity with the formation of thermal boundary layers near the thermal zones. Thus, it is seen that convection mode of heat transfer dominates as increasing the density inversion parameter. It is also observed that the vertical temperature stratification is found on heat transfer inside the cavity for all locations when T_m=0 and 1.

Figure 5.

Isotherms for different density inversion parameters with Ha = 25 and Gr = 10⁶. BB: bottom–bottom; MM: middle–middle; TB: top–bottom.

Heat transfer rate

In heat transfer analysis, a significant quantity that measures the rate of heat dissipation is the determination of local Nusselt number and the average Nusselt number. The local Nusselt number for different density inversion parameters and thermally active zones with $H a = 25$ is depicted in Figure 6(a) to (e). The results from the figure depicts that analysis on heat transfer rate can be done significantly on changing the location of thermal zones and values of density inversion parameter. The local Nusselt number values seem to be high near the lower edge of the heater whereas poor heat transfer rate is observed near the upper edge of the heater while $T_{m} \leq 0.5$ . But, for $T_{m} > 0.5$ , high rate of local heat transfer is observed near the upper edge of the heater in contrast to the lower edge for all locations and Hartmann numbers. That is, the rate of heat transfer is high near the lower end of the heater and it reduces along the heater length and it reaches its minimum near the upper edge of the heater when $T_{m} \leq 0.5$ . The opposite trend on local heat transfer rate is observed when $T_{m} > 0.5$ for all locations. When heating zone is at the bottom or middle location and $T_{m} > 0.5$ , the high local heat transfer rate is observed near the trailing edge of the heater due to some unheated space near the heater which supports the buoyancy force inside the cavity. The similar trend is observed near the leading edge of the heater when heating zone is at the top or middle location and $T_{m} \leq 0.5$ . Comparing the results obtained from the figures, it is observed that, while increasing the density inversion parameter, the heat transfer decreases first and then increases.

Figure 6.

Local Nusselt number for different density inversion parameter and thermally active zones with Ha = 25. BB: bottom–bottom; BT: bottom–top; MM: middle–middle; TB: top–bottom; TT: top–top.

The effects of five thermally active zones on average heat transfer rate $(\bar{N u})$ for different Hartmann numbers and density inversion parameters are depicted in Figure 7(a) to (d). First, in the absence of magnetic field, that is when $H a = 0$ , for the locations BB, MM, TT, average heat transfer rate decreases on increasing $T_{m}$ from 0 to $0.5$ (Figure 7(a)) and when $T_{m} = 0.5$ , the heat transfer rate is poor because the maximum density plane is at the center of the cavity. For the values of $T_{m}$ from $0.6$ to $1$ , heat transfer rate increases significantly. Poor heat transfer is observed for the BT case when $T_{m} \geq 0.4$ and for TB case when $T_{m} < 0.4$ for all values of Hartmann number. The average heat transfer rate decreases on increasing the density inversion parameter for BT location. The average heat transfer rate decreases initially and then increases on increasing the density inversion parameter for TB location. The similar thermal locations (TT, MM, BB) show a similar trend on average Nusselt number with density inversion parameter. However, the dissimilar locations (TB, BT) show opposite trend on average Nusselt number with density inversion parameter. Thus, it is established that the increase in Hartmann number declines the heat transfer rate inside the cavity. Hence, from the observations, the heat transfer in the cavity is found higher for MM thermal zone when $T_{m} < 0.5$ or TB thermal zone when $T_{m} \geq 0.5.$ A weak rate of heat transfer is observed for the case of BT thermally active location when $T_{m} > 0.2$ .

Figure 7.

Average Nusselt number versus density inversion parameter for different thermally active zones and Hartmann numbers. BB: bottom–bottom; BT: bottom–top; MM: middle–middle; TB: top–bottom; TT: top–top.

The averaged heat transfer rate is decreased 3, 58, 0.9% for BB, BT, TT thermal zones when changing density inversion parameter from 0 to 1.0 with Ha = 0. However, the averaged heat transfer rate is increased 1.6 and 292.8% for MM and TB thermal zones when changing density inversion parameter from 0 to 1.0 with Ha = 0. The averaged heat transfer rate is decreased 3.3 and 63.1% for BB and BT thermal zones when changing density inversion parameter from 0 to 1.0 with Ha = 50. But, the averaged heat transfer rate is increased 1.5, 8.6, 306.6% for MM, TT, TB thermal zone when changing density inversion parameter from 0 to 1.0 with Ha = 50. The averaged heat transfer rate is decreased 56.6, 49.7, 49.9, 58.2% for MM, BB, BT, TT thermal zones when changing density inversion parameter from 0 to 0.5 with Ha = 0. But, the averaged heat transfer rate is increased 87.5% for TB thermal zone when changing density inversion parameter from 0 to 0.5 with Ha = 0. The averaged heat transfer rate is decreased 50.2, 46.6, 61.4, 56.8% for MM, BB, BT, TT thermal zones when changing density inversion parameter from 0 to 0.5 with Ha = 50. But, the averaged heat transfer rate is increased 80.5% for TB thermal zone when changing density inversion parameter from 0 to 0.5 with Ha = 50.

Conclusion

The magneto-convection of cold water near its maximum density in a cavity with partially thermally active zones located at various positions is studied numerically. The analysis is mainly focused on the significant changes in flow pattern and temperature distribution inside the cavity when the thermal zones are altered to different positions under the presence of uniform magnetic field. The study is carried out for five different locations and the following observations are made from the analysis:

When the density inversion parameter is increased, it is seen that the formation of bicellular structure disappears and the cold cell occupies the entire cavity and this is due to the moving away of density inversion plane from the cooler toward the heater.

The rate of heat transfer within the cavity is higher for MM thermal zone when $T_{m} < 0.5$ or TB thermal zone when $T_{m} \geq 0.5$ .

It is observed that the heat transfer rate behaves nonlinearly when increasing the density inversion parameter. The poor heat transfer rate is found when the maximum density plane is at the center of the cavity.

The convective heat transfer rate inside the cavity declines on increasing the Hartmann number significantly in the presence of maximum density.

Footnotes

Declaration of conflicting interests

The authors declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The authors received no financial support for the research, authorship, and/or publication of this article.

R Bindhu is currently working as Assistant Professor in the Department of Mathematics, PSG College of Technology, Coimbatore, Tamil Nadu, India. She has 15 years of teaching experience. She completed her Master degree in Mathematics from Bharathiar University, Coimbatore and M.Phil degree from PSG College of Arts and Science, Coimbatore, India.

G Sai Sundarakrishnan is a Professor in the Department of Applied Mathematics and Computational Sciences, PSG College of Technology, India. He has more than 23 years of teaching experience. His research focuses on digital topology and algebraic topology. He has published 30 papers in reputed International Journals.

S Sivasankaran received his MSc, MPhil, and PhD degrees from Bharathiar University, India. Then, he received Post-Doctoral Fellowship in National Cheng Kung University and National Taiwan University, Taiwan. He was a Research Professor in Yonsei University, South Korea. He was an Assistant Professor in Sungkyunkwan University, South Korea and Senior Lecturer in University of Malaya, Malaysia. Presently, he is working as an Associate Professor at King Abdulaziz University, Saudi Arabia. He is an Associate Editor in two journals and member of editorial board in several international journals. His areas of interest are convective heat and mass transfer, CFD, Nanofluids, Micro-channel heat sinks and Porous Media. He has published more than 120 research papers in reputed international journals.

M Bhuvaneswari received MSc, MPhil, and PhD degrees from Bharathiar University, Coimbatore, India. She received the Post-Doctoral Fellowship in Mechanical Engineering twice from National Cheng Kung University, Taiwan and twice from Sungkyunkwan University, South Korea. Then, she worked as a research fellow in University of Malaya, Malaysia. Her areas of interest are numerical and analytical methods for PDE, radiation, heat and mass transfer.

Appendix

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