^{1}

^{2}

^{3}

A numerical research on magnetohydrodynamic mixed convection flow in a lid-driven trapezoidal enclosure at non-uniform heating of bottom wall has been studied numerically. The enclosure consists of insulated top wall and cold side walls, too. It also contains a heated triangular block (
*Rot* = 0
° - 90
°) located somewhere inside the enclosure. The boundary top wall of the enclosure is moving through uniform speed
*U*
_{0}. The geometry of the model has been represented mathematically by coupled governing equations in accordance with proper boundary conditions and then a two-dimensional Galerkin finite element based numerical approach has been adopted to solve this paper. The numerical computations have been carried out for the wide range of parameters Prandtl number (0.5 ≤
*Pr* ≤ 2), Reynolds number (60 ≤
*Re* ≤ 120), Rayleigh number (
*Ra* = 10
^{3}) and Hartmann number (
*Ha* = 20) taking with different rotations of heated triangular block. The results have been shown in the form of streamlines, temperature patterns or isotherms, average Nusselt number and average bulk temperature of the fluid in the enclosure at non-uniform heating of bottom wall. It is also indicated that both the streamlines, isotherm patterns strongly depend on the aforesaid governing parameters and location of the triangular block but the thermal conductivity of the triangular block has a noteworthy role on the isotherm pattern lines. Moreover, the variation of
*Nu*
_{av} of hot bottom wall and
*θ*
_{av} in the enclosure is demonstrated here to show the characteristics of heat transfer in the enclosure.

Mixed convection heat transfer within a closed enclosure akin to geometry may occur on account of the combined effect of buoyancy along with shear and nowadays has received substantial attention under the lid-driven enclosure problems. The investigation of flow and heat transfer in lid-driven cavities is the vast studied problems in thermo-fluids area, where configuration of the lid-driven enclosure is encountered in many realistic engineering and industrial applications, such as, contains cooling of electronic devices, materials processing, flow and heat transfer in solar ponds, thermal-hydraulics of nuclear reactors and metal coating, etc. But the intriguing circumstances may occur when an electrically conducting fluid contained by a shear along with buoyancy driven enclosure has been externally applied the magnetic field whereas the magnetic field presents the stability to the flow and to stifles, the heat transfer. The analysis of the above phenomena including a heated obstacle may extend its uses to different other practical situations, while several authors studied recently heat transfer within enclosures, discussed in the following literature.

Transport mechanism of mixed convection within a shear as well as buoyancy-driven cavity had been investigated by Aydin [_{2}O_{3}-water nanofluid in a lid driven trapezoidal cavity having the effect of an disposed magnetic field numerically. It was determined that the idyllic magnitude of magnetic penchant angle is depending on the model of electrical conductivity. Mehmood [

According to the above literature review, much studies have been found on mixed convection within a lid-driven square cavity but no attention has been paid to the problem of magneto-hydrodynamic mixed convection flow within a trapezoidal cavity embedded inside heated triangular block of different rotations (Rot = 0˚ - 90˚) with practical applications which are also necessary to investigate. The trapezoidal cavity is considered here for an inclination angle ф = 30˚. For this, the present work has been arranged by entire thoughtful about the problem geometry, solution procedure and detailed inspection to provide the results via the fluid flow and heat transfer. In the present investigation, transport phenomena for different rotations of heated triangular obstacle will be explained by using the various non-dimensional parameters. These types of parameters are Prandtl number, Reynolds number, Rayleigh number and Hartmann number. Here Reynolds number is varied from 60 to 120 to simulate forced convection, mixed convection and natural convection dominated flow within the enclosure. The Rayleigh number and Hartmann number are taken as 10^{3} and 20 respectively. The Prandtl number has been considered from 0.5 to 2 for fluids.

The physical situation of a two-dimensional lid-driven trapezoidal cavity of length L and height H has been depicted schematically in _{0}) with the left wall inclined at an angle ф = 30˚ along with Y axis entrenched inside for various rotations (0˚ ≤ rot ≤ 90˚) of heated (T_{h}) triangular block. The velocity components correspondingly are u and v along x-direction and y-direction. The top wall of the trapezoidal enclosure is moving with a constant velocity U_{0} along the positive direction of x-axis. No-slip boundaries are simultaneously considered for three left, right and bottom walls. The top lid is kept at insulated (T_{i}) while bottom wall is non-uniformly heated (T_{h}) and side walls are cold (T_{c}) and T_{h} > T_{c}_{.}_{ }

The fluid is considered for different Prandtl number (0.5 ≤ Pr ≤ 2), Reynolds number (60 ≤ Re ≤ 120), keeping fixed Hartmann number (Ha = 20) and Rayleigh number (Ra = 10^{3}). The fluid flow is supposed to be laminar, incompressible and Newtonian.

To solve the problems of fluid flow and heat transfer, the basic laws are utilized ignoring the viscous dissipation term, joule heating term in the energy equation. In the momentum equation, Boussinesq approximation is employed for the variations of density as a function of temperature. Also, the buoyancy force is included as a body force in the y-momentum equation for laminar incompressible thermal flow. The non-dimensional principal equations for two dimensional steady-state MHD mixed convection thermal flow in a lid-driven trapezoidal cavity are stated as follows:

∂ U ∂ X + ∂ V ∂ Y = 0 (1)

U ∂ U ∂ X + V ∂ U ∂ Y = − ∂ P ∂ X + 1 R e ( ∂ 2 U ∂ X 2 + ∂ 2 U ∂ Y 2 ) (2)

U ∂ V ∂ X + V ∂ V ∂ Y = − ∂ P ∂ Y + 1 R e ( ∂ 2 V ∂ X 2 + ∂ 2 V ∂ Y 2 ) + R a P r θ − H a 2 R e V (3)

U ∂ θ ∂ X + V ∂ θ ∂ Y = 1 R e P r ( ∂ 2 θ ∂ X 2 + ∂ 2 θ ∂ Y 2 ) (4)

Non-dimensional variables are used for making the dimensionless governing Equations (1)-(4) stated as follows:

X = x L , Y = y L , U = u U 0 , V = v U 0 , P = p ρ U 0 2 , θ = T − T c T h − T c

The dimensionless governing parameters in the above Equations (1) to (4) are the Reynolds number Re, Grashof number Gr, Prandtl number Pr and Rayleigh number Ra. and defined as follows:

R e = U 0 L ν , P r = ν α , G r = β g Δ T L 3 ν 2 , H a 2 = σ B 0 2 L 2 μ , R a = G r × P r

where Δ T = T h − T c and α = κ ρ C p are the temperature difference and thermal diffusivity of the fluid respectively.

The dimensionless boundary conditions under consideration can be written as follows:

On the bottom wall: U = 1 , V = 0 , θ = sin ( π X ) , ∀ X = [ 0 , 1 ] , Y = 0

On the left wall: U = 0 , V = 0 , θ = 0 , ∀ Y = [ 0 , 1 ]

On the right wall: U = 0 , V = 0 , θ = 0 , ∀ Y = [ 0 , 1 ]

On the top wall: U = 0 , V = 0 , ∂ θ ∂ N = 0 , ∀ Y = 1

On the different orientations of triangular block (obstacle) surface:

U = 0 , V = 0 , θ = 1

where N is the non-dimensional distances either along X or Y direction acting normal to the surface. According to Singh and Sharif [

N u a v = ∫ 0 1 ( ∂ θ ∂ Y ) y = 0 d X

where N u local = − ∂ θ ∂ Y and the average bulk temperature is defined as

θ a v = ∫ θ d V ¯ / V ¯ ,

where V ¯ is the cavity volume.

A grid size sensitivity test has been conducted for a triangular obstructed lid-driven trapezoidal enclosure (ф = 30˚) being there with magnetic field at relevant values of Pr = 0.7, Re = 100, rot = 0˚, Ha = 20 and Ra = 10^{3} to attain the appropriate grid size. The following seven different meshes has been taken for the grid size sensitivity test through the following nodes and elements: 5085 nodes, 750 elements; 8415 nodes, 1244 elements; 13,249 nodes, 1978 elements; 14,601 nodes, 2186 elements; 14,757 nodes, 2210 elements; 18,316 nodes, 2757 elements; 21,277 nodes, 3212 elements. The mesh size of 14,757 nodes 2210 elements are selected to find the satisfactory solution for the base case problem as presented in

In order to check the accuracy, validation is the necessary part for the numerical investigation. Since the code validation against experimental data is not probable for the present study, for this, the numerical code is compared within a trapezoidal enclosure (ф = 30˚) for Pr = 0.7 and. Ra = 10^{5} as reported by Basak et al.

Nodes (Elements) | 5085 (750) | 8415 (1244) | 13,249 (1978) | 14,601 (2186) | 14,757 (2210) | 18,316 (2757) | 21,277 (3212) |
---|---|---|---|---|---|---|---|

Nu | 6.841483 | 8.025347 | 8.043388 | 8.042868 | 8.042316 | 8.045789 | 8.04712 |

Time (s) | 4.717 | 7.18 | 11.206 | 12.227 | 12.838 | 15.833 | 19.328 |

[

The solution of the dimensionless governing equations associated with its proper boundary conditions is explained by the Galerkin finite element formulation. The continuum domain is alienated into small units of non-overlapping areas, defined as elements. Six node triangular elements with quadratic interpolation functions are used to improve the finite element equations. All nodes are associated with velocity and temperature and then linear interpolation functions for pressure are employed to discretize the domain. Furthermore, interpolation functions, such as, local normalized elements have been used to estimate the contingent variables in each element. By substituting the acquired approximations in the governing equations along with boundary conditions, residuals for each of the conservation equations are yielded and then reduced to zero in a weighted sense over each element volume using the Galerkin method. The description of this method is already discussed in detail by Dechaumphai [

Ra | Numerical Data Corvaro and Paroncini [ | Experimental Data Corvaro and Paroncini [ | Present Numerical Data (current code) | Error (%) (Experimental and Present data) |
---|---|---|---|---|

7.56 × 10^{4} | 5.12 | 4.80 | 5.29 | −10.21 |

1.38 ×10^{5} | 5.86 | 5.859 | 6.1 | −4.09 |

1.71 × 10^{5} | 6.2 | 6.30 | 6.38 | −1.26 |

1.98 × 10^{5} | 6.28 | 6.45 | 6.55 | −1.55 |

2.32 × 10^{5} | 6.4 | 6.65 | 6.79 | −2.11 |

2.5 × 10^{5} | 6.5 | 6.81 | 6.91 | −1.47 |

A numerical study on the flow of mixed convection in presence of magnetic field (MHD) in a lid-driven trapezoidal enclosure (ф = 30˚) enclosing different rotations of heated triangular block or obstacle (Rot = 0˚ - 90˚) has been presented here. The analysis has been conducted for the various ranges of non-dimensional parameters such as Rayleigh number (Ra = 10^{3}), Hartmann number (Ha = 20), Prandtl number (0.5 ≤ Pr ≤ 2), and Reynolds number (60 ≤ Re ≤ 120) as well as various rotations of heated triangular block within trapezoidal cavity. The numerical results have been displayed in terms of both streamlines and isotherms and the rate of heat transfer characteristics such as average Nusselt numbers and average bulk temperature as well as different rotations of heated block or obstacle.

The numerical results have been presented the impact of Reynolds number on the structure of fluid flow and isotherm distributions of MHD mixed convection flow in a lid-driven trapezoidal enclosure in Figures 3-7 for the assorted values of Reynolds number (Re = 60 - 120) varying different rotations or orientations of heated triangular obstacle (Rot = 0˚ - 90˚) keeping fixed Rayleigh

number (Ra = 10^{3}), Hartmann number (Ha = 20) and Prandtl number (Pr = 0.7).

At relatively little Reynolds number (Re = 60), it has been seen from ^{3} and Pr = 0.7. Within these cells, a symmetrical clockwise (CW) and anti-clockwise (ACW) cells developed near the left and right walls inside the enclosure correspondingly when Rot = 0˚. But when the rotations of the heated triangular block changes its position inside the enclosure, an anti-clockwise cell formed in the right side of the wall taking maximum part and clockwise cells developed near the left wall and also left-top wall of the enclosure for Rot = 60˚. When Rot = 90˚ then it is viewed that a clockwise cells occupy the more part of the enclosure where two recirculation cells are present and another anti-clockwise cell developed near the right wall of the enclosure. By this, a well mixture fluid within the enclosure has been implied. Besides, the secondary vortices are found inside the primary cells of the enclosures. It is also seen from

The corresponding isotherm distributions have been shown in the ^{3}. From these figures, it has been found compacted isotherm lines at the heated wall of the enclosure in accordance with around the heated wall of the triangular block, testifies the noticeable strong influence of convective heat exchange. Isotherm lines are also parallel near the side walls. It is also viewed that isotherm distributions around the heated triangular block and too from the non-uniform heating of the base wall which explains pure conduction heat transfer. Besides isotherm distributions are getting shifted towards the side (left and right) walls of the enclosures symmetrically on account of slight stronger convection.

The numerical results of average Nusselt number (Nu_{av}) at the non-uniform heating of bottom wall and average bulk temperature (θ_{av}) of the enclosure varying assorted rotations or orientations of heated triangular block (Rot = 0˚ - 90˚) have been displayed in

number. It is noticed from _{av}) for different orientations of heated triangular block (Rot = 0˚ - 90˚) increases by increasing the values of Re which shows the convective heat transfer Phenomenon. It is also noticed that Nu_{av} is higher for Rot = 90˚ and almost lesser for Rot = 60˚ means that heat transfer rate is changing by the effect of change of locations of heated triangular obstacle. On the other hand, it is observed from

The numerical results on the streamlines as well as isotherm distributions for the considered different orientations of heated triangular block (Rot = 0˚ - 90˚) are illustrated in Figures 9-12 for the effect of Prandtl number (Pr = 0.5 - 2) while Ha = 20, Ra = 10^{3} and Re = 60.

From ^{3} and Re = 60. This means that the flow is influenced by the buoyancy force. One oval vortex is seen inside the left eddy circulations while one tiny vortex inside the right eddy circulations. It is also found dense circulations near the left wall of the enclosure. When the rotations of heated triangular block changes its location (Rot = 60˚, 90˚), it is noticed that number of eddy circulation cells reduce and then found two recirculation cells. It has been noticed that right eddy circulations occupy the maximum part of the cavity and the left eddy circulation cell occupies the rest part of the cavity. Moreover dense circulations also reduce by the effect of heated triangular

obstacle. Owing to increase of Prandtl number (Pr = 0.7 - 2), it has been perceived from Figures 10-12 that similar types of recirculation cells like

The isotherm distributions with the rotations of heated triangular obstacle (Rot = 0˚ - 90˚) are illustrated in Figures 9-12 for different Pr (Pr = 0.5 - 2) by the way of temperature field in the separate flow area where while Ha = 20, Ra = 10^{3} and Re = 60. At low Pr = 0.5, it is seen that isotherm lines are smooth curves at the heated bottom wall as well as around the heated triangular obstacle (Rot = 0˚ - 90˚) and isotherm lines is pushed into near the side walls by the effect of a slighter convection of flow. It is observed that isotherm lines are compacted near the side walls. With the aid of increasing Pr, it is also observed linear isotherm lines in the enclosure and then impenetrable to the respective side walls. It can easily be noticed that isotherms for various Pr are seen to be clustered around the heated triangular block of the enclosure, near the heated base wall as well as cooled side walls of the enclosure, implying the shear temperature gradient beside the upright direction within this field. Furthermore the thinner thermal boundary layer is seen near the heated wall of the enclosure with the increasing Pr.

_{av}) at the heated surface and average bulk temperature (θ_{av}) of the cavity having different orientations of heated triangular block (Rot = 0˚ - 90˚) for various Pr where Ha = 20, Ra = 10^{3} and Re = 60. It has been observed from

for Rot = 0˚, 90˚ but minimum at Rot = 60˚ for large Pr. From

In this paper, Magneto-hydrodynamic mixed convection flow in a lid-driven trapezoidal enclosure (ф = 30˚) for the effect of the rotations of heated triangular block or obstacle (Rot = 0˚ - 90˚) has been performed with the aid of governing non-dimensional parameters such as Rayleigh number, Hartmann number, Prandtl number and Reynolds number within trapezoidal cavity. The précis conclusions are as follows:

• Due to the rotations of heated triangular obstacle, a little effect of Reynolds number and Prandtl number being there with magnetic field is observed on the streamlines and isotherm structures within the enclosure. The eddy circulation cell’s size decreases by the effect of heated triangular obstacle and clustered isotherm lines are observed around the heated triangular obstacle as well as the heated wall of the enclosure where Ha is fixed.

• Average Nusselt number varying rotations of triangular block increases owing to increase of Re and fixed Ha and also is higher and lesser for Rot = 90˚ and Rot = 60˚ respectively.

• Average bulk temperature enhances with the raise of Re and is found higher for Rot = 0˚. Moreover, average bulk temperature behaves conduction-like distribution for Re = 80 and then starts convection flow for Re > 80.

• The rate of heat transfer, that is, Nu_{av} rises harshly by increasing Pr and maximum at Rot = 0˚, 90˚ but minimum at Rot = 60˚ for large Pr.

• Average bulk temperature goes up steadily for increasing Pr and is the highest for Rot = 0˚ with increasing all Pr and minimum at low Pr = 0.5.

• Furthermore, heat conducting block, block with joule heating, nanofluids and also hybrid nanofluids can be used in a trapezoidal cavity as a future research work.

The authors like to state their thanks to the Department of Arts and Sciences, Ahsanullah Uiniversity of Science and Technology, Dhaka-1208, for giving the computer facility during this work.

The authors declare no conflicts of interest regarding the publication of this paper.

Hossain, M.S., Alim, Md.A. and Andallah, L.S. (2020) Finite Element Analysis of Magnetohydrodynamic Mixed Convection in a Lid-Driven Trapezoidal Enclosure Having Heated Triangular Block. American Journal of Computational Mathematics, 10, 441-459. https://doi.org/10.4236/ajcm.2020.103025

B_{0}: Magnetic induction

C_{p}: Specific heat at constant pressure (J/kg∙K)

g: Gravitational acceleration (m/s^{2})

h: Convective heat transfer coefficient (W/m^{2}∙K)

k: Thermal conductivity of fluid(W/m K)

L: Langth or base of trapezoidal cavity (m)

H: Height of trapezoidal cavity (m)

N: Total number of nodes

Nu_{av}: Average Nusselt number

θ_{av}: Average bulk temperature

P: Non-dimensional pressure

p: Pressure, Pa

Pr: Prandtl number

Ra: Rayleigh number

Ha: Hartmann number

Re: Reynolds number

T: Non-dimensional temperature

T_{i}: Temperature of insulated top wall (K)

T_{h}: Temperature of hot bottom wall (K)

T_{c}: Temperature of cold side wall (K)

U: x component of dimensionless velocity

u: x component of velocity (m/s)

V: y component of dimensionless velocity

v: y component of velocity (m/s)

U_{0}: Lid velocity

x, y: Distance along Cartesian coordinates

X, Y: Dimensionless distance along Cartesian Coordinates

α: Thermal diffusivity (m^{2}/s)

β: Coefficient of thermal expansion (K^{−1})

ρ: Density of the fluid (kg/m^{3})

∆θ: Temperature difference

Θ: Dimensionless fluid temperature

Μ: Dynamic viscosity of the fluid (Pa∙s)

Ψ: Stream function

ф Inclination angles

ν: Kinematic viscosity of the fluid (m^{2}/s)

σ: Fluid electrical conductivity (Ω^{−1}∙m^{−1})

av: average

Rot: Different rotations of heated triangular block