Effect of Hydromagnetic Mixed Convection Double Lid-Driven Square Cavity with Inside Elliptic Heated Block

Mixed convection heat transfer in a two-dimensional the effect of hydromagnetic mixed convection in a double lid driven square cavity with inside elliptic heated block is studied numerically. The left wall of the square cavity and inside the elliptic block was kept at Th while the right wall of the square cavity at a cold temperature Tc with Th > Tc. The lid is assumed to be upper wall moving in left to right and lower wall right to left directions respectively. The magnetic field of strength B is applied parallel to xaxis. Result is presented firstly for different Reynolds number ( ), Prandtl number Pr = 0.733 and presented for different Grashof number . The numerical results studied the effect of Reynolds number, Grashof number and buoyancy ratio on the local values. It is found that direction of lid and different elliptic heated block are more effective on heat and mass transfer on fluid flow with increasing magnetic field for all studied parameters.


Introduction
Mixed convection in enclosures is encountered in many engineering systems such as cooling of electronic components, ventilation in building and fluid movement in solar energy collectors, astrophysics, geology, biology, and chemical processes, as well as in many engineering applications such as solar ponds, crystal manufacturing, and metal solidifications processes. Also mixed convection involving the combined effect of forced and natural convection has been the focus of research due to its occurrence in numerous technological, engineering, and natural applications such as: cooling of electronic devices, lubrication technologies, drying technologies, food processing, float glass production, etc. [1][2][3]. Al-Amiri et al. [4] investigated numerically steady mixed convection in a square lid-driven cavity under the combined buoyancy effects of thermal and mass diffusion. The results demonstrate the range where high heat and mass transfer rates can be attained for a given Richardson number. Sharif [5] studied numerically laminar mixed convective heat transfer in two-dimensional shallow rectangular driven cavities of aspect ratio 10. The top moving lid of the cavity is at a higher temperature than the bottom wall. The effects of inclination of the cavity on the flow and thermal fields are investigated. The stream line and isotherm plots and the variation of the local and average Nusselt numbers at the hot and cold walls are presented. Chen and Cheng [6] investigated numerically the periodic behavior of the mixed convective flow in a rectangular cavity with a vibrating lid. The periodic flow patterns and heat transfer characteristics found are discussed with attention being focused on the interaction between the frequency of the lid velocity vibration and the frequency of the natural periodic flow. Khanafer et al. [7] investigated numerically unsteady laminar mixed convection heat transfer in a lid driven cavity. Teamah et al. [8] analyzed the numerical simulation of double-diffusive mixed convective flow in rectangular enclosure with insulated moving lid. Saha et al. [9] have performed the numerical effect of internal heat generation or absorption on MHD (magnetohydrodynamics) mixed convection flow in a lid driven cavity. The significant reduction in the average Nusselt number were produced as the strength of the applied magnetic field was increased. In addition, heat generation predicated to decrease the average Nusselt number whereas heat absorption increases it. Dawood et al. [10] investigated hydro-magnetic mixed convection double diffusive in a Lid Driven Square Cavity. Hussein [11] investigated the mixed convection in Square Lid-driven with Eccentric Circular Body. Munshi et al. [12] analyzed a numerical study of Mixed Convection in Square Lid-driven with internal elliptic body and constant flux heat source on the bottom wall. Nasrin [13] carried out aspect ratio effect of vertical liddriven chamber having a centered conducting solid on mixed magneto convection. Billah et al. [14] investigated the numerical analysis of fluid flow due to mixed convection in a lid-driven cavity having an heated circular hollow cylinder. Md Rahman et al. [15] studied unsteady mixed convection in a porous media filled liddriven cavity heated by semi-circular heaters.
The main aim of this work is to examine the effect of hydromagnetic mixed convection double lid-driven square cavity with inside elliptic heated block. Results will be presented by streamlines, isotherms, Nusselt number, and velocity profiles.

Mathematical Formulation
The governing equations describing the problem under consideration are based on the laws of mass, linear momentum and energy with buoyancy forces. The energy equation is written using the Boussinesq approximation. This means that all thermophysical properties of the fluid at a reference temperature are taken constant except in the buoyancy term of the momentum equation. In addition, the radition heat exchange is negligible in this study. The non-dimensional governing equations can be written as follows. The equations are non-dimensionalized by using the following dimensionless quantities where ν = µ/ρ, is the reference kinematic viscosity and θ is the non-dimensional temperature. After substitution of dimensionless variable we get the non-dimensional governing equations are:

Nusselt Number calculation
Equating the heat transfer by convection to the heat transfer by conduction at hot wall: Introducing the dimensionless variables, defined in equation (1), into equation (6), gives: The average Nusselt number is obtained by integrating the above local Nusselt number over the vertical hot wall:

Numerical Technique
The coupled governing partial differential equations, i.e., mass, momentum and energy equations are transferred into a system of integral equations by using the Galerkin weighted residual finite-element method. The integration involved in each term of these equations is performed with the aid Gauss quadrature method. The nonlinear algebraic equations so obtained are modified by imposition of boundary conditions. These modified nonlinear equations are transferred into linear algebraic equations with the aid of Newton's method. Lastly, Triangular factorization method is applied for solving those linear equations. For numerical computation and post-processing, the software COMSOL Multiphyics is used.

Program validation and comparison with previous work
The present numerical code is valided against the problem by Al-Amiri et al. [4] was modified and used for the computations in the study. The working fluid is chosen Prandtl number Pr = 0.733. The left wall is kept heated at T h and right wall is kept at cold T c . The upper surface move to right and lower surface move to left. Streamlines and isotherms plotted in Fig. 2 are showing good agreement.

Results and Discussion
The finite element simulation is performed to examine the laminar mixed convection flow and heat transfer in a square cavity having a centered elliptic cylinder. The mixed convection phenomenon in the cavity is influenced by Richardson number Ri ranging from 10 3 to 10 6 , Reynolds number Re varing from 50 to 150 and Prandtl number Pr = 0.733 respectively. Numerical computations are carried out and a parametric study is performed to illustrate the influence of the physical parameters on the resulting streamlines and isotherms as well as the velocity components at the enclose midsection, Nusselt number, velocity and temperature along the heat sources. Streamlines for Re = 50 and Pr = 0.733 are presented in Fig. 3 to understand the effect of Richardson number, Ri on flow field and temperature distribution. Most of the heat transfer occurs due to conduction except near the sliding right wall. Elliptic shape eyes formed at bottom wall of the cavity. For higher Richardson number, Ri elliptic shaped eyes are formed in the right wall of the cavity and also flow strength increases which are shown Fig. 3.
Conduction dominant heat transfer is obtained from the isotherms in Fig. 4 at Ri = 10 3 to Ri = 10 6 . With increases in Richardson number, Ri, isotherms concentrates near the right wall so isotherm lines are more bending which means increasing heat transfer through convection.
Variation of Local Nusselt number, velocity, temperature and average Nusselt number with Pr = 0.733 and Re = 50 inside the various elliptic diameter are shown in Fig. 5. When Richardson number Ri is small the local Nusselt number flow line near the bottom wall and Richardson number Ri increasing flow line increasing near the right wall. Fig. 5 shows the variation of the temperature and velocity profiles along the mid-sections of the cavity and the temperature distribution in most of the cavity is shown to retain similar behavior to that of a pure conduction regime except near the sliding top wall. In addition, Fig. 5 illustrates that the velocity components U and V

Conclusion
Effect of hydro-magnetic mixed convection in a double lid driven square cavity with inside elliptic heated block has been performed. Results have been presented in terms of streamlines, isotherms, average Nusselt number at the heated block average temperature of the fluid and the temperature at the elliptic cylinder to analyze the effect of Grashof number, Gr and Reynolds number, Re on the fluid flow and heat transfer in the cavity for the aforementioned Prandtl number, Pr. In view of obtained results, the following findings have been summarized: a. Grashof number, Gr has a great significant effect on the streamlines and isotherms at the three convective regimes, Buoyancy-induced vorex in the streamlines increased and thermal layer near the heated surface become thin and concentrated with increasing Reynolds number, Re.