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Download Download International Journal of Mathematics and Physics 10, №2, 36 (2019) IRSTI 27.35.46 https://doi.org/10.26577/ijmph-2019-i2-6 A. Abdibekova , D. Zhakebayev , A. Zhumali Al-Farabi Kazakh National University, Almaty, Kazakhstan, e-mail: [email protected], Stuart number effect on 3-d mhd convection in a cubic area Abstract. In this paper mathematical modeling of magnetohydrodynamics natural convection in three dimensional area at different Stuart numbers has been considered. The magnetic field is considered vertically and results have been shown at different planes of 3-D enclosure. The modeling of natural convection is based on the solution of a filtered unsteady three - dimensional Navier- Stokes equation and the equation for temperature. The problem is solved numerically: the equations of motion and temperature – by a finite-difference method in combination with penta-diagonal matrix using the Adams-Bashfort scheme, the equation for pressure – by spectral Fourier method with combination of matrix factorization. Change the dynamic of natural convection is gained over the time depending on the different values of Stuart numbers. As result of modelling, isothermal surfaces, velocity and temerature contoures, also profiles for different Sturart numbers are obtained. Key words. Natural convection, magnetohydrodynamics, finite difference method, spectral method. Introduction the transfer rate decreases with the Hartmann number. Natural convection is a phenomenon that occurs In [10], the Boltzmann MRT double lattice in many engineering applications, resulting in method was applied to simulate three-dimensional airflow near surfaces of solid particles or liquids, MHD of natural convection flow in a cubic cavity. such as airflow in double-glazed windows, airflow Two different populations with models D3Q19 and in double-glazed doors of refrigerated display cases D3Q7 were used to determine the flow field and and airflow in gaps or cavities building walls. To temperature, respectively. The effect of the clearly understand, many researchers have devoted Hartmann and Grashof numbers on the projection of themselves to the study of this phenomenon in order the flow trace and the heat transfer rate on various to enhance or reduce this heat transfer mode. surfaces of the cavity, where the flow structure and Natural convection of the flow is one of the most isotherms in different planes of the casing change important problems in fluid mechanics and [1,2]. sharply due to an increase in the Hartmann and Magnetic field convection has been developed Grashof numbers, since the magnetic field is strong, and has been used in recent decades [3-7]. In [8], the rates are suppressed. two-dimensional mixed convection in a chamber Three-dimensional nanofluidic non-Darsian was solved using the finite volume method. They natural convection is presented in the presence of examined the sinusoidal boundary condition and the Lorentz forces [11]. The lattice Boltzmann method effect of the ratio of amplitudes, phase deviation, is selected for mesoscopic analysis. The simulation Richardson number and Hartmann number on the results are presented for various amounts of Darcy, heat transfer rate. Their results show that the Nusselt Rayleigh, and Hartmann numbers, and the volume number increases in amplitude ratio. In addition, the fraction of Al2O3. The results show that convection Nusselt number increases with the phase deviation dominates at large Darcy and Rayleigh numbers; to / 2 , and then decreases. In [9], the results therefore, distorted isotherms are observed at high for a laminar mixed convection flow in the presence Darcy and Rayleigh numbers. The motion of the of a magnetic field in the upper cavity controlled by nanofluid increases with increasing volume fraction, a cover with a set of Graskoff and Hartmann the Rayleigh and Darcy numbers, but decreases with numbers are presented. They used the finite volume increasing Hartmann number. The temperature method to model the equations and concluded that gradient on a hot surface decreases with increasing © 2019 al-Farabi Kazakh National University Int. j. math. phys. (Online) A. Abdibekova et al. 37 Hartmann number, while it increases with number decreases by 64% and 70% for the left and increasing Darcy number, Rayleigh number. The right walls, respectively. influence of the use of nanoparticles reaches a In this paper, we consider a mathematical model maximum degree for the maximum Hartmann value of the problem of natural convection under the and the minimum value of the Darcy and Rayleigh influence of a vertical magnetic field, where the numbers. effect of the Start number on convection of the In [12], convective flows and heat transfer in a MHD flow was obtained. magnetic field were studied. They also use the finite The applied magnetic field B H 0 j effect volume method and report that the heat transfer rate in the Navier-Stokes equations is the inclusion of is increasing. In [13], the LBM method was used to the Lorentz force to the momentum equations solve a two-dimensional MHD flow in an inclined F J B , where J (E V B) -is electric cavity with four heat sources. They thought that the l double model of multiple time relaxation models the current density, E - is electric field strength, which we set equal to zero, and is electric equations of momentum and energy, and explores the effect of the Hartmann number on fluid flow and conductivity, V u1i u2 j u3k velocity of heat transfer. They show that the average Nusselt fluid, and all of these in combination we obtain number decreases due to an increase in the Fl (V B) B – Lorentz force, where Hartmann number for all Rayleigh numbers. In [14], MHD natural convection in a Fl (u1i u2 j u3k ) (H 0 j) (H 0 j) is three-dimensional square cavity with a sinusoidal in detail, after using the properties of the temperature distribution on one side wall was multiplication of unit vectors, we obtain investigated using the new Boltzmann lattice F (u H k u H i )(H j), or method with a double relaxation time model using l 1 0 3 0 0 2 2 nano-liquid copper-water. The influence of various Fl (u1H 0 i u3 H 0 k ) , and parameters, such as the Rayleigh and Hartmann Fl F1 F2 F3 , where numbers, the volume fraction of nanoparticles, and 2 2 the phase deviation on heat transfer, was considered. F1 u1H 0 , F2 0, F3 u3H 0 . Concerning the present results, the following The problem is based on solving non-stationary conclusions are drawn: Convection heat transfer equations of magnetohydrodynamics with filtration decreases with increasing Hartmann number, and in combination with the continuity equation, the average Nusselt number decreases for both the equations for temperature, equations of motion of left and right walls, but the decrease for the right charged particles, taking into account the continuity wall is greater than for the left. When the Hartmann equation in a Cartesian coordinate system in number increases from 0 to 50, the average Nusentt dimensionless form u (u u ) p 1 u Ra i i j i F , i 2 t x j xi Re x j x j Re Pr u i 0, (1) xi (u ) 1 i , t x j Re Pr x j x j where u (i 1,2,3) are the velocity components, i Stuart number, where Ha H0L / – F1 Nu1, F2 0, F3 Nu3 -non-dimensional Hartmann number, H - magnetic field strength, 2 2 is the conductivity of the medium, which is LH Ha Lorentz force [10], N 0 is the determinedfrom plasma physics. 0 Re Int. j. math. phys. (Online) International Journal of Mathematics and Physics 10, №2, 36 (2019) 38 Stuart number effect on 3-d mhd convection in a cubic area U D(T T )L – characteristically velocity, domain. The boundary conditions imposed for 0 1 0 3 temperature is Dirichlet on the right and left p is the full pressure, t is the time, boundary, and Neumann on the other directions of the domain. The velocity components are equal to 0 (TT0 )/( TT 10 ) – non-dimensional in all directions. temperature in ionosphere, where T0 and T1 are the respective temperatures of the minimum and g(T T )L3 maximum of the area, Ra 1 0 3 vD -Rayleigh, where is volumetric thermal expansion coefficient, g – acceleration due to Ra gravity, Re is the Reynolds number, Pr v Pr – Prandtl number, D – diffusion D coefficient, L is the typical length, is the kinematic viscosity coefficient, is the density of the flow. A schematic picture of the computational Figure 1 – Illustration of the problem statement domain is shown in Figure 1, where the left wall - indicated by the blue color, corresponds to the low temperature of flow. The right wall layer - Numerical method highlighted in red, corresponds to high temperature of the flow. To solve the problem of homogeneous Initial conditions for temperature, velocity incompressible MHD turbulence, a scheme of components are set zero in all directions of the splitting by physical parameters is used: * n1 n (u ) (u) 1 1 3 1 I. 2 (u*)n 1 2 (u)n K n K n 1, t 2Re 2Re 2 2 (u*)n 1 II. p , (u)n 1 (u*)n 1 III. p, t n1 n 1 1 3 1 IV. 2 n1 2 n G n G n1 t 2Pe 2Pe 2 2 where Ra convective and diffusion terms of the intermediate n n n n n K (u )u F 2 , velocity field a finite-difference method in Re Pr combination with penta-diagonal matrix is used,which allowed to increase the order of n n n G (u ) where Pe Re Pr – Peclet accuracy in space. The numerical algorithm for the number.
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