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 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 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 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 , . 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 , Pr v Pr  – , 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:

* n1  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  n1  n 1 1  3 1  IV.  2 n1  2 n   G n  G n1  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. solution of incompressible MHD turbulence is During the first stage, the full magneto considered at [15]. hydrodynamic equation system is solved without At the second step, the pressure Poisson the pressure consideration. For approximation of the equation is solved, which ensures that the continuity

International Journal of Mathematics and Physics 10, №2, 36 (2019) Int. j. math. phys. (Online) A. Abdibekova et al. 39 equation is satisfied. The Poisson equation is nn1 3t n transformed from the physical space into the i,, jk i ,, jk []hr i,, jk  spectral space by using a Fourier transform. To 2 solve the three-dimensional Poisson equation, the tttnn1 1 []hr  [] ar   (3) spectral conversion in combination with matrix 2 ijk,, 2 ijk,, 2 Re Pr sweeping algorithm is developed [15]. The resulting nn11n 1 222   pressure field in the third stage is used to recalculate    the final velocity field [16]. 222   xxx1  23 At the fourth stage, the equation for temperature  i,, jk i,, jk i,, jk is solved by using Adams-Bashforth scheme. Consider the temperature distribution in the where n1 horizontal direction at the pointi, j,k : ()u  n  1  []hr i,, jk  x1 i,, jk   n1 n1  ()uu12 () ()u3 ()u  ()u    2 3 tx123  x  x xx (2) 23i,, jk i,, jk 1 222   n 1 222 []ar 1   Re Pr xxx i ,, jk 1 23 2 RePr nnn 222     When using the explicit Adams-Bashfort 222   xxx1  23 scheme for convective terms and the implicit  i,, jk i,, jk i,, jk Crank-Nicholson scheme for viscous terms, equation (2) takes the form: Discretization of convective expressions looks like this:

     (u )  27(u )  27(u )  (u )  u1  1 i2, j,k 1 i1, j,k 1 i, j,k 1 i1, j,k 4   i, j,k   O(x1 ),  x1  24x1

u 27 u  27 uu    u2 2i1, j 2, k 2ij, 1, k 22ijk,, ij, 1, k 4 i,, jk   Ox()2 xx2224 24  x 2

u 27 u  27 uu    u3 3i,, jk 2 3i,, jk 1 33i,, jk i,, jk 1 4 i,, jk   Ox(3 ). xx3324 24  x 3

Discretization of diffusion conditions looks like this:

2        16      30    16          i 2, j,k i 1, j,k i, j,k i 1, j,k i 2, j,k  2  i, j,k 2 ;  x1  12x1

 2    16    30   16       i, j2,k i, j1,k i, j,k i, j1,k i, j2,k 2 1 2 ;    i, j ,k   x2  2 12 x2

Int. j. math. phys. (Online) International Journal of Mathematics and Physics 10, №2, 36 (2019) 40 Stuart number effect on 3-d mhd convection in a cubic area

 2    16    30   16       i, j,k2 i, j,k1 i, j,k i, j,k1 i, j,k2 2 1 2 ;    i, j,k   x3  2 12 x3

where

  u1  9u1  9u1  u1      9  9     u    i1, j,k i, j,k i1, j,k i2, j,k   i 1, j,k i, j,k i 1, j,k i 2, j,k ; 1 i, j,k      16   16 

   9  9   i, j1,k i, j,k i1, j1,k   u  9u  9u  u   i, j2,k  2i2, j,k 2i1, j,k 2i, j,k 2i1, j,k u    ; 2 i, j,k  16   16     

    9   9       u3  9u3  9u3  u3  u    i, j,k 2 i, j,k 1 i, j,k i 1, j,k 1   i2, j,k i1, j,k i, j,k i1, j,k ; 3 i, j,k      16   16 

Then the left side of equation (3) is denoted by tq1 2 q    qi, j,k i,, jk 2 2 Re Pr x1 i,, jk 2 q   n1   n (4) tq1  i, j,k i 1, j,k i, j,k    2 RePr x2 2 i,, jk n1 2 From equation (4) we find i, j,k t13 qt    []hr n  2 i,, jk 2 RePr x3 i,, jk 2  n1  q   n i, j,k i, j,k i, j,k t nn1 n1 []hri,, jk  t [] ar i,, jk Replacing all i, j,k from the equations (13), 2 we obtain Equation (5) is converted to

 t 1  2 t 1  2 t 1  2            1 2 2 2 qi, j,k di, j,k (6)  2 Re Pr x1 2 Re Pr x2 2 Re Pr x3 

where 3t t d   [hr]n  [hr]n1  t[ar]n i, j,k 2 i, j,k 2 i, j,k i, j,k

Assuming that equation (6) has second-order accuracy in time, we can instead solve the following equation:

International Journal of Mathematics and Physics 10, №2, 36 (2019) Int. j. math. phys. (Online) A. Abdibekova et al. 41

 t 1 2  t 1 2  t 1 2           *  (7) 1 2 1 2 1 2 q i, j,k di, j,k  2 Re Pr x1  2 Re Pr x2  2 Re Pr x3 

We can show that equation (7) is an  t 1 2  4     approximation О t to equation (6). 1 2 qi, j,k Bi, j,k (10)  2 Re Pr x3  * Since the difference between qi, j,k and qi, j,k has a higher order, we return to the same notation At the first stage, the Ai, j,k search is carried out and just use qi, j,k . in the direction of the x1 coordinates: To determine q 1 the equation (7) is solved i , j,k 2 in 3 stages:  t 1 2      1 2  Ai, j,k di, j,k  2 Re Pr x1   t 1 2      1 2  Ai, j,k di, j,k (8)  2 Re Pr x1  n1 t 1  2 A 2       t 1   Ai, j,k  2  di, j,k 1   Bi, j,k  Ai, j,k (9) 2 RePr  x1  2 Re Pr x2 i, j,k  2 

t 1  A  16 A   30 A 16 A   A     i 2, j,k i 1, j,k i, j,k i 1, j,k i 2, j,k  Ai, j,k 2 di, j,k 2 RePr 12x1

s  A 16 s  A  (1 30 s )  A 16 s  A  s  A  d (11) 1 i2, j,k 1 i1, j,k 1 i, j,k 1 i1, j,k 1 i2, j,k i, j,k

t Once we have determined the value qi, j,k , we where s1  . 24 Re Pr x n1 1 find  This equation (11) is solved by the method of i, j,k n1 n i, j,k  qi, j,k i, j,k the penta-diagonal matrix, which determines Ai, j,k . n1 The same procedure is repeated further for The other components of temperatureijk are

directions x2 in the second stage, namely, Bi, j,k is solved in a similar way. determined by solving equation (9), and the solution from the first stage, as the coefficient on the right, Simulation results and the s1coefficient in the penta-diagonal matrix t are replaced by s  . Finally, in the The results of modeling the imposition of a 2 vertical magnetic field is obtained,where the lateral 24 Re Pr x2 distribution of the temperature field is pronounced. third stage, qi, j,k is solved using a similar The is chosen Gr  20000 , the penta-diagonal system shown in equation (9). Prandtl number Pr =0.09, the Stuart number has the

Int. j. math. phys. (Online) International Journal of Mathematics and Physics 10, №2, 36 (2019) 42 Stuart number effect on 3-d mhd convection in a cubic area

following values: 1) N = 0; 2) N = 0.09; 3) N = 2.16, shown at Figure 2. It is seen that isothermal surfaces kinematic viscosity   0.013 diffusion changeconsiderably and gradient of the boundary coefficient equal to D  0.14 For calculations, the layer declines with increasing of Stuartnumber, so mesh size is 34  34  34 . The size of the heat transfer rate, which depends on the temperature computational domain is equal to gradient, graduallydecreases with increasing L  2 , L  2 , L  2 , which corresponds to magnetic field, which indicates a weakening of the 1 2 3 overallheat transfer effect. These trends were also the directions x1, x2 and x3 . discovered [17–19], who also studiednatural In this paper effect of Stuart number on convection or Rayleigh Bernard convection under isothermal surfaces for differentStuart numbers is the influence of a magneticfield.

а) b) c)

Figure 2 – Isothermal surfaces for various Stuart coefficients а) N  0 ; b) N  0.09 ;c) N  2.16 at Gr  20000

The contours of vertical velocity effect of the magnetic field. This is recognized by components, and temperature contours on the the universal effect of magnetic fields, which is different planes of the enclosure are very theoretically interpreted in [20]. Moreover, another important for understanding the trend offlow, so important characteristic of the effect of the vertical present results for the different locations of the magnetic field is that when the magnetic fields are cavity have been shown infigures 4 and 5. It is stronger, the vortex structures will be more regular shown at figures 4-5 that, increasing Stuart and will be shown parallel to each other. number isothermallines become parallel to the Consequently, thermal convection caused by the walls and temperature gradient on the wall movement of the vortex cells will decrease due to declines, therefore heat transfer rate decreases. the amplification of magnetic fields. As for the physics of the influence of MHD on Figures 6-7 show a longitudinal dimensionless the structure of natural convection flows and heat temperature and velocity profile, respectively, for a transfer, this is due to the fact that in MHD flows the different number of Stuart 1) N  0 ; 2) N  0.09 motion of vortex structures perpendicular to 3) N  2.16 . It is observed that the solution has a magnetic fields, i.e. horizontally oriented vortex linear velocity dependence along the transverse cells, strongly suppressed due to the anisotropic direction.

International Journal of Mathematics and Physics 10, №2, 36 (2019) Int. j. math. phys. (Online) A. Abdibekova et al. 43

а) b) c)

Figure 3 – Contours of x3 vertical velocity components on x2  0.5 plane for different Stuart numbersа) N  0 ; b) N  0.09 ;c) N  2.16 at Gr  20000

а) b) c)

Figure 4 – Temperature contours on x1  0.5 plane for different Stuart numbersа) N  0 ; b) N  0.09 ;c) N  2.16 at Gr  20000

а) b) c)

Figure 5 – Temperature contours on x2  0.5 plane for different Stuart numbersа) N  0 ; b) N  0.09 ;c) N  2.16 at Gr  20000

Int. j. math. phys. (Online) International Journal of Mathematics and Physics 10, №2, 36 (2019) 44 Stuart number effect on 3-d mhd convection in a cubic area

Figure 6 – Temperature profile for different values of the Stuart number 1) N  0 ; 2) N  0.09 ; 3) N  2.16 at Gr  20000 .

Figure 7 – Velocity profile for different Stuart values 1) N  0 ; 2) N  0.09 ; 3) N  2.16 at Gr  20000 .

International Journal of Mathematics and Physics 10, №2, 36 (2019) Int. j. math. phys. (Online) A. Abdibekova et al. 45

Conclusion natural convection flow in a square cavity using Large-Eddy Simulation in Lattice MHD natural convection in a three dimensional Boltzmann.”Iran.J. Sci. Technol., Trans. Mech. area at different Stuart numbers with temperature Eng. 35(2011):133–142. distribution on side wall has been considered by 3. Kefayati, G.H.R., Gorji, M., Sajjadi, H., finite difference method with spectral method. Ganji, D.D.“Lattice Boltzmann simulation of MHD To solve the equations of flow motion and mixed convection in a lid-driven square cavity with temperature, the finite difference method is used in linearly heated wall.”Scientia Iranica, Trans. B – combination with a pentadiagonal matrix, and Mech. Eng. 19(2012):1053–1065. solved by using the Adams-Bashfort scheme. The 4. Hamid Reza Ashorynejad, Kurosh Sedighi, Poisson equation is solved by spectral method using Mousa Farhadi, Ehsan Fattahi."Simulating the fast Fourier transform. magnetohydrodynamic natural convection ow in a Thus, the following conclusions are drawn: horizontal cylindrical annulus using the lattice isothermal surfaces change considerably and Boltzmann method".Heat Transf. Asian Res. 41 gradient of the boundary layer declines with (2012): 468–483. increasing of Stuart number, so heat transfer rate, 5. Ashorynejad, H.R., Abdulmajeed, A., which depends on the temperature gradient, Mohsen Sheikholeslami, M.“Magnetic field effects gradually decreases with increasing magnetic field, on natural convection flow of a nanofluid in a which indicates a weakening of the overall heat horizontal cylindrical annulus using Lattice transfer effect. As for the physics of the influence of Boltzmann method.”Int. J. Therm. Sci. MHD on the structure of natural convection flows 64(2013):240–250. and heat transfer, this is due to the fact that in MHD 6. Xu, B.Q. Li, D.E. Stock.“An experimental flows the motion of vortex structures perpendicular study of thermally induced convection of molten to magnetic fields, i.e. horizontally oriented vortex gallium in magnetic fields.”Int. J. Heat Mass cells, strongly suppressed due to the anisotropic Transf. 49(2006): 2009–2019. effect of the magnetic field. The effect of the 7. Grosan, T., Revnic, C., Pop, I., Ingham, vertical magnetic field is that when the magnetic D.B. “Magnetic field and internal heat generation fields are stronger, the vortex structures will be effects on the free convection in a rectangular cavity more regular and will be shown parallel to each filled with a porous medium.”Int. J. Heat Mass other. Consequently, thermal convection caused by Transfer. 52(2009):5691–5700. the movement of the vortex cells will decrease due 8. Sivasankaran, S., Malleswaran, A., Lee, J., to the amplification of magnetic fields. Sundar, P.“Hydro-magnetic combined convection As result of modelling, isothermal surfaces, in a lid-driven cavity with sinusoidal boundary velocity and temerature contoures, also profiles for conditions on both sidewalls.” Int. J. Heat Mass different Sturart numbers are obtained. Transf. 54(2011):512–525. 9. Oztop, H.F., Al-Salem, K., Pop, I.“MHD Acknowledgement mixed convection in a lid-driven cavity with corner heater.” Int. J. Heat Mass Transf. Theauthors were supported by the Ministry of 54(2011):3494–3504. Education and Science of the Republic of 10. Saijadi, H., Amri Delouei, A., Sheikho- Kazakhstan Grant No. AP05133516. leslami, M., Atashafrooz., M., Succi, S. “Simulation of the dimensional MHD natural References convection using double MRT Lattice Boltzmann method.” J.Physica A, 515(2019): 474-496. 1. Abouei Mehrizi, Farhadi, M., Hassanzadeh 11. Sheikholeslami, M.“Lattice Boltzmann Afrouzi, H., Shayamehr, S.,Lotfizadeh, H.“Lattice method simulation for MHD non-Darcy nanofluid Boltzmann simulation of natural convection flow free convection.” J. Physica B, 516(2017): 55–71. around a horizontal cylinder located beneath an 12. Bhuvaneswari, M., Sivasankaran, S., Kim, insulation plate.”J. Theoret. Appl. Mech. 51(2013): Y.J.“Magneto convection in a square enclosure with 729–739. sinusoidal temperature distributions on both side 2. Sajjadi, H., Hosseinizadeh, S.F., Gorji, M., walls.” Numer. Heat Transf. Part A, Kefayati, Gh.R.“Numerical analysis of turbulent 59(2011):167–184.

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International Journal of Mathematics and Physics 10, №2, 36 (2019) Int. j. math. phys. (Online)