## Abstract

A numerical study of magneto-hydrodynamic mixed convection in a cavity has been conducted to investigate the influence of magnetic field on integrated flux of thermal energy, linear momentum, and kinetic energy. Shear force through lid motion establishes the forced convection effect and buoyancy force due to differential heating of the moving lid and the stationary interface ensures the natural convection phenomenon. Additionally, conduction through the solid slab with prescribed temperature at the outer surface attached to the cavity induces conjugate heat transfer. Further, the top and bottom walls throughout the domain are kept insulated and a uniform horizontal magnetic field is applied on the interface toward left. Fluid flow and heat transfer characteristics are examined for a range of Hartmann number (Ha): 0, 10, 50, and 120 at fixed values of Reynolds number, Grashof number, and Prandtl number of 300, 9 × 104 and 0.71, respectively. The result shows that the transport of heat in the near wall regions of the fluid domain is primarily governed by diffusion, whereas advection appears stronger in the central region of the cavity. Increase in magnetic field strength from Ha = 0 to 120 gradually suppresses the recirculating structure of the flow signifying a reduction in advective strength as depicted by the decrease in the value of total integrated heat flux from $25.15×10-3$ to $6.0×10-3$. The reduction in heat flux, momentum fluxes, and kinetic energy fluxes with increase in magnetic field has been well correlated in the range of $0≤Ha≤120$.

## References

References
1.
Agarwal
,
R.
,
1981
, “
A Third-Order-Accurate Upwind Scheme for Navier–Stokes Solutions at High Reynolds Numbers
,”
19th Aerospace Sciences Meeting
, St. Louis, MO, Jan. 12–15.10.2514/6.1981-112
2.
Ghia
,
U.
,
Ghia
,
K. N.
, and
Shin
,
C.
,
1982
, “
High-Re Solutions for Incompressible Flow Using the Navier–Stokes Equations and a Multigrid Method
,”
J. Comput. Phys.
,
48
(
3
), pp.
387
411
.10.1016/0021-9991(82)90058-4
3.
Thompson
,
M.
, and
Ferziger
,
J. H.
,
1989
, “
An Adaptive Multigrid Technique for the Incompressible Navier–Stokes Equations
,”
J. Comput. Phys.
,
82
(
1
), pp.
94
121
.10.1016/0021-9991(89)90037-5
4.
Koseff
,
J.
, and
Street
,
R.
,
1984
, “
The Lid-Driven Cavity Flow: A Synthesis of Qualitative and Quantitative Observations
,”
ASME J. Fluids Eng.
,
106
(
4
), pp.
390
398
.10.1115/1.3243136
5.
Pilkington
,
L. A. B.
,
1969
, “
Review Lecture: The Float Glass Process
,”
Proc. R. Soc. London Ser. A
,
314
(
1516
), pp.
1
25
.10.1098/rspa.1969.0212
6.
Imberger
,
J.
, and
Hamblin
,
P. F.
,
1982
, “
Dynamics of Lakes, Reservoirs, and Cooling Ponds
,”
Annu. Rev. Fluid Mech.
,
14
(
1
), pp.
153
187
.10.1146/annurev.fl.14.010182.001101
7.
Moallemi
,
M.
, and
Jang
,
K.
,
1992
, “
Prandtl Number Effects on Laminar Mixed Convection Heat Transfer in a Lid-Driven Cavity
,”
Int. J. Heat Mass Transfer
,
35
(
8
), pp.
1881
1892
.10.1016/0017-9310(92)90191-T
8.
,
A. K.
, and
Koseff
,
J. R.
,
1996
, “
Combined Forced and Natural Convection Heat Transfer in a Deep Lid-Driven Cavity Flow
,”
Int. J. Heat Fluid Flow
,
17
(
5
), pp.
460
467
.10.1016/0142-727X(96)00054-9
9.
Khanafer
,
K. M.
, and
Chamkha
,
A. J.
,
1999
, “
Mixed Convection Flow in a Lid-Driven Enclosure Filled With a Fluid-Saturated Porous Medium
,”
Int. J. Heat Mass Transfer
,
42
(
13
), pp.
2465
2481
.10.1016/S0017-9310(98)00227-0
10.
Aydm
,
O.
,
1999
, “
Aiding and Opposing Mechanisms of Mixed Convection in a Shear- and Buoyancy-Driven Cavity
,”
Int. Commun. Heat Mass Transfer
,
26
(
7
), pp.
1019
1028
.10.1016/S0735-1933(99)00091-3
11.
Manca
,
O.
,
Nardini
,
S.
, and
Vafai
,
K.
,
2006
, “
Experimental Investigation of Mixed Convection in a Channel With an Open Cavity
,”
Exp. Heat Transfer
,
19
(
1
), pp.
53
68
.10.1080/08916150500318380
12.
Rahman
,
M. M.
,
Parvin
,
S.
,
Rahim
,
N.
,
Islam
,
M.
,
Saidur
,
R.
, and
Hasanuzzaman
,
M.
,
2012
, “
Effects of Reynolds and Prandtl Number on Mixed Convection in a Ventilated Cavity With a Heat-Generating Solid Circular Block
,”
Appl. Math. Modell.
,
36
(
5
), pp.
2056
2066
.10.1016/j.apm.2011.08.014
13.
,
A.
,
Pandit
,
S. K.
,
Sarma
,
S. S.
, and
Pop
,
I.
,
2016
, “
Mixed Convection in a Double Lid-Driven Sinusoidally Heated Porous Cavity
,”
Int. J. Heat Mass Transfer
,
93
, pp.
361
378
.10.1016/j.ijheatmasstransfer.2015.10.010
14.
Oreper
,
G.
, and
Szekely
,
J.
,
1983
, “
The Effect of an Externally Imposed Magnetic Field on Buoyancy Driven Flow in a Rectangular Cavity
,”
J. Cryst. Growth
,
64
(
3
), pp.
505
515
.10.1016/0022-0248(83)90335-4
15.
Alchaar
,
S.
,
Vasseur
,
P.
, and
Bilgen
,
E.
,
1995
, “
Natural Convection Heat Transfer in a Rectangular Enclosure With a Transverse Magnetic Field
,”
ASME J. Heat Transfer
,
117
(
3
), pp.
668
673
.10.1115/1.2822628
16.
Rudraiah
,
N.
,
Barron
,
R.
,
Venkatachalappa
,
M.
, and
Subbaraya
,
C.
,
1995
, “
Effect of a Magnetic Field on Free Convection in a Rectangular Enclosure
,”
Int. J. Eng. Sci.
,
33
(
8
), pp.
1075
1084
.10.1016/0020-7225(94)00120-9
17.
Chamkha
,
A. J.
,
2002
, “
Hydromagnetic Combined Convection Flow in a Vertical Lid-Driven Cavity With Internal Heat Generation or Absorption
,”
Numer. Heat Transfer, Part A: Appl.
,
41
(
5
), pp.
529
546
.10.1080/104077802753570356
18.
Chatterjee
,
D.
,
2013
, “
MHD Mixed Convection in a Lid-Driven Cavity Including a Heated Source
,”
Numer. Heat Transfer, Part A: Appl.
,
64
(
3
), pp.
235
254
.10.1080/10407782.2013.779191
19.
Chatterjee
,
D.
,
Halder
,
P.
,
Mondal
,
S.
, and
Bhattacharjee
,
S.
,
2013
, “
Magnetoconvective Transport in a Vertical Lid-Driven Cavity Including a Heat Conducting Square Cylinder With Joule Heating
,”
Numer. Heat Transfer, Part A: Appl.
,
64
(
12
), pp.
1050
1071
.10.1080/10407782.2013.811955
20.
Chatterjee
,
D.
, and
Gupta
,
S. K.
,
2014
, “
Hydromagnetic Mixed Convective Transport in a Nonisothermally Heated Lid-Driven Square Enclosure Including a Heat-Conducting Circular Cylinder
,”
Ind. Eng. Chem. Res.
,
53
(
51
), pp.
19775
19787
.10.1021/ie501080y
21.
Chatterjee
,
D.
, and
Halder
,
P.
,
2014
, “
MHD Mixed Convective Transport in Square Enclosure With Two Rotating Circular Cylinders
,”
Numer. Heat Transfer, Part A: Appl.
,
65
(
8
), pp.
802
824
.10.1080/10407782.2013.846687
22.
Chatterjee
,
D.
,
Mondal
,
B.
, and
Halder
,
P.
,
2014
, “
Hydromagnetic Mixed Convective Transport in a Vertical Lid-Driven Cavity Including a Heat Conducting Rotating Circular Cylinder
,”
Numer. Heat Transfer, Part A: Appl.
,
65
(
1
), pp.
48
65
.10.1080/10407782.2013.812399
23.
Chatterjee
,
D.
, and
Halder
,
P.
,
2016
, “
Magnetoconvective Transport in a Lid-Driven Square Enclosure With Two Rotating Circular Cylinders
,”
Heat Transfer Eng.
,
37
(
2
), pp.
198
209
.10.1080/01457632.2015.1044416
24.
Chatterjee
,
D.
, and
Kumar Gupta
,
S.
,
2017
, “
Magnetohydrodynamic Natural Convection in a Square Enclosure With Four Circular Cylinders Positioned at Different Rectangular Locations
,”
Heat Transfer Eng.
,
38
(
17
), pp.
1449
1465
.10.1080/01457632.2016.1255078
25.
Chatterjee
,
D.
, and
Mishra
,
R.
,
2018
, “
Numerical Investigation of Transient Magnetohydrodynamic Mixed Convection in a Ventilated Cavity Containing Two Heated Circular Cylinders
,”
Heat Transfer Eng.
,
39
(
12
), pp.
1052
1066
.10.1080/01457632.2017.1358487
26.
Ray
,
S.
, and
Chatterjee
,
D.
,
2014
, “
MHD Mixed Convection in a Lid-Driven Cavity Including Heat Conducting Circular Solid Object and Corner Heaters With Joule Heating
,”
Int. Commun. Heat Mass Transfer
,
57
, pp.
200
207
.10.1016/j.icheatmasstransfer.2014.07.029
27.
Ray
,
S.
, and
Chatterjee
,
D.
,
2014
, “
MHD Mixed Convection in a Lid-Driven Cavity Including Heat Conducting Solid Object and Corner Heaters With Joule Heating
,”
Numer. Heat Transfer, Part A: Appl.
,
66
(
5
), pp.
530
550
.10.1080/10407782.2014.892399
28.
Bansal
,
S.
, and
Chatterjee
,
D.
,
2015
, “
Magneto-Convective Transport of Nanofluid in a Vertical Lid-Driven Cavity Including a Heat-Conducting Rotating Circular Cylinder
,”
Numer. Heat Transfer, Part A: Appl.
,
68
(
4
), pp.
411
431
.10.1080/10407782.2014.986361
29.
Li
,
B.-W.
,
Wang
,
W.
, and
Zhang
,
J.-K.
,
2016
, “
Combined Effects of Magnetic Field and Thermal Radiation on Fluid Flow and Heat Transfer of Mixed Convection in a Vertical Cylindrical Annulus
,”
ASME J. Heat Transfer
,
138
(
6
), p.
062501
.10.1115/1.4032609
30.
Sudhagar
,
P.
,
Kameswaran
,
P. K.
, and
Kumar
,
B. R.
,
2016
, “
Magnetohydrodynamics Mixed Convection Flow of a Nanofluid in an Isothermal Vertical Cone
,”
ASME J. Heat Transfer
,
139
(
3
), p.
034503
.10.1115/1.4035039
31.
,
A.
,
Chamkha
,
A. J.
,
Ismael
,
M. A.
, and
Salah
,
T.
,
2018
, “
Magnetohydrodynamics Natural Convection in a Triangular Cavity Filled With a Cu-Al2O3/Water Hybrid Nanofluid With Localized Heating From Below and Internal Heat Generation
,”
ASME J. Heat Transfer
,
140
(
7
), p.
072502
.10.1115/1.4039213
32.
Cramer
,
K.
, and
Pai
,
S.
,
1973
,
Magnetofluid Dynamics for Engineers and Applied Physicists
,
McGraw-Hill Book Company
,
New York
.
33.
Hayase
,
T.
,
Humphrey
,
J.
, and
Greif
,
R.
,
1992
, “
A Consistently Formulated Quick Scheme for Fast and Stable Convergence Using Finite-Volume Iterative Calculation Procedures
,”
J. Comput. Phys.
,
98
(
1
), pp.
108
118
.10.1016/0021-9991(92)90177-Z
34.
Muralidhar
,
K.
, and
Sundarajan
,
T.
,
2003
,
Computational Fluid Flow and Heat Transfer: IIT Kanpur Series of Advanced Texts
,
Alpha Science International
,
Pangbourne, UK
.
35.
Luikov
,
A.
,
1974
, “
Conjugate Convective Heat Transfer Problems
,”
Int. J. Heat Mass Transfer
,
17
(
2
), pp.
257
265
.10.1016/0017-9310(74)90087-8