An analysis is made of the linear stability of wide-gap hydromagnetic (MHD) dissipative Couette flow of an incompressible electrically conducting fluid between two rotating concentric circular cylinders in the presence of a uniform axial magnetic field. A constant heat flux is applied at the outer cylinder and the inner cylinder is kept at a constant temperature. Both types of boundary conditions viz; perfectly electrically conducting and electrically nonconducting walls are examined. The three cases of $μ<0$ (counter-rotating), $μ>0$ (co-rotating), and $μ=0$ (stationary outer cylinder) are considered. Assuming very small magnetic Prandtl number $Pm$, the wide-gap perturbation equations are derived and solved by a direct numerical procedure. It is found that for given values of the radius ratio $η$ and the heat flux parameter $N$, the critical Taylor number $Tc$ at the onset of instability increases with increase in Hartmann number $Q$ for both conducting and nonconducting walls thus establishing the stabilizing influence of the magnetic field. Further it is found that insulating walls are more destabilizing than the conducting walls. It is observed that for given values of $η$ and $Q$, the critical Taylor number $Tc$ decreases with increase in $N$. The analysis further reveals that for $μ=0$ and perfectly conducting walls, the critical wave number $ac$ is not a monotonic function of $Q$ but first increases, reaches a maximum and then decreases with further increase in $Q$. It is also observed that while $ac$ is a monotonic decreasing function of $μ$ for $N=0$, in the presence of heat flux $(N=1)$, $ac$ has a maximum at a negative value of $μ$ (counter-rotating cylinders).

1.
Taylor
,
G. I.
, 1923, “
Stability of a Viscous Liquid Contained Between Two Rotating Cylinders
,”
Philos. Trans. R. Soc. London, Ser. A
0962-8428,
223
, pp.
289
343
.
2.
Chandrasekhar
,
S.
, 1961,
Hydrodynamic and Hydromagnetic Stability
,
Clarendon Press
,
Oxford, UK
.
3.
Donnelly
,
R. J.
, and
Ozima
,
M.
, 1962, “
Experiments on the Stability of Flow Between Rotating Cylinders in the Presence of a Magnetic Field
,”
Proc. R. Soc. London, Ser. A
1364-5021,
226
, pp.
272
286
.
4.
Donnelly
,
R. J.
, and
Caldwell
,
D. R.
, 1963, “
Experiments on the Stability of Hydromagnetic Couette Flow
,”
J. Fluid Mech.
0022-1120,
19
, pp.
257
263
.
5.
Roberts
,
P. H.
, 1964, “
The Stability of Hydromagnetic Couette Flow
,”
Proc. Cambridge Philos. Soc.
0068-6735,
60
, pp.
635
651
.
6.
Hollerbach
,
R.
, and
Skinner
,
S.
, 2001, “
Instabilities of Magnetically Induced Shear Layers and Jets
,”
Proc. R. Soc. London, Ser. A
1364-5021,
457
, pp.
785
802
.
7.
Chen
,
C. K.
, and
Chang
,
M. H.
, 1998, “
Stability of Hydromagnetic Dissipative Couette Flow With Nonaxisymmetric Disturbance
,”
J. Fluid Mech.
0022-1120,
366
, pp.
135
158
.
8.
Chang
,
T. S.
, and
Sartory
,
W. K.
, 1967, “
On the Onset of Instability by Oscillatory Modes in Hydromagnetic Couette Flow
,”
Proc. R. Soc. London, Ser. A
1364-5021,
301
, pp.
451
471
.
9.
Niblett
,
E. R.
, 1958, “
The Stability of Couette Flow in an Axial Magnetic Field
,”
Can. J. Phys.
0008-4204,
36
, pp.
1509
1525
.
10.
Kurzweg
,
U. H.
, 1963, “
The Stability of Couette Flow in the Presence of an Axial Magnetic Field
,”
J. Fluid Mech.
0022-1120,
17
, pp.
52
60
.
11.
Harris
,
D. L.
, and
Reid
,
W. H.
, 1964, “
On the Stability of Viscous Flow Between Rotating Cylinders, Part 2 Numerical Analysis
,”
J. Fluid Mech.
0022-1120,
20
, pp.
95
101
.
12.
Takhar
,
H. S.
,
Ali
,
M. A.
, and
Soundalgekar
,
V. M.
, 1989, “
Stability of MHD Couette Flow in a Narrow Gap Annulus
,”
Appl. Sci. Res.
0003-6994,
46
, pp.
1
24
.
13.
Ali
,
M. A.
,
Takhar
,
H. S.
, and
Soundalgekar
,
V. M.
, 1992, “
Stability of Flow Between Two Rotating Cylinders in the Presence of a Constant Heat Flux at the Outer Cylinder
,”
J. Appl. Mech.
0021-8936,
59
, pp.
464
465
.
14.
Soundalgekar
,
V. M.
,
Ali
,
M. A.
, and
Takhar
,
H. S.
, 1994, “
Hydromagnetic Stability of Dissipative Couette Flow: Wide-Gap Problem
,”
Int. J. Energy Res.
0363-907X,
18
, pp.
689
695
.
15.
Velikhov
,
E. P.
, 1959, “
Stability of an Ideally Conducting Liquid Flowing Between Cylinders Rotating in a Magnetic Field
,”
Sov. Phys. JETP
0038-5646,
36
, pp.
1398
1404
.
16.
Balbus
,
S. A.
, and
Hawley
,
J. F.
, 1991, “
A Powerful Local Shear Instability in Weakly Magnetized Disks. I. Linear Analysis
,”
Astrophys. J.
0004-637X,
376
, pp.
214
222
.
17.
Rüdiger
,
G.
, and
Zhang
,
Y.
, 2001, “
MHD Instability in Differentially-Rotating Cylindric Flows
,”
Astron. Astrophys.
0004-6361,
378
, pp.
302
308
.
18.
Willis
,
A. P.
, and
Barenghi
,
C. F.
, 2002, “
Magnetic Instability in a Sheared Azimuthal Flow
,”
Astron. Astrophys.
0004-6361,
388
, pp.
688
691
.
19.
Sparrow
,
E. M.
,
Goldstein
,
R. J.
, and
Jonsson
,
U. K.
, 1964, “
Thermal Instability in a Horizontal Fluid Layer: Effect of Boundary Conditions and Non-linear Temperature Profiles
,”
J. Fluid Mech.
0022-1120,
18
, pp.
513
528
.
20.
Mutabazi
,
I.
,
,
A.
, and
Dumouchel
,
F.
, 2001, “
,”
Proceedings 12th International Couette-Taylor Workshop
, Evanston, IL, September 6–8.
21.
Gailitis
,
A.
et al.
, 2001, “
Magnetic Field Saturation in the Riga Dynamo Experiment
,”
Phys. Rev. Lett.
0031-9007,
86
(
14
), pp.
3024
3027
.
22.
Stieglitz
,
R.
, and
Müller
,
U.
, 2001, “
Experimental Demonstration of a Homogeneous Two-scale Dynamo
,”
Phys. Fluids
1070-6631,
13
(
3
), pp.
561
564
.
23.
Takhar
,
H. S.
,
Ali
,
M. A.
, and
Soundalgekar
,
V. M.
, 1989, “
Stability of the Flow Between Rotating Cylinders-Wide Gap Problem
,”
J. Fluids Eng.
0098-2202,
111
, pp.
97
99
.