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Journal of Fluids Engineering | Volume 137 | Issue 3 | research-article
Research Papers: Fundamental Issues and Canonical Flows

Effect of Radial Temperature Gradient on the Stability of Magnetohydrodynamic Dean Flow in an Annular Channel at High Magnetic Parameter

[+] Author and Article Information
S. Dholey

Department of Mathematics,
T.D.B. College,
Raniganj 713 347, India
e-mail: sdholey@gmail.com

Contributed by the Fluids Engineering Division of ASME for publication in the JOURNAL OF FLUIDS ENGINEERING. Manuscript received March 17, 2014; final manuscript received July 13, 2014; published online October 21, 2014. Assoc. Editor: Shizhi Qian.

J. Fluids Eng 137(3), 031203 (Oct 21, 2014) (12 pages) Paper No: FE-14-1139; doi: 10.1115/1.4028598 History: Received March 17, 2014; Revised July 13, 2014
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The effect of a radial temperature gradient on the stability of Dean flow of an electrically conducting fluid in an annular channel is investigated. A strong constant magnetic field is imposed in the axial direction. Finite-difference method is used to solve the eigenvalue problem. For given values of gap width d, between the cylinders, and magnetic parameter Q, electrically nonconducting (NC) walls are found to be more destabilizing than perfectly conducting (PC) walls when the temperature parameter N < 0. This trend persists even for small positive values of N but when N (>0) exceeds a critical value depending on Q, PC walls are slightly more destabilizing than NC walls. The critical value of the radii ratio η (0 < η < 1) beyond which the first unstable mode becomes nonaxisymmetric is determined for various values of N and Q.
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Figures

Grahic Jump Location
Fig. 1

Vertical cross section of the flow configuration placed in a horizontal position. The uniform magnetic field H is along the common axis z of the cylinders and k0 = (∂p/∂θ)0.

+Vertical cross section of the flow configuration placed in a horizontal position. The uniform magnetic field H is along the common axis z of the cylinders and k0 = (∂p/∂θ)0.
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Vertical cross section of the flow configuration placed in a horizontal position. The uniform magnetic field H is along the common axis z of the cylinders and k0 = (∂p/∂θ)0.
Grahic Jump Location
Fig. 2

(a) and (b) Variation of ac with N for several values of Q in the cases of both NC and PC walls for (a) a narrow gap width with η = 0.95 and (b) a wide gap width with η = 0.40

+(a) and (b) Variation of ac with N for several values of Q in the cases of both NC and PC walls for (a) a narrow gap width with η = 0.95 and (b) a wide gap width with η = 0.40
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(a) and (b) Variation of ac with N for several values of Q in the cases of both NC and PC walls for (a) a narrow gap width with η = 0.95 and (b) a wide gap width with η = 0.40
Grahic Jump Location
Fig. 3

(a)–(d) Variation of critical wave number ac with N for two fixed values of η ( = 0.95 and 0.40) and several values of magnetic parameter Q in the cases of both NC and PC walls

+(a)–(d) Variation of critical wave number ac with N for two fixed values of η ( = 0.95 and 0.40) and several values of magnetic parameter Q in the cases of both NC and PC walls
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(a)–(d) Variation of critical wave number ac with N for two fixed values of η ( = 0.95 and 0.40) and several values of magnetic parameter Q in the cases of both NC and PC walls
Grahic Jump Location
Fig. 4

Variation of critical wave number ac with η for several values of N when Q = 25 for NC and PC walls

+Variation of critical wave number ac with η for several values of N when Q = 25 for NC and PC walls
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Variation of critical wave number ac with η for several values of N when Q = 25 for NC and PC walls
Grahic Jump Location
Fig. 5

(a) and (b) Variation of critical Dean number Rec(d/R1)1/2 with N for several values of Q in the cases of both NC and PC walls for (a) a narrow gap width with η = 0.95 and (b) a wide gap width with η = 0.40

+(a) and (b) Variation of critical Dean number Rec(d/R1)1/2 with N for several values of Q in the cases of both NC and PC walls for (a) a narrow gap width with η = 0.95 and (b) a wide gap width with η = 0.40
View Large  |  Save Figure  |  Download Slide (.ppt)
(a) and (b) Variation of critical Dean number Rec(d/R1)1/2 with N for several values of Q in the cases of both NC and PC walls for (a) a narrow gap width with η = 0.95 and (b) a wide gap width with η = 0.40
Grahic Jump Location
Fig. 6

(a) and (b) Variation of critical Dean number Rec(d/R1)1/2 with η for several values of Q in the cases of both NC and PC walls for (a) N = −1 and (b) N = 1

+(a) and (b) Variation of critical Dean number Rec(d/R1)1/2 with η for several values of Q in the cases of both NC and PC walls for (a) N = −1 and (b) N = 1
View Large  |  Save Figure  |  Download Slide (.ppt)
(a) and (b) Variation of critical Dean number Rec(d/R1)1/2 with η for several values of Q in the cases of both NC and PC walls for (a) N = −1 and (b) N = 1
Grahic Jump Location
Fig. 7

Variation of critical wave number ac with Q at a fixed value of N = −1 and several values of η for NC walls. The solid curve is the common asymptote for the curves η = 0.90 and η = 0.30 for Q → ∞.

+Variation of critical wave number ac with Q at a fixed value of N = −1 and several values of η for NC walls. The solid curve is the common asymptote for the curves η = 0.90 and η = 0.30 for Q → ∞.
View Large  |  Save Figure  |  Download Slide (.ppt)
Variation of critical wave number ac with Q at a fixed value of N = −1 and several values of η for NC walls. The solid curve is the common asymptote for the curves η = 0.90 and η = 0.30 for Q → ∞.
Grahic Jump Location
Fig. 8

Variation of critical wave number ac with Q at a fixed value of N = −1 and several values of η for PC walls

+Variation of critical wave number ac with Q at a fixed value of N = −1 and several values of η for PC walls
View Large  |  Save Figure  |  Download Slide (.ppt)
Variation of critical wave number ac with Q at a fixed value of N = −1 and several values of η for PC walls
Grahic Jump Location
Fig. 9

Variation of critical wave number ac versus Q with fixed values of η = 0.30 and N = 1 for NC and PC walls. The solid curves, labeled a and b are the asymptotes for the curves of PC and electrically NC walls for Q → ∞.

+Variation of critical wave number ac versus Q with fixed values of η = 0.30 and N = 1 for NC and PC walls. The solid curves, labeled a and b are the asymptotes for the curves of PC and electrically NC walls for Q → ∞.
View Large  |  Save Figure  |  Download Slide (.ppt)
Variation of critical wave number ac versus Q with fixed values of η = 0.30 and N = 1 for NC and PC walls. The solid curves, labeled a and b are the asymptotes for the curves of PC and electrically NC walls for Q → ∞.
Grahic Jump Location
Fig. 10

Variation of critical Dean number Rec(d/R1)1/2 with magnetic parameter Q for several values of η and N for NC and PC walls. The solid curves, labeled a and b are the asymptotes for the curves of η = 0.30 and η = 0.90 for electrically NC walls with a fixed value of N = −1, and c and d are the asymptotes for the curves of η = 0.30 for electrically NC walls and PC walls with a fixed value of N = 1 for Q → ∞.

+Variation of critical Dean number Rec(d/R1)1/2 with magnetic parameter Q for several values of η and N for NC and PC walls. The solid curves, labeled a and b are the asymptotes for the curves of η = 0.30 and η = 0.90 for electrically NC walls with a fixed value of N = −1, and c and d are the asymptotes for the curves of η = 0.30 for electrically NC walls and PC walls with a fixed value of N = 1 for Q → ∞.
View Large  |  Save Figure  |  Download Slide (.ppt)
Variation of critical Dean number Rec(d/R1)1/2 with magnetic parameter Q for several values of η and N for NC and PC walls. The solid curves, labeled a and b are the asymptotes for the curves of η = 0.30 and η = 0.90 for electrically NC walls with a fixed value of N = −1, and c and d are the asymptotes for the curves of η = 0.30 for electrically NC walls and PC walls with a fixed value of N = 1 for Q → ∞.

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