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Abstract
The problem of twophase unsteady MHDflow between two concentric cylinders of infinite length has been analysed when the outer cylinder is impulsively started. The system of partial differential equations describing the flow problem is formulated taking the viscosity of the particle phase into consideration. Unified closed form expressions are obtained for the velocities and the skin frictions for both cases of the applied magnetic field being fixed to either the fluid or the moving outer cylinder. The problem is solved using a combination of the Laplace transform technique, D’Alemberts and the Riemannsum approximation methods. The solution obtained is validated by comparisons with the closed form solutions obtained for the steady states which has been derived separately. The governing equations are also solved using the implicit finite difference method to verify the present proposed method. The variation of the velocity and the skin friction with the dimensionless parameters occuring in the problem are illustrated graphically and discussed for both phases.
I. INTRODUCTION
II. MATHEMATICAL ANALYSIS
III. SEMIANALYTICAL SOLUTION
IV. NUMERICAL SOLUTION
V. RESULT AND DISCUSSION
VI. CONCLUSION
Key Topics
 Magnetic fields
 13.0
 Viscosity
 13.0
 Frictions
 12.0
 Couette flows
 10.0
 Multiphase flows
 10.0
Figures
Schematic diagram of the problem
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Profile of the fluid phase velocity u _{1} showing the effect of t with G = 0 and G = 1 represented by a and b respectively (K = 50.0, R _{ν} = 1.0, R _{ f } = 0.25 and M = 2).
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Profile of the fluid phase velocity u _{1} showing the effect of t with G = 0 and G = 1 represented by a and b respectively (K = 50.0, R _{ν} = 1.0, R _{ f } = 0.25 and M = 2).
Profile of the particle phase velocity u _{2} showing the effect of t with G = 0 and G = 1 represented by a and b respectively (K = 50.0, R _{ν} = 1.0, R _{ f } = 0.25 and M = 2).
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Profile of the particle phase velocity u _{2} showing the effect of t with G = 0 and G = 1 represented by a and b respectively (K = 50.0, R _{ν} = 1.0, R _{ f } = 0.25 and M = 2).
Profile of the fluid phase velocity u _{1} showing the effect of M with G = 0 and G = 1 represented by a and b respectively (K = 50.0, R _{ν} = 1.0, R _{ f } = 0.25 and t = 0.5).
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Profile of the fluid phase velocity u _{1} showing the effect of M with G = 0 and G = 1 represented by a and b respectively (K = 50.0, R _{ν} = 1.0, R _{ f } = 0.25 and t = 0.5).
Profile of the particle phase velocity u _{2} showing the effect of M with G = 0 and G = 1 represented by a and b respectively (K = 50.0, R _{ν} = 1.0, R _{ f } = 0.25 and t = 0.5).
Click to view
Profile of the particle phase velocity u _{2} showing the effect of M with G = 0 and G = 1 represented by a and b respectively (K = 50.0, R _{ν} = 1.0, R _{ f } = 0.25 and t = 0.5).
Profile of the fluid phase velocity u _{1} showing the effect of R _{ν} and t with G = 0 and G = 1 represented by a and b respectively (K = 50.0, r = 1.5, R _{ f } = 0.25, M = 2.0 and t = 0.5).
Click to view
Profile of the fluid phase velocity u _{1} showing the effect of R _{ν} and t with G = 0 and G = 1 represented by a and b respectively (K = 50.0, r = 1.5, R _{ f } = 0.25, M = 2.0 and t = 0.5).
Profile of the particle phase velocity u _{2} showing the effect of R _{ν} and t with G = 0 and G = 1 represented by a and b respectively (K = 50.0, r = 1.5, R _{ f } = 0.25, M = 2.0 and t = 0.5).
Click to view
Profile of the particle phase velocity u _{2} showing the effect of R _{ν} and t with G = 0 and G = 1 represented by a and b respectively (K = 50.0, r = 1.5, R _{ f } = 0.25, M = 2.0 and t = 0.5).
Profile of the fluid phase velocity u _{1} showing the effect of R _{ f } and t with G = 0 and G = 1 represented by a and b respectively (K = 50.0, r = 1.5, R _{ν} = 1.0, M = 2.0 and t = 0.2).
Click to view
Profile of the fluid phase velocity u _{1} showing the effect of R _{ f } and t with G = 0 and G = 1 represented by a and b respectively (K = 50.0, r = 1.5, R _{ν} = 1.0, M = 2.0 and t = 0.2).
Profile of the particle phase velocity u _{2} showing the effect of R _{ f } and t with G = 0 and G = 1 represented by a and b respectively (K = 50.0, r = 1.5, R _{ν} = 1.0, M = 2.0 and t = 0.2).
Click to view
Profile of the particle phase velocity u _{2} showing the effect of R _{ f } and t with G = 0 and G = 1 represented by a and b respectively (K = 50.0, r = 1.5, R _{ν} = 1.0, M = 2.0 and t = 0.2).
Variation of the fluid phase skin friction on the outer surface of the inner cylinder (r = 1) showing the effect of R _{ν} and t with G = 0 and G = 1 represented by a and b respectively (K = 50.0, M = 2.0 and R _{ f } = 0.25).
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Variation of the fluid phase skin friction on the outer surface of the inner cylinder (r = 1) showing the effect of R _{ν} and t with G = 0 and G = 1 represented by a and b respectively (K = 50.0, M = 2.0 and R _{ f } = 0.25).
Variation of the particle phase skin friction on the outer surface of the inner cylinder (r = 1) showing the effect of R _{ν} and t with G = 0 and G = 1 represented by a and b respectively (K = 50.0, M = 2.0 and R _{ f } = 0.25).
Click to view
Variation of the particle phase skin friction on the outer surface of the inner cylinder (r = 1) showing the effect of R _{ν} and t with G = 0 and G = 1 represented by a and b respectively (K = 50.0, M = 2.0 and R _{ f } = 0.25).
Variation of the fluid phase skin friction on the inner surface of the outer cylinder (r = 2) showing the effect of R _{ν} and t with G = 0 and G = 1 represented by a and b respectively (K = 50.0, M = 2.0 and R _{ f } = 0.25).
Click to view
Variation of the fluid phase skin friction on the inner surface of the outer cylinder (r = 2) showing the effect of R _{ν} and t with G = 0 and G = 1 represented by a and b respectively (K = 50.0, M = 2.0 and R _{ f } = 0.25).
Variation of the particle phase skin friction on the inner surface of the outer cylinder (r = 2) showing the effect of R _{ν} and t with G = 0 and G = 1 represented by a and b respectively (K = 50.0, M = 2.0 and R _{ f } = 0.25).
Click to view
Variation of the particle phase skin friction on the inner surface of the outer cylinder (r = 2) showing the effect of R _{ν} and t with G = 0 and G = 1 represented by a and b respectively (K = 50.0, M = 2.0 and R _{ f } = 0.25).
Tables
Numerical values of the velocity obtained using the Riemannsum approximation method and that obtained using finite difference method for both the fluid and the particle phases when t = 0.2, G = 1, K = 50, R _{ f } = 0.25 and M = 2.
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Numerical values of the velocity obtained using the Riemannsum approximation method and that obtained using finite difference method for both the fluid and the particle phases when t = 0.2, G = 1, K = 50, R _{ f } = 0.25 and M = 2.
Numerical values of the steady state skin friction obtained using the Riemannsum approximation method and that obtained using finite difference method when K = 50, R _{ f } = 0.25 and M = 2.
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Numerical values of the steady state skin friction obtained using the Riemannsum approximation method and that obtained using finite difference method when K = 50, R _{ f } = 0.25 and M = 2.
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Abstract
The problem of twophase unsteady MHDflow between two concentric cylinders of infinite length has been analysed when the outer cylinder is impulsively started. The system of partial differential equations describing the flow problem is formulated taking the viscosity of the particle phase into consideration. Unified closed form expressions are obtained for the velocities and the skin frictions for both cases of the applied magnetic field being fixed to either the fluid or the moving outer cylinder. The problem is solved using a combination of the Laplace transform technique, D’Alemberts and the Riemannsum approximation methods. The solution obtained is validated by comparisons with the closed form solutions obtained for the steady states which has been derived separately. The governing equations are also solved using the implicit finite difference method to verify the present proposed method. The variation of the velocity and the skin friction with the dimensionless parameters occuring in the problem are illustrated graphically and discussed for both phases.
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