A Numerical Study of MHD Laminar Flow in a Rotating Curved Pipe with Circular Cross Section

Affiliation(s)

Discipline of Mathematics, School of Engineering and Technology, Khulna University, Khulna, Bangladesh.

Discipline of Mathematics, School of Engineering and Technology, Khulna University, Khulna, Bangladesh.

ABSTRACT

The incompressible viscous steady flow through a rotating curved pipe of circular cross-section with magnetic field is investigated numerically to examine the combined effects of rotation (Coriolis force), magnetic field and curvature (centrifugal force) on the flow. The curvature of the pipe has been assumed to be small, that is, the radius of the circle in which the central line of the pipe is coiled is large in comparison with the radius of the cross section. Spectral method is applied as a main tool for the numerical technique, where Fourier series, Chebyshev polynomials, Collocation methods, and Iteration method are used as secondary tools. The flow depends on the Taylor number (*Tr*), Dean Number (*Dn*), Magnetic Parameter (*M*) and the dimensionless curvature of the pipe δ. When *Tr* > 0, the rotation is in the direction so that the Coriolis force enforces the curvature effect. When *Tr* < 0, the rotation is in the direction so that the Coriolis force exhibits an opposite effect to that of the curvature. The calculations are carried out for −1500 ≤ *Tr* ≤ 1500, *Dn* ≥ 1000 (large Dean number), *M* ≥ 0 and* δ* = 0.01. Due to high magnetic field four-vortex solution is observed in a rotating curved pipe system. Visualization is attained with MAPLE software.

The incompressible viscous steady flow through a rotating curved pipe of circular cross-section with magnetic field is investigated numerically to examine the combined effects of rotation (Coriolis force), magnetic field and curvature (centrifugal force) on the flow. The curvature of the pipe has been assumed to be small, that is, the radius of the circle in which the central line of the pipe is coiled is large in comparison with the radius of the cross section. Spectral method is applied as a main tool for the numerical technique, where Fourier series, Chebyshev polynomials, Collocation methods, and Iteration method are used as secondary tools. The flow depends on the Taylor number (

Cite this paper

Hoque, M. and Alam, M. (2015) A Numerical Study of MHD Laminar Flow in a Rotating Curved Pipe with Circular Cross Section.*Open Journal of Fluid Dynamics*, **5**, 121-127. doi: 10.4236/ojfd.2015.52014.

Hoque, M. and Alam, M. (2015) A Numerical Study of MHD Laminar Flow in a Rotating Curved Pipe with Circular Cross Section.

References

[1] Dean, W.R. (1927) Note on the Motion of Fluid in a Curved Pipe. Philosophical Magazine and Journal of Science, 4, 208-223.

http://dx.doi.org/10.1080/14786440708564324

[2] Dean, W.R. (1928) Note on the Motion of Fluid in a Curved Pipe. Philosophical Magazine and Journal of Science, 5, 673-695.

http://dx.doi.org/10.1080/14786440408564513

[3] Adler, M. (1934) Stromung in Gekrummten Rohren. Zeitschrift for Angewandte Mathematik und Mechanics (ZAMM), 14, 257-275.

http://dx.doi.org/10.1002/zamm.19340140502

[4] Winters, K.H. (1987) A Bifurcation Study of Laminar Flow in a Curved Tube of Rectangular Cross-Section. Journal of Fluid Mechanics, 180, 343-369.

http://dx.doi.org/10.1017/S0022112087001848

[5] Yanase, S., Goto, N. and Yamamoto, K. (1989) Dual Solutions of the Flow through a Curved Tube. Fluid Dynamics Research, 5, 191-201.

http://dx.doi.org/10.1016/0169-5983(89)90021-X

[6] Chang, C.C. and. Lundgren, T.S. (1961) Duct Flow in Magnetohydrodynamics. Zeitschrift für Angewandte Mathematik und Physik, 12, 100.

http://dx.doi.org/10.1007/BF01601011

[7] Shercliff, J.A. (1956) The Flow of Conducting Fluids in Circular Pipes under Transverse Magnetic Fields. Journal of Fluid Mechanics, 1, 644.

http://dx.doi.org/10.1017/S0022112056000421

[8] Walker, J.S. (1986) Liquid-Metal Flow in a Thin Conducting Pipe near the End of a Region of Uniform Magnetic Field. Journal of Fluid Mechanics, 167, 199-217.

http://dx.doi.org/10.1017/S0022112086002793

[9] Masud, M.A., Islam, Md.R., Sheikh, Md.R. and Alam, M.M. (2010) Stable Solution Zone for Fluid Flow through Curved Pipe with Circular Cross-Section. JNAME, 7, 19-26.

[10] Hoque, M.M., Anika, N.N. and Alam, M.M. (2013) Magnetic Effects on Direct Numerical Solution of Fluid Flow through a Curved Pipe with Circular Cross Section. Europien Journal of Scientific Research, 101, 343-361.

[11] Hoque, M.M., Anika, N.N. and Alam, M.M. (2013) Numerical Analysis of Magnetohydrodynamics Flow in a Curved Duct. International Journal of Scientific and Engineering Research, 7, 607-617.

[12] Hoque, M.M. and Alam, M.M. (2013) Effects of Dean Number and Curvature on Fluid Flow through a Curved Pipe with Magnetic Field. Procedia Engineering, 56, 245-253.

http://dx.doi.org/10.1016/j.proeng.2013.03.114

[13] Williams, G.S., Hubbell, C.W. and Fenkell, G.M. (1902) Experiments of Detroit, Michigan, on the Effect of Curvature upon the Flow of Water in Pipes. Transactions of the ASCE, 47, 1-196.

[14] Miyazaki, H. (1971) Combined Free- and Forced-Convective Heat Transfer and Fluid Flow in Rotating Curved Circular Tube. International Journal of Heat Mass Transfer, 14, 1295-1309.

http://dx.doi.org/10.1016/0017-9310(71)90179-7

[15] Ito, H. and Nanbu, K. (1971) Flow in Rotating Straight Pipes for Circular Cross-Section. AMSE Journal of Basic Engineering, 93, 383-394.

http://dx.doi.org/10.1115/1.3425260

[16] Daskopoulos, P. and Lenhoff, A.M. (1989) Flow in Curved Ducts: Bifurcation Structure for Stationary Ducts. Journal of Fluid Mechanics, 203, 125-148.

http://dx.doi.org/10.1017/S0022112089001400

[17] Hoque, M.M., Alam, M.M., Ferdows, M. and Beg, O.A. (2013) Numerical Simulation of Dean Number and Curvature Effects on Mahneto-Biofluid Flow through a Curved Conduit. Proceedings of the Institution of Mechanical Engineers, Part H: Journal of Engineering in Medicine, 227, 1155-1170.

http://dx.doi.org/10.1177/0954411913493844

[18] Beg, O.A., Hoque, M.M., Wahiduzzaman, M., Alam, M.M. and Ferdows, M. (2014) Spectral Numerical Simulation of Magneto-Physiological Laminar Dean Flow. Journal of Mechanics in Medicine and Biology, 14, Article ID: 1450047.

http://dx.doi.org/10.1142/S021951941450047X

[19] Busby, D.E. (1967) Biomagnetics Considerations Relevant to Manned Space Flight. NASA-CR-889, NASA, Washington DC, 63 p.

[20] Tripathi, D. and Bég, O.A. (2012) A Numerical Study of Oscillating Peristaltic Flow of Generalized Maxwell Viscoelastic Fluids through a Porous Medium. Transport in Porous Media, 95, 337-348.

http://dx.doi.org/10.1007/s11242-012-0046-5

[1] Dean, W.R. (1927) Note on the Motion of Fluid in a Curved Pipe. Philosophical Magazine and Journal of Science, 4, 208-223.

http://dx.doi.org/10.1080/14786440708564324

[2] Dean, W.R. (1928) Note on the Motion of Fluid in a Curved Pipe. Philosophical Magazine and Journal of Science, 5, 673-695.

http://dx.doi.org/10.1080/14786440408564513

[3] Adler, M. (1934) Stromung in Gekrummten Rohren. Zeitschrift for Angewandte Mathematik und Mechanics (ZAMM), 14, 257-275.

http://dx.doi.org/10.1002/zamm.19340140502

[4] Winters, K.H. (1987) A Bifurcation Study of Laminar Flow in a Curved Tube of Rectangular Cross-Section. Journal of Fluid Mechanics, 180, 343-369.

http://dx.doi.org/10.1017/S0022112087001848

[5] Yanase, S., Goto, N. and Yamamoto, K. (1989) Dual Solutions of the Flow through a Curved Tube. Fluid Dynamics Research, 5, 191-201.

http://dx.doi.org/10.1016/0169-5983(89)90021-X

[6] Chang, C.C. and. Lundgren, T.S. (1961) Duct Flow in Magnetohydrodynamics. Zeitschrift für Angewandte Mathematik und Physik, 12, 100.

http://dx.doi.org/10.1007/BF01601011

[7] Shercliff, J.A. (1956) The Flow of Conducting Fluids in Circular Pipes under Transverse Magnetic Fields. Journal of Fluid Mechanics, 1, 644.

http://dx.doi.org/10.1017/S0022112056000421

[8] Walker, J.S. (1986) Liquid-Metal Flow in a Thin Conducting Pipe near the End of a Region of Uniform Magnetic Field. Journal of Fluid Mechanics, 167, 199-217.

http://dx.doi.org/10.1017/S0022112086002793

[9] Masud, M.A., Islam, Md.R., Sheikh, Md.R. and Alam, M.M. (2010) Stable Solution Zone for Fluid Flow through Curved Pipe with Circular Cross-Section. JNAME, 7, 19-26.

[10] Hoque, M.M., Anika, N.N. and Alam, M.M. (2013) Magnetic Effects on Direct Numerical Solution of Fluid Flow through a Curved Pipe with Circular Cross Section. Europien Journal of Scientific Research, 101, 343-361.

[11] Hoque, M.M., Anika, N.N. and Alam, M.M. (2013) Numerical Analysis of Magnetohydrodynamics Flow in a Curved Duct. International Journal of Scientific and Engineering Research, 7, 607-617.

[12] Hoque, M.M. and Alam, M.M. (2013) Effects of Dean Number and Curvature on Fluid Flow through a Curved Pipe with Magnetic Field. Procedia Engineering, 56, 245-253.

http://dx.doi.org/10.1016/j.proeng.2013.03.114

[13] Williams, G.S., Hubbell, C.W. and Fenkell, G.M. (1902) Experiments of Detroit, Michigan, on the Effect of Curvature upon the Flow of Water in Pipes. Transactions of the ASCE, 47, 1-196.

[14] Miyazaki, H. (1971) Combined Free- and Forced-Convective Heat Transfer and Fluid Flow in Rotating Curved Circular Tube. International Journal of Heat Mass Transfer, 14, 1295-1309.

http://dx.doi.org/10.1016/0017-9310(71)90179-7

[15] Ito, H. and Nanbu, K. (1971) Flow in Rotating Straight Pipes for Circular Cross-Section. AMSE Journal of Basic Engineering, 93, 383-394.

http://dx.doi.org/10.1115/1.3425260

[16] Daskopoulos, P. and Lenhoff, A.M. (1989) Flow in Curved Ducts: Bifurcation Structure for Stationary Ducts. Journal of Fluid Mechanics, 203, 125-148.

http://dx.doi.org/10.1017/S0022112089001400

[17] Hoque, M.M., Alam, M.M., Ferdows, M. and Beg, O.A. (2013) Numerical Simulation of Dean Number and Curvature Effects on Mahneto-Biofluid Flow through a Curved Conduit. Proceedings of the Institution of Mechanical Engineers, Part H: Journal of Engineering in Medicine, 227, 1155-1170.

http://dx.doi.org/10.1177/0954411913493844

[18] Beg, O.A., Hoque, M.M., Wahiduzzaman, M., Alam, M.M. and Ferdows, M. (2014) Spectral Numerical Simulation of Magneto-Physiological Laminar Dean Flow. Journal of Mechanics in Medicine and Biology, 14, Article ID: 1450047.

http://dx.doi.org/10.1142/S021951941450047X

[19] Busby, D.E. (1967) Biomagnetics Considerations Relevant to Manned Space Flight. NASA-CR-889, NASA, Washington DC, 63 p.

[20] Tripathi, D. and Bég, O.A. (2012) A Numerical Study of Oscillating Peristaltic Flow of Generalized Maxwell Viscoelastic Fluids through a Porous Medium. Transport in Porous Media, 95, 337-348.

http://dx.doi.org/10.1007/s11242-012-0046-5