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Thin-film flow in helically-wound rectangular channels of arbitrary torsion and curvature

Published online by Cambridge University Press:  23 December 2014

D. J. Arnold ,
Y. M. Stokes  and
J. E. F. Green
D. J. Arnold
Affiliation:
School of Mathematical Sciences, The University of Adelaide, Adelaide, SA 5005, Australia
Y. M. Stokes*
Affiliation:
School of Mathematical Sciences, The University of Adelaide, Adelaide, SA 5005, Australia
J. E. F. Green
Affiliation:
School of Mathematical Sciences, The University of Adelaide, Adelaide, SA 5005, Australia
*
†Email address for correspondence: [email protected]
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Abstract

Laminar helically-symmetric gravity-driven thin-film flow down a helically-wound channel of rectangular cross-section is considered. We extend the work of Stokes et al. (Phys. Fluids, vol. 25 (8), 2013, 083103) and Lee et al. (Phys. Fluids, vol. 26 (4), 2014, 043302) to channels with arbitrary curvature and torsion or, equivalently, arbitrary curvature and slope. We use a non-orthogonal coordinate system and, remarkably, find an exact steady-state solution. We find that the free-surface shape and flow have a complicated dependence on the curvature, slope and flux down the channel. Moderate to large channel slope has a significant effect on the flow in the region of the channel near the inside wall, particularly when the curvature of the channel is large. This work has application to flow in static spiral particle separators used in mineral processing.

JFM classification

Low-Reynolds-number flows: Lubrication theory Interfacial Flows (free surface): Thin films Waves/Free-surface Flows: Waves/Free-surface Flows
Type
Papers
Information
Journal of Fluid Mechanics , Volume 764 , 10 February 2015 , pp. 76 - 94
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Copyright
© 2014 Cambridge University Press 

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References

Das, S. K., Godiwakka, K. M., Panda, L., Bhattacharya, K. K., Singh, R. & Mehrotra, S. P. 2007 Mathematical modeling of separation charactersistics of a coal-washing spiral. Intl J. Miner. Process. 85, 118–132.CrossRefGoogle Scholar
Dean, W. R. 1927 Note on the motion of fluid in a curved pipe XVI. Phil. Mag. 4 (20), 208–223.CrossRefGoogle Scholar
Dean, W. R. 1928 Fluid motion in a curved channel. Proc. R. Soc. Lond. A 121 (787), 402–420.Google Scholar
Germano, M. 1982 On the effect of torsion on a helical pipe flow. J. Fluid Mech. 125, 1–8.CrossRefGoogle Scholar
Germano, M. 1989 The Dean equations extended to a helical pipe flow. J. Fluid Mech. 203, 289–305.CrossRefGoogle Scholar
Holland-Batt, A. B. 1975 A quantitative model of the motion of particles in the RSM/Mintek on-stream particle size analyser. Powder Technol. 11, 11–25.CrossRefGoogle Scholar
Holland-Batt, A. B. 1989 Spiral separation: theory and simulation. Trans. Inst. Mining Met. C 98, C46–C60.Google Scholar
Holland-Batt, A. B. 1995 Some design considerations for spiral separators. Miner. Engng 8 (11), 1381–1395.CrossRefGoogle Scholar
Holland-Batt, A. B. 2009 A method for the prediction of the primary flow on large diameter spiral troughs. Miner. Engng 22 (4), 352–356.CrossRefGoogle Scholar
Holland-Batt, A. B. & Holtham, P. N. 1991 Particle and fluid motion on spiral separators. Miner. Engng 4 (3/4), 457–482.CrossRefGoogle Scholar
Holtham, P. N. 1990 Flow visualisation of secondary currents on spiral separators. Miner. Engng 3 (3/4), 279–286.CrossRefGoogle Scholar
Holtham, P. N. 1992 Primary and secondary fluid velocities on spiral separators. Miner. Engng 5 (1), 79–91.CrossRefGoogle Scholar
Lee, S., Stokes, Y. M. & Bertozzi, A. L. 2014 Behaviour of a particle-laden flow in a spiral channel. Phys. Fluids 26 (4), 043302.CrossRefGoogle Scholar
Lynch, D. G., Waters, S. L. & Pedley, T. J. 1996 Flow in a tube with non-uniform time-dependent curvature: governing equations and simple examples. J. Fluid Mech. 323, 237–265.CrossRefGoogle Scholar
Manoussaki, D. & Chadwick, R. S. 2000 Effects of geometry on fluid loading in a coiled cochlea. SIAM J. Appl. Maths 61 (2), 369–386.CrossRefGoogle Scholar
Matthews, B. W., Fletcher, C. A. J. & Partiridge, A. C. 1998 Computational simulation of fluid and dilute particulate flows on spiral concentrators. Appl. Math. Model. 22, 965–979.CrossRefGoogle Scholar
Matthews, B. W., Fletcher, C. A. J., Partiridge, A. C. & Vasquez, S. 1999 Computations of curved free surface water flow on spiral concentrators. J. Hydraul. Engng 125, 1126–1139.CrossRefGoogle Scholar
Siggers, J. H. & Waters, S. L. 2005 Steady flows in pipes with finite curvature. Phys. Fluids 17, 077102.CrossRefGoogle Scholar
Siggers, J. H. & Waters, S. L. 2008 Unsteady flows in pipes with finite curvature. J. Fluid Mech. 600, 133–165.CrossRefGoogle Scholar
Stokes, Y. M. 2001 Flow in spiral channels of small curvature and torsion. In IUTAM Symposium on Free Surface Flows (ed. King, A. C. & Shikhmurzaev, Y. D.), Fluid Mechanics and its Applications, vol. 62, pp. 289–296. Kluwer.CrossRefGoogle Scholar
Stokes, Y. M., Duffy, B. R., Wilson, S. K. & Tronnolone, H. 2013 Thin-film flow in helically wound rectangular channels with small torsion. Phys. Fluids 25 (8), 083103.CrossRefGoogle Scholar
Stokes, Y. M., Wilson, S. K. & Duffy, B. R. 2004 Thin-film flow in open helically-wound channels. In Proceedings of the 15th Australiasian Fluid Mechanics Conference (ed. Behnia, M., Lin, W. & McBain, G. D.), The University of Sydney.Google Scholar
Thomson, J. 1876 On the origin of windings of rivers in alluivial plains with remarks on the flow of water round bends in pipes. Proc. R. Soc. Lond. A 25 (171–178), 5–8.Google Scholar
Thomson, J. 1877 Experimental demonstration in respect to the origin of windings of rivers in alluvial plains, and to the mode of flow of water round bends of pipes. Proc. R. Soc. Lond. A 26 (179–184), 356–357.Google Scholar
Wang, J. & Andrews, J. R. G. 1994 Numerical simulations of liquid flow on spiral concentrators. Miner. Engng 7 (11), 1363–1385.CrossRefGoogle Scholar
Zabielski, L. & Mestel, A. J. 1998 Steady flow in a helically symmetric pipe. J. Fluid Mech. 370, 297–320.CrossRefGoogle Scholar