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Instability and transition of a vertical ascension or fall of a free sphere affected by a vertical magnetic field

Published online by Cambridge University Press:  16 November 2018

Jun-Hua Pan
Affiliation:
School of Engineering Science, University of Chinese Academy of Sciences, Beijing 101408, China
Nian-Mei Zhang
Affiliation:
School of Engineering Science, University of Chinese Academy of Sciences, Beijing 101408, China
Ming-Jiu Ni*
Affiliation:
School of Engineering Science, University of Chinese Academy of Sciences, Beijing 101408, China
*
Email address for correspondence: mjni@ucas.ac.cn

Abstract

When the Galileo number is below the first bifurcation, the instability and transition of a vertical ascension or the fall of a free sphere affected by a vertical magnetic field are investigated numerically. A compact model is used to explain that the magnetic field can destabilize the fluid–solid system. When the interaction parameter exceeds a critical value, the sphere trajectory is transitioned from a steady vertical trajectory to a steady oblique one. Furthermore, the trajectory will remain vertical at a sufficiently large magnetic field because of a double effect of the magnetic field on the fluid–solid system. Under the influence of an external vertical magnetic field, four wake patterns at the rear of the sphere are found and the physical behaviour of the free sphere is independent of the density ratio. The wake or trajectory of the free sphere is only determined by the Galileo number $G$ and the interaction parameter $N$. A close relationship between the streamwise vorticity and the sphere motion is found. An interesting ‘agglomeration phenomenon’ is also found, which shows that the vertical velocities are agglomerated into a point for a certain magnetic field regardless of the Galileo number and satisfy a scaling law $V_{z}\sim N^{-1/4}$, when $N>1$. The principal results of the present work are summarized in a map of regimes in the $\{G,N\}$ plane.

Type
JFM Papers
Copyright
© 2018 Cambridge University Press 

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References

Cuevas, S., Smolentsev, S. & Abdou, M. A. 2006 On the flow past a magnetic obstacle. J. Fluid Mech. 553, 227252.Google Scholar
Davidson, P. A. 2001 An Introduction to Magnetohydrodynamics. Cambridge University Press.Google Scholar
Dullweber, A., Leimkuhler, B. & McLachlan, R. 1997 Symplectic splitting methods for rigid body molecular dynamics. J. Chem. Phys. 107 (15), 58405851.Google Scholar
El-Kaddah, N., Patel, A. D. & Natarajan, T. T. 1995 The electromagnetic filtration of molten aluminum using an induced-current separator. JOM J. Miner. Met. Mater. Soc. 47 (5), 4649.Google Scholar
Ern, P., Risso, F., Fabre, D. & Magnaudet, J. 2012 Wake-induced oscillatory paths of bodies freely rising or falling in fluids. Annu. Rev. Fluid Mech. 44, 97121.Google Scholar
Fabre, D., Tchoufag, J. & Magnaudet, J. 2012 The steady oblique path of buoyancy-driven disks and spheres. J. Fluid Mech. 707, 2436.Google Scholar
Giacobello, M., Ooi, A. & Balachandar, S. 2009 Wake structure of a transversely rotating sphere at moderate Reynolds numbers. J. Fluid Mech. 621, 103130.Google Scholar
Goldsworthy, F. A. 1961 Magnetohydrodynamic flows of a perfectly conducting, viscous fluid. J. Fluid Mech. 11 (4), 519528.Google Scholar
Horowitz, M. & Williamson, C. H. K. 2010 The effect of Reynolds number on the dynamics and wakes of freely rising and falling spheres. J. Fluid Mech. 651, 251294.Google Scholar
Hu, H. H., Patankar, N. A. & Zhu, M. Y. 2001 Direct numerical simulations of fluid–solid systems using the arbitrary Lagrangian–Eulerian technique. J. Comput. Phys. 169 (2), 427462.Google Scholar
Jenny, M., Bouchet, G. & Dušek, J. 2003 Nonvertical ascension or fall of a free sphere in a Newtonian fluid. Phys. Fluids 15 (1), L9L12.Google Scholar
Jenny, M. & Dušek, J. 2004 Efficient numerical method for the direct numerical simulation of the flow past a single light moving spherical body in transitional regimes. J. Comput. Phys. 194 (1), 215232.Google Scholar
Jenny, M., Dušek, J. & Bouchet, G. 2004 Instabilities and transition of a sphere falling or ascending freely in a Newtonian fluid. J. Fluid Mech. 508, 201239.Google Scholar
Johnson, T. A. & Patel, V. C. 1999 Flow past a sphere up to a Reynolds number of 300. J. Fluid Mech. 378, 1970.Google Scholar
Kanaris, N., Albets, X., Grigoriadis, D. & Kassinos, S. 2013 Three-dimensional numerical simulations of magnetohydrodynamic flow around a confined circular cylinder under low, moderate, and strong magnetic fields. Phys. Fluids 25 (7), 074102.Google Scholar
Lahjomri, J., Capéran, P. & Alemany, A. 1993 The cylinder wake in a magnetic field aligned with the velocity. J. Fluid Mech. 253, 421448.Google Scholar
Lighthill, M. J. 1963 Boundary layer theory. In Laminar Boundary Layers (ed. Rosenhead, L.), pp. 46113. Oxford University Press.Google Scholar
Mathai, V., Zhu, X., Sun, C. & Lohse, D. 2017 Mass and moment of inertia govern the transition in the dynamics and wakes of freely rising and falling cylinders. Phys. Rev. Lett. 119 (5), 054501.Google Scholar
Mathai, V., Zhu, X., Sun, C. & Lohse, D. 2018 Flutter to tumble transition of buoyant spheres triggered by rotational inertia changes. Nat. Commun. 9 (1), 1792.Google Scholar
Moreau, R. J. 2013 Magnetohydrodynamics. Springer Science & Business Media.Google Scholar
Mück, B., Günther, C., Müller, U. & Bühler, L. 2000 Three-dimensional MHD flows in rectangular ducts with internal obstacles. J. Fluid Mech. 418, 265295.Google Scholar
Mutschke, G., Gerbeth, G., Shatrov, V. & Tomboulides, A. 2001 The scenario of three-dimensional instabilities of the cylinder wake in an external magnetic field: a linear stability analysis. Phys. Fluids 13 (3), 723734.Google Scholar
Ni, M.-J., Munipalli, R., Huang, P., Morley, N. B. & Abdou, M. A. 2007 A current density conservative scheme for incompressible MHD flows at a low magnetic Reynolds number. Part II. On an arbitrary collocated mesh. J. Comput. Phys. 227 (1), 205228.Google Scholar
Pan, J.-H., Zhang, N.-M. & Ni, M.-J. 2018 The wake structure and transition process of a flow past a sphere affected by a streamwise magnetic field. J. Fluid Mech. 842, 248272.Google Scholar
Schwarz, S., Kempe, T. & Fröhlich, J. 2015 A temporal discretization scheme to compute the motion of light particles in viscous flows by an immersed boundary method. J. Comput. Phys. 281, 591613.Google Scholar
Tchoufag, J., Fabre, D. & Magnaudet, J. 2014 Global linear stability analysis of the wake and path of buoyancy-driven disks and thin cylinders. J. Fluid Mech. 740, 278311.Google Scholar
Veldhuis, C. H. J. & Biesheuvel, A. 2007 An experimental study of the regimes of motion of spheres falling or ascending freely in a Newtonian fluid. Intl J. Multiphase Flow 33 (10), 10741087.Google Scholar
Yonas, G. 1967 Measurements of drag in a conducting fluid with an aligned field and large interaction parameter. J. Fluid Mech. 30 (4), 813821.Google Scholar
Zhang, C., Eckert, S. & Gerbeth, G. 2005 Experimental study of single bubble motion in a liquid metal column exposed to a DC magnetic field. Intl J. Multiphase Flow 31 (7), 824842.Google Scholar
Zhang, J. & Ni, M.-J. 2017 What happens to the vortex structures when the rising bubble transits from zigzag to spiral? J. Fluid Mech. 828, 353373.Google Scholar
Zheng, T. X., Zhong, Y. B., Lei, Z. S., Ren, W. L., Ren, Z. M., Debray, F., Beaugnon, E. & Fautrelle, Y. 2015 Effects of high static magnetic field on distribution of solid particles in BiZn immiscible alloys with metastable miscibility gap. J. Alloys Compounds 623, 3641.Google Scholar
Zhou, W. & Dušek, J. 2015 Chaotic states and order in the chaos of the paths of freely falling and ascending spheres. Intl J. Multiphase Flow 75, 205223.Google Scholar