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Oscillatory flow between concentric spheres driven by an electromagnetic force

Published online by Cambridge University Press:  04 June 2021

Aldo Figueroa Open the ORCID record for Aldo Figueroa [Opens in a new window] ,
Michel Rivero Open the ORCID record for Michel Rivero [Opens in a new window] ,
José Núñez Open the ORCID record for José Núñez [Opens in a new window] ,
Jaziel A. Rojas Open the ORCID record for Jaziel A. Rojas [Opens in a new window]  and
Iván Rivera Open the ORCID record for Iván Rivera [Opens in a new window]
Aldo Figueroa*
Affiliation:
CONACYT-Centro de Investigación en Ciencias-Universidad Autónoma del Estado de Morelos, Av. Universidad No. 1001, Col. Chamilpa, Cuernavaca, Morelos62209, México
Michel Rivero*
Affiliation:
Instituto de Investigaciones en Materiales, Unidad Morelia, Universidad Nacional Autónoma de México, 58190Morelia, Michoacán, México
José Núñez
Affiliation:
Escuela Nacional de Estudios Superiores, Unidad Morelia, Universidad Nacional Autónoma de México, 58190Morelia, Michoacán, México
Jaziel A. Rojas
Affiliation:
Instituto de Investigación en Ciencias Básicas y Aplicadas-Universidad Autónoma del Estado de Morelos, Av. Universidad No. 1001, Col. Chamilpa, Cuernavaca, Morelos62209, México
Iván Rivera
Affiliation:
Instituto de Investigación en Ciencias Básicas y Aplicadas-Universidad Autónoma del Estado de Morelos, Av. Universidad No. 1001, Col. Chamilpa, Cuernavaca, Morelos62209, México
*
†Email addresses for correspondence: [email protected], [email protected]
†Email addresses for correspondence: [email protected], [email protected]
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Abstract

The time-dependent flow driven by electromagnetic forcing of an electrolytic fluid in the gap of a concentric spheres set-up is studied experimentally and theoretically. The driving Lorentz force is generated by the interaction of an alternating current radially injected through electrodes located at the equatorial zone of the spheres and a dipolar magnetic field produced by a permanent magnet inside the inner sphere. Experimentally, the time-dependent flows were explored in the laminar regime with a Reynolds number ${Re} = 640$ and different forcing frequencies, which resulted in oscillatory Reynolds numbers ranging from $28$ to $2820$. Velocity profiles in the equatorial line between spheres were obtained with particle image velocimetry. Given the symmetry of the problem at the equatorial plane, asymptotic and approximate solutions for the azimuthal velocity are obtained for the limiting cases of low-${Re}_{\omega }$ (in real arguments) and high-${Re}_{\omega }$ (in complex arguments). Furthermore, a general methodology is proposed in such a way that an exact solution for the problem is obtained. The analytical solutions reproduce the main characteristic behaviour of the flow. An estimation of the oscillatory boundary layer due to the electromagnetic forcing is obtained through the exact solution. A full three-dimensional numerical model, that introduces the dipolar magnetic field and the radial dependency of the applied current, is able to quantitatively reproduce both the analytical solutions and the experimental measurements. Additionally, numerical results show a resonant behaviour of the flow when the forcing frequency is approximately ${Re}_{\omega } \approx 560$.

JFM classification

Boundary Layers: Boundary layer structure Geophysical and Geological Flows: Rotating flows Instability: Taylor-Couette flow
Type
JFM Papers
Information
Journal of Fluid Mechanics , Volume 920 , 10 August 2021 , A5
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Copyright
© The Author(s), 2021. Published by Cambridge University Press

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References

REFERENCES

Ahadzi, G.M., Liston, A.D., Bayford, R.H. & Holder, D.S. 2004 Neuromagnetic field strength outside the human head due to impedance changes from neuronal depolarization. Physiol. Meas. 25 (1), 365–378.CrossRefGoogle ScholarPubMed
Beals, R. & Szmigielski, J. 2013 Meijer G-functions: a gentle introduction. Not. Am. Math. Soc. 60 (7), 866–872.CrossRefGoogle Scholar
Box, F., Thompson, A.B. & Mullin, T. 2015 Torsional oscillations of a sphere in a Stokes flow. Exp. Fluids 56 (12), 209.CrossRefGoogle Scholar
Cuevas, S., Smolentsev, S. & Abdou, M.A. 2006 On the flow past a magnetic obstacle. J. Fluid Mech. 553, 227–252.CrossRefGoogle Scholar
Davidson, P.A. 2001 An Introduction to Magnetohydrodynamics. Cambridge University Press.CrossRefGoogle Scholar
Drazin, P.G. & Riley, N. 2006 The Navier–Stokes equations: a classification of flows and exact solutions. Lecture Note Series, vol. 334. Cambridge University Press.CrossRefGoogle Scholar
Figueroa, A., Cuevas, S. & Ramos, E. 2011 Electromagnetically driven oscillatory shallow layer flow. Phys. Fluids 23 (1), 013601.CrossRefGoogle Scholar
Figueroa, A., Demiaux, F., Cuevas, S. & Ramos, E. 2009 Electrically driven vortices in a weak dipolar magnetic field in a shallow electrolytic layer. J. Fluid Mech. 641, 245–261.CrossRefGoogle Scholar
Figueroa, A., Meunier, P., Cuevas, S., Villermaux, E. & Ramos, E. 2014 Chaotic advection at large Péclet number: electromagnetically driven experiments, numerical simulations, and theoretical predictions. Phys. Fluids 26 (1), 013601.CrossRefGoogle Scholar
Figueroa, A., Rojas, J.A., Rosales, J. & Vázquez, F. 2016 Electromagnetically driven flow between concentric spheres: Experiments and simulations. In Recent Advances in Fluid Dynamics with Environmental Applications (ed. J. Klapp, L.D.G. Sigalotti, A. Medina, A. López and G. Ruiz-Chavarría), pp. 253–264. Springer International Publishing.CrossRefGoogle Scholar
Figueroa, A., Schaeffer, N., Nataf, H.-C. & Schmitt, D. 2013 Modes and instabilities in magnetized spherical Couette flow. J. Fluid Mech. 716, 445–469.CrossRefGoogle Scholar
Garcia, F., Seilmayer, M., Giesecke, A. & Stefani, F. 2020 Chaotic wave dynamics in weakly magnetized spherical Couette flows. Chaos 30 (4), 043116.CrossRefGoogle ScholarPubMed
Garcia, F. & Stefani, F. 2018 Continuation and stability of rotating waves in the magnetized spherical Couette system: secondary transitions and multistability. Proc. R. Soc. Lond. A 474, 20180281.Google ScholarPubMed
Gissinger, C., Ji, H. & Goodman, J. 2011 Instabilities in magnetized spherical Couette flow. Phys. Rev. E 84, 026308.CrossRefGoogle ScholarPubMed
Griebel, M., Dornseifer, T. & Neunhoeffer, T. 1998 Numerical Simulation in Fluid Dynamics. SIAM.CrossRefGoogle Scholar
Hollerbach, R. 2009 Non-axisymmetric instabilities in magnetic spherical Couette flow. Proc. R. Soc. Lond. A 465 (2107), 2003–2013.Google Scholar
Hollerbach, R., Wei, X., Noir, J. & Jackson, A. 2013 Electromagnetically driven zonal flows in a rapidly rotating spherical shell. J. Fluid Mech. 725, 428–445.CrossRefGoogle Scholar
Hollerbach, R., Wiener, R.J., Sullivan, I.S., Donnelly, R.J. & Barenghi, C.F. 2002 The flow around a torsionally oscillating sphere. Phys. Fluids 14 (12), 4192–4205.CrossRefGoogle Scholar
Kaplan, E., Nataf, H.-C. & Schaeffer, N. 2018 Dynamic domains of the Derviche Tourneur sodium experiment: simulations of a spherical magnetized Couette flow. Phys. Rev. Fluids 3 (3), 34608.CrossRefGoogle Scholar
Kasprzyk, C., Kaplan, E., Seilmayer, M. & Stefani, F. 2017 Transitions in a magnetized quasi-laminar spherical Couette flow. Magnetohydrodynamics 53 (2), 393–401.CrossRefGoogle Scholar
Lemoff, A.V. & Lee, A.P. 2000 An AC magnetohydrodynamic micropump. Sensors Actuators B 63 (3), 178–185.CrossRefGoogle Scholar
Mizerski, K.A. & Bajer, K. 2007 On the effect of mantle conductivity on the super-rotating jets near the liquid core surface. Phys. Earth Planet. Inter. 160 (3–5), 245–268.CrossRefGoogle Scholar
Ogbonna, J., Garcia, F., Gundrum, T., Seilmayer, M. & Stefani, F. 2020 Experimental investigation of the return flow instability in magnetized spherical Couette flows. Phys. Fluids 32 (12), 124119.CrossRefGoogle Scholar
Proudman, I. 1956 The almost-rigid rotation of viscous fluid between concentric spheres. J. Fluid Mech. 1 (5), 505–516.CrossRefGoogle Scholar
Rivero, M. & Cuevas, S. 2012 Analysis of the slip condition in magnetohydrodynamic (MHD) micropumps. Sensors Actuators B 166–167, 884–892.CrossRefGoogle Scholar
Rivero, M. & Cuevas, S. 2018 Magnetohydrodynamic pumps for sensor applications. In Advances in Sensors: Reviews’ Book Series, 5., chap. 12. International Frequency Sensor Association.Google Scholar
Rivero, M., Garzón, F., Núñez, J. & Figueroa, A. 2019 Study of the flow induced by circular cylinder performing torsional oscillation. Eur. J. Mech. B/Fluids 78, 245–251.CrossRefGoogle Scholar
Schrauf, G. 1986 The first instability in spherical Taylor–Couette flow. J. Fluid Mech. 166, 287–303.CrossRefGoogle Scholar
Sisan, D.R., Mujica, N., Tillotson, W.A., Huang, Y.M., Dorland, W., Hassam, A.B., Antonsen, T.M. & Lathrop, D.P. 2004 Experimental observation and characterization of the magnetorotational instability. Phys. Rev. Lett. 93 (11), 8–11.CrossRefGoogle ScholarPubMed
Soward, A.M. & Dormy, E. 2010 Shear-layers in magnetohydrodynamic spherical Couette flow with conducting walls. J. Fluid Mech. 645, 145–185.CrossRefGoogle Scholar
Stewartson, K. 1966 On almost rigid rotations. Part 2. J. Fluid Mech. 26 (1), 131–144.CrossRefGoogle Scholar
Stokes, G.G. 1851 On the effect of the internal friction of fluids on the motion of pendulums. Trans. Camb. Phil. Soc. 9, 8–106.Google Scholar
Suslov, S.A., Pérez-Barrera, J. & Cuevas, S. 2017 Electromagnetically driven flow of electrolyte in a thin annular layer: axisymmetric solutions. J. Fluid Mech. 828, 573–600.CrossRefGoogle Scholar
Thielicke, W. & Stamhuis, E.J. 2014 Towards user-friendly, affordable and accurate digital particle image velocimetry in Matlab. J. Open. Res. Softw. 2, pe30.CrossRefGoogle Scholar
Travnikov, V., Eckert, K. & Odenbach, S. 2011 Influence of an axial magnetic field on the stability of spherical Couette flows with different gap widths. Acta Mechanica 219 (3–4), 255–268.CrossRefGoogle Scholar
Wicht, J. 2014 Flow instabilities in the wide-gap spherical Couette system. J. Fluid Mech. 738, 184–221.CrossRefGoogle Scholar
Wimmer, M. 1976 Experiments on a viscous fluid flow between concentric rotating spheres. J. Fluid Mech. 78 (2), 317–335.CrossRefGoogle Scholar