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Stability of plane-parallel flow of magnetic fluids under external magnetic fields

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Abstract

In this work, we present a theoretical study on the stability of a two-dimensional plane Poiseuille flow of magnetic fluids in the presence of externally applied magnetic fields. The fluids are assumed to be incompressible, and their magnetization is coupled to the flow through a simple phenomenological equation. Dimensionless parameters are defined, and the equations are perturbed around the base state. The eigenvalues of the linearized system are computed using a finite difference scheme and studied with respect to the dimensionless parameters of the problem. We examine the cases of both the horizontal and vertical magnetic fields. The obtained results indicate that the flow is destabilized in the horizontally applied magnetic field, but stabilized in the vertically applied field. We characterize the stability of the flow by computing the stability diagrams in terms of the dimensionless parameters and determine the variation in the critical Reynolds number in terms of the magnetic parameters. Furthermore, we show that the superparamagnetic limit, in which the magnetization of the fluids decouples from hydrodynamics, recovers the same purely hydrodynamic critical Reynolds number, regardless of the applied field direction and of the values of the other dimensionless magnetic parameters.

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Abbreviations

B :

magnetic induction field (T)

H :

applied magnetic field (A/m)

H 0 :

magnetic field intensity (A/m)

ℓ :

half distance between plates (m)

M :

magnetization of magnetic fluid (A/m)

M 0 :

equilibrium magnetization of magnetic fluid (A/m)

p :

pressure (Pa)

u :

velocity field (m/s)

U :

characteristic velocity scale (m/s)

η :

dynamic viscosity of fluid (Pa·s)

μ 0 :

vacuum magnetic permeability (H/m)

ρ :

fluid density (kg/m3)

σ :

stress tensor of magnetic fluid (Pa)

τ :

magnetic relaxation time (s)

\({\cal A},{\cal B}\) :

functions appearing on perturbed magnetization equations

B :

magnetic induction

c :

propagation velocity of perturbation

c i :

imaginary part of propagation velocity of perturbation

c r :

real part of propagation velocity of perturbation

e x :

unit vector on horizontal direction

H :

magnetic field

i:

imaginary unity, \({\rm{i}} = \sqrt {- 1} \)

j :

index of mesh points

M :

magnetization of magnetic fluid

n :

unitary normal vector

N :

number of mesh points

P :

block matrix appearing in generalized eigenvalue problem

Q :

block matrix appearing in generalized eigenvalue problem

u :

horizontal velocity component

\({\tilde w}\) :

combined vector of amplitudes for generalized eigenvalue problem

α :

wave number of perturbation

Δy :

mesh size

ϵ :

Levi-Civita permutation tensor

C pm :

magnetic pressure coefficient, \({C_{pm}} = {{{\mu _0}H_0^2} \over {\rho {U^2}}}\)

Pe :

Péclet number, \(Pe = {{\tau U} \over \ell}\)

Re :

Reynolds number, \(Re = {{\rho U\ell} \over \eta}\)

Re c :

critical Reynolds number

Re hc :

critical Reynolds number for purely hydrodynamical problem

γ :

exponent of power law relation between Rec and Cpm

χ0:

magnetic susceptibility

{·}b :

base state variables

{·}1 :

horizontal component of vector

{·}2 :

vertical component of vector

\(\widetilde{\left\{\cdot \right\}}\) :

amplitudes of perturbations

{·}′:

perturbations of variables

\({{\cal L}_{{\rm{OS}}}}\) :

Orr-Sommerfeld operator.

References

  1. ROSENSWEIG, R. E. Ferrohydrodynamics, Dover Publications Inc., New York (1997)

    Google Scholar 

  2. ODENBACH, S. Ferrofluids: Magnetically Controllable Fluids and Their Applications, SpringerVerlag, New York (2002)

    Book  Google Scholar 

  3. PAPELL, S. S. Low Viscosity Magnetic Fluid Obtained by the Colloidal Suspension of Magnetic Particles, US Patent number 3215572, The United States of America (1963)

  4. SCHERER, C. and FIGUEIREDO NETO, A. M. Ferrofluids: properties and applications. Brazilian Journal of Physics, 35, 718–727 (2005)

    Article  Google Scholar 

  5. LÜBBE, A. S., ALEXIOU, C., and BERGEMANN, C. Clinical applications of magnetic drug targeting. Journal of Surgical Research, 95(2), 200–206 (2001)

    Article  Google Scholar 

  6. BRUSENTSOV, N. A., NIKITIN, L. V., BRUSENTSOVA, T. N., KUZNETSOV, A. A., BAYBURTSKIY, F. S., SHÜMAKOV, L. I., and JURCHENKO, N. Y. Magnetic fluid hyperthermia of the mouse experimental tumor. Journal of Magnetism and Magnetic Materials, 252, 378–380 (2002)

    Article  Google Scholar 

  7. CUNHA, F. R. and SOBRAL, Y. D. Characterization of the physical parameters in a process of magnetic separation and pressure-driven flow of a magnetic fluid. Physica A, 343, 36–64 (2004)

    Article  Google Scholar 

  8. RINALDI, C., CHAVES, A., ELBORAI, S., HE, X., and ZAHN, M. Magnetic fluid rheology and flows. Current Opinion in Colloid & Interface Science, 10, 141–157 (2005)

    Article  Google Scholar 

  9. SHLIOMIS, M. I. Effective viscosity of magnetic suspensions. Soviet Physics JETP, 34, 1291–1294 (1972)

    Google Scholar 

  10. SCHUMACHER, K. R., SELLIEN, I., KNOKE, G. S., CADER, T., and FINLAYSON, B. A. Experiment and simulation of laminar and turbulent ferrofluid pipe flow in an oscillating magnetic field. Physical Review E, 67, 026308 (2003)

    Article  Google Scholar 

  11. CHEN, T. S. and EATON, T. E. Magnetohydrodynamic stability of the developing laminar flow in a parallel-plate channel. Physics of Fluids, 15, 592–596 (1972)

    Article  Google Scholar 

  12. RADWAN, A. E. Stability of a force-free magnetic field of a conducting fluid cylinder. Zeitschrift für Angewandte Mathematik und Physik, 48, 827–839 (1997)

    Article  MathSciNet  Google Scholar 

  13. ELDABEA, N. T. M., EL-SABBAGHB, M. F., and EL-SAYED, M. A. S. Hydromagnetic stability of plane Poiseuille and Couette flow of viscoelastic fluid. Fluid Dynamics Research, 38, 699–715 (2006)

    Article  MathSciNet  Google Scholar 

  14. VISLOVICH, A. N., SINITSYN, A. K., and TYMANOVICH, V. V. Instability of plane-parallel Couette flow of a magnetic liquid in a homogeneous magnetic field. Magnitinaya Gidrodinamika, 2, 32–37 (1984)

    MATH  Google Scholar 

  15. STILES, P. J. and BLENNERHASSETT, P. J. Stability of cylindrical Couette flow of a radially magnetized ferrofluids in a radial temperature gradient. Journal of Magnetism and Magnetic Materials, 122, 207–209 (1993)

    Article  Google Scholar 

  16. HART, J. E. Ferromagnetic rotating Couette flow: the role of magnetic viscosity. Journal of Fluid Mechanics, 453, 21–38 (2002)

    Article  MathSciNet  Google Scholar 

  17. ALTMEYER, S., DO, Y., and LAI, Y. C. Transition to turbulence in Taylor-Couette ferrofluidic flow. Scientific Reports, 5, 10781 (2015)

    Article  Google Scholar 

  18. COWLEY, M. D. and ROSENSWEIG, R. E. The interfacial stability of a ferromagnetic fluid. Journal of Fluid Mechanics, 30, 671–688 (1967)

    Article  Google Scholar 

  19. ELHEFNAWY, A. R. F. Nonlinear Rayleigh-Taylor instability in magnetic fluids with uniform horizontal and vertical magnetic fields. Zeitschrift für Angewandte Mathematik und Physik, 44, 495–509 (1993)

    Article  MathSciNet  Google Scholar 

  20. YECKO, P. Stability of layered channel flow of magnetic fluids. Physics of Fluids, 21, 034102 (2009)

    Article  Google Scholar 

  21. LI, M. and ZHU, L. Interfacial instability of ferrofluid flow under the influence of a vacuum magnetic field. Applied Mathematics and Mechanics (English Edition), 42(8), 1171–1182 (2021) https://doi.org/10.1007/s10483-021-2758-7

    Article  MathSciNet  Google Scholar 

  22. MUKHERJEE, A., VAIDYA, A., and YECKO, P. Laminar shear in a ferrofluid: stability studies. Magnetohydrodynamics, 49, 505–511 (2013)

    Article  Google Scholar 

  23. SCHMID, P. J. and HENNINGSON, D. S. Stability and Transition in Shear Flows, SpringerVerlag, New York (2001)

    Book  Google Scholar 

  24. CUNHA, F. R., SOBRAL, Y. D., and GONTIJO, R. G. Stabilization of concentration waves in fluidized beds of magnetic particles. Powder Technology, 241, 219–229 (2013)

    Article  Google Scholar 

  25. ENTOV, V. M., BARSOUM, M., and SHMARYAN, L. E. On capillary instability of jets of magneto-rheological fluids. Journal of Rheology, 40, 727–739 (1996)

    Article  Google Scholar 

  26. SCHMID, P. J. Nonmodal stability theory. Annual Review of Fluid Mechanics, 39, 129–162 (2007)

    Article  MathSciNet  Google Scholar 

  27. KERSWELL, R. R. Nonlinear nonmodal stability theory. Annual Review of Fluid Mechanics, 50, 319–345 (2018)

    Article  MathSciNet  Google Scholar 

  28. ZAHN, M. and PIOCH, L. L. Ferrofluid flows in AC and traveling wave magnetic fields with effective positive, zero or negative dynamic viscosity. Journal of Magnetism and Magnetic Materials, 201, 144–148 (1999)

    Article  Google Scholar 

  29. RINALDI, C. and ZAHN, M. Effects of spin viscosity on ferrofluid flow profiles in alternating and rotating magnetic fields. Physics of Fluids, 14, 2847–2870 (2002)

    Article  Google Scholar 

  30. KORLIE, M. S., MUKHERJEE, A., NITA, B. G., STEVENS, J. G., TRUBATCH, A. D., and YECKO, P. Analysis of flows of ferrofluids under simple shear. Magnetohydrodynamics, 44, 51–60 (2008)

    Article  Google Scholar 

  31. ROSA, A. P. Microstructure and Magnetorheology of Ferrofluids in Shear: Theory and Simulation (in Portuguese), Ph. D. dissertation, Universidade de Brasiilia (2018)

  32. ROSA, A. P., ABADE, G. C., and CUNHA, F. R. Computer simulations of equilibrium magnetization and microstructure in magnetic fluids. Physics of Fluids, 29, 092006 (2017)

    Article  Google Scholar 

  33. CARVALHO, D. D. and GONTIJO, R. G. Magnetization diffusion in duct flow: the magnetic entrance length and the interplay between hydrodynamic and magnetic timescales. Physics of Fluids, 32, 072007 (2020)

    Article  Google Scholar 

  34. DE VICENTE, J., KLINGENBERG, D. J., and HIDALGO-ALVAREZ, R. Magnetorheological fluids: a review. Soft Matter, 7, 3701–3710 (2011)

    Article  Google Scholar 

  35. RAMOS, D. M., CUNHA, F. R., SOBRAL, Y. D., and RODRIGUES, J. L. A. F. Computer simulations of magnetic fluids in laminar pipe flows. Journal of Magnetism and Magnetic Materials, 289, 238–241 (2005)

    Article  Google Scholar 

  36. CUNHA, F. R. and SOBRAL, Y. D. Asymptotic solution for pressure-driven flows of magnetic fluids in pipes. Journal of Magnetism and Magnetic Materials, 289, 314–317 (2005)

    Article  Google Scholar 

  37. SINGH, C., DAS, A. K., and DAS, P. K. Flow restrictive and shear reducing effect of magnetization. Physics of Fluids, 28, 087103 (2016)

    Article  Google Scholar 

  38. ORSZAG, S. A. Accurate solution of the Orr-Sommerfeld stability equation. Journal of Fluid Mechanics, 50, 659–703 (1971)

    Article  Google Scholar 

  39. PEREIRA, I. D. O. Rheology of Ferrofluids in Shear Flows, M. Sc. dissertation, Universidade de Brasília (2019)

  40. PAPADOPOULOS, P. K., VAFEAS, P., and HATZIKONSTANTINOU, P. M. Ferrofluid pipe flow under the influence of the magnetic field of a cylindrical coil. Physics of Fluids, 24, 122002 (2012)

    Article  Google Scholar 

  41. TZIRTZILAKIS, E. E. and XENOS, M. A. Biomagnetic fluid flow in a driven cavity. Meccanica, 48, 187–200 (2013)

    Article  MathSciNet  Google Scholar 

  42. SINZATO, Y. Z. and CUNHA, F. R. Modeling and experiments of capillary flow of non-symmetric magnetic fluids under uniform field. Journal of Magnetism and Magnetic Materials, 508, 166867 (2020)

    Article  Google Scholar 

  43. SINZATO, Y. Z. and CUNHA, F. R. Capillary flow of magnetic fluids with effect of hydrodynamic dispersion. Physics of Fluids, 33, 102006 (2021)

    Article  Google Scholar 

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Acknowledgements

P. Z. S. PAZ is grateful for the financial support provided by Coordination for the Improvement of Higher Education Personnel-Brazil (CAPES) (Finance Code 001) and National Council for Scientific and Technological Development-Brazil (CNPq) during the course of this research. F. R. CUNHA acknowledges the financial support of CNPq (No. 305764/2015-2). Y. D. SOBRAL acknowledges the financial support of University of Brasilia (Call DPI/DPG No. 02/2021).

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Authors and Affiliations

  1. Department of Mathematics, University of Brasília, Campus Universitário Darcy Ribeiro, Brasília-DF, 70910-900, Brazil

    P. Z. S. Paz & Y. D. Sobral

  2. Laboratory of Microhydrodynamics and Rheology-Vortex Group, Department of Mechanical Engineering, University of Brasília, Campus Universitário Darcy Ribeiro, Brasília-DF, 70910-900, Brazil

    F. R. Cunha

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  1. P. Z. S. Paz
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Paz, P.Z.S., Cunha, F.R. & Sobral, Y.D. Stability of plane-parallel flow of magnetic fluids under external magnetic fields. Appl. Math. Mech.-Engl. Ed. 43, 295–310 (2022). https://doi.org/10.1007/s10483-022-2813-9

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