1. An Accurate Mode-Selection Mechanism for Magnetic Fluids in a Hele-Shaw Cell David P. Jackson Dickinson College, Carlisle, PA USA José A. Miranda Universidade Federal de Pernambuco, Recife, Brazil Slide 2

2. Aug. 15, 2007 Mar del Plata, ArgentinaD.P. Jackson 2/20 The Birthday Girl!

3. Aug. 15, 2007 Mar del Plata, ArgentinaD.P. Jackson 3/20 What is a Ferrofluid? Colloidal suspension of tiny magnets (  10 nm) coated with a molecular surfactant Thermal motion keeps the dipoles uniformly distributed and randomly oriented unless there is a magnetic field present The dipoles align in a magnetic field For details, see Ferrohydrodynamics, Ronald E. Rosensweig (Cambridge University Press, 1985), (Dover, 1997)

4. Aug. 15, 2007 Mar del Plata, ArgentinaD.P. Jackson 4/20 Basic Physical Situation Ferrofluid is confined between two closely spaced glass plates and placed in a perpendicular magnetic field

5. Aug. 15, 2007 Mar del Plata, ArgentinaD.P. Jackson 5/20 Experimental Setup Hele-Shaw cell Light Video Camera Helmholtz Coils

6. Aug. 15, 2007 Mar del Plata, ArgentinaD.P. Jackson 6/20 Qualitative Description No magnetic fieldUniform magnetic field Parallel-plate capacitorCurrent Ribbon Uniform magnetization collinear with field Outward magnetic pressure competes with surface tension that results in a fingering instability

7. Aug. 15, 2007 Mar del Plata, ArgentinaD.P. Jackson 7/20 Sample Evolution Single drop experimental example

8. Aug. 15, 2007 Mar del Plata, ArgentinaD.P. Jackson 8/20 Controlling the Instability How can we control the fingering instability? Add an azimuthal field that falls off with distance

9. Aug. 15, 2007 Mar del Plata, ArgentinaD.P. Jackson 9/20 Essential Physics Outward force caused by a magnetic pressure due to dipole alignment from normal field Inward force caused by surface tension that tends to minimize surface area Inward force caused by the radial gradient of the azimuthal field

10. Aug. 15, 2007 Mar del Plata, ArgentinaD.P. Jackson 10/20 Governing Equations Navier-Stokes: Hele-Shaw Approximations: Laplace’s Equation: Interfacial BC:

11. Aug. 15, 2007 Mar del Plata, ArgentinaD.P. Jackson 11/20 Conformal Mapping Solve Laplace’s Eq. on unit disk (Poisson integral formula) Map exists from complex (simply connected) domain to unit disk Interfacial BC gives evolution equation for domain boundary Equation looks like: plane Bensimon et. al., Rev. Mod. Phys. 58, 977 (1986)

12. Aug. 15, 2007 Mar del Plata, ArgentinaD.P. Jackson 12/20 Numerical Evolution Destabilizing (normal) field only!

13. Aug. 15, 2007 Mar del Plata, ArgentinaD.P. Jackson 13/20 Linear Stability Analysis Specifying and linearizing the equation of motion leads to growth rates where

14. Aug. 15, 2007 Mar del Plata, ArgentinaD.P. Jackson 14/20 Growth Rates I Unstable Stable Destabilizing (normal) field only!

15. Aug. 15, 2007 Mar del Plata, ArgentinaD.P. Jackson 15/20 Growth Rates II Unstable Stable Single unstable mode! Possible mode- selection mechanism!

16. Aug. 15, 2007 Mar del Plata, ArgentinaD.P. Jackson 16/20 Stability Phase Portrait Solid lines are neutral stability curves Gray areas denote regions where a particular mode is the fastest growing Diamonds denote specific values used for simulations n=3 n=2 n=4 n=5 Single Unstable Modes

17. Aug. 15, 2007 Mar del Plata, ArgentinaD.P. Jackson 17/20 Precisely Selected Modes Simulations run with identical initial conditions - Bond numbers chosen so that there is only a single unstable mode

18. Aug. 15, 2007 Mar del Plata, ArgentinaD.P. Jackson 18/20 Simulations with N B  =1.5 Initial condition is left-right n=2 mode As N B increases, more modes become stable When only a single mode is unstable, the initial condition is drown out

19. Aug. 15, 2007 Mar del Plata, ArgentinaD.P. Jackson 19/20 Simulations with N B  =2.5

20. Aug. 15, 2007 Mar del Plata, ArgentinaD.P. Jackson 20/20 Summary An azimuthal magnetic field can be used to control the normal field fingering instability of a magnetic fluid in a Hele-Shaw cell By tuning the azimuthal and normal fields, one can produce a situation in which a single unstable mode exists Numerical simulations demonstrate that mode growth can be accurately selected Large enough azimuthal fields completely stabilize the interface

21. An Accurate Mode-Selection Mechanism for Magnetic Fluids in a Hele-Shaw Cell David P. Jackson Dickinson College, Carlisle, PA USA José A. Miranda Universidade Federal de Pernambuco, Recife, Brazil Slide 2

22. David P. Jackson Dickinson College, Carlisle, PA USA An Accurate Mode-Selection Mechanism for Magnetic Fluids in a Hele-Shaw Cell

23. Aug. 15, 2007 Mar del Plata, ArgentinaD.P. Jackson 23/20 Acknowledgements PASI 2007 Organizers –Great workshop - job well done! –Accepting my application Funding –Dickinson College –PASI 2007 sponsors NSF, US DOE, UNICEN, NJIT, CONICET, CLAF, ICTP, SIAM

24. Aug. 15, 2007 Mar del Plata, ArgentinaD.P. Jackson 24/20 Talk Outline Magnetic Fluid Primer Analysis of Crossed Field Configuration Theoretical Predictions Simulation Results Summary

25. Aug. 15, 2007 Mar del Plata, ArgentinaD.P. Jackson 25/20 Final-State Configurations Various final-state configurations are possible, from a simple finger (left) to a complex labyrinth (right)

26. Aug. 15, 2007 Mar del Plata, ArgentinaD.P. Jackson 26/20 Governing Equation The Navier-Stokes equation for a viscous magnetic liquid is “convective” derivative:

27. Aug. 15, 2007 Mar del Plata, ArgentinaD.P. Jackson 27/20 Hele-Shaw Flow Standard Hele-Shaw approximations yields D’Arcy’s law with generalized pressure

28. Aug. 15, 2007 Mar del Plata, ArgentinaD.P. Jackson 28/20 Interface Equation To get an equation of motion for the interface, we use the interfacial boundary condition Note that only the normal velocity is specified here. We can choose any tangential velocity we want! We will make use of this freedom to use the conformal mapping algorithm.

29. Aug. 15, 2007 Mar del Plata, ArgentinaD.P. Jackson 29/20 Conformal Mapping The Poisson integral formula allows one to solve the Dirichlet problem on the unit disk. The Riemann mapping theorem guarantees that a simply connected domain can always be mapped to the unit disk. The interfacial boundary condition then gives an an evolution equation for the map itself (which includes the boundary) plane

30. Aug. 15, 2007 Mar del Plata, ArgentinaD.P. Jackson 30/20 Evolution Equation In terms of complex variables, the evolution equation becomes Here, A is an integral operator that takes a real valued function (in this case,  ) and returns a complex function that is analytic within the unit disk and has the specified real-valued function (  ) on the boundary Bensimon et. al., Rev. Mod. Phys. 58, 977 (1986)