Planar Geometry Ferrofluid Flows in Spatially Uniform Sinusoidally Time-varying Magnetic Fields

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1 Presented at the 11 COMSOL Conference in Boston Planar Geometry Ferrofluid Flows in Spatially Uniform Sinusoidally Time-varying Magnetic Fields Shahriar Khushrushahi, Alexander Weddemann, Young Sun Kim and Markus Zahn Massachusetts Institute of Technology, Cambridge, MA, USA

2 Ferrofluids Ferrofluids Nanosied particles in carrier liquid (diameter~1nm) Super-paramagnetic, single domain particles Coated with a surfactant (~nm) to prevent agglomeration Applications Hermetic seals (hard drives) Magnetic hyperthermia for cancer treatment permanently magnetied core d N M d solvent molecule R p S adsorbed dispersant S. Odenbach, Magnetoviscous Effects in Ferrofluids: Springer,.

3 Motivation Prior ferrofluid problems solved in COMSOL are usually in spherical and cylindrical geometries Ferrofluid pumping in planar geometry subjected to perpendicular and tangential magnetic fields Well posed problem with analytical solutions Traditionally solved using mathematical packages such as Mathematica Can COMSOL replicate these results? 3

4 Planar Geometry Setup y v v x i, ω x i y 4

5 How to impose B x field? (a) DC Current source gives H=NI/s (b) V=Λ δ(t)->b=λ /A X. He, "Ferrohydrodynamic flows in uniform and non-uniform rotating magnetic fields," Ph.D thesis, Dept. of Electrical Engineering and Computer Science, MIT, Cambridge, MA, 6. S. Khushrushahi, "Ferrofluid Spin-up Flows in Uniform and Non-uniform Rotating Magnetic Fields," PhD, Dept. of Electrical Engineering and Computer Science, MIT, Cambridge, 1. 5

6 Governing Equations Extended Navier-Stokes Equation v t = = v v M H v v gi ( ) p ( ) ( ) ( ) ( ) Neglecting Inertia Boundary condition on v, Conservation of internal angular momentum = = J ( v ) ( ) ( ) ( ' ' ' t Neglecting Inertia v( r R wall ) M H v ) ( ) Boundary condition on ω unless η =, Incompressible flow ω( r R wall ) ρ [kg/m 3 ] is the ferrofluid mass density, p [N/m ] is the fluid pressure, ζ [Ns/m ] is the vortex viscosity, η [Ns/m ] is the dynamic shear viscosity, λ [Ns/m ] is the bulk viscosity, ω [s 1 ] is the spin velocity of the ferrofluid, v is the velocity of the ferrofluid, J [kg/m] is the moment of inertia density, η [Ns] is the shear coefficient of spin viscosity and λ [Ns] is the bulk coefficient of spin viscosity, φ[%] is the magnetic particle volume fraction x 3 6

7 Magnetic Field Equations Maxwell s equations for non-conducting fluid M,, jt B B jt H H jt M Re e Re e Re e dbx B Bx constant dx dh H H constant dx Assumption B H M M eq H fluid Magnetic Relaxation Equation M 1 v M M ( M M) t eff Langevin Equation M 1 HM dv Ms[coth( a) ], a a kt eff 1 KV B 3 a p B Vh, N exp N kt f kt M s [Amps/m] represents the saturation magnetiation of the material,m d [Amps/m] is the domain magnetiation (446kA/m for magnetite), V h is the hydrodynamic volume of the particle,v p is the magnetic core volume per particle, T is the absolute temperature in Kelvin, k = [J/K] is Boltmann s constant, f [1/s] is the characteristic frequency of the material and K a is the anisotropy constant of the magnetic domains p 7

8 Substituting in Relaxation Equation M χ jm M H τ τ x x y x M χ jm M H τ τ y x B B H M H M M M x x ( x x ) x x x χ H y ( j 1) B x / y j 1 ( j 1 χ) χ H j 1 χ B xy / y j 1 ( j 1 χ) B Re[ Bxi B( x) i ] e x H Re[ H x( x) i H i ] e x ( jt) ( jt) 8

9 Force and Torque Densities μ * d μ F Re M H F x M x,f dx 4 μ 1 * * * T Re M H Ty Re M Bx μm x M H Linear and Angular Momentum Eqns p ' dy d v dx dx dv d y Ty y ' dx dx ' μ p p M x ρgx 4 9

10 Normaliation and Substitution Hˆ Mˆ Bˆ x v, H, M, B, x, v, y y, H H H d d T y T ' p d p,,,, H H H d H H y M x y yh j1b x j 1 j 1 1 x y j 1 j 1 j H B M y 1 T Re M B M H M * * y x x p ' dy 1 d v dx dx T y dv d y y dx dx 1

11 Torque Density Analytical Torque Density Small spin limit Torque Density lim Ty T 1 T y 1 * Re j 1 H B x B x y 1 H

12 COMSOL Setup Linear Momentum Equation D Incompressible Navier Stokes Module p ' dy 1 d v dx dx COMSOL Subdomain quantities Value ρ η Fx,Fy 1 dy dx Inlet BC Pressure, No viscous Stress p ' po p', No Slip BC v Outlet BC, Normal Stress f = 1

13 COMSOL Setup Angular Momentum Equation General PDE Equation dv d y T y y dx dx COMSOL Subdomain quantities Value Boundary Conditions COMSOL Quantities All walls (if ) Dirichlet boundary condition R= -, G= y All walls (if ) Neumann boundary condition G= Г, F dv d y T y y dx dx e a,d a, 13

14 COMSOL Setup Magnetic Relaxation Equation D Perpendicular Induction Currents, Vector Potential Boundary Conditions All walls H = COMSOL Quantities H, H B M x x x COMSOL Subdomain quantities Value M y yh j1b x j 1 j 1 1 x y j 1 j 1 j H B y, 14

15 η Results, Weak Rotating Fields p ' Parameters used 1, 1, 1,.1, 1, '.1 S. Khushrushahi, "Ferrofluid Spin-up Flows in Uniform and Non-uniform Rotating Magnetic Fields," PhD, Dept. of Electrical Engineering and Computer Science, MT, Cambridge, 1. 15

16 η Results, Weak Rotating Fields p ' Parameters used 1, 1, 1,.1, 1, '.1 S. Khushrushahi, "Ferrofluid Spin-up Flows in Uniform and Non-uniform Rotating Magnetic Fields," PhD, Dept. of Electrical Engineering and Computer Science, MT, Cambridge, 1. 16

17 η = Results, Strong Rotating Fields p ' Parameters used 1, 1, 1,.1, 1, ' S. Khushrushahi, "Ferrofluid Spin-up Flows in Uniform and Non-uniform Rotating Magnetic Fields," PhD, Dept. of Electrical Engineering and Computer Science, MT, Cambridge, 1. 17

18 η = Results, Strong Rotating Fields Small Spin Velocity limit does not hold p ' Parameters used 1, 1, 1,.1, 1, ' S. Khushrushahi, "Ferrofluid Spin-up Flows in Uniform and Non-uniform Rotating Magnetic Fields," PhD, Dept. of Electrical Engineering and Computer Science, MT, Cambridge, 1. 18

19 Kinks for special parameters p ' Parameters used 1,.59, 1, 5, ' L. L. V. Pioch, "Ferrofluid flow & spin profiles for positive and negative effective viscosities," Masters of Engineering, Dept. of Electrical Engineering and Computer Science, MIT, M. Zahn and L. Pioch, "Ferrofluid flows in AC and traveling wave magnetic fields with effective positive, ero or negative dynamic viscosity," J. Magn. Magn. Mater., vol. 1, p. 144, M. Zahn and L. L. Pioch, "Magnetiable fluid behaviour with effective positive, ero or negative dynamic viscosity," Indian Journal of Engineering & Materials Sciences, vol. 5, pp. 4-41, 1998.

20 Conclusions Ferrohydrodynamic flows are difficult to model Coupling of five vector equations Linear and angular momentum equations Gauss s law for magnetic flux density Ampere s law with no free current Ferrofluid magnetic relaxation equation Solving the basic planar geometry ferrofluid pumping problem is valuable before moving to cylindrical and spherical geometries COMSOL gives identical results to prior software of choice - Mathematica