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## Material Information- Title:
- Flow of Immiscible Ferrofluids in a Planar Gap in a Rotating Magnetic Field
- Creator:
- Sule, Bhumika Shrikar
- Place of Publication:
- [Gainesville, Fla.]
Florida - Publisher:
- University of Florida
- Publication Date:
- 2013
- Language:
- english
- Physical Description:
- 1 online resource (97 p.)
## Thesis/Dissertation Information- Degree:
- Master's ( M.S.)
- Degree Grantor:
- University of Florida
- Degree Disciplines:
- Chemical Engineering
- Committee Chair:
- RINALDI,CARLOS
- Committee Co-Chair:
- BUTLER,JASON E
- Graduation Date:
- 12/13/2013
## Subjects- Subjects / Keywords:
- Boundary conditions ( jstor )
Ferrofluids ( jstor ) Fluid flow ( jstor ) Magnetic fields ( jstor ) Magnetism ( jstor ) Mathematical constants ( jstor ) Pressure gradients ( jstor ) Velocity ( jstor ) Velocity distribution ( jstor ) Viscosity ( jstor ) Chemical Engineering -- Dissertations, Academic -- UF ferrofluids - Genre:
- bibliography ( marcgt )
theses ( marcgt ) government publication (state, provincial, terriorial, dependent) ( marcgt ) born-digital ( sobekcm ) Electronic Thesis or Dissertation Chemical Engineering thesis, M.S.
## Notes- Abstract:
- We have obtained analytical solutions for the flow of two layers of immiscible ferrofluids of different thickness between two parallel plates. The flow is mainly driven by the generation of antisymmetric stresses and couple stresses in the ferrofluids due to the application of a uniform rotating magnetic field. The translational velocity and spin velocity profiles were obtained for the zero spin viscosity and non-zero spin viscosity cases and the effect of applied pressure gradient on the flow was studied. The interfacial linear and internal angular momentum balance equations derived for the air-ferrofluid interface case are extended for the case when there is a ferrofluid-ferrofluid interface to obtain the velocity profiles. The magnitude of the translational velocity is directly proportional to the frequency of the applied magnetic field and the square of the magnetic field amplitude. The spin velocity is in the direction of the rotating magnetic field and its direction remains the same at lower values of applied pressure gradient. The direction of translational velocity depends on the balance between the magnitudes of vorticity, body torque, spin velocity and diffusion of internal angular momentum, however at higher values of applied pressure gradient, the pressure gradient dominates the flow. This work shows the importance of surface stresses in driving flows in ferrofluids. ( en )
- General Note:
- In the series University of Florida Digital Collections.
- General Note:
- Includes vita.
- Bibliography:
- Includes bibliographical references.
- Source of Description:
- Description based on online resource; title from PDF title page.
- Source of Description:
- This bibliographic record is available under the Creative Commons CC0 public domain dedication. The University of Florida Libraries, as creator of this bibliographic record, has waived all rights to it worldwide under copyright law, including all related and neighboring rights, to the extent allowed by law.
- Thesis:
- Thesis (M.S.)--University of Florida, 2013.
- Local:
- Adviser: RINALDI,CARLOS.
- Local:
- Co-adviser: BUTLER,JASON E.
- Electronic Access:
- RESTRICTED TO UF STUDENTS, STAFF, FACULTY, AND ON-CAMPUS USE UNTIL 2014-06-30
- Statement of Responsibility:
- by Bhumika Shrikar Sule.
## Record Information- Source Institution:
- UFRGP
- Rights Management:
- Copyright Sule, Bhumika Shrikar. Permission granted to the University of Florida to digitize, archive and distribute this item for non-profit research and educational purposes. Any reuse of this item in excess of fair use or other copyright exemptions requires permission of the copyright holder.
- Embargo Date:
- 6/30/2014
- Resource Identifier:
- 907474140 ( OCLC )
- Classification:
- LD1780 2013 ( lcc )
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PAGE 1 1 FLOW OF IMMISCIBLE FERROFLUIDS IN A PLANAR GAP IN A ROTATING MAGNETIC FIELD By BHUMIKA SHRIKAR SULE A THESIS PRESENTED TO THE GRADUATE SCHOOL OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF SCIENCE UNIVERSITY OF FLORIDA 2013 PAGE 2 2 2013 Bhumika Shrikar Sule PAGE 3 3 To my parents Shrikar Sule and Neha Sule PAGE 4 4 ACKNOWLEDGMENTS I would like to express my sincere gratitude to my advisor, Dr. Carlos Rinaldi, for his great support and advice. Thank you very much for always encouraging me to do my best, and believing I could do it. I would like to specially thank Post doctoral Research Associate, Isaac Torres Diaz for his great help and support during my research. Also I would like to thank my group members Rohan, Tapomoy, Ana, Lorena, Melissa and Maria for their invaluable friendship and support I t hank my family, e s pecially my parents for their motivation and support throughout my graduate studies. Finally I would like to thank the Chemical Engineering Department for g ivin g me the opportunity to study at University of Florida. PAGE 5 5 TABLE OF CONTENTS page ACKNOWLEDGMENTS ................................ ................................ ................................ .. 4 LIST OF TABLES ................................ ................................ ................................ ............ 7 LIST OF FIGURES ................................ ................................ ................................ .......... 8 ABSTRACT ................................ ................................ ................................ ................... 10 CHAPTER 1 INTRODUCTION ................................ ................................ ................................ .... 12 2 ANALYTICAL SOLUTION ................................ ................................ ....................... 16 2.1 Definition of the Problem ................................ ................................ ................... 16 2.2 Governing Equations ................................ ................................ ........................ 17 2.3 Magnetic Field Problem ................................ ................................ .................... 22 2.4 Zero Spin Viscosity Solution ................................ ................................ ............. 25 2.5 Non zero Spin Viscosity Solution ................................ ................................ ...... 26 2.6 Relation between Pressure Gradients and Boundary Conditions ..................... 27 3 PREDICTED VELOCITY PROFILES ................................ ................................ ...... 31 3.1 Zero Spin Viscosity Case ................................ ................................ .................. 32 3.1.1 Zero Spin Viscosity without Pressure Gradient ................................ ....... 32 3.1.2 Zero Spin Viscosity with Pressure Gradient ................................ ............ 35 3.2 Non zero Spin Viscosity Case ................................ ................................ .......... 38 3.2.1 Non zero Spin Viscosity without Pressure Gradient ................................ 38 3.2.2 Non zero Spin Viscosity with Pressure Gradient ................................ ..... 43 3.3 Comparison between Zero Spin Viscosity and Non zero Spin Viscosity Cases ................................ ................................ ................................ .................. 44 4 CONCLUSIONS ................................ ................................ ................................ ..... 46 APPENDIX A CON STANTS EVALUATED ................................ ................................ ................... 48 A 1 Zero Spin Viscosity Case ................................ ................................ ................. 48 A 2 Non zero Spin Viscosity Case ................................ ................................ .......... 49 B VELOCITY PROFILES FOR ZERO SPIN VISCOSITY CASE ................................ 87 C VELOCITY PROFILES FOR NON ZERO SPIN VISCOSITY CASE ....................... 89 PAGE 6 6 C 1 Ferrofluid 1 Ferrofluid A, Ferrofluid 2 Ferrofluid C ................................ ...... 89 C 2 Ferrofluid 1 Ferrofluid A, Ferrofluid 2 Ferrofluid B ................................ ...... 93 LIST OF REFERENCES ................................ ................................ ............................... 95 BIOGRAPHICAL SKETCH ................................ ................................ ............................ 97 PAGE 7 7 LIST OF TABLES Table page 3 1 Physical properties of ferrofluids ................................ ................................ ......... 31 PAGE 8 8 LIST OF FIGURES Figure page 2 1 Schematic illustration for the flow of two immiscible ferrofluids of thickness and between two parallel plates. The uniform rotating ma gnetic field is generated by an imposed axial magnetic field and a transverse magnetic flux density ................................ ................................ ................................ ... 16 3 1 Body torque in ferrofluids at different magnetic field frequencies. ...................... 32 3 2 Dimensional velocity profiles at different field frequencies at a constant field amplitude of H = 2 mT for Ferrofluid A and Ferrofluid C. ................................ .... 33 3 3 Dimensional velocity profiles at different magnetic field amplitudes at a constant field frequency of 150 Hz for Ferrofluid A and Ferrofluid C. ................. 34 3 4 Dimensional velocity profiles at different field frequencies at a constant field amplitude of H = 2 mT for Ferrofluid A and Ferrofluid B. ................................ .... 35 3 5 Dimensional velocity profiles at different magnetic field amplitudes at a constant field frequency of 150 Hz for Ferrofluid A and Ferroflu id B. ................. 35 3 6 Variation of dimensional translational velocity with field frequency at a constant field amplitude of H = 2mT and at different values of applied pressure gradient. ................................ ................................ ............................... 36 3 7 Effect of pressure gradient on translational velocity at a constant field frequency of 150 Hz and field amplitude of H = 2 mT. ................................ ........ 37 3 8 Non dimensional velocity profiles at different values without the application of pressure gradient with ferrofluid/non ferrofluid interface.. ............................... 38 3 9 Non dimensional velocity profiles at different values without the application of pressure gradient for Ferrofluid A and Ferrofluid C.. ................................ ...... 39 3 10 Dimensional velocity profiles at different field frequencies at a constant field amplitude of H = 2 mT for ferrofluids with different spin viscosity values Ferrofluid A and Ferrofluid C.. ................................ ................................ ............ 40 3 11 Dependence of direction of translational velocity when ferrofluids of different spin viscosity values are considered Ferrofluid A and Ferrofluid C.. ................ 41 3 12 Dependence of direction of translational velocity when ferrofluids of similar spin viscosity values are considered Ferrofluid A and Ferrofluid B. ................. 42 PAGE 9 9 3 13 Dimensional velocity profiles at different field frequencies at a constant field amplitude of H = 2 mT for ferrofluids with similar spin viscosity values.. ............ 42 3 14 Effect of pressure gradient on translational velocity at a constant field frequency of 150 Hz and field amplitude of H = 2 mT for different ferrofluid combinations. ................................ ................................ ................................ ..... 43 3 15 Non dimension al translational velocity profiles for zero spin viscosity and non zero spin viscosity cases for different combination of ferrofluids.. ............... 45 B 1 Dimensional velocities at different field amplitudes at a field frequency of 150 Hz and pressure gradient of 0.1 Pa/m. ................................ ............................... 87 B 2 Effect of positive pressure gradient applied on spin velocity at a field frequency of 150 Hz and field amplitude of H = 2 mT.. ................................ ....... 87 B 3 Effect of negative pressure gradient applied on spin velocity at a field frequency of 150 Hz and field amplitude of H = 2 mT. ................................ ........ 88 B 4 Variation of dimensional spin velocity with field frequency at a constant field amplitude of H = 2mT at different values of applied pressure gradient. .............. 88 C 1 Effect of pressure gradient applied on spin velocity profiles at a field frequency of 150 Hz and field amplitude of H = 2 mT. ................................ ........ 89 C 2 Dimensional velocities at different field amplitudes at a field frequency of 150 Hz and a pressure gradient of 0.1 Pa/m for Ferrofluid A and Ferrofluid C. ......... 90 C 3 Variation of dimensional translational velocity with field frequency at a constant field amplitude of H = 2mT at different values of applied pressure gradient for ferrofluids with widely different spin viscosity values Ferrofluid A and Ferrofluid C. ................................ ................................ ............................. 91 C 4 Variation of dimensional spin velocity with field frequency at a constant field amplitude of H = 2mT at different values of applied pressure gradient for ferrofluids with widely different spin viscosity values Ferrofluid A and Ferrofluid C. ................................ ................................ ................................ ........ 92 C 5 Variation of dimensional translational velocity with field frequency at a constant field amplitude of H = 2mT at different values of applied pressure gradient for similar spin viscosity ferrofluids Ferrofluid A and Ferrofluid B. ..... 93 C 6 Variation of d imensional spin velocity with field frequency at a constant field amplitude of H = 2mT at different values of applied pressure gradient for similar spin viscosity ferrofluids Ferrofluid A and Ferrofluid B. ......................... 94 PAGE 10 10 Ab stract of Thesis Presented to the Graduate School of the University of Florida in Partial Fulfillment of the Requirements for the Degree of Master of Science FLOW OF IMMISCIBLE FERROFLUIDS IN A PLANAR GAP IN A ROTATING MAGNETIC FIELD By Bhumika Shrikar Sule December 2013 Chair: Carlos Rinaldi Major: Chemical Engineering We have obtained analytical solutions for the flow of two layers of immiscible ferrofluids of different thi ckness between two parallel plates. The flow is mainly driven by the generation of antisymmetric stresses and couple stresses in the ferrofluids due to the application of a uniform rotating magnetic field. The translational velocity and spin velocity profiles were obtained for the zero spin viscosity and non zero spin viscosity cases and the effect of applied pressure gradient on the flow was studied. The interfacial linear and internal angular momentum balance equations derived for the air ferrofluid interface case are extended for the case when there is a ferro fluid ferro fluid interface to obtain the velocity profiles. The magnitude of the t ranslational velocity is directly proportional to the frequency of the applied magnetic field and the square of the magnetic field amplitude. The spin velocity is in the direction of the rotating magnetic field and its direction remains the same at lower v alues of applied pressure gradient. The direction of translational velocity depends on the balance between the magnitudes of vorticity, body torque, spin velocity and diffusion of internal angular momentum, however at higher values of applied pressure grad ient, the pressure gradient dominates PAGE 11 11 the flow. This work shows the importance of surface stresses in driving flow s in ferrofluids. PAGE 12 12 CHAPTER 1 INTRODUCTION Ferrofluids are stable colloidal suspensions of single domain magnetic nanoparticles in a non magnetic carrier liquid such as water or oil. 1 They are composed of ferrimagnetic particles such as magnetite (Fe 3 O 4 Fe 2 O 3 ), or cobalt ferrite (CoO Fe 2 O 3 ) and have diameters usually between 5 20 nm. There are a large number of magnetic particles in a ferrofluid per unit volume due to the extremely small size of the particles. 2 The particles in a ferrofluid are randomly oriented and the fluid has no net magnetization in the absence of an applied field. On application of a magnetic field of m oderate strength the individual dipole moments of the particles are aligned in the direction of the field, resulting in a net magnetization in the ferrofluid. The magnetic field energy in ferrofluids can be converted into motion without using external mec hanical parts. Ferrofluids have a variety of applications in engineering, microfluidics, and biomedical fields. They are used in fluid seals, inertia l dampers, as heat transfer fluids in loudspeakers and in stepper motors. 3 Ferrofluids also have applications in microfluidic pumps and valves 4 6 and in microfluidic actuators and devices wh ere they can actuate flow. 7,8 The ability to change the magnetic and optical properties of ferrofluids have made it possible for ferrofluids to be used in magneto optic sensors, 9 flow sensors, 10 and in temperature sens ors using thin ferrofluid films. 11 F lows can be generated in ferrofluids in response to magnetic fields, due to generation of antisymmetric stresses, and because of couple stresses upon application Mos kowitz and Rosensweig in 1967 12 for the fl ow of ferrofluid in a stationary cylindrical PAGE 13 13 container subjected to a uniform rotating magnetic field. Various explanations have been proposed for the phenomenon of spin up flow. The spin diffusion theory by Zaitsev and Shliomis, 13 which assumes that the magnetic field is uniform in the ferrofluid region and the fluid magnetization is proportional to the magnetic field, is based on the structured continuum theory. 14 Zaitsev and Shliomis predicted that the fluid would rotate in rigid body motion and a thin boundary layer would form near the cylinder wall. The spin diffusion theory includes the effect of spin vis cosity the dynamic coefficient in the constitutiv e equation of the couple stress representing the short range transfer of internal angular momentum. Rosensweig et al. 15 demonstrated that the observed spin up flow s in ferrof luids were due to a magnetic field driven interfacial phenomenon as they observed counter rotation of fluid and field for a concave meniscus and co rotation for a convex meniscus in a capilla ry. O bservations where the fluid co rotates or counter rotates w ith the magnetic field depending on the magnetic field frequency and amplitude were also made by Brown and Horsnell 16 and Kagan et al. 17 Rosensweig et al 15 concluded that surface stresses rather than volumetric stresses generated due to body couple s in the bulk of the fluid we re responsible for the spin up phenomenon. It was demonstrated later by the experi ments of Chaves et al 18 that the surface flow driven by the application of the rotating magnetic field coexists with a bulk flow and can be suppressed by covering the cylinder eliminating the ai r ferrofluid interface. Krau ss et al 5,19 performed experiments in a free surface geometry consisting of a circular duct with square cross section with a pool of ferrofluid in contact with air, and in which a uniform rotating magnetic field is applied with rotation axis parallel to the PAGE 14 14 ferrofluid air interface This geometry was used t o evaluate the existence of surface flows in ferrofluids. Their experiments showed that the magnitude of flow increases with the square of magnetic field amplitude, is proportional to the thickness of the ferrofluid layer, and has a maximum at a fluid spe cific frequency T he ir flow measurements were consistent with the effects of magn etic stresses at the ferrofluid air interface as indicated by numerical calculations. The equations governing the interfacial stress balance in ferrofluids have also been rec ently derived by Rosensweig 20 However the analyses of Krau ss et al. 19 and of Rosensweig 20 did not consider the potential role of spin viscosity, related to the couple stresses, in driving the flow in ferrofluids. Rinaldi and collaborators 18,21 25 showed qualitative agreement between experimental translational velocity profiles for cylindrical, annular and spherical geometries with the theoretical predictio ns of the spin diffusion theory The value of the spin viscosity estimated from experimental measurement 18,22,25 w as in the range of 10 12 to 10 8 kg m s 1 which is several orders of magnitude higher than the value estimated by Zaitsev and Shliomis 13 and Feng et a l 26 on the basis of dimensional arguments in the infinite dilution limit. These studies prov ide evidence for the existence of couple stresses in ferrofluids and the role of spin viscosity in driving the ferrofluid flow in the presence of rotating magnetic fields Past analyses for the flow driven by the application of a uniform rotating magnetic field were made considering the presence of a single ferrofluid with uniform body torque in the entire fe rrofluid region and neglecting stresses at the air ferrofluid in terface. Motivated by the results of Krau ss et al., 19 we consider ed a ferrofluid non ferrofluid and ferrofluid ferrofluid interface to study the flows driven by surface stress balance. In this PAGE 15 15 contribution, analytical sol utions are obtained for the general case of flow of two layers of immiscible ferrofluids between two plates driven by the application of a uniform rotating magnetic field. The interfacial linear momentum balance and internal angular momentum balance equations including the effects of vortex viscosity, spin viscosity, and surface tension derived by Chaves and Rinaldi 27 are extended to the analysis of two immiscible f errofluids to obtain the translational and spin velocity profiles. The existence of tangential stress and couple stress at the interface and the difference in the magnitudes of body torques generated in t he two ferrofluids is studied. V elocity profiles are obtained for the zero spin viscosity and non zero spin viscosity cases considering different ferrofluid combinations and the effect of applied pressure gradient on the flows is analyzed. PAGE 16 16 C HAPTER 2 ANALYTICAL SOLUTION 2.1 De finition of the P roblem The flow geometry analyzed is illustrated in Figure 2 1. It consists of two layers of immiscible ferrofluids of thickness and between two parallel horizontal plates. The flow in the ferrofluids is induced by the application of a uniform rotating magnetic field. The field is generated by an imposed uniform magnetic field and uniform transverse magnetic flux density represented by the functions (2 1) (2 2) In Equations ( 2 1 ) and ( 2 2 ) and are the complex amplitudes of magnetic field and magnetic flux density respectively, which are independent of position, is the radian frequency and is the imaginary number Figure 2 1 Schematic illustration for the flow of two immiscible ferrofluids of thickness and between two parallel plates. The uniform rotating magnetic field is generated by an imposed axial m agnetic field and a transverse magnetic flux density PAGE 17 17 The thickness of the ferrofluids and is small compared to the length and width of the plate, thus the vectors of translational velocity and spin velocity are unidirectional and dependent only on the coordinate (2 3) The flow direction and the direction of spin of the particles depend on the direction of rotation of the magnetic field. 27 For the purpose of this analysis, we assume the magnetic field to be rotating in the counterclockwise direction. 2.2 Governing Equations The ferrohydrodynamic equations governing the flow as given by Rosensweig 1 are (2 4) (2 5) (2 6) (2 7) Equation ( 2 4 ) is the equation of continuity for an incompressible fluid, Equation ( 2 5 ) represents the conservation of linear momentum, Equation ( 2 6 ) is the internal angular momentum equation and Equation ( 2 7 ) is the magnetization relaxation equation. In these equations, is the local mass average velocity, is the spin velocity vector, is the fluid density, is the gravitational acceleration, is the permeability of free space is the flu id pressure, is the suspension scale shear PAGE 18 18 viscosity, is the vortex viscosity, is the coefficient of spin viscosity, is the moment of inertia density of the suspension, is the relaxation time, is the magnetization vector of the suspension, and is the magnetic field vector. The second term on the right hand side of the linear momentum equation, Equation ( 2 5 ) represents the magnetic body force due to field inhomogeneities and the fourth represents the antisymmetric component of the Cauchy stress which occurs when there is difference between the rate of rotation of the particles and half the local vorticity of the flow. 28 In the internal angular momentum bal ance equation, Equation ( 2 6 ) the first tem on the right hand side represents the external body torque acting on the ferrofluid whenever the local magnetization is not aligned to the applied field, the second repres en ts the interchange of momentum between internal angular and macroscopic linear forms and the third term represents the diffusion of internal angular momentum between contiguous material elements. 28 The first term on the right hand side of Equation ( 2 7 ) is the spin magnetization coupling term and the second term represents the orientational diffusion of towards an equilibrium value. The Langevin relation 1 gives the equilibrium magnetization for a superparamagnetic ferrofluid as (2 8) where is the Langevin function, is the Langevin parameter which is a measure of the relative magnitudes of magnetic and thermal energy, is the domain magnetization of the magnetic nanoparticles, is the absolute temperature and is the volume of magnetic cores. I t is commonly assumed PAGE 19 19 that the magnetic relaxation can simultaneously occur by the Brownian and Nel mechanisms, (2 9) where is the Brownian relaxation time and is the Ne l relaxation time. The vortex viscosity in the dilute limit for monodisperse nanoparticle suspensions in Newtonian fluids is given as 22 (2 10) where is the shear viscosity of the suspending fluid and is the hydrodynamic volume fraction of suspended particles. magnetoquasistatic limit 29 are (2 11) The interfacial boundary conditions for the magnetic field the continuity of the normal component of magnetic induction and the jump in the tangential magnetic field due to surface currents are (2 12) (2 13) in which is a unit vector, locally normal to the interface and pointing from phase 2 to 1 and is the surface current density. Scaling of the g overning e quations The problem is solved for the two ferrofluids separately using a regular perturbation expansion to decouple the ferrohydrodynamic equations and to obtain the magnetic field and flow field solutions. The ferrohydrodynamic equations are considered under the assumptions of PAGE 20 20 incompressible flow, low Reynolds number and small amplitude of the magnetic field in order to formulate the regular perturbation problem. The perturbation parameter is defined as ( 2 14) and the conditions which usually apply in ferrofluids are used as shown by Chaves et al. 22 The following scaled variables are introduced (2 15) (2 16) (2 17) as given by Chaves et al. 22 f ferrofluid 1 and ferrofluid 2 respectively and is the initial magnetic susceptibility. The resulting scaled variables are assumed to be of order unity and they are denoted by an over tilde. The details of the scales for pressure, sp in and linear velocity are given by Chaves et al. 22 is the dynamic pressure 30 obtained by taking and The equilibrium magnetization in Equation ( 2 8 ) reduces to the form (2 18) by expanding Equation ( 2 8 ) in powers of and taking the terms with power zero in as shown by Chaves et al. 22 PAGE 21 21 Similarly, using the scaled variables given in Equations ( 2 15 ) ( 2 16 ) and ( 2 17 ) the dimensionless scaled ferrohydrodynamic equations for the zeroth order problem of the regular perturbation expansion for f errofluid 1 and ferrofluid 2 are (2 19) (2 20) (2 21) (2 22) (2 23) (2 24) (2 25) (2 26) (2 27) In Equations ( 2 15 ) to ( 2 17 ) and Equations ( 2 19 ) to ( 2 27 ) is the frequency of the applied magnetic field and the dimensionless frequency if defined as (2 28) Additionally, the effective viscosity of the ferrofluids is (2 29) PAGE 22 22 and the dimensionless parameter is defined as ( 2 30) 2.3 Magnetic Field Problem Under the assumption that the fluid layers are infinitely long in the and (2 31) (2 32) which shows that the components are independent of However, the magnetization generated by the imposed fields makes the components potentially dependent on the coordinate 31 We assume the vectors have the functional form (2 33) (2 34) (2 35) where the symbol (^) is used to denote the dimensionless complex components. The magnetization equation for the two ferrofluids Equations ( 2 24 ) and ( 2 25 ) under these assumptions reduces to the components (2 36) PAGE 23 23 (2 37) Introducing Equations ( 2 33 ) and ( 2 35 ) in the Equations ( 2 36 ) and ( 2 37 ) the complex components of the magnetization are obtained as (2 38) (2 39) To obtain the complex components of the magnetic field vector, we introduce Equations ( 2 33 ) to ( 2 35 ) and Equation ( 2 38 ) in the magnetic flux density vector, (2 40) resulting in (2 41) (2 42) Substituting Equation ( 2 41 ) in Equation ( 2 38 ) yields, (2 43) Equations ( 2 41 ) and ( 2 43 ) show that are uniform for the zeroth order analysis. From the equations of magnetic field and magnetization, we conclude that the magnetic field is uniform, therefore the magnetic body force in both the ferrofluids is zero PAGE 24 24 (2 44) The component of the body couple is given by (2 45) (2 46) where the subscript (k) is used to indicate ferrofluid 1 or ferrofluid 2. Equation ( 2 46 ) is simplified by using the relation of the real part of the product of two complex functions and given by ( 2 47) where the superscript (*) denotes the complex conjugate. Thus the e xpression for the component of the body torque is obtained as (2 48) (2 49) To generate a uniform rotating magnetic field in the counterclockwise direction, the imposed z component of the magnetic field and the component of the magnetic induction are taken as 27 which gives the magnetic body couple i n the ferrofluids as (2 50) PAGE 25 25 (2 51) The above expressions for the body couple indicate that the body couple is constant within each ferrofluid. 2.4 Zero Spin Viscosity Solution In this case the spin viscosity in both the ferrofluids is taken to be zero that is, couple stresses are neglected. This condition reduces the linear momentum balance and internal angular momentum balance equations, Equatio ns ( 2 20 ) to ( 2 23 ) in component form, to (2 52) (2 53) (2 54) ( 2 55) Differentiating Equations ( 2 54 ) and ( 2 55 ) and substituting the result in Equations ( 2 52 ) and ( 2 53 ) respectively and integrating the resulting differential equations, we find that the translational velocity profiles a re given by (2 56) (2 57) PAGE 26 26 Equations ( 2 56 ) and ( 2 57 ) are differentiated with respect to and the result is substituted in Equations ( 2 54 ) and ( 2 55 ) respectively to get the spin velocity profiles as (2 58) (2 59) In the above equations, are constants which depend on the properties of the ferrofluids the applied magnetic field and the relevant boundary conditions 2.5 Non zero Spin Viscosity Solution By considering the existence of spin viscosity, the linear momentum balance equations reduce to the component forms given in Equations ( 2 52 ) and ( 2 53 ) and the internal angular momentum balance equations, Equations ( 2 22 ) and ( 2 23 ) reduce to the component forms given by (2 60) (2 61) We integrate Equations ( 2 52 ) and ( 2 53 ) once and introduce the result in Equations ( 2 60 ) and ( 2 61 ) respectively to yield a non homogeneous differential equation for the component of the spin veloc ity, with the solution (2 62) PAGE 27 27 (2 63) The translational velocity profiles are determined by substituting Equations ( 2 62 ) and ( 2 63 ) in Equations ( 2 52 ) and ( 2 53 ) and integrating to obtain (2 64) (2 65) In the above equations, and are constants which depend on the properties of the ferrofluids the applied magnetic field and the relevant boundary c onditions. 2.6 Relation b etween Pressure Gradients and Boundary Conditions The total linear and angular momentum balance is required to determine the relation between pressure gradients in the two ferrofluids and to obtain the boundary conditions at the fluid fluid interface. The total interfacial linear momentum balance at the fluid fluid interface is derived in 27 as (2 66) where is the unit normal, is the radius of curvature, and is the surface tension. The interface is considered flat and the un it PAGE 28 28 normal is in the direction. Using the scaled variables as defined in Equation ( 2 15 ) the normal component of the interfacial stress balance is given by the equation (2 67) The Cauchy stress tensor in ferrofluids is given by the relation 1 (2 68) Using the normal component of the Ca uch y stress tensor in Equation ( 2 68 ) and substituting the expressions for magnetic field and magnetic flux density from Equations ( 2 33 ) ( 2 34 ) and ( 2 41 ) and considering Equation ( 2 67 ) is solved to obtain the pressure relation between the two ferrofluids as (2 69) where (2 70) From the above relation, it is seen that the static pressure difference between the two fluids depends only on the properties of the ferrofluids and the frequency of the magnetic field. The pressure difference is independent of the z direction, which means that has the same constant value for both fluids. 30 As given by the relation and considering gravity acting perpendicular to the flow PAGE 29 29 we obtain that the dynamical pressure gradient also has the same constant value throughout both ferrofluids, such that (2 71) Scaled b oun dary c onditions To evaluate the constants in the zero spin viscosity case, we consider the no slip condition for translational velocity at the plates given by (2 72) The boundary cond itions at the interface include the continuity of tangential component of translational velocity (2 73) and the tangential component of total linear momentum balance, Equation ( 2 66 ) (2 74) the deta iled derivation for Equation ( 2 74 ) is given by Chaves and Rinaldi. 27 In order to evaluate the constants in the non zero spin viscosity case, we use the boundary conditions for t he spin velocities in addition to the boundary conditions given in Equations ( 2 72 ) to ( 2 74 ) The no slip condition for the spin velocity at the walls is given by (2 75) we consider the boundary condition for spin velocities at the interface as (2 76) PAGE 30 30 analogous to the continuity of tangential velocity at the fluid fluid interface. The interfacial internal angular momentum balance, derived in detail by Chaves and Rinaldi, 27 is extended to the case of two ferrofluids and the boundary condition at the interface is given by (2 77) Using the boundary conditions mentioned, the constants for the zero spin viscosity and non zero spin viscosity cases were evaluated using Mathematica and the velocity profiles are plotted in Chapter 3. The constants are given in Appendix A. PAGE 31 31 CHAPTER 3 PREDICTED VELOCITY PROFILES To calculate the translational and spin velocity profiles, we considered ferrofluids with representative properties as ferrofluid 1 and ferrofluid 2. The properties of the representative ferrofluids are summarized in Table 3 1. The thickness of ferrofluid 1 is considered to be 2 mm and the aspect ratio is taken to be 2. To show the dependence of flow profiles on properties of ferrofluids, we consider Ferrofluid A and Ferrofluid B with similar properties and Ferrofluid C with properties different a s compared to Ferrofluid A and F errofluid B. Ferrofluid C has a spin viscosity which is two orders of magnitude higher than that of Ferrofluid A and Ferrofluid B. Table 3 1 Physical properties of ferrofluids Physical Properties Ferrofluid A Ferrofluid B Ferrofluid C (kg/m 3 ) 1080 1030 1030 (m Pa s) 5.1 4.5 1.03 (m Pa s) 1.64 1.64 1.02 (s) 1.7 x 10 6 1.9x 10 6 1.67 x 10 5 0.9 1.2 0.1 0.05 0.04 0.002 (m Pa s) 0.123 0.0984 0.00306 (kg m/s) 5.6 x 10 10 6 x 10 10 3.6 x 10 8 In Figure 3 1 non dimensional values of torque generated in the ferrofluids are plotted against the frequency of the applied magnetic field. The body torque d epends on the properties of the ferrofluid such as initial magnetic susceptibility and relaxation time and on the direction of the rotating magnetic field. For counterclockwise rotating magnetic field, positive torque is generated in each of the ferrofluids. The torque PAGE 32 32 generated is constant with in each ferrofluid. Figure 3 1 shows that the magnitude of torque increases with increase in the frequency of the magnetic field. It is also seen that the difference between the magnitudes of torques in the ferrofluids increases with increase in the magnet ic field frequency. Figure 3 1 Body torque in ferrofluids at different magnetic field frequencies 3.1 Zero Spin Viscosity Case To plot the velocity profiles for the zero spin viscosity case, we have taken ferrofluid 1 as Ferrofluid A and ferrofluid 2 as Ferrofluid C or Ferrofluid B and set for both. 3.1.1 Zero Spin Viscosity without Pressure Gradient Figures 3 2 and 3 3 show the predicted dimensional translational and spin velocity profiles for several frequencies and amplitudes of the applied magnetic field for Ferrofluid A and Ferrofluid C It is seen that the translational velocity varies linearly with distance from the plates in both the ferrofluids and the spin velocity is constant within each ferrofluid. The spin velocity is in the positive y direction, that is, in the direction of rotating magnetic field. The magnitude of the body torque generated in ferrofluid 2 is PAGE 33 33 higher than in ferrofluid 1, thus driving the translat ional velocity in the positive z direction i n both ferrofluids. It was also seen that if the torque generated in ferrofluid 1 is higher than that generated in ferrofluid 2, the translational velocity is in the negative z direction. The existence of translational velocity in the zero spin viscosity c ase is in contrast with the prediction of no flow in planar, cylindrical, annular and spherical geometries in the absence of a free surface 18,21 25 These cases concluded that flow in ferrofluids is due to the presence of couple stresses, related to the diffusive transport of internal angular momentum which are neglected in the zero spin viscosity ca se. In the case of a ferrofluid non fer rofluid interface or ferrofluid ferrofluid interface, there is an imbalance of the tangential component of the antisymmetric stress tensor, which drives the flow. Also as seen from Figure 3 1, the magnitude of the body torques generated in the two ferrofluids is different, driving the flow in ferrofluids. Figure 3 2 Dimensional velocity profiles at different field frequencies at a constant field amplitude of H = 2 mT for Ferrofluid A and Ferrofluid C A) Translational velocity profiles, B) spin velocity profiles A B PAGE 34 34 Figure 3 3 Dimensional velocity profiles at different magnetic field amplitudes at a constant field frequency of 1 50 Hz for Ferrofluid A and Ferrofluid C A) Translational velocity profiles, B) spin velocity profiles. The magnitude of the translational and spin velocities increases with increasing frequency of the applied magnetic field at a constant field amplitude as seen in Figure 3 2. This is because the difference betw een the magnitudes of torques generated in the two ferrofluids increases with the field frequency as shown in Figure 3 1. Figure 3 3 shows that the magnitude of dimensional velocities increases with the square of the magnetic field amplitude at a constant field frequency. The predicted dimensional translational and spin velocity profiles at different field frequencies and field amplitudes for Ferrofluid A and Ferrofluid B are plotted in Figures 3 4 and 3 5 As the properties of Ferrofluid A and Ferrofluid B are similar, the difference between the magnitude of torques generated in the two ferrofluids is small as seen from Figure 3 1, and thus the magnitude of translational velocity obtained is less compared to the case when Ferrofluid A and Ferrofluid C are considered. It is therefore concluded that for flows to be significant, properties of the two immiscible ferrofluids should be widely different. A B PAGE 35 35 Figure 3 4 Dimensional velocity profiles at different field frequencies at a constant field amplitude of H = 2 mT for Ferrofluid A and Ferrofluid B. A) Translational velocity profiles, B) spin velocity profiles Figure 3 5 Dimensional velocity profiles at differ ent magnetic field amplitudes at a constant field frequency of 150 Hz for Ferrofluid A and Ferrofluid B A) Translational velocity profiles, B) spin velocity profiles 3.1.2 Z ero Spin Viscosity with Pressure Gradient The application of a pressure gradient c auses the translational velocity to acquire a parabolic shape as seen in Figure 3 6 The direction of velocity depends on the magnitude and direction of the applied pressure gradient. A positive pressure gradient drives the flow in the negative z directio n. Figure s 3 6 show that this effect is seen the most at lower values of applied field frequency. As noted, at a higher magnitude of PAGE 36 36 positive pressure gradient, the pressure gradient dominates the flow and the translational velocity is in the negative z di rection in both ferrofluids. The application of a negative pressure gradient causes the ferrofluids to flow in the positive z direction as seen in Figure s 3 6 C, D that is, in the same direction as the flow generated by the rotating magnetic field fo r the particular properties of Ferrofluid A and Ferrofluid C Thus the negative pressure gradient assists the flow and the magnitude of velocity is higher for a particular field frequency with a higher pressure gradient applied. Also it is seen that at hi gher applied pressure gradient values, the dependence of velocity magnitude on the field frequency reduces. Figures 3 7 A, B clearly show the effect of applied pressure gradient on the magnitude and direction of translational velocity, as explained before Figure 3 6 Variation of dimensional translational velocity with field frequency at a constant field amplitude of H = 2mT and at different values of applied pressure gradient. A) (dP/dz) = 0.1 Pa/m, B) (dP/dz) = 1 Pa/m, C) (dP/dz) = 0.1 Pa /m, D) (dP/dz) = 1 Pa/m A B PAGE 37 37 Figure 3 6. Continued Figure 3 7 Effect of pressure gradient on translational velocity at a constant field frequency of 150 Hz and field amplitude of H = 2 mT A) Application of negative pressure gradient, B) application of positive pressure gradient. The spin velocity varies linearly with when a pressure gradient is applied. Low values of applied pressure gradient have minimal effect on the magnitude of spin velocity and the spin velocity remains in the direction of the r otating magnetic field ( positive y di rection) A t higher values of pressure gradient there is an effect on the magnitude of spin velocity. The graphs showing the effect of pressure gradient on spin velocity are given in Appendix B. Also t he graphs for the variation of translational and C D A B PAGE 38 38 spin velocity profiles with field amplitude at a constant field frequency o f 150 Hz and pressure gradient of 0.1 Pa/m are given in Appendix B. 3.2 Non zero Spin Viscosity Case 3.2.1 Non zero Spin Viscosity without Pressure Gradient Using the appropriate solutions we plot the non dimensional velocity graphs for the case of fer rofluid non ferrofluid interface. We consider ferrofluid 1 as Ferrofluid A and the non ferrofluid as water. As seen from Figure 3 8 the translational velocity is not linearly varying with and the solution tends to the zero spin vi scosity case as Figure 3 8 A shows that the magnitude of flow increases with increasing which is opposite to what is observed in spin up flow in a cylindrical container. I ncreasing spin viscosity acts to resist rotation of the particles, leading to a lower spin velocity and concomitantly, a decreased mismatch in the interfacial momentum balance. This is evident in Figure 3 8 B, where t he spin velocity is not constant and tends to fo rm a boundary layer close to the wall at high values of similar to the case of spin up flow in a cylindrical container. Figure 3 8 Non dimensional velocity profiles at different values without th e a pplication of pressure gradient with ferrofluid/non ferrofluid interface. A) Translational velocity profiles, B) spin velocity profiles. A B PAGE 39 39 Figure 3 9 Non dimensional velocity profiles at different values without the a pplication of pressure gradient for Ferrofluid A and Ferrofluid C. A) Translational velocity profiles, B) spin velocity profiles. Figure 3 9 shows the non dimensional translational and spin velocity profiles at different values of considering both as ferrofluids. It is seen that the magnitude of the translational velocity increases with increasing value as seen in the ferrofluid non ferrofluid case, however the behavior observed in case of spin velocity is different. Also it was observed that the solution tends to infinity at higher values of so the non dimensional velocity plots are made only at lower values. Interestingly t he solution does not tend to the zero spin viscosity solution as This may be due to the spin velocity being required to satisfy a boundary condition at the interface which is taken to be analogous to the condition of continuity of tangential velocity at the fluid fluid interface. Hence, appears to lead to a singular problem. From Figure 3 10 it is seen that the magnitude of dimensional velocity increases with increase in frequency of the applied magnetic field, as observed for the zero spin viscosity case. The spin velocity is in the positive y direction in both ferrofluids, however the translational velocity is in the negative z direction, opposite to that observed when the spin visco sity is zero. A B PAGE 40 40 Figure 3 10 Dimensional velocity profiles at different field frequencies at a constant field amplitude of H = 2 mT for ferrofluids with different spin viscosity values Ferrofluid A and Ferrofluid C A) Translational velocity profiles, B) spin velocity profiles. The flow reversing in the non zero spin viscosity case is explained by the internal angular momentum balance equations, (3 1) (3 2) We consider two cases different spin viscosity values in both ferrofluids and similar spin viscosity values in both ferrofluids. Figure 3 11 shows non dimensional plots of translational velocity, slope of the velocity, spin velocity and second derivative of spin velocity for Ferrofluid A and Ferrofluid C. Ferrofluid C has a spin viscosity value two orders of magnitude higher than Ferrofluid A. From Figure 3 11, it is seen that the translational velocity is in the negative Z direction in both ferrofluids e xcept very close to the wall in Ferrofluid A where the velocity is in the positive z direction. This region is not seen clearly due to the magnitude of velocity in the order of 10 6 The spin velocity is B A PAGE 41 41 close to zero near each wall to satisfy the boundar y condition and the body couple dominates the diffusion term in Equations (3 1) and (3 2). In order to balance the positive torque generated by the counter clockwise rotating field, the slope of the velocity is positive and the vorticity is negative very c lose to the wall in Ferrofluid A and in the entire region in Ferrofluid C. This implies a positive velocity very close to the wall in Ferrofluid A and negative velocity in the entire Ferrofluid C region. In the region away from the wall in Ferrofluid A, th e spin velocity and the spin diffusion dominate the torque and the velocity has a negative slope as shown in Figure 3 11 A, thus the translational velocity is along the negative z axis. Figure 3 11 Dependence of direction of translational velocity when ferrofluids of different spin viscosity values are considered Ferrofluid A and Ferrofluid C A) Non dimensional translational velocity and slope of velocity, B) n on dimensional spin velocity and sec ond order derivative of spin velocity. When ferrofluids with similar spin viscosity values are considered, as shown in Figure 3 12, the body couples dominate the flow near the walls and the slope of velocity is positive near the walls, resulting in transl ational velocity in the positive z direction in Ferrofluid A and in the negative z direction in Ferrofluid B. In the region near the interface, the magnitude of spin velocity increases as seen in Figure 3 12 B, so the slope of the velocity becomes negativ e. This is shown by the decrease in magnitude of A B PAGE 42 42 translational velocity after the peak in Ferrofluid A and the flow going in the negative z direction in Ferrofluid B. Figure 3 1 2 Dependence of direction of translational velocity when ferrofl uids of similar spin viscosity values are considered Ferrofluid A and Ferrofluid B A) Non dimensional translational velocity and slope of velocity, B) Non dimensional spin velocity and second order derivative of spin. Figure 3 1 3 Dimensional velocity profiles at different field frequencies at a constant field amplitude of H = 2 mT for ferrofluids with similar spin viscosity values. A) Translational velocity for Ferrofluid A and F errofluid B B) spin velocity for Ferrofluid A and F errofluid B. A B A B PAGE 43 43 Plots for dimensional velocity profiles for Ferrofluid A and Ferrofluid B are shown in Figure 3 13 at different values of field frequencies. They show the same behavior as in the zero spin viscosity case. 3.2.2 Non zero Spin Viscosity with Pre ssure Gradient To show the effect of applied pressure gradient on the translational velocity profiles for the non zero spin viscosity case, we have plotted graphs at different pressure gradient values at a field frequency of 150 Hz and field amplit ude of 2 mT as shown in Figures 3 1 4 The nature of the graphs is similar to th ose observed for the zero spin viscosity case and it is clearly seen that the flow is in the direction opposite the pressure gradient at higher values of applied pressure gradient Figure 3 14 Effect of pressure gradient on translational velocity at a constant field frequency of 150 Hz and field amplitude of H = 2 mT for different ferrofluid combinations. A) Positive pressure gradient for Ferrofluid A and Ferrofluid C, B) negative pressure gradient for Ferrofluid A and Ferrofluid C, C) positive pressure gradient for Ferrofluid A and Ferrofluid B, D) negative pressure gradient for Ferrofluid A and Ferrofluid B A B PAGE 44 44 Figure 3 14. Continued As in the zero spin viscosity case, the spin velocity remains in the positive y direction at low values of applied pressure gradient and the plots are given in Appendix C. The plots for dimensional velocities at various field frequencies and field amplitud es for different fer rofluid combination s and pressure gradient values are give n in Appendix C and they show similar behavior as observed for the zero spin viscosity case. 3.3 Comparison between Zero Spin Viscosity and Non zero Spin Viscosity Cases Figure 3 15 shows the non dimensional translational velocity profiles for zero spin viscosity and non zero spin viscosity cases for Ferrofluid A and Ferrofluid C and Ferrofluid A and Ferrofluid B c ombination s Comparing the velocity profiles in the zero spin viscosity and non zero spin viscosity cases, it is seen that the flow direction changes when the effect of spin viscosity is considered. Such a stark qualitative change in the flow profile could be used as an experimental verification of the existence of spin viscosity neglected in the analyse s by Rosensweig et al. 15 and Krau ss et al. 19 As such, experiments where two immiscible ferrofluids are in contact and subjected to a rotating magneti c fie ld, in the manner of Krau ss et al 19 could test the existence of the spin C D PAGE 45 45 viscosity and complement experiments carried out with a single ferroflui d in a cylindrical geometry. 22 Figure 3 15 Non dimensional translational velocity profiles for zero spin viscosity and non zero spin viscosity cases for different combination of ferrofluids. A) Ferrofluid A and Ferrofluid C, B) Ferrofluid A and Ferrofluid B. An interesting feature of the solutions obtained here is that the non zero spin viscosity solution does not tend to the zero spin viscosity case as This discrepancy may be due to the boundary condition of contin uity of spin velocity at the ferroflui d ferrofluid interface, which at the moment is ad hoc. As such, experimental measurements of the velocity profile could also be used to test whether this boundary condition is correct. When a ferrofluid non ferrofluid interface was considered as shown in Figure 3 8, the condition of continuity of spin velocity at the interface was not required and the solution tends to the zero spin viscosity case as A B PAGE 46 46 C HAPTER 4 CONCLUSIONS In this work we have obtained analytical solutions for the translational and spin velocity profiles for the flow of two immiscible ferrofluids driven by the application of a uniform rotating magnetic field, extending past work of modeling the flow of ferrofluid with a ferrofluid air interface. The analysis predicts the existence of translational velocity in the zero spin viscosity ca se. This is in contrast with analyses that predict no flow when the spin viscosity is neglected for the bulk flow of ferrofluid in a cylin drical container. It therefore underscores the importance of surface stresses responsible for flow in ferrofluids as implied by Rosensweig et al 15 The interfacial linear and internal angular momentum balance conditions show that a flow can be generated by an imbalance of the tangential component of the antisymmetric stress tensor at the fluid fluid interface in addition to the couple stresses. The analyse s using different combination s of ferrofluids in the zero spin viscosity and non zero spin viscosity cases show that for flow to be significant, the properties of the ferrofluids must be significantly different. The magnitude of the dimensional translational and spin velocity in the two ferrofluids increases with the frequency of the applied magnetic field and also increases with the square of the magnetic field amplitude as seen from the velocity profiles plotted for the different cases. The spin velocity is in the direction of rotating magnetic field, direction, and low values of applied pressure gradient have minimal effect on the spin velocity. The direction of translational velocity depends on the balance between the terms in the internal angular momentum balance equation. At higher values o f applied pressure gradient, the PAGE 47 47 pressure gradient dominates the flow generated by the rotating magnetic field and the ferrofluids flow in the direction of applied pressure gradient. This work of modeling the flow of ferro fluids in a planar channel has po tential applications in microfluidic and nanofluidic devices Ferrofluids are being considered as components in microfluidic devices where they could actuate flow. The parameters of the rotating magnetic field and the magnitude and direction of the pressur e gradient can be varied to get the desired flow magnitude and direction. PAGE 48 48 APPENDIX A CONSTANTS EVALUATED The constants for zero spin viscosity and non zero spin viscosity cases are evaluated using Mathematica A 1 Zero Spin Viscosity Case PAGE 49 49 A 2 Non zero Spin Viscosity Case PAGE 50 50 PAGE 51 51 PAGE 52 52 PAGE 53 53 PAGE 54 54 PAGE 55 55 PAGE 56 56 PAGE 57 57 PAGE 58 58 PAGE 59 59 PAGE 60 60 PAGE 61 61 PAGE 62 62 PAGE 63 63 PAGE 64 64 PAGE 65 65 PAGE 66 66 PAGE 67 67 PAGE 68 68 PAGE 69 69 PAGE 70 70 PAGE 71 71 PAGE 72 72 PAGE 73 73 PAGE 74 74 PAGE 75 75 PAGE 76 76 PAGE 77 77 PAGE 78 78 PAGE 79 79 PAGE 80 80 PAGE 81 81 PAGE 82 82 PAGE 83 83 PAGE 84 84 PAGE 85 85 PAGE 86 86 PAGE 87 87 A PPENDIX B VELOCITY PROFILES FOR ZERO SPIN VISCOSITY CASE Figure B 1 Dimensional velocities at different field amplitudes at a field frequency of 150 Hz and pressure gradient of 0.1 Pa/m. A) Translational velocity profiles, B) Spin velocity profiles The magnitude of velocitiy increases with increasing magnetic fi el d amplitude at a constant field freque ncy and pressure gradient value as shown in F igure B 1 Figure B 2 Effect of positive pressure gradient applied on spin velocity at a field frequency of 150 Hz and field amplitude of H = 2 mT. A) Spin velocity of f errofluid 1, B) spin velocity of ferrofluid 2. From Figure B 2, it is seen that the magnitude of spin velocity increases with increasing value of positive pressure gradient applied in the entire ferrofluid 1region and in ferrofluid 2 upto a certain point after which the magnitude of spin velocity decreases. A B A B PAGE 88 88 Figure B 3 shows that the magnitude of spin velocity decreases with increasing magnitude of negative pressure gradient applied in entire ferrofluid 1 region and in ferrofluid 2 upto a certain point after which the magnitude of spin velocity increases. Figure B 3 Effect of negative pressure gradient applied on spin velocity at a field frequency of 150 Hz and field amplitude of H = 2 mT. A) Spin velocity of f errofluid 1, B) spin velocity of ferrofluid 2. The effect of field frequency on spin velocity profiles a t different values of applied pressure gradient is given in Figure B 4. Figure B 4 Variation of dimensional spin velocity with field frequency at a constant field amplitude of H = 2mT at different values of applied pressure gradient. A) (dP/d z) = 0.1 Pa/m, B) (dP/dz) = 0.1 Pa/m B A A B PAGE 89 89 A PPENDIX C VELOCITY PROFILES FOR NON ZERO SPIN VISCOSITY CASE C 1 Ferrofluid 1 Ferrofluid A, Ferrofluid 2 Ferrofluid C As seen in Figure C 1, t he magnitude of spin velocity increases with increasing value of positive pressure gradient applied and decreases with increasing value of negative pressure gradient applied in both ferrofluids for the non zero spin viscosity case. Figure C 1 Effect of pressure gradient applied on spin velocity profiles at a field frequency of 150 Hz and field amplitude of H = 2 mT. A) Positive pressure gradient values, B) negative pressure gradient values. A B PAGE 90 90 Figure C 2 shows that the magnitude o f translational and spin velocity increases with increasing magnetic field amplitude similar to the zero spin viscosity case Figure C 2 Dimensional velocities at different field amplitudes at a field frequency of 150 Hz and a pressure gradient of 0.1 Pa/m for Ferrofluid A and Ferrofluid C. A) Translational velocity profiles, B) Spin velocity profiles A B PAGE 91 91 Graphs showing the effect of field frequency on translational and spin velocity profiles at different values of app lied pressure gradient are given in Figures C 3 and C 4 for Ferrofluid A and Ferrofluid C combination. Figure C 3 Variation of dimensional translational velocity with field frequency at a constant field amplitude of H = 2mT at different values of applied pressure gradient for ferrofluids with widely different spin viscosity values Ferrofluid A and Ferrofluid C. A) (dP/dz) = 0.1 Pa/m, B) (dP/dz) = 1 Pa/m, C) (dP/dz) = 0.1 Pa/m, D) (dP/dz) = 1 Pa/m A B C D PAGE 92 92 Figure C 4 Variation of dimensional spin velocity with field frequency at a constant field amplitude of H = 2mT at different values of applied pressure gradient for ferrofluids with widely different spin viscosity values Ferrofluid A and Ferrofl uid C. A) (dP/dz) = 0.1 Pa/m, B) (dP/dz) = 1 Pa/m, C) (dP/dz) = 0.1 Pa/m, D) (dP/dz) = 1 Pa/m A B C D PAGE 93 93 C 2 Ferrofluid 1 Ferrofluid A, Ferrofluid 2 Ferrofluid B Graphs showing the effect of field frequency on translational and spin velocity pro files at different values of applied pressure gradient are given in Figures C 5 and C 6 f or Ferrofluid A and Ferrofluid B combination. Figure C 5 Variation of dimensional translationa l velocity with field frequency at a constant field amplitude of H = 2mT at different values of applied pressure gradient for similar spin viscosity ferrofluids Ferrofluid A and Ferrofluid B. A) (dP/dz) = 0.1 Pa/m, B) (dP/dz) = 1 Pa/m, C) (dP/dz) = 0.1 Pa/m, D) (dP/dz) = 1 Pa/m A B C D PAGE 94 94 Figure C 6 Variation of dimensional spin velocity with field frequency at a constant field amplitude of H = 2mT at different values of applied pressure gradient for similar spin viscosity ferrofluids Ferrofluid A and Ferrofluid B. 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Melche r, Continuum Electromechanics (MIT, Cambridge, 1981). 30 W. M. Deen, Analysis of Transport Phenomena (Oxford U niversity Press, New York, 1998). 31 C. Rinaldi and M. Zahn, Effects of spin viscosity on ferrofluid flow profiles in alternating and rotating ma gnetic fields," Phys Fluids 14 2847 (2002). PAGE 97 97 BIOGRAPHICAL SKETCH Bhumika Sule received the degree of Bachelor of Chemical Engineering from Institute of Chemical Technology (formerly UDCT), Mumbai, India in May 2012 In fall 2012, she joined University of Florida to pursue a Master of Science degree in chemical e ngineering xml version 1.0 encoding UTF-8 REPORT xmlns http:www.fcla.edudlsmddaitss xmlns:xsi http:www.w3.org2001XMLSchema-instance xsi:schemaLocation http:www.fcla.edudlsmddaitssdaitssReport.xsd INGEST IEID E8X076AVN_HL0MXN INGEST_TIME 2014-05-13T18:46:23Z PACKAGE UFE0046361_00001 AGREEMENT_INFO ACCOUNT UF PROJECT UFDC FILES |