# Micropolar Electromagnetic Fluids: Theory and Simulation

TABLE OF CONTENTS

ACKNOWLEDGEMENT … ……………………………………...……… …………… iii

ABSTRACT … ……………………………………...……… …………………………... iv

TABLE OF CONTENT S ..………………………...……………. ……………………... vi

LIST OF FIGURES ………… ………………...…………………………… ………....... ix

LIST OF TABLES …………………………...………………… …………………....... xiii

LIST OF ACRONYMS ……………………...……………………… ……………....... xiv

LIST OF SYMBOLS ……………………...……………………… ……………............ xv

CHAPTER 1

INTRODUCTION …………………………...… ……….………… … .. … 1

1.1

I ntroduction …………………………. ..… ………. ………………… ……………...….1

1.2

R ecent D eve lop ments

in M icro /N ano - scale F luid P hysics ………… …………………2

1.3

K ey Elements of this Dissertation…..............................................................................4

1.4

S ummary

of D issertation O rganization ..… ………. ……………… … ………………...5

CHAPTER 2

THEORY OF MICROPOLAR FLUID DYNAMICS ..………….… ...11

2.1

W hat

I s a F luid ? . ………………...… ………. ………………… ………….………….11

2.2

K inematics …………………...… ………. ………………… …………………………12

2.3

B alance L aws ………………...… ………. ………………… ………………...………14

2.3.1

El ectromagnetic Balance Laws ... ………………...………….………… …… ...14

2.3.2

Thermodynamic Balance Laws ………………...………….………… ….. … ...16

2.4

C onstitutive E quations …...… ………. ………………… …………..…………….…..18

2.4.1

Wang’s Representation Theorem …………………………..………… …... ….. 21

vii

2.4.2

Onsager Theory for Linear Constitutive Equations ………..………… …... ….. 29

2.4.3

Extension of Nonli near Onsager Theory of Irreversibility ..………… . … …... .. 31

2.5

C onnections

to C ouple S tress T heory

and N ewtonian F luid M echanics …… …... ….. 34

2.6

M agneto - M icropolar F luid D ynamics (M 2 FD) …….…...………….…..…… …... ….. 37

CHAPTER 3

COMPUTATIONAL MICROP OLAR FLUID DYNAMICS –

INCOMPRESSIBLE FLOW …….…..…….…..………….…..…….…. ...…….…..… . 40

3.1

F inite D ifference (FD) M ethod …….…..…….…..………….…..…….…..… …... …. 40

3.1.1

Projection Method …….…..…….…..………….…..…….…..…… …… …..… 40

3.1.2

Time - Centered Split Method (TCSM) …….…..…….…..……… ……. ….….. 41

3.1.3

Examples …….…..…….…..………….…..…….….. …….…..… ….. ….…..… 43

3.1.3 .1 Couette Flow …….…..…….…..………….…..…….… ……. .…….… .43

3 .1 .3 .2 Hagen - Poiseuille Flow …….…..…….…..……… ……... .…..…….….. 45

3.1.3 .3 Numerical Accuracy Study …….…..…….…..… ……. ……..…….….. 46

3.1.3 .4 Lid - Driven Cavity Flow …….…..…….… …... ………….…..…….….. 46

CHAPTER 4

COMPUTATIONAL MICROP OLAR FLUID DYNAMICS

–

COMPRESSIBLE FLOW …….…..…….…..…….…..…….…..………….…. .…. ….. 60

4.1

S pectral D ifference

(SD) M ethod …….…..…….…..…….…..…….…..… …. ….….. 60

4.1.1

Conservative f orm of MFD equations …… . …..…… . …..…… . … …… ..…… .60

4.1.2

Generalized Stokes’ Hypothes is …… . …..…… . …..…… . …..… ……… … . ….. 63

4.1.3

Introduction to Spectral Difference (SD) Method …… . …..…… …... …..…… .64

4.2

N umerical R esults …….…..…….…..…….…..…….…..…….…..… …. ….…..……. 68

viii

4.2.1

Compressible Plane Couette Flow …… . …..…… . …..…… ……. … . ..…… . ….. 68

4 .2.1.1 Analytical and Exact Solut ion s …… . …..…… . … … ..…… .…… … . ….. 68

4 .2.1.2 Error Analysis …… . …..…… . …..…… . …..…… . …..…… …. …..…… .70

4.2.2

Flow Past a Cylinder …… . …..…… . …..…… . …..…… . …..…… …... …..…… .70

4 .2.2.1 Effect of Coupling Coefficient (Coupling Effect) …… . …..…… …. … 71

4 .2.2.2 Demonstration of 4 th Order SD Method …… . …..…… . …..…… ….. … 73

4 .2.2.3 Mesh Refinement …. …… .… …… . …..…… . …..…… . …..… … … .. ….. 73

4.2.3

Magneto - Micropoalr Fluid Dynamics (M 2 FD) …… . …..…… . …..… ….. … . … 74

CHAPTER 5

CONCLUDING REMARKS AND DISCUSSION ..…….…..…….....101

5.1

Conclusions…………………….. …….…..…….…..…….…..…….…..… ….….…101

5.2

F uture Works…… …….…..…….…..…….…..…….…..…….…..… ….…..…..…..105

ix

LIST OF FIGURES

Figure 1 - 1 Comparison of experimental data with the model predictions for water flow in microchannel [10 ] …………………………………………………………………………6

Figure 1 - 2 Comparison of translational velo cities in NEMD simulation (square dots), MFD solutions (solid line) and NS solution (dot line) [4].

………………………………7

Figure 1 - 3

Zoom - in of the c omparison of translational velocities in NEMD simulation (square dots), MFD solutions (solid line) and NS solution (dot line) [4]…………………8

Figure 1 - 4 Comparison of molecular rotation in NEMD simulation (square dots) and gyration in MFD (dot line for MFD solution with no slip condition and solid line for those with slip boundary condition) [4]…………………………………………………...9

F igure 1 - 5 NEMD simulation [4], MFD solution and NS solution of translational velocities with no - slip boundary condition. (Reproduced from Ref. [4])………………..10

Figure 2 - 1 Deformation of Microelement………………………………………………..39

Figure 3 - 1 The comparison of angula r velocity in Navier - Stokes equation and microgyration in MFD. (Couette flow) …………………………………………………..49

Figure 3 - 2 Time evolution of velocity in Couette flow while the arrow indicates the transient process as time marches on …………………………………………………….50

Fig ure 3 - 3 The comparison of velocity profiles in Navier - Stokes equations and MFD. (Poiseuille flow) ………………………………………………………………………….51

x

Figure 3 - 4 The comparison of angular velocity in Navier - Stokes equation and microgyration in MFD. (Poiseuille flow) …………………………………………… …..52

Figure 3 - 5 Error analysis of the numerical scheme ……………………………………...53

Figure 3 - 6 Center Velocity in Cavity Flow ………………………………………………54

Figure 3 - 7 Pressure Distribution in Cavity flow …………………………………………55

Figure 3 - 8 G yration in cavity flow……………………………………… ………………56

Figure 3 - 9 Vorticity in cavity flow………………………………………………………57

Figure 3 - 10 Total Velocity in Cavity Flow ………………………………………………58

Figure 4.1 Distribution of flux and solution points for the fourth order SD scheme …….77

Figure 4 - 2 Illustration of flow pa st a cylinder…………………………………………...78

Figure 4 - 3 Time history of lift coefficient …………………………...…………………..79

Figure 4 - 4 Time history of drag coefficient …………………….………………………..80

Figure 4 - 5 Instantaneous gyration contour of Case I . The upstream velocity at the in let is 0.2…………………………………………………………………..…………………….81

Figure 4 - 6 Instantaneous gyration contour of Case II . The upstream velocity at the inlet is 0.2………………………………………………………………………………………...82

Figure 4 - 7 Instantaneous gyration contour of Case III . The upstream vel ocity at the inlet is 0.2……………………………………………………………………………………...83

xi

Figure 4 - 8

Instantaneous gyration contour of Case II solved by 4 th order SD method . The upstream velocity at the inlet is 0.2……………………………………………………...84

Figure 4 - 9 Instantaneous gyration contour of Case II solved by a finer mesh (40 elements around the cylinder) . The upstream velocity at the inlet is 0.2…………………………..85

Figure 4 - 10 Illustration of flow past a cylinder with transverse uniform magnetic field..86

Figure 4 - 11 Instantaneous gyration c ontour of Case II with Ha=3 . The upstream velocity at the inlet is 0.2………………………………………………………………………….87

Figure 4 - 12 Z oom - in of instantaneous gyration contour of Case II with Ha=3 . The upstream velocity at the inlet is 0.2……………………………………………………...88

Figure 4 - 13 Streamline contour of Case II with Ha=3 …………………………………..89

Figure 4 - 14 Instantaneous gyration contour of Case II with Ha=5 . The upstream velocity at the inlet is 0.2………………………………………………………………………….90

Figure 4 - 15 Z oom - in of instantaneous gyration contour of Case II with Ha=5 . The upstream velocity at the inlet is 0.2……………………………………………………...91

Figure 4 - 16 Streamline contour of Case II with Ha=5 …………………………………..92

Figure 4 - 17 Instantaneous gyration contour of Case II with Ha=7 . The upstream velocity at the inle t is 0.2………………………………………………………………………….93

Figure 4 - 18 Z oom - in of instantaneous gyration contour of Case II with Ha=7 . The upstream velocity at the inlet is 0.2……………………………………………………...94

Figure 4 - 19 Streamline contour of Case II with Ha=7 …………………………………..95

xii

Fi gure 4 - 20 Time history of drag coefficient in M 2 HD cases……………………………96

Figure 4 - 21 Time history of lift coefficient in M 2 HD cases……………………………..97

xiii

LIST OF TABLES

Table 3 - 1 Error Analysis of Poiseuille Flow……………………………………………..59

Table 4 - 1 Velocity Error Analy sis of Compressible Plane Couette Flow………………..98

Table 4 - 2 Temperature Error Analysis of Compressible Plane Couette Flow…………...99

Table 4 - 3 Strouhal number and vortex shedding period of flow past a cylinder with and ……………………………………………………………..100

xiv

LIST OF ACRONYMS

Atomistic Field Theory – AFT

Computatinoal Micropolar Fluid Dynamics – CMFD

Electromagnetic – EM

Finite Difference – FD

Generalized Atomistic Finite Element Method – GAFEM

Magneto - Micropola r Fluid Dynamics – M 2 FD

Magneto - Micropolar - hydrodynamics – M 2 HD

Micropolar Fluid Dynamics – MFD

Molecular Dynamics – MD

Navier - Stokes – NS

Non - equilibrium Molecular Dynamics – NEMD

Spectral Difference – SD

Time - Centered Split Method – TCSM

xv

LIST OF SYMBOLS

: Eulerian Coordinate

: Lagrangian Coordinate

: microdeformation

: velocity

: gyration

: dielectric displacemen t

: magnetic flux

: electric field

: electric current

: free charge density

: speed of light

: polarization

: magnetization

: magnetic flux

: density

: Cauchy stress

: body force density

: EM force

: moment stress

: EM body couple

: internal energy density

: entropy density

: heat flux

: heat source

: EM energy

: absolute temperature

: pressure

: microinertia

: generalized Helmholtz free energy density

: radius

: permut ation symbol

: Kronecker delta function

: thermodynamic force

: dissipative function

: thermodynamic flux

: total energy

: constant pressure specific heat

: constant volume specific heat

1

CHAPTER 1

INTRODUCTION

1.1

INTRODUCTION

Research activities aiming to explore fluid physics at nano and micro scales have been increasing

over the past 20 years. There are existing literatures, which analyzed fluid mechanics in microchannels and micromachined fluid systems (e.g. pumps and valves) using Navier - Stokes equations [1]. The validity of Navier - Stokes equations

for flow physics at micro/nano scale has been a deb ate for decades [2,3]. Part of the studies show that when the channel size is comparable to the molecular size, the spin ning of molecule, which was observed in Molecu lar Dynamics (MD) simulations [4,5 ], significantly

affects the flow field. This effect of molecular s pin is not taken into account by the Navier - Stokes equations. A novel approach, Microc ontinuum Theory, consisting of Micropolar, Microstretch and M icromorphic (3M) theori es, developed by Eringen [6 - 9 ] provides a mathematical foundation to captur e this

effect . In 3M theories, each particle has a finite size and contains a microstructure, which can rotate and deform independently regardless of the motion of the centroid of the pa rticle. In the Micropolar Theory, there are three additional degrees o f freedom , gyration, to determine the rotation of the microstructure. Hence , addition al governing equations are required for gyration. The additional equation s introduce a

mathematically rigorous mechanism to take the effect of particle spin

into account . The Micropolar T heory thus represents a promising alternative approach, to numerically solving micro scale fluid dynamics, which can be much more computationally efficient than MD simulations.

2

1.2

RECENT DEVE LOP MENTS OF MICRO/NANO - SCALE FLUID PHYSICS

Papautsk y et. al. [10] was the first one, who adopt ed the M icropolar fluid model to explain the experimental observation of volume flow rate reduction for the flow in a rectangul ar microchannel as shown in Figure

( 1 - 1 ) . In Ref. [10], the authors concluded that the numerical model of Micropolar fluid dynamics (MFD) predicted experimental data better than the classical Navier - Stokes theory. In addition, G ad - El - Hak [11] explicitly states that microscale flows are essentially different from flows in macroscale. The Nav ier - Stokes description is incapable of explaining the se

observed effects. The calculated hydrodynamic quantities for a fluid as a classical continuous medium (from Navier - Stokes equations) differ significantly from those obtained experimentally, and the di fference increases with the decrease of the diameter of the channel through which the fluid flows. Kucaba - Pietal et. al. [4] utilized MD simulation to study the limitation and applicability of Micropolar theory. Their work concluded that at channel width not smaller than 10 diameters of molecular, Micropolar theory is in reasonable agreement with MD simulation results. Delhommelle and Evans [5 ] compared the Non - equilibrium MD (NEMD) simulation results with the solutions of Poiseuille flow in Micropolar flu id dynamics and Newtonian fluid . Their study concluded that the translational streaming velocity profile deviates from the classical parabolic profile in Newtonian fluid

owing to molecular rotation . Such deviations, shown in Figures

( 1 - 2 ) and (1 - 3), are du e to

neither molecular orientation nor packing effects. It is attributed to the molecular

rotation according to their study . In addition, Delhommelle and Evans also compared the molecular rotation profile in NEMD and the gyration pro file in MFD as shown in

3

Figure (1 - 4) . Such spinning motion cannot be observed in Newtonian fluid dynamics.

Figure (1 - 5) is reproduced from Ref.[4]. It shows the comparison of translational velocities in NEMD simulation [4], MFD solution and NS solution with no - slip conditions. I t is seen that MFD solution fits closer to NEMD solution than NS solution.

The other application of Micropolar fluid theory is associated with micro/nanofluidic systems under the presence of Lorentz force, e.g. magnetohydrodynamic (MHD) micropump [12]. MH D is the study of flow of electrically conducting liquids in electric and magnetic fields. In 1832, Ritchie discovered that the applying an electric field, which is perpendicular to a magnetic field, to a conducting fluid could pump the fluid [13]. From NE MD simulation [3,4], it has been proven that MFD is a better option for modeling microchanel flow. Hence with electromagnetic interaction, Magneto - Micropolar fluid dynamics (M 2 FD) or Magneto - Micropolar - hydrodynamics (M 2 HD) naturally becomes a promising can didate for modeling the flow physics at micro/nano scale.

There a re many recent developments of M icropolar theory, which focused on the numerical analysis of Hagen - Poiseuille flow and applications on nano/microfludics , including Papautsky[10], Ye[14] and Hansen[15]. However the se studies only solve for steady state solution , which does not include pressure. The complete M 2 HD theory is still out of reach.

4

Based on the above literature survey, this dissertation proposes an extension of Micropolar electromag netic theory for fluids and development of corresponding numerical methods.

1.3

KEY ELEMENTS OF THIS DISSERTATION

This dissertation can be divided into two main parts, theory and simulation.

In the theoretical part, the complete set of constitutive equations for electromagnetic fluids are derived through two approaches: (1) Wang’s representation theorem and (2) Onsager’s theory. The constitutive equations of fluids are always required to satisfy the axiom of objectivity. The axiom of objectivity allows the uti lization of Wang’s representation theorem. The constitutive equations are, therefore, obtained through Wang’s representation theorem and later linearlized for practical applications. The linear constitutive equations can also be obtained through Onsager’s theory. These two sets of results are identical. One new mechanism, curl of gyration, is found in the induction of electric current. The nonlinear Onsager’s theory for constitutive equations was originally derived by Edelen. It is now further extended for fluids through the integration with Wang’s representation theorem in this dissertation. The connections to Couple Stress theory and Navier - Stokes equations are discussed.

In the simulation part, a second order Finite Difference (FD) method is integrated w ith time - centered split method (TCSM). It is successfully developed for incompressible fluids. A higher order Spectral Difference (SD) method is further developed for compressible Micropolar fluid. The analytical and exact solutions,

5

including velocity, gy ration and temperature, for incompressible and compressible plane Couette flow and incompressible Hagen - Poiseuille flow, are given. These analytical and exact solutions are used to verify the order of numerical accuracy for aforementioned numerical solvers . Based on numerical results, the physical meanings of material coefficients are clearly described. Flows past a cylinder with and without imposed transverse magnetic field are also studied. Imposing magnetic field is demonstrated as an effective approach for flow control.

1.4

SUMMARY OF DISSERTAT ION ORGANIZATION

This dissertation is organized as follows: In Chapter 2, we re - formulate the Micropolar theory for electromagnetic (EM) fluids. Constitutive equations are derived from Wang’s representation theorem an d Onsager’s theory. The extensions of nonlinear Onsager’s theory for fluid are derived. A set of constitutive equations for Magneto - Micropolar fluid dynamics (M 2 FD) is given to conclude Chapter 2. In Chapter 3, the Finite Difference (FD) method is develope d with time - centered split method (TCSM) and projection method for incompressible flow problem, including the plane Couette flow, Hagen - Poiseuille flow and lid - driven cavity flow. In Chapter 4, the higher order Spectral Difference (SD) method is utilized t o solve unsteady and compressible viscous flow problem. The order of numerical accuracy at desired third and fourth order is also studied and proven. Flow past a cylinder is solved and gyration contour plots are given. One of the M 2 HD flow problems, flow p ast a cylinder under a transverse uniform magnetic field, is studied. The results are shown and discussed. Chapter 5 concludes this dissertation.

6

FIGURES

Figure 1 - 1 Comparison of experimental data with the model predictions for water flow in microchannel [10 ]

7

Figure 1 - 2 Comparison of translational velocities in NEMD simulation (square dots), MFD solutions (solid line) and NS solution (dot line) [4] .

8

Figure 1 - 3

Zoom - in of the c omparison of translational velocities in NEMD simulati on (square dots), MFD solutions (solid line) and NS solution (dot line) [4] .

9

Figure 1 - 4 Comparison of molecular rotation in NEMD simulation (square dots) and gyration in MFD (dot line for MFD solution with no slip condition and solid line for those with slip boundary condition) [4]

10

Figure 1 - 5 NEMD simulation [4], MFD solution and NS solution of translational velocities with no - slip boundary condition. (Reproduced from Ref. [4])

11

CHAPTER 2

THEORY OF MICROPOLAR FLUID DYNAMICS

The electromagnetic behavior of a conducting fl uid has been extensively studied

recently, especially in the applications of magnetohydrodynamics [1 6] electrohydrodynamics [17] solar wind and solar corona [18 ], geophysics [ 19] and plasma physics [20]. Such flui ds include plasma s , liquid metal s , the ferrofluid f luid core of the Earth, etc. The theoretical background for those fluids and applications are currently mainly based on Navier - Stokes equation coupled with Maxwell's equations. However this classical theor y cannot describe the orientation of fluid element s , which is essential when observing the physical picture of fluid flow at micro/nano scale under the influence of a Lorentz force field, e.g. blood, or electrically - conducting fluid flow in a microchannel [21].

2.1

WHAT IS A FLUID ?

A Fluid can be formally defined as follows: [22,23 ]

Definition A body is called a fluid if every configuration of the body leaving de nsity unchanged can be taken as the reference configuration.

This definition is in agreement with

classical descriptions that (1) fluid doe s not have a preferred shape [24 ] and (2) fluid cannot withstand shearing forces, however sma ll, without sustained motion [25]. If every configuration is to be a reference configuration, then one may have with the density

12

unchanged (cf. Fig. 2 - 1), i.e.,

(2 - 1)

Here is the shifter, which is the directional cosine between the Lagrangian and Euler ian coordinates.

Hence , the difference between Eulerian coordinate and

Lagrangian coordinate vanishes . Therefore ,

the axiom of objectivity [22,23 ] should be always considered and obeyed. Obedience to objectivity implies that the material property of fluid satisfies isotropy.

Axiom of Objectivity The constitutive equations must be form - invariant with respect to rigid body motion of the spatial frame of reference.

2.2

KINEMATICS

A Micropolar continuum is a collection of continuously distributed finite size

part icles that can rotate. A material point, P , in the reference configuration is identified by a position vector, X K , K=1,2,3 , and three directors attached to the point P . These three directors in Micropolar continuum are rigid.

The motion, at time t , carries the finite - size particle to a spatial point and rotates the th ree directors to a new orientation . Thus, such motion is similar

to the motion of the Earth. It can

not only revolve around the Sun, which is macromotion, but also

can

spin on its own axis, which is micromotion. These motions and their inverse motion for Micropolar continuum can be ( cf. Figure (2 - 1)) expressed by [5 - 9,26 ]

13

(2 - 2)

and

(2 - 3)

It is straightforward to prove that

(2 - 4)

Consequently, the later part of eq. (2 - 3) becomes

(2 - 5)

Here and throughout, an index followed by a common denot es a partial derivative, e.g.

(2 - 6)

For fluid flow, defo rmation - rate tensors are crucial to the characteriz ation of the viscous resistance. Deformation - rate tensors may be deduced by calculating the materia l time derivative of the spatial deformation tensors. For Micropolar fluid, two objective deformation - rate tensors are [5 - 9,26 ]

(2 - 7)

is the velocity of the centroid of the parti cle;

is velocity gradient; is gyration vector, the addi tional rotating degrees of freedom of the particle. The mean free path of a fluid is larger than that of a solid, i.e., a fluid molecule has more space

to move around before colliding into another fluid molecule. Hence when a fluid

14

particle in Micropolar continuum, which can be either a fluid molecule or a group of fluid molecules, rotates, the effect of gyration appears and such phenomena cannot be ob se rved in classical c o ntinuum theory. On the contrary, the gyration vector in Micropolar fluid d ynamics (MFD) is a natural choice to reveal such effect.

2.3

BALANCE LAWS

Balance laws of the electromagnetic theory of a Micropolar continuum consist of the electro magnetic balance laws (Maxwell ’ s equations) and the thermodynamic balance laws.

2.3.1

Elemctromagnetic Balance Laws

The EM balance laws are the well - known Maxwell’s equations, which can be written in Lorentz - Heavisi de system as [27 - 28 ]

Gauss’ Law

(2 - 8)

Faraday’s Law

(2 - 9)

Magnetic Flux

(2 - 10)

Ampere’s Law

15

(2 - 11)

where D is dielectric displacement vector, B is magnetic flux vector, E is electric field vector, H is magnetic field vector, J is current vector, q e is free charge density and c is

the speed of light in vacuum. The divergence of Ampere’s Law with the use of G a u s s’ Law leads to the equation of conservation of charge.

(2 - 12)

Polarization vector, P , and magnetization vector, M , are defined as

(2 - 13)

The EM vectors, D , E , H , M and J , are all referred to a fixed laboratory frame R C . The Galilean transformation of inertial frames from a group consists of time - independent spatial rotations and pure Galilean transformation, i.e.

(2 - 14)

where

(2 - 15)

The requirement of the form - invariance of the Maxwell’s equations under the Galilean transformations gives the following transformation:

(2 - 16)

(2 - 17)

(2 - 18)

16

(2 - 19)

where the quantities, E * , J * , M * and H * , are re ferred to a co - moving frame R G with material body having a velocity of v . The electric filed, E * , on a co - moving frame is usually called electromotive intensity [22] .

2.3.2

Thermodynamic Balance Laws

The thermodynamic balance laws of Micropolar continuum with E M intera ction can be expressed as [5 - 9,26 ]

Conservation of Mass

(2 - 20)

Conservation of Linear Momentum

t l k , l + ρ f k − v k ( ) + F k = 0

(2 - 21)

Conservation of Angular Momentum

m l k , l + e k i j t i j + ρ l k − j ω k ( ) + L k = 0 (2 - 22)

Conservation of Energy

(2 - 23)

Clausius - Duhem Inequality