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All Theses Theses
12-2018
Asynchronous propulsion of the three spheremicro-swimmer using perturbed magnetic fieldArnab MitraClemson University, [email protected]
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Recommended CitationMitra, Arnab, "Asynchronous propulsion of the three sphere micro-swimmer using perturbed magnetic field" (2018). All Theses. 2998.https://tigerprints.clemson.edu/all_theses/2998
Asynchronous propulsion of the three spheremicro-swimmer using perturbed magnetic field
A Thesis
Presented to
the Graduate School of
Clemson University
In Partial Fulfillment
of the Requirements for the Degree
Masters of Science
Mechanical Engineering
by
Arnab Mitra
December 2018
Accepted by:
Dr. Phanindra Tallapragada, Committee Chair
Dr. Yue Wang
Dr. Xiangchun Xuan
Abstract
In recent years much effort have been placed on development of microscale devices capable
of propulsion in low Reynold number environment. These devices have potential in biomedicine,
micro-fabrication and sensing fields. One of the most promising device that has been extensively
studied is magnetic micro-swimmers. Due to small size of the swimmer they operate in low Reynolds
number regime. In this case the hydrodynamics is governed by the viscosity rather than inertia.
Since the swimmer is so small any kind of motor or other propulsion system is not feasible so we
are using magnetic field to remotely control the swimmer. The model used in this work is a simple
3-bead swimmer with a permanent magnetic dipole. Most of the work done using this model shows
propulsion in synchronous ”in-sync” regime where the dipole of the swimmer is able to follow the
applied magnetic field. The nature of motion of the swimmer changes with change in frequency of
the applied field. It has been proved that propulsion decreases beyond ”step-out” frequency of the
applied field. Our work is mainly in the out of sync regime when frequency of applied field is too
high for the moment of the swimmer to follow. The existing publication utilizes a non-inertial model
(neglects the mass of the swimmer) to predict the locomotion of the swimmer, we also use a similar
model for our work. By using a perturbed magnetic field we found propulsion exists in asynchronous
regime.
ii
Dedication
“As there are a number of beliefs, there are a number of ways .. ” - Sri Ramakrishna
“You have a right to perform your prescribed duty, but you are not entitled to the fruits
of action. Never consider yourself to be the cause of the results of your activities, and never be
attached to not doing your duty.” - Bhagavad Gita, Chapter II, Verse 47
This work is dedicated to God and my parents
iii
Acknowledgments
I want to thank my advisor Dr. Phanindra Tallapragada for having faith in me and guiding
me through my research. I would not have been able to complete my M.S and research without
his support and aid. Next, I would like to thank Clemson University for supporting my dream for
higher education and letting me use the wonderful Palmetto Cluster, without which it would have
been impossible to carry out the extensive amount of simulation needed for this work. Next, I want
to thank my lab-mates Senbagaraman Sudrasanam and Jake Buzhardt for helping me whenever I
had any problems with my codes. Lastly, I would like to thank my friends Saptarshi Charaborty,
Ghanshyam Sharma, Monsur Islam and Angshuman Goswami who went above and beyond to help
me make my life bearable so far away from home.
iv
Table of Contents
Title Page . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . i
Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ii
Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ii
Dedication . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iii
Dedication . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iii
Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iv
Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iv
List of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vii
1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.1 Types of micro-swimmers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31.3 Objectives and Contributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
2 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62.1 Low Reynolds number hydrodynamics . . . . . . . . . . . . . . . . . . . . . . . . . . 62.2 Magnetism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
3 Problem Setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
Problem Setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93.1 Swimmer Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93.2 Reference frames and kinematics of the swimmer . . . . . . . . . . . . . . . . . . . . 93.3 Mobility Matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103.4 Special Case - No External Force . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
4 Simulation Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
Simulation Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
5 Conclusion and Scope . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70
Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70
v
Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72
vi
List of Figures
3.1 Model of the swimmer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
4.1 Representation of Rotating Magnetic field for omega = 33rad/s . . . . . . . . . . . . 174.2 Single sided amplitude spectrum of the X component of magnetic field described in
case 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 184.3 Single sided amplitude spectrum of the Y component of magnetic field described in
case 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 194.4 Single sided amplitude spectrum of the Z component of magnetic field described in
case 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 204.5 Propulsion Velocity(µm/s) versus driving Frequnecy(rad/s) . . . . . . . . . . . . . . 214.6 Angle between magnetic moment of the swimmer and the magnetic field in body
coordinate System vs time for driving frequency 33 rad/s . . . . . . . . . . . . . . . 224.7 Angle between magnetic moment of the swimmer and the magnetic field in body
coordinate System vs time for driving frequencies 35 rad/s . . . . . . . . . . . . . . 234.8 Trajectory of the swimmer for driving frequency 33 rad/s . . . . . . . . . . . . . . . 244.9 Trajectory of the swimmer for driving frequency 35 rad/s . . . . . . . . . . . . . . . 244.10 ∆ for field described in case 1 vs driving frequency ω . . . . . . . . . . . . . . . . . . 254.11 Single sided amplitude spectrum of λ for propulsion under magnetic field in case 1
for ω = 11 rad/s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 274.12 Single sided amplitude spectrum of λ for propulsion under magnetic field in case 1
for ω = 19 rad/s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 284.13 Single sided amplitude spectrum of λ for propulsion under magnetic field in case 1
for ω = 31 rad/s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 294.14 Single sided amplitude spectrum of λ for propulsion under magnetic field in case 1
for ω = 41 rad/s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 304.15 External torque on the swimmer in inertial frame under magnetic field described in
Case 1 for ω = 33 rad/s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 314.16 External torque on the swimmer in inertial frame under magnetic field described in
Case 1 for ω = 41 rad/s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 324.17 Representation of rotating magnetic field described in case 2 where ω = 33rad/s . . 334.18 Single sided amplitude spectrum of the X component of magnetic field described in
case 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 344.19 Single sided amplitude spectrum of the Y component of magnetic field described in
case 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 354.20 Single sided amplitude spectrum of the Z component of magnetic field described in
case 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 364.21 Propulsion Velocity(µm/s) versus driving Frequnecy(rad/s) . . . . . . . . . . . . . . 374.22 Cosine of angle between magnetic moment of the swimmer and the magnetic field in
body coordinate System vs time for driving frequency 33 rad/s . . . . . . . . . . . . 384.23 Cosine of angle between magnetic moment of the swimmer and the magnetic field in
body coordinate System vs time for driving frequency 35 rad/s . . . . . . . . . . . . 39
vii
4.24 Trajectory of the swimmer for driving frequency 33 rad/s . . . . . . . . . . . . . . . 394.25 Trajectory of the swimmer for driving frequency 35 rad/s . . . . . . . . . . . . . . . 404.26 ∆ for field described in case 2 vs driving frequency ω . . . . . . . . . . . . . . . . . . 414.27 Single sided amplitude spectrum of λ for propulsion under magnetic field in case 2
for ω = 11 rad/s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 424.28 Single sided amplitude spectrum of λ for propulsion under magnetic field in case 2
for ω = 19 rad/s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 434.29 Single sided amplitude spectrum of λ for propulsion under magnetic field in case 2
for ω = 31 rad/s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 444.30 Single sided amplitude spectrum of λ for propulsion under magnetic field in case 2
for ω = 41 rad/s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 454.31 External torque on the swimmer in inertial frame under magnetic field described in
Case 2 for ω = 33 rad/s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 464.32 External torque on the swimmer in inertial frame under magnetic field described in
Case 2 for ω = 41 rad/s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 474.33 Representation of the magnetic field described in Case 3 . . . . . . . . . . . . . . . . 484.34 Single sided amplitude spectrum of the X component of magnetic field described in
case 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 494.35 Single sided amplitude spectrum of the Y component of magnetic field described in
case 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 504.36 Single sided amplitude spectrum of the Z component of magnetic field described in
case 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 514.37 Cosine of angle between magnetic moment of the swimmer and the magnetic field in
body coordinate System vs time for driving frequency 33 rad/s . . . . . . . . . . . . 524.38 Cosine of angle between magnetic moment of the swimmer and the magnetic field in
body coordinate System vs time for driving frequency 41 rad/s . . . . . . . . . . . . 524.39 Trajectory of the swimmer for driving frequency 33 rad/s . . . . . . . . . . . . . . . 534.40 Trajectory of the swimmer for driving frequency 41 rad/s . . . . . . . . . . . . . . . 534.41 Variation in Propulsion Velocity(µm/s) with oscillation amplitude (degrees) and os-
cillation factor ζ for driving frequencies 15 rad/s . . . . . . . . . . . . . . . . . . . . 544.42 Variation in Propulsion Velocity(µm/s) with oscillation amplitude (degrees) and os-
cillation factor ζ for driving frequencies 30 rad/s . . . . . . . . . . . . . . . . . . . . 544.43 Variation in propulsion velocity(µm/s) with driving frequnecy(rad/s) and oscillation
factor ζ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 554.44 Propulsion Velocity(µm/s) versus driving frequency(rad/s) . . . . . . . . . . . . . . 564.45 Cosine of angle between magnetic moment of the swimmer and the magnetic field in
body coordinate System vs time for ε = 0.02, ω = 33rad/s . . . . . . . . . . . . . . . 574.46 Cosine of angle between magnetic moment of the swimmer and the magnetic field in
body coordinate System vs time for ε = 0.02,ω = 41rad/s . . . . . . . . . . . . . . . 584.47 Cosine of angle between magnetic moment of the swimmer and the magnetic field in
body coordinate System vs time for ε = 1, ω = 33rad/s . . . . . . . . . . . . . . . . 594.48 Cosine of angle between magnetic moment of the swimmer and the magnetic field in
body coordinate System vs time for ε = 1, ω = 41rad/s . . . . . . . . . . . . . . . . 604.49 Trajectory of the swimmer for ε = 0.01, ω = 33rad/s . . . . . . . . . . . . . . . . . . 614.50 Trajectory of the swimmer for ε = 0.01, ω = 41rad/s . . . . . . . . . . . . . . . . . . 614.51 Trajectory of the swimmer for ε = 0.02, ω = 33rad/s . . . . . . . . . . . . . . . . . . 624.52 Trajectory of the swimmer for ε = 0.02, ω = 41rad/s . . . . . . . . . . . . . . . . . . 624.53 ∆ versus driving frequency(rad/s) for different values of ε . . . . . . . . . . . . . . . 634.54 Major frequencies in λ vs ω for different values for ε = 0.01 . . . . . . . . . . . . . . 644.55 Major frequencies in λ vs ω for different values for ε = 0.02 . . . . . . . . . . . . . . 654.56 Major frequencies in λ vs ω for different values for ε = 0.5 . . . . . . . . . . . . . . . 66
viii
4.57 Major frequencies in λ vs ω for different values for ε = 1 . . . . . . . . . . . . . . . . 674.58 External torque on the swimmer in inertial frame under magnetic field described in
Case 3 for ω = 41 rad/s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 684.59 External torque on the swimmer in inertial frame under magnetic field described in
Case 3 for ω = 55 rad/s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69
ix
Chapter 1
Introduction
Mobility is important for survival of many living organisms. Due to small length scale of
microorganisms like bacteria their swimming characteristics is different from swimming character-
istics of a macroscopic organism like a blue whale. Most microorganisms live in fluid environments
where they experience a viscous force that is many orders of magnitude stronger than inertial forces.
This is known as the low Reynolds number (Re) regime (Re < 1)characterized by instantaneous and
time-reversible flows that are described by the time-independent Stokes equation. Purcell stated in
his ”Scallop Theorem” in his 1976 paper on ‘Life at low Reynolds number’[10] if a low-Reynolds
number swimmer executes geometrically reciprocal motion, that is a sequence of shape changes that
are identical when reversed, then the net displacement of the swimmer must be zero, if the fluid
is incompressible and Newtonian[6]. In Purcell’s own words, ‘Fast, or slow, it exactly retraces its
trajectory, and it’s back where it started’[10].
1.1 Types of micro-swimmers
1.1.1 Natural micro-swimmers
Researchers have identified in nature, microorganisms break time-reversal symmetry with
rotating helices[12]. They found that Prokaryotes such as E. Coli has a rigid helical and passive
flagella. A rotary motor embedded in the cell rotates the flagella at the point of attachment which
in turn causes propulsion. They have compared the motion to that of a cork screw pulling itself
1
through the cork. They also stated propulsion is not equal to the wavelength per turn. In contrast to
this propulsion Eukaryotes such as spermatozoa has a flexible actively deforming flagella. Molecular
motors distributed along the flagellum’s length produce bending moments, and the coordinated
action of those motors generates a wave-like motion. The wave’s frequency can range from a few
to 100 Hz and its shape can take one of two forms. The flagellar waveform resembles a sinusoidal
traveling wave. It pushes fluid in the direction of the wave propagation and the micro-organism
in the other. Also there are few micro-organism have cilia that show flexible oar-like beats[1] and
can propel themselves. Inspired by this method of locomotion similar propulsion strategies has
been utilized to propel artificial micro-swimmers. Micro-scale robots have been extensively studied
for a long time due to their numerous applications like micro-manipulation and micro-fabrication.
Some of the better known possible applications are drug delivery systems, tissue manipulation using
magnetized particles and MRI machines.
1.1.2 Artificial micro-swimmers
Inspired by this method of locomotion in natural micro-swimmers similar propulsion tech-
niques has been utilized to propel artificial micro-swimmers. Micro-scale robots have been ex-
tensively studied for a long time due to their numerous applications like micro-manipulation and
micro-fabrication. Some of the better known possible applications are drug delivery systems, tissue
manipulation using magnetized particles and MRI machines. The artificial micro-swimmers can be
classified in different categories. Some of the major categories are as follows:
1.1.2.1 Methods of propulsion
Propulsion using rigid helical micro-propellers. A helical propeller attached to a payload
when subjected to external torque by external magnetic field propels along the direction of its long
axis[4],[15]. As a helix rotates about its long axis, the coupling between rotational and translational
motion leads to propulsion at low Re. The external torque subjects the swimmer to a viscous drag
from the fluid. Due to shape of the swimmer there is component of the viscous drag which remains
unbalanced propels the swimmer along the direction of the long axis of the helix. Experimental
research on a few flexible micro-swimmers have also been demonstrated, including a microswimmer
that is based on a chain of superparamagnetic beads and actuated by a magnetic field [3], a biohybrid
elastic microswimmer made of elastic filament and actuated by cardiomyocytes [13], other models
2
for flexible tails were also explored [11], [5]. Flexible swimmers are modeled as an elastic beam
and the differential equations of beam deflection were solved to find acting force and displacements.
They found that these flagellas pushes the fluid in the direction of the wave and as result of force
balance by Newtons third law the swimmer in the opposite direction.
1.1.2.2 Method of Actuation
Different types of micro-robots that are being studied are using electrically and optically
controlled bacteria, eg. Micro-robots developed by Steager et al [14] are controlled by Ultraviolet
light and electric currents.
Research has been done on micro-swimmers which use phoretic propulsive force eg. Paxton
et al.[9], they use chemical reaction to cause oxygen gradient to induce propulsion.
Many studies have been done on magnetically actuated micro-robots using various model
that depends on the type of the swimmer. They can be classified under three main groups: flexible
swimmers, helical swimmers. These swimmers are remotely actuated by external magnetic field
generated by electromagnetic coil or pairs of coil to propel the swimmer in desired direction.
1.1.2.3 Directionality of the swimmer
Studies have shown that some strategies described before can propel the swimmer in one
direction based on their configuration. These swimmers are called uni-directional swimmers, e.g.
Helical swimmer, Flagellar swimmer. These swimmers can exactly trace their path in the opposite
direction when the actuating force is reversed.
Some swimmers are also classified by the position of the payload. When the propeller is
placed behind the payload during propulsion it is called pushing system and when it is in front its
called pulling system.
1.2 Motivation
The micro-swimmers studied in these papers are advantageous in some aspect of particle
manipulation but they have some inherent disadvantages. First, they are very difficult to manu-
facture and requires expertise in micro-fabrication. Secondly, chemically actuated robots can swim
faster but they risk exposing the subject to potentially poisonous chemicals[16]. Dielectrophoret-
3
ically manipulated robots can be controlled easily, but they use a very high electromagnetic field
which requires very high power input, they are invasive, you have insert electrodes into the subject
to effectively propel the swimmers.
Only magnetically actuated micro-swimmers are least invasive to the subject. Because the
actuation by remote magnetic field does not interfere with the subjects when the field strength is
limited to permissible limit. In this method of propulsion probes or any other chemicals is not
introduced into the system. However the magnetic field strength should be regulated.
This thesis covers in details modeling and simulation of magnetically actuated micro-swimmers
that are capable of swimming in Newtonian fluid. The model described in this thesis can be easily
manufactured with out any sophisticated fabrication equipments. The fabrication process for this
swimmer is described in Cheang, Meshkati, Kim (2014) [2], its very simple and provides good results.
This model is based on the work done by Kim, Meshkati, Fu [2], [7] and Morozov, Mirzae
[8]. For future reference we will abbreviate these source [2] as KMF, [7] as MF and [8] as MM.
This model is magnetically actuated at very low magnetic field strength and produced
significant propulsion. Advantages of this model are:
• It is very easy to fabricate. No specialized fabrication skill is necessary.
• It is propelled by low strength external magnetic field and hence very no-invasive and has less
side effects.
• It is not an uni-directional swimmer. The swimmer itself can be functionalized as payload.
1.3 Objectives and Contributions
1.3.1 Objectives
In the work done by these groups they have used a rotating magnetic field with a single
frequency to actuate the robots. They have found out that as the driving frequency increases the
propulsion increases up to a certain frequency and then propulsion stops. This frequency is called
step-out frequency. They have also noticed that beyond step out frequency the swimmer enters
asynchronous zone where the body fixed magnetic moment of swimmer can not follow the applied
field and as a result they concluded that propulsion cease to exist in asynchronous. In this work
4
we used a variety of magnetic field which has multiple frequencies have proved that propulsion is
possible in asynchronous zone and the speed is comparable with the propulsion under rotating field
with single frequency in synchronous regime.
1.3.2 Contributions
In previous literature they have used a magnetic field with fixed axis of rotation. In this work
we have studied the effects of a perturbed magnetic field on the propulsion of the micro-swimmer.
In ?? they have stated that sustained propulsion is not possible in asynchronous zone. We have
proved that it is field specific and it does not cover every possible field configuration that can be
used to control the swimmer. In this work we have proved that sustained propulsion is possible in
asynchronous zone by using a perturbed magnetic field configuration.
5
Chapter 2
Background
The background necessary for development of the model is explained in this chapter. The
model incorporates principles of low-Reynolds-number hydro-dynamics and magnetism. These two
topics are discussed below.
2.1 Low Reynolds number hydrodynamics
In fluid mechanics the characteristics of the flow are described using a dimensionless quantity
called Reynolds number. It represents the ratio of inertial forces to viscous forces acting on the
swimmer in the flow.
ReynoldsNumber = Re =ρ.v.L
µ(2.1)
Where ρ is the density of the fluid, v is the velocity of the swimmer, L is the characteristic length
of the swimmer and µ is the viscosity of the fluid. When Re > 1 then inertial forces are dominant
and when Re < 1 viscous forces are dominant. In case of a large body such as human swimming
the Re ∼ 105 and in case of bacteria (e.g. E. Coli.) the Re ∼ 10−4 to 10−5 range. So swimming in
Nano- or micro- scale is fundamentally different from swimming in macro scale. The motion of the
micro-swimmer has traditionally been considered as a special case of the Navier-Stokes equation for
an incompressible fluid,
ρ.
(∂u
∂t+ u.∇u
)= −∇p+ µ∇2u + b (2.2)
6
where u is th velocity of the fluid in R3, p is the pressure, b is the body force and µ is the viscosity
of the fluid.
If the Reynolds Number of the flow is small, the convective acceleration of the fluid u.∇u
is negligible and if the motion of the fluid is steady or nearly so, then the equation reduces to time
invariant Stokes flow,
µ∇2u = ∇p (2.3)
where the pressure is rescaled to eliminate the body force. The boundary conditions on the surface
of the body on which a reference point (such as center of the connecting sphere) xc is translating
with velocity U and spinning with angular velocity Ω,
u = U + Ω× (x− xc) (2.4)
The stress tensor, σ, associated with the velocity field of the body is,
σ = −pI + µ(∇u + (∇u)
T)
(2.5)
The resistive force that the fluid exerts on the body is obtained by integrating the traction on the
surface of the body,
Fh =
∫S
σ.ndS (2.6)
The drag Torque is
Th =
∫S
(x− xc)× σ.ndS (2.7)
2.2 Magnetism
In this section some concepts of magnetism applicable to our work are presented. We discuss
some relevant definitions and briefly explains Low Reynolds Number Hydrodynamics. NOTE: The
material in this chapter follows the presentation of the books “Introduction to magnetism and
magnetic materials” by David Jiles, and “Introduction to magnetic materials” by Bernard Cullity.
7
2.2.1 Magnetic field
The magnetic field is one of the most important concepts in electromagnetism. When a
magnetic field is present, it can be identified by the force it exerts on moving electric charges, the
force acted on a current-carrying object, the torque acted on a magnetic dipole or by reorientation of
electron spins in certain atoms. Magnetic fields can be produced by moving charges or by permanent
magnets. In either case, there are moving electrons that cause magnetic field. Magnetic field has
both direction and magnitude and can represented as a vector field. The units of magnetic field are
ampere per meter (Am−1) in SI and oersteds in EMU system.
2.2.2 Magnetic dipole
The two fundamental entities in magnetization are the current loop (the limit when the
radius of a current loop goes to zero) and the magnetic dipole (the limit when a pair of magnetic
poles get infinitely close to each other). In both cases, a magnetic moment m can be associated
with the entity. For the case of the current loop, the magnetic moment is given by the product of
the area of the loop, A and electric current in the loop, I. For the magnetic dipole, the moment is
given by the product of the pole strength, p and separation distance between the poles, d. It can be
shown that torque on a magnetic dipole is given by
τ = m×B (2.8)
The units of magnetic moment in Sommerfield convention are A.m2.
The above equation shows that the magnetic induction tends to align the dipole such that
m lies parallel to B, hence when they are not parallel there will be a potential energy (Ep) in the
system.
EP = m.B (2.9)
8
Chapter 3
Problem Setup
3.1 Swimmer Model
Our swimmer can simply be described as two spheres connected to a third sphere of equal
radius by two massless rigid rods. Rods are slender hence it experiences no drag force.
Φ
α
Figure 3.1: Model of the swimmer
3.2 Reference frames and kinematics of the swimmer
Consider the swimmer as a rigid body moving in a viscous fluid. The motion of the swimmer
in the fluid will be described with reference to both spatially fixed frame of reference with axes X-Y-Z
and a body fixed frame of reference with axes e1−e2−e3 attached to center of the connecting sphere
9
of the 3 bead swimmer. The configuration manifold of the body is SE3 consisting of the group of
rigid body translations and the group of rigid body rotations. The transformation of a vector χb in
the body frame to a vector χ in the spatial frame is given through the rotation R ε SO3,
χ = R.χb (3.1)
The rotations shall be locally parametrized by the classical Euler angles (θ, φ, ψ) and the rotation
matrix is a composition of the rotations around the body e3 − e1 − e3 axes respectively,
R = Rz(ψ).Rx(θ).Rz(φ) (3.2)
The configuration variables will be denoted by,
q = (x, y, z, θ, φ, ψ) (3.3)
The velocity of the center of the connecting sphere in the spatially fixed frame will be denoted by,
U = (U1, U2, U3) and the velocity of the center of the connecting sphere in the body fixed frame will
be denoted by, V = (V1, V2, V3). They are related as, U = R.V .
3.3 Mobility Matrix
It is generally proven in [4, 5, 6] that Force and torque experienced by a body in low
Reynold number regime can be linearly related to the velocity and angular velocity of the swimmer
by a matrix which is called ”Mobility Tensor”.
Fh
Th
=
A ξ
ξT B
V
ω
= R
V
ω
(3.4)
Where the Resistance Tensor, R is a 6 × 6 matrix with each of the sub matrices A,B, ξ are
of size 3 × 3 . If the external Force and Moment Fe and Te respectively are prescribed, then the
steady velocity of the body is such that the drag force and moment due to the fluid exactly cancel
10
out applied force. This velocity and angular velocity of the body are given by the equation,
V
ω
=
K C
CT M
.Fe
Te
= G
Fe
Te
(3.5)
where G is the ”Mobility Tensor”. The Lorentz reciprocal theorem implies that the resistance
and mobility tensors are symmetric [5, 7, 4]. Furthermore the submatrices A,B,K,M are symmetric
and positive definite. Moreover for the bodies with a uniform distribution of mass with any planes of
symmetry G = R−1. The resistance and mobility tensors are a function solely of the body geometry
and scale linearly with the viscosity of the fluid.The drag force and moment are related linearly to
the velocity and angular velocity of the swimmer in the body frame of reference.The center of the
mobility, the center of resistance and the center of mass do not have to coincide for a body of an
arbitrary shape.
The low Reynolds number world is Aristotelian : a body is in motion if and only if a force
is exerted on it. Equations 3.4 and 3.5 relate the forces to the velocities and not the accelerations of
the body. This phenomenon arises because the viscous drag force that the fluid exerts on the body
when it moves equals the external force (if it is steady). Equilibrium between the external force and
drag force is attained in negligible time. Prescribing an external force and moment on the swimmer
as control input, the motion of the swimmer is described purely through a kinematic model.
3.4 Special Case - No External Force
This is the case when the external torque matches hydrodynamic resistance/drag torque
Te = Th. Then
ω =MTe (3.6)
and
V = CTe (3.7)
The equation 3.6 can be solved independent of the equation 3.7. The equation 3.7 can be solved
subsequently. Finally the position of the swimmer with respect to spatial reference frame is obtained
11
by solving the equation
x
y
z
= RV (3.8)
12
Literature Review
In this section we will discuss briefly the concepts and the equations described in the paper
MM.
An arbitrary shaped body with a permanent magnetic moment mo was subjected to a
rotating magnetic field. Two frames are used lab coordinate system (LCS) and body coordinate
system (BCS) which is rigidly fixed to the body.
Mganetic Field H is described in LCS. H is a rotating magnetic field in (XY) plane.
H = H(cos(ωt), sin(ωt), 0) (3.9)
mBCS = m(n1, n2, n3) (3.10)
n = m/|m| = (sin(Φ) cos(α), sin(Φ) sin(α), cos(Φ)) (3.11)
where Φ and α are correspondingly the spherical polar and azimuthal magnetization angles.
Magnetic torque is the source of external torque Te = m×H.
C andM are the coupling and rotational viscous mobility tensors respectively. In this system
principle axes of rotation whose bases are (e1, e2, e3) are eigenvectors of M are chosen as BCS. M
has a diagonal form in this system. Eigenvalues and corresponding eigenvectors in ascending order.
Where M∞ ≤M∈ ≤M3, so we can say that e3 is the easy axis.
Easy axis means the axis on which it takes minimum amount of torque to cause same amount
of rotation.
Orientation of BCS with respect to LCS can be found using rotation matrix R which is
13
function of the three Euler angles (θ, φ, ψ) in ‘3 – 1 – 3 ‘ parametrization.
Hence we can write;
HBCS = R.HLCS (3.12)
The components of ω in BCS can be expressed through Euler angles as follows
ω1 = φ sin(θ) sin(ψ) + θ cos(ψ), ω2 = φ sin(θ) cos(ψ)− θ sin(ψ), ω3 = φ cos(θ) + ψ (3.13)
Using equation 3.13 and plugging into equation 3.6 and after simple algebraic reduction we get,
1 + ε
ω0
(φ sin(θ) sin(ψ) + θ cos(ψ)
)= n2 sin(θ) sin(φ) + n3
(cos(φ) sin(ψ) + sin(φ)cos(ψ) cos(θ)
)(3.14)
1− εω0
(φ sin(θ) cos(ψ)− θ sin(ψ)
)= −n1 sin(θ) sin(φ) + n3
(cos(φ) cos(ψ)− sin(φ) sin(ψ) cos(θ)
)(3.15)
1
pω0
(φ cos(θ) + ψ
)= n⊥
(sin(ψ + α) cos(φ) + sin(φ) cos(ψ + α) cos(θ)
)(3.16)
Where, φ = φ− ωt, n⊥ = sin(Φ).
Characteristic frequency, ω0 = mHM⊥, where M⊥ is the harmonic average of the minor
mobilites, M⊥ = 2 M1.M2
M1+M2and longitudinal p =
(M3
M⊥
)≥ 1 and transverse rotational anisotropy
ε = M2−M1
M1+M2≥ 0. For example slender objects p 1, ε = 0 for bodies of revolution (e.g. Cylinder
and spheroid) or compact isotropic objects (Sphere, Cube).
This paper discusses mostly about a particular dynamics regime which is called synchronous
dynamics. Synchronous dynamics means when the magnetic moment which is fixed to the body can
follow the applied magnetic field.
This kind of dynamics is observed when the Euler angles are independent of time.
φ = const., ψ = const., θ = const. (3.17)
Throughout the paper the rotational problem has been solved by integrating the dynamical
system numerically.
14
The solution of the rotational problem depends on anisotropy parameters ε and p as well as
orientation of m given by Φ and α.
In low-frequency tumbling regime the net magnetic moment m = m⊥ + m‖ where m⊥,m‖
are the components perpendicular and parallel to the rotation easy axis, rotates in the plane of
applied magnetic field H in order to catch up with it and reduce the magnetic energy Em = −m.H.
We denote angle between easy axis and magnetic moment as β which is a function of the
other configuration angle Φ and α described in fig. 3.1. At low frequency input we observe tumbling
motion but as we increase driving frequency at critical frequency ωtw = cos(β) tumbling regime
divides into two high frequency wobbling states.
ωso =
√cos2(β) + sin2(β)p2 (3.18)
Above step out frequency ωso, the viscous drag torque can no longer be balanced by the magnetic
torque propulsion cease to exist and the motion moves into asynchronous regime.
15
Chapter 4
Simulation Results
There is an infinite possible configuration of magnetic moment but we considered the follow-
ing configuration (Φ = 90 and α = 20) stated in MM paper who have calculated these configuration
from experimental results of KMF paper. We found out that step-out frequency for our chosen con-
figuration is ∼ 33.5rad/s. Which we will see is very close to our simulation results for the 3 bead
swimmer. It has been stated in experimental paper like MKF that vertex angle of 3 bead swimmers
are most likely to fall in the range of π2 to 2π
3 . We chose the mobility matrix values from MF for
a 3 bead micro-swimmer with vertex angle π2 . The values of the submatrix C and M which relates
velocity and angular velocity to applied magnetic torque are,
C =
0 0 0
0 0 −1.16
0 1.94 0
× 1012N−1s−1 (4.1)
M =
5.84 0 0
0 8.96 0
0 0 5.11
× 1017radN−1s−1m−1 (4.2)
1. Rotating Magnetic field about 1-axis: We express the Inertial frame by X-Y-Z coordinate
16
-0.01
-0.005
0
0.005
0.01
Hx (mT)-0.01
-0.005
0
0.005
0.01
Hy (mT)
-1
-0.5
0
0.5
1
Hz
(mT
)
Figure 4.1: Representation of Rotating Magnetic field for omega = 33rad/s
system. The magnetic field can be expressed as follows.
HX = A.cos(ωt) (4.3)
HY = A.sin(ωt) (4.4)
HZ = 0 (4.5)
We computed results when A = 10 mT. We chose this value because similar magnetic field
strength has been used in the experimental result based paper of KMF. We performed a Fourier
analysis on the X, Y, Z components of the magnetic field where ω = 33rad/s. From fig. 4.2,
4.3, 4.4 we observe that X- component has frequency 33 rad/s, Y-component has frequency
33 rad/s, Z-component has no frequency because there is no magnetic field in that axis.
In fig. 4.5 the swimmer clearly shows tumbling regime at low frequencies when there is no
propulsion then propulsion increases till step-out frequency beyond which the propulsion along
z-axis which coincides with direction of axis of the field rotation decreases. In fig. 4.6, 4.7
we can also observe synchronous and asynchronous zones by checking how the angle between
magnetic moment fixed to the swimmer and the magnetic field in body fixed frame varies with
time. We observe beyond step-out frequency the angle varies with time but below step-out
17
0 20 40 60 80 100 120 140 160 180 200f (rad/s)
0
1
2
3
4
5
6
7
|P1(
f)|
10-3
Figure 4.2: Single sided amplitude spectrum of the X component of magnetic field described in case1
18
0 20 40 60 80 100 120 140 160 180 200f (rad/s)
0
1
2
3
4
5
6
7
|P1(
f)|
10-3
Figure 4.3: Single sided amplitude spectrum of the Y component of magnetic field described in case1
19
0 20 40 60 80 100 120 140 160 180 200f (rad/s)
-1
-0.8
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
0.8
1
|P1(
f)|
Figure 4.4: Single sided amplitude spectrum of the Z component of magnetic field described in case1
20
5 10 15 20 25 30 35 40 45
(rad/s)
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
Vz (
m/s
)
Figure 4.5: Propulsion Velocity(µm/s) versus driving Frequnecy(rad/s)
frequency it reaches a constant value.
λ =m.HBCS
|m||HBCS |(4.6)
Where λ is the cosine of the angle between magnetic moment of the swimmer and the magnetic
field in body coordinate System. m and HBCS are magnetic moment vector and magnetic
field vector in body coordinate System. Figure 4.8, 4.9 are the plots for trajectories of the
swimmer at frequency below and beyond step-out frequency respectively for 75 cycles of field
rotation. Beyond step-out frequency it does not have significant propulsion. Beyond step-out
frequency the propulsion or lack of is called asynchronous regime. In previous literature they
have not quantified the asynchronous behavior of the swimmer. We tried to quantify that
with two parameters. One, the maximum and minimum value λ is reaching is clearly showing
the extent of angle between magnetic moment and magnetic field. Two, the rate at which
the angle between magnetic moment and field is changing. So in fig. ?? we plot the variable
∆ = max(λ) −min(lambda) vs driving frequency ω. We observe that ∆ = 0 in synchronous
21
0 5 10 15 20 25 30t (s)
-1
-0.8
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
0.8
1
Figure 4.6: Angle between magnetic moment of the swimmer and the magnetic field in body coor-dinate System vs time for driving frequency 33 rad/s
22
0 5 10 15 20 25 30t (s)
-1
-0.8
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
0.8
1
Figure 4.7: Angle between magnetic moment of the swimmer and the magnetic field in body coor-dinate System vs time for driving frequencies 35 rad/s
23
Figure 4.8: Trajectory of the swimmer for driving frequency 33 rad/s
Figure 4.9: Trajectory of the swimmer for driving frequency 35 rad/s
24
5 10 15 20 25 30 35 40 45
(rad/s)
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
2
Figure 4.10: ∆ for field described in case 1 vs driving frequency ω
25
regime. Which means that the angle between magnetic moment and the field is not changing.
We observe beyond step-out frequency in asynchronous zone the angle is changing. We also
performed a Fourier analysis on λ. Since λ is cosine of the angle between magnetic moment and
field it’s Fourier analysis provides us with rate of change of the angle. From fig. 4.13 we can
see that below step-out frequency 34rad/s there is no frequency in spectrum of λ because the
angle is time invariant. But beyond step-out frequency in fig. 4.14 we see multiple frequency
at which the angle is changing. There are many frequencies between 0 to 4 rad/s with peaks at
1, 1.5, 2 and 3 rad/s. Which means the moment is slipping in and out of phase with magnetic
field.
To explain this further we calculated the magnetic torque experienced by the swimmer in
inertial frame. In fig. 4.15 we can see below step-out frequency the torque in X, Y direction
are changing sign so no net motion in those direction but torque in Z direction reaches a steady
value and we get propulsion in Z direction because torque is linearly related to velocity. In fig.
4.16 we can see beyond step-out frequency torque in Z diretion changes direction with time so
the velocity change direction with it hence propulsion is not possible.
2. Rotating Magnetic Field with Oscillating axis: We simulated the results for the swimmer when
subjected to a magnetic field with oscillating axis described as follows. Using 4.3 in 4.7.
H =
HX
HY
HZ
(4.7)
R =
1 0 0
0 cos(Γ) sin(Γ)
0 −sin(Γ) cos(Γ)
(4.8)
Γ = γ. sin(ζωt) (4.9)
H2 = R.H (4.10)
Where ω is the driving frequency of the magnetic field. ζ is a constant factor. The field
26
0 2 4 6 8 10 12 14 16 18 20f (rad/s)
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
2
|P1(
f)|
Figure 4.11: Single sided amplitude spectrum of λ for propulsion under magnetic field in case 1 forω = 11 rad/s
27
0 2 4 6 8 10 12 14 16 18 20f (rad/s)
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
|P1(
f)|
Figure 4.12: Single sided amplitude spectrum of λ for propulsion under magnetic field in case 1 forω = 19 rad/s
28
0 2 4 6 8 10 12 14 16 18 20f (rad/s)
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
2
|P1(
f)|
Figure 4.13: Single sided amplitude spectrum of λ for propulsion under magnetic field in case 1 forω = 31 rad/s
29
0 2 4 6 8 10 12 14 16 18 20f (rad/s)
0
0.5
1
1.5
2
2.5
3
3.5
4
4.5
5
|P1(
f)|
Figure 4.14: Single sided amplitude spectrum of λ for propulsion under magnetic field in case 1 forω = 41 rad/s
30
0 2 4 6 8 10 12t (s)
-5
-4
-3
-2
-1
0
1
2
3
4
Tor
que
(N-m
)
10-17
Tx
Ty
Tz
Figure 4.15: External torque on the swimmer in inertial frame under magnetic field described inCase 1 for ω = 33 rad/s
31
0 2 4 6 8 10 12t (s)
-5
-4
-3
-2
-1
0
1
2
3
4
5
Tor
que
(N-m
)
10-17
Tx
Ty
Tz
Figure 4.16: External torque on the swimmer in inertial frame under magnetic field described inCase 1 for ω = 41 rad/s
32
-0.01
0.01
-0.005
0.005 0.01
0H
z (m
T)
0.005
0.005
Hy (mT)
0
Hx (mT)
0
0.01
-0.005-0.005
-0.01 -0.01
Figure 4.17: Representation of rotating magnetic field described in case 2 where ω = 33rad/s
described in the eqn. 4.10 can be graphically represented by the fig. 4.17. We did a fourier
transform analysis of the magnetic field described by eqn. 4.10, where ω = 33rad/s, ζ = 0.2
and γ = π4 and found out that X-component of the field is oscillating with frequency 33 rad/s,
Y-component of the field is oscillating with 3 frequencies where 33 rad/s is the major frequency
and 22 and 45 rad/s are the minor frequencies, Z-component of the field is oscillating with 2
frequencies of equal contribution they are 26 and 39 rad/s. The following figures 4.18, 4.19,
4.20 shows the Fourier analysis of the field. We observed that So when we simulated results
for a rotating magnetic field with oscillating axis about X-axis in the inertial X-Y-Z frame
described above and observed the following results. We observe propulsion velocity is lower in
the direction of the Z-axis compared to case 1. We also observe propulsion in asynchronous
regime. We can see that from fig. ??, ?? which shows angle between magnetic moment and
the field is time-variant, asynchronous.
We also did similar analysis of the extent of asynchronous behavior in which significant propul-
sion is occurring. We observe in fig. 4.26 that δ has non zero values even when ω is small.
When means that swimmer enters asynchronous regime at a very low frequency. But we
also observe that beyond a certain value of ω the ∆ value increases to a very high value and
propulsion decreases. It means that beyond a certain value of ω the phase slip between the
magnetic moment and the field is too extreme and the external magnetic torque is failing to
33
0 20 40 60 80 100 120 140 160 180 200f (rad/s)
0
1
2
3
4
5
6
7
|P1(
f)|
10-3
Figure 4.18: Single sided amplitude spectrum of the X component of magnetic field described incase 2
34
0 20 40 60 80 100 120 140 160 180 200f (rad/s)
0
1
2
3
4
5
6
|P1(
f)|
10-3
Figure 4.19: Single sided amplitude spectrum of the Y component of magnetic field described incase 2
35
0 20 40 60 80 100 120 140 160 180 200f (rad/s)
0
0.5
1
1.5
2
2.5
3
3.5
4
|P1(
f)|
10-3
Figure 4.20: Single sided amplitude spectrum of the Z component of magnetic field described in case2
36
5 10 15 20 25 30 35 40 45
(rad/s)
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
0.45
0.5
Vel
oci
ty (
m/s
)
Figure 4.21: Propulsion Velocity(µm/s) versus driving Frequnecy(rad/s)
overcome the drag forces. We also did a Fourier analysis of ∆. In fig. 4.27, 4.28, 4.29, 4.30
we see that angle between moment and the field is varying with few discrete frequencies with
varying contribution but beyond 33rad/s we observe a band width of frequencies of varying
contribution with which the angle is changing. We also not that none of the frequencies have
any correlations with their corresponding driving frequencies. Which means it is not locking
with the magnetic frequency at all.
We can see from fig. 4.31, 4.32 that below 33 rad/s even though the swimmer is out of sync
with the field propulsion is observed because the torque in Z direction does not change direction
but beyond that it does so no propulsion is observed.
3. Magnetic field used is super imposition of field described in 1 and 2 We superimposed two field
so that we can observe the characteristics of both field in the trajectories of the swimmer. We
superimposed field described in case in case 1 but we used different magnitudes for the field
37
0 10 20 30 40 50 60 70 80 90 100t (s)
-0.1
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
Figure 4.22: Cosine of angle between magnetic moment of the swimmer and the magnetic field inbody coordinate System vs time for driving frequency 33 rad/s
38
0 10 20 30 40 50 60 70 80 90
t (s)
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
Figure 4.23: Cosine of angle between magnetic moment of the swimmer and the magnetic field inbody coordinate System vs time for driving frequency 35 rad/s
Figure 4.24: Trajectory of the swimmer for driving frequency 33 rad/s
39
Figure 4.25: Trajectory of the swimmer for driving frequency 35 rad/s
in case 2. The field used can be described as follows:
H3 = H + εH2 (4.11)
Where H and H2 are the magnetic field described in case 1 and 2. ε is the contribution factor
of the field described in Case 2. ε is a constant (time invariant) factor. The field expressed by
the eqn. 4.11 can be graphically represented by fig. 4.33 We did a Fourier transform analysis of
the magnetic field described by eqn. 4.10, where ω = 33 rad/s, ε = 0.1,ζ = 0.2 and γ = π4 and
found out that X-component of the field is oscillating with frequency 33 rad/s, Y-component of
the field is oscillating with frequencies major 33 rad/s and minor frequencies 22 and 45 rad/s
w, Z-component of the field is oscillating with 2 frequencies of equal contribution they are 26
and 39 rad/s. The following figures 4.34, 4.35, 4.36 shows the Fourier analysis of the field.
We simulated results for ε = 0.1. We observe that the swimmer enters asynchronous regime
at lower driving frequency and the propulsion is comparable to synchronous regime propulsion
described in Case 1. Fig. 4.37,4.38 and fig. 4.39, 4.40 shows that swimmer is in asynchronous
regime and propulsion is still possible respectively
We varied the parameters γ and ζ in eqn. 4.9 and found out that when ζ > 1 the propulsion
40
5 10 15 20 25 30 35 40 45
(rad/s)
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
2
Figure 4.26: ∆ for field described in case 2 vs driving frequency ω
41
0 2 4 6 8 10 12 14 16 18 20f (rad/s)
0
0.5
1
1.5
2
2.5
3
|P1(
f)|
Figure 4.27: Single sided amplitude spectrum of λ for propulsion under magnetic field in case 2 forω = 11 rad/s
42
0 2 4 6 8 10 12 14 16 18 20f (rad/s)
0
0.5
1
1.5
2
2.5
3
3.5
4
4.5
5
|P1(
f)|
Figure 4.28: Single sided amplitude spectrum of λ for propulsion under magnetic field in case 2 forω = 19 rad/s
43
0 2 4 6 8 10 12 14 16 18 20f (rad/s)
0
1
2
3
4
5
6
|P1(
f)|
Figure 4.29: Single sided amplitude spectrum of λ for propulsion under magnetic field in case 2 forω = 31 rad/s
44
0 2 4 6 8 10 12 14 16 18 20f (rad/s)
0
1
2
3
4
5
6
7
8
9
10
|P1(
f)|
Figure 4.30: Single sided amplitude spectrum of λ for propulsion under magnetic field in case 2 forω = 41 rad/s
45
0 2 4 6 8 10 12t (s)
-5
-4
-3
-2
-1
0
1
2
3
4
Tor
que
(N-m
)
10-17
Tx
Ty
Tz
Figure 4.31: External torque on the swimmer in inertial frame under magnetic field described inCase 2 for ω = 33 rad/s
46
0 2 4 6 8 10 12t (s)
-5
-4
-3
-2
-1
0
1
2
3
4
Tor
que
(N-m
)
10-17
Tx
Ty
Tz
Figure 4.32: External torque on the swimmer in inertial frame under magnetic field described inCase 2 for ω = 41 rad/s
47
-1
0.015
-0.5
0.010.015
0
Hz
(mT
)
10-3
0.005 0.01
0.5
Hy (mT)
0 0.005
Hx (mT)
0
1
-0.005-0.005
-0.01-0.01
-0.015 -0.015
Figure 4.33: Representation of the magnetic field described in Case 3
cease to exist and when ζ < 1 we observe the following results. We varied oscillating factor ζ
from 0.2 to 0.02 and found that propulsion is happening in asynchronous regime. But effects
on magnitude of the propulsion velocity for same ω is not significant. We varied γ oscillation
amplitude and oscillation factor ζ keeping driving frequency ω of the fixed axis field constant
and observed following distribution of propulsion velocity in fig. 4.41, 4.42. We observe that
for ε = 0.1 the velocity vs ω plot has same characteristics as that of field described in case 1.
Propulsion velocity increases upto a certain driving frequency and then it drops. But from fig.
?? we see that propulsion of comparable magnitude is possible in asynchronous regime. We
varied the contribution factor ε and observed variation in Propulsion Velocity as we change
the driving frequency.
We can see in fig. 4.47, 4.48, 4.45, 4.46 that the field described in case 2 is introducing
the asynchronous nature in the swimmer’s motion. As we reduce the contribution of H2 in
H3, ε, we observe that λ reaches a constant value and velocity vs driving plot shows similar
characteristic to the plot for field described in case 1 and there is no propulsion in asynchronous
zone. But when ε is increased we observe that asynchronous nature starts at a lower driving
frequency and significant propulsion exists. We also observe in fig. 4.44 that as contribution
factor ε is increased the magnitude of the propulsion increases and propulsion exists at very
high driving frequency ω. We also performed Fourier analysis and check the effect of ε and
48
0 20 40 60 80 100 120 140 160 180 200f (rad/s)
0
1
2
3
4
5
6
7
8
|P1(
f)|
10-3
Figure 4.34: Single sided amplitude spectrum of the X component of magnetic field described incase 3
49
0 20 40 60 80 100 120 140 160 180 200f (rad/s)
0
1
2
3
4
5
6
7
8
|P1(
f)|
10-3
Figure 4.35: Single sided amplitude spectrum of the Y component of magnetic field described incase 3
50
0 20 40 60 80 100 120 140 160 180 200f (rad/s)
0
0.5
1
1.5
2
2.5
3
3.5
4
|P1(
f)|
10-4
Figure 4.36: Single sided amplitude spectrum of the Z component of magnetic field described in case3
51
0 10 20 30 40 50 60 70 80 90 100
t (s)
-0.05
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
0.45
Figure 4.37: Cosine of angle between magnetic moment of the swimmer and the magnetic field inbody coordinate System vs time for driving frequency 33 rad/s
0 10 20 30 40 50 60 70 80
t (s)
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
Figure 4.38: Cosine of angle between magnetic moment of the swimmer and the magnetic field inbody coordinate System vs time for driving frequency 41 rad/s
52
Figure 4.39: Trajectory of the swimmer for driving frequency 33 rad/s
Figure 4.40: Trajectory of the swimmer for driving frequency 41 rad/s
53
0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2 0.22
10
20
30
40
50
60
70
80
90
Figure 4.41: Variation in Propulsion Velocity(µm/s) with oscillation amplitude (degrees) and oscil-lation factor ζ for driving frequencies 15 rad/s
0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2 0.22
10
20
30
40
50
60
70
80
90
Figure 4.42: Variation in Propulsion Velocity(µm/s) with oscillation amplitude (degrees) and oscil-lation factor ζ for driving frequencies 30 rad/s
54
0.2
0.15
0.1
0.05
0
(rad/s)
05 10 15 20 25 30 35 40 45
0.2
0.4
Vel
oci
ty (
-m/s
) 0.6
0.8
Figure 4.43: Variation in propulsion velocity(µm/s) with driving frequnecy(rad/s) and oscillationfactor ζ
ω on asynchronous behavior. We can see from fig. 4.53 that it shows similar characteristics
with fig. 4.10 when ε values are small. Which means the contribution of the field described in
case 2 is small. Hence it shows similar properties as case 1. As we increase ε we see that the
asynchronous behavior does not ramp up rapidly. We also compiled the frequencies at which
the angle between magnetic moment and magnetic field is changing for different values of ε in
fig. 4.54, , , . We can observe that there is no direct correlation between the frequencies
of the driving magnetic field and the frequency with which the angle between the magnetic
moment and the field is changing.
We calculated magnetic torque experienced by the swimmer for magnetic fields described in
Case 3. From figures 4.58,4.59 we can see beyond step-out frequeny for the torque in Z direction
has periodic direction change hence propulsion decreases.
55
5 10 15 20 25 30 35 40 45 50 55
(rad/s)
0
0.2
0.4
0.6
0.8
1
1.2
Vel
ocity
(-m
/s)
= 1 = 0.5 = 0.02 = 0.01
Figure 4.44: Propulsion Velocity(µm/s) versus driving frequency(rad/s)
56
0 10 20 30 40 50 60 70 80 90 100t (s)
-0.1
0
0.1
0.2
0.3
0.4
0.5
Figure 4.45: Cosine of angle between magnetic moment of the swimmer and the magnetic field inbody coordinate System vs time for ε = 0.02, ω = 33rad/s
57
0 10 20 30 40 50 60 70 80t (s)
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
Figure 4.46: Cosine of angle between magnetic moment of the swimmer and the magnetic field inbody coordinate System vs time for ε = 0.02,ω = 41rad/s
58
0 10 20 30 40 50 60 70 80 90 100t (s)
-0.2
0
0.2
0.4
0.6
0.8
1
Figure 4.47: Cosine of angle between magnetic moment of the swimmer and the magnetic field inbody coordinate System vs time for ε = 1, ω = 33rad/s
59
0 10 20 30 40 50 60 70 80t (s)
-0.1
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
Figure 4.48: Cosine of angle between magnetic moment of the swimmer and the magnetic field inbody coordinate System vs time for ε = 1, ω = 41rad/s
60
Figure 4.49: Trajectory of the swimmer for ε = 0.01, ω = 33rad/s
Figure 4.50: Trajectory of the swimmer for ε = 0.01, ω = 41rad/s
61
Figure 4.51: Trajectory of the swimmer for ε = 0.02, ω = 33rad/s
Figure 4.52: Trajectory of the swimmer for ε = 0.02, ω = 41rad/s
62
5 10 15 20 25 30 35 40 45 50 55
(rad/s)
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
2
= 1 = 0.5 = 0.1 = 0.02 = 0.01
Figure 4.53: ∆ versus driving frequency(rad/s) for different values of ε
63
10 15 20 25 30 35 40 45
(rad/s)
0
0.5
1
1.5
2
2.5
3
3.5
4
4.5
5
(rad
/s)
Figure 4.54: Major frequencies in λ vs ω for different values for ε = 0.01
64
10 15 20 25 30 35 40 45
(rad/s)
0
0.5
1
1.5
2
2.5
3
3.5
4
4.5
5
(rad
/s)
Figure 4.55: Major frequencies in λ vs ω for different values for ε = 0.02
65
10 15 20 25 30 35 40 45
(rad/s)
0
1
2
3
4
5
6
7
8
(rad
/s)
Figure 4.56: Major frequencies in λ vs ω for different values for ε = 0.5
66
10 15 20 25 30 35 40 45
(rad/s)
0
1
2
3
4
5
6
7
8
(rad
/s)
Figure 4.57: Major frequencies in λ vs ω for different values for ε = 1
67
0 2 4 6 8 10 12t (s)
-8
-6
-4
-2
0
2
4
Tor
que
(N-m
)
10-17
Tx
Ty
Tz
Figure 4.58: External torque on the swimmer in inertial frame under magnetic field described inCase 3 for ω = 41 rad/s
68
0 2 4 6 8 10 12t (s)
-8
-6
-4
-2
0
2
4
6
8
Tor
que
(N-m
)
10-17
Tx
Ty
Tz
Figure 4.59: External torque on the swimmer in inertial frame under magnetic field described inCase 3 for ω = 55 rad/s
69
Chapter 5
Conclusion and Scope
Previous work has shown that propulsion cease to exist as we increase the driving frequency
of the field and it enters the asynchronous zone. They have inferred that there is no sustained
propulsion in this regime. But this conclusion is field specific. By using different magnetic field
specifically with multiple frequency of oscillation we have shown in this work that a swimmer can
enter asynchronous zone at even lower driving frequency and can show significant propulsion. We
have also quantified the asynchronous behavior of the swimmer with the applied field and found out
that swimming is possible for certain amount of asynchronous behavior. We have also shown that
propulsion decreases beyond step-out frequency because of periodic direction change of the applied
torque.
This work shows in details the complex dynamics of propulsion of a magnetic micro-swimmer
controlled by external perturbed magnetic field. We have shown that sustained propulsion of the
swimmer in the asynchronous zone. This result will be crucial during design of signal to be used
during control of the swimmer while performing complex trajectories.
The special cases in which this model of the magnetic field becomes relevant is when we do
experiments. The environmental vibrations can be transferred to the magnetic field which in turn
will affect the propulsion of the swimmer. We may observe propulsion under such perturbed field
to be different than the propulsion that has been predicted using an unperturbed field. Another
aspect that can be explored is while sending complex signals to the magnetic coils that produce the
magnetic field there may be some level of noise in that signal which in turn will affect the propulsion
of magnetic field by shifting the working frequency regime that has been estimated based on previous
70
research. By modeling the perturbed magnetic field we can get a more realistic simulation of the
propulsion of the micro-swimmers.
These swimmer models and simulations were done for propulsion in Newtonian fluid. But
for propulsion in non-Newtonian fluid we will have to modify our model of the swimmer to provide
more accurate simulation of the trajectories.
71
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