In this section we will present the general setup for the problem and the important non-dimensional parameters. All the experiments with the discussed cilium design used iron
fil-5.2 Setup 127 ings, which are ferromagnetic [102]. However, in this analysis, we will consider embedded permanently magnetised powder.
The mathematics for a ferromagnetic cilium contains a singularity. The moment on an iron filing in a fixed magnetic field pushes the filing to align with the magnetic field. Mathematic-ally the moment isµ×B,forµ the magnetic dipole of the iron filing andBthe magnetic field.
Due to the geometry of an iron filing (long and thin), it can only be magnetised along its length, but which way along its length will change so that the magnetic dipole always has a positive component in the direction of the magnetic field. A cross product is theoretically maximal when the two vectors are perpendicular, so when the iron filing is perpendicular to the mag-netic field. If the iron filing is perturbed clockwise away from this perpendicular position , the direction of the magnetic dipole will be in a direction such that the filing will continue rotating clockwise to align with the magnetic field, and if it is perturbed anti-clockwise the magnetic dipole will be in the opposite direction so the filing will continue rotating anti-clockwise to align with the magnetic field. Thus there is a singularity in the numerics whenever any of the iron filings in the cilium are perpendicular to the magnetic field., as such permanently mag-netised powder (which has no such singularity) is a simpler first step, and future work could include extending this model to the experimental ferromagnetic cilium.
5.2.1 Cilium parametrisation
We will model our cilium in 2D, in the plane of the magnetic field rotation. We model a cilium of lengthLand of negligible width and depth, see Fig.5.2. We assume that the metal filings are at an additional anticlockwise angleχ to the cilium tangent, and that this angle is fixed for any one cilium. We assume the cilium has uniform elastic properties, despite it being a composite of metal filings and PDMS, with constant Young’s modulusE. The moment of inertia of the cilium cross-section,I, is also assumed to be constant along the length of the cilium.
We define the vertical direction, ey, as the direction parallel to the cilium base and we define the horizontal direction,ex,as the direction perpendicular to the cilium base. These are shown in Fig.5.2.
The magnetic field B will force the cilium. At any time t it will be at an angle Φ(t) to the horizontal and it rotates with a fixed angular frequencyω. The magnetic field remains at constant magnitude and at any fixed time is taken to be uniform. The magnetic dipole,µ of the magnetic filings in the cilium, is in the direction of the metal filings, which isχ+α, and has strengthµ0so
µ =µ0 cos(α+χ) sin(α+χ)
!
. (5.1)
We will then defineH =µ0|B|/L, which is the quantity determining the total strength of the magnetic field acting on our cilium.
This cilium is working at low Reynolds number so the hydrodynamic drag from the fluid can be estimated usingCN andCT, the normal and tangential drag coefficients [103].
We parametrise the distance along the cilium by s and we defineα(s,t) as the angle of the cilium tangent to the horizontal at timet. The position of the cilium given by r(s,t) = (x(s,t),y(s,t))is determined byα,where
x(s,t) = Z s
0
cos(α(s′,t))ds′, (5.2)
y(s,t) = Z s
0
sin(α(s′,t))ds′, (5.3)
where we assume the base of the filament is at the origin. Then, taking derivatives, gives the tangent
t= ∂r
∂s = cos(α) sin(α)
!
, (5.4)
and normal
n= −sin(α) cos(α)
!
, (5.5)
to the cilium, which are both shown in Fig.5.2. This parametrisation with respect toα is useful since it satisfies the condition of no-extension of the cilium length, sot·t=1.
5.2.2 Non-dimensional Parameters
There are three non-dimensional parameters important to this problem:ζ is the ratio of elastic to magnetic forces
ζ = EI
HL2; (5.6)
the Sperm number is the ratio of elastic to drag forces Sp=L
CNω EI
14
; (5.7)
5.2 Setup 129
Figure 5.2: A cilium of lengthLand clamped at its base, parametrised byα, being bent under a magnetic fieldB which is at an angle Φto the horizontal (at any fixed time) and rotating at an angular frequencyω. The coordinate system is defined by the normaln and the tangent t to the cilium.
and finally, the ratio of the normal to tangential drag coefficient CNT =CN
CT. (5.8)
Throughout this chapter we will keepCNT =2. This is a standard approximation for a long thin filament [29,103], although other values as low as 1.4 have also been used [30,104].
5.2.3 Effect of the angle of the magnetic filings
The torque exerted on each magnetic filing individually by the magnetic field is given by µ×B. Multiplying out the cross product, rearranging and factorising gives
µ×B=µ0 cos(α+χ) sin(α+χ)
!
× |B|cos(Φ)
|B|sin(Φ)
!
(5.9)
=µ0 cos(α) sin(α)
!
× |B|cos(Φ−χ)
|B|sin(Φ−χ)
!
. (5.10)
Therefore, having the metal filings at an angleχ to the cilium when a magnetic forceBis ap-plied at an angleΦis equivalent to having the metal filings parallel to the cilium and applying the magnetic field at an angle Φ−χ. So having the metal filings at an angle only causes a phase lag on the magnetic field direction. Since we are rotating the magnetic field at a constant speed and are interested in the steady state position of the cilium we will now assumeχ =0 so the metal filings are aligned with the tangent of the cilium.
We are aware that the phase delay would also have affected the relative starting potion of the magnetic field when it is turned on so affects our initial condition, which could affect the speed and which steady state the cilium approaches. At small Sperm number we do not believe alternative steady states exist due to the equilibrium validation we will discuss in §5.4, and at high Sperm number it takes many rotations to settle to a steady state, so again we do not anticipate other stable states. Although it is a possibility we feel it is unlikely other steady states exist even at middle Sperm number and the existence of alternative cilium steady states is not considered here.