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9 RR protected version o f British history.
num-ber where the hydrodynamic drag contributes to asymmetries (see Fig.5.3c). The cilium steady state does not always centre around its starting vertical position (see Fig.5.3cand5.3d). The cilium can also become wrapped around itself, maintaining a permanent, sometimes signi-ficant bend throughout the entire stroke (see Fig.5.3d). (However, at this extreme, issues of physical accuracy, and the accuracy of the model due to our assumption of small curvature should be noted.) In general, the recovery stroke is faster than the effective stroke, being par-ticularly fast when the cilium becomes tightly wrapped (see Fig.5.3b and 5.3d) but there is a much smaller difference in speed when the effective and recovery stroke are similar (see Fig.5.3a).
5.4 Equilibrium model and asymptotics
In order to validate the force model and understand the behaviour of the cilium in the limit of low Sperm number we will compare our force model to an equilibrium model. This equilib-rium model will balance magnetic forces against elastic forces to identify the minimal energy position and thus where the cilium would settle given an infinite amount of time and a fixed direction of the uniform magnetic field.
5.4.1 Equilibrium model governing equation
5.4.1.1 Elastic and magnetic energy formulations
The elastic energy held when the cilium is bent, assuming bending in one axis, no twist and small strain, is
1 2
Z L 0
EIκ2(s)ds, (5.38)
whereκ is the radius of curvature [105].
Magnetic energy is held when the magnetic dipoles (corresponding to the magnetic filings in the cilium) are not aligned with the magnetic field and the total magnetic energy of the cilium is
−1 L
Z L
0
µ·B
ds. (5.39)
(We note this quantity can be negative due to ignoring an additional constant which will not affect our energy minimisation.) This energy is minimised when the magnetic dipole and magnetic field are parallel and pointing in the same direction. It is largest when they are parallel but in opposite directions.
5.4.1.2 Magnetic field equivalent to a point force
It is interesting to note that Eq.5.39can be directly integrated sinceµ·B=µ0t·B=µ0(∂r/∂s)· Bgiving the magnetic energy as
−µ0
L r(L)·B, (5.40)
if the base of the cilium is at the origin. As such the magnetic energy contained in a cilium in a uniform magnetic field is equivalent to the energy of a cilium with a point force on its end pointing in the same direction.
5.4.1.3 Minimising energy equation
The total energy contained in the cilium is Z L
0
−µ0 L
∂r
∂s·B+EI 2
∂2r
∂s2
· ∂2r
∂s2
ds. (5.41)
This non-dimensionalises to Z 1
0
−∂r∗
∂s∗·B∗+ζ 2
∂2r∗
∂s∗2
· ∂2r∗
∂s∗2
ds∗, (5.42)
where stars represent non-dimensional quantities. Similar to previously, the stars will now be dropped but all quantities in the rest of this chapter will be non-dimensional.
Substituting the dependence onΦandα this equation reduces to Z 1
0
−
"
cos(α−Φ) +ζ 2
∂ α
∂s 2#
ds. (5.43)
This is the energy function, which we are to minimise. We minimise it by applying standard calculus of variational methods, so the energy minima (and maxima) positions of the cilium solve
ζ∂2α
∂s2 −sin(α−Φ) =0, (5.44)
subject to the boundary conditions that the cilium end is force free
∂ α
∂s(1) =0, (5.45)
and the cilium is vertical at its base
α(0) = π
2. (5.46)
5.4 Equilibrium model and asymptotics 139 We numerically solved this boundary value problem by using ‘bvp4c’ in MATLAB. We take initial conditions first as the vertical cilium to obtain global energy minima.We also stead-ily increasedΦ(the angle of the magnetic field) solving the boundary value problem for each Φbut using the position of the cilium at the previous slightly smaller value ofΦas the initial condition.
5.4.2 Comparison to the force model
The comparison of the cilium position from the force model and the equilibrium model are shown in Fig.5.4. In the force model the magnetic field direction is rotated in discrete in-crements, and the cilium is plotted, just prior to increasing the magnetic field by an increment.
The two are in good agreement for the forward stroke of the cilium so the force model follows the local energetic minima of the cilium, as would be expected physically, thus validating the force model.
At high ζ the entire stroke follows the energetic global minima (see Fig.5.4a). This is because there is a continuous change in the position of the global minima as the magnetic field is rotated. This changes atζ =0.417 where there is supercritical pitchfork bifurcation in the position of the cilium when the magnetic field is vertically downwards. For this magnetic field direction, at high ζ the energetically optimal cilium position is unbent. But as ζ is decreased, at ζ =0.417 this position becomes unstable and two global minima appear, one bent to the left and one to the right. Therefore, at lowζ, as the magnetic field is rotated past the vertically downwards position, the global minimum jumps from being bent to the left to being bent to the right. The existence of this bifurcation is well known since for a vertically downwards magnetic field our energy equation is the well known problem of a point weight on the end of a flexible rod [105].
At lowerζ the global minima are separated, see Fig.5.4b, but initially the jump between global minima occurs very soon after the angle of the magnetic field is rotated past vertically downwards. But at lower ζ a local energetic minimum becomes important as seen as in Fig.5.4c. Rather than immediately jumping between global minima there is a curled local minimum, which the cilium follows. This behaviour must be a local energy minimum due to the asymmetry, since the global minimum would give symmetric cilium positions. Therefore, the cilium stays in the local minimum till the magnetic field has rotated far enough, then it jumps to the global minimum for that magnetic field direction.
During the jump between energetic minima hydrodynamic forces become significant, al-lowing the forward and recovery strokes to become very different, see Fig.5.4c. Most of the cilia in the recovery stroke have not been plotted in Fig.5.4band5.4csince the stroke happens
(a)ζ =1
(b)ζ=0.3
(c)ζ=0.1
Figure 5.4: Comparison between the force and equilibrium model during the stroke for in-creasing angle of the magnetic field. For the force model Sp=0.32. For the equilibrium model the previous position of the cilium at a small magnetic field angle is used to update the position at the new magnetic field angle. Blue cilia are part of the forward anticlockwise stroke and red cilia the clockwise recovery stroke. The equilibrium model is represented by crosses and the force model by solid lines.
5.4 Equilibrium model and asymptotics 141 very quickly so the recovery stroke only represents a very small change of the magnetic field direction.
5.4.3 Asymptotics
In this limit of low Sperm number we can additionally study the the two cases ζ →0 and ζ →∞analytically, to more explicitly determine the shape of the cilium in each limit and the total energy contained by the bent cilium.
5.4.3.1 Limit ofζ →∞ In this limit Eq.5.44reduces to
∂2α
∂s2 =0, α(0)=π2, ∂ α
∂s(1) =0, (5.47)
which has the simple solution
α = π
2, (5.48)
so the cilium remains vertical at leading order.
We can calculate a first order correction by assumingα =π2+ζ−1α1+O(ζ−2)so at next order inζ
∂2α1
∂s2 −cos(Φ) =0, α1(0)=0, ∂ α1
∂s (1) =0, (5.49) with the solution
α(s) = π 2+ζ−1
cos(Φ)
2 s2−cos(Φ)s
+O(ζ−2), (5.50)
and therefore the cilium position a distance ˆsalong the cilium is y=
Z sˆ
0
sin(α(s))ds=sˆ+O(ζ−2), (5.51)
x= Z sˆ
0
cos(α)ds=−ζ−1
cos(Φ)
6 sˆ3−cos(Φ) 4 sˆ2
+O(ζ−2). (5.52) Therefore, the cilium remains vertical in this regime with the first order correction being of O(ζ−1)and determined by the magnitude of the horizontal magnetic force.
The natural logarithm of the total bending energy this cilium has is
ln
"
1 2
Z 1
0
∂ α
∂s 2
ds
#
=ln
(ζ−1cos(Φ))2 2
Z 1
0
(s2−2s+1)ds
=−2ln(ζ) +ln
cos(Φ)2 6
. (5.53)
5.4.3.2 Limit ofζ →0 In this limit (5.44) reduces to
sin(α−Φ) =0, (5.54)
with the solutionα=Φ+nπi.e. the cilium aligns with the magnetic field. However, although this obeys the boundary condition ∂ α
∂s(1) =0, it does not obeyα(0) =π2 . As such we require a boundary layer abouts=0.
We take a boundary layer length scale λ and a new boundary length-scale ˜s so s=λs˜ which reduces Eq.5.44to
ζ λ2
∂2α
∂s˜2(s)˜ −sin(α(s)˜ −Φ) =0, (5.55) where requiring both terms to be of the same order of magnitude in the boundary layers gives λ ∼ζ
1
2. Therefore, in the boundary layer
∂2α
∂s˜2 −sin(α−Φ) =0, α(0) =π
2, α(∞) =Φ. (5.56) This equation is similar to Eq.5.44but the changed form of the boundary conditions means an analytic solution exists. Shiftingα by defining ˆα =α−Φ−π2 gives
∂2αˆ
∂s˜2 −cos ( ˆα)=0, αˆ(0) =−Φ, αˆ(∞) =−π
2. (5.57)
This equation has the known exact solution [105] of αˆ = π
2−4 tan−1
tanh
s˜+s˜0 2
, (5.58)
5.4 Equilibrium model and asymptotics 143 where
˜
s0=2 tanh−1
tan
π/2+Φ 4
. (5.59)
Therefore,
α =π+Φ−4 tan−1
tanh
s˜+s˜0 2
. (5.60)
Eq.5.60can be integrated using results from [105] to give thexandyposition of the cilium in this sharply bent section
x(s) =ζ
1 2
cosπ
2+Φ
x0−2
1+ 1
cosh(ζ1/2s+s˜0)
−sin π
2 +Φ y0+2 tanh(ζ1/2s+s˜0)−(s+s˜0) i
, (5.61)
y(s) =ζ
1 2
sinπ
2+Φ
x0−2
1+ 1
cosh(ζ1/2s+s˜0)
−π
2 +Φ y0+2 tanh(ζ1/2s+s˜0)−(s+s˜0)i
, (5.62) wherex0andy0are constants of integration chosen to fix the base of the rod. Therefore, over a small region the cilium bends significantly to line up with the magnetic field and the remainder of the cilium is straight.
The total bending energy this cilium generates can be approximated by only considering bending within the boundary layer, since outside the boundary layer the cilium is straight.
Therefore, the natural logarithm of the total bending energy is ln ζ−1/2
2 Z ∞
0
∂ α
∂s˜ 2
ds˜
!
=−1
2ln(ζ) +ln
"
2−4 tanh(s˜20) 1+tanh2(s˜20)
!#
. (5.63)
5.4.4 Comparison of asymptotics to numerics
Fig.5.5validates the equilibrium model against the asymptotic solutions for the two magnetic field directions ofΦ=±3π/8. At both extremes ofζ the bending energy of the asymptotics and equilibrium model agree well and the shapes of the cilium obtained from the equilibrium model matches the shapes predicted by the asymptotics. At highζ the x component of the magnetic field for bothΦ=3π/8 andΦ=−3π/8 tends towards the same cilium shape and
bending energy value, as was predicted by our asymptotics since the x component of the magnetic field is the same in both cases.