CAPÍTULO 2: CARACTERÍSTICAS DEL SISTEMA
2.3 Modelamiento del negocio
Numerical estimates for the eccentricity of a bottom-heavy spheroid, whose centre-of-mass is offset from its centre-of-buoyancy by a factor h, may be obtained through solution of the mobility problem.
10−2 100 102 104 106 0
0.5 1 1.5x 10−3
Exact Numerical
angularvelocity(rads−1)
δh
Figure 2.13: The angular velocity of a sphere close to a wall, computed using the numerical method and an asymptotic approach, against the distance from the wall to the centre of the sphere.
The spheroid is placed at the centre of a shear box, which imposes a shear flow with rate-of-strain magnitude e, and is free to rotate but not translate, see Figure 2.14. The spheroid is assumed to have negligible sedimentation force resulting in no external force. However, the bottom-heaviness implies there is a gravitational torque acting on the cell, thus Lext= mgh(p × k), where k is the vertical axis, in a right-handed co-ordinate system, that makes an angle θ to the spheroid’s principal axis p, and m and g are the spheroids mass and the constant of gravitational acceleration, respectively.
The abscissa for the spheroid are generated in the same manner as the cell body, discussed in § 2.4, and are static with respect to the centre-of-buoyancy. We restrict motion to within the xz-plane and use (2.16) to compute the angular velocity Ω. Rotation occurs solely around the y-axis, pointing into the page in Figure 2.14, thus we have that ˙θ = Ωy. We can then obtain a numerical estimate by fitting the numerical data to (2.22).
Exact solutions can also be obtained, with α0 given by (2.21) and β by,
β = hmg µvα⊥
, (2.29)
where v is the volume of the spheroid and µ is the fluid viscosity [81]. The viscous torque parameter α⊥ for a spheroid is given by [81]
α⊥= (α21− 1)2(α21+ 1)4 (2 + γ(2α41− 3α21− 1))α21
,
where γ = cosh−1(α1)/α1qα21− 1 and α1 is the ratio of semi-major to semi minor axes.
W G C
Ω i
k
Figure 2.14: A spheroid whose centre-of-mass, G, offset from its centre-of-buoyancy C. The offset causes a gravita-tional torque which re-orients the spheroid with angular velocity Ω. The spheroid is placed in the centre of a shear flow.
The exact and numerical estimates for a fixed rate-of-strain e = 5 × 10−2 s−1 and fixed dis-cretisation size are shown in Table 2.7, for the properties listed in Table 2.3 and m = 5.2 × 10−10g (spheroid mass), and h = 10−5 cm (gravity offset). The numerical estimates are obtained by fitting all variables in (2.22). Fitting for α0 alone increases the numerical estimate by 9%. To fit α0 alone we consider first the spheroid rotating in no external flow and compute β using (2.23). We then use β and the imposed rate-of-strain e to find α0. When we perform a two paramter fit for α0 and β we obtain the same increase in the estimate. However, if we fit for e and α0 or for all three parameters then we get the estimates shown in Table 2.7. This is due to small errors in the numerics causing a slightly higher shear in the plane the spheroid sits in than the value introduced when attempting to impose the shear flow. The spheroids have an approximate discretisation size ∆s = 0.0682 and regularisation parameter ǫ = 0.19∆s0.66, chosen based upon analysis similar to that discussed in
§ 3.2.4. The shear box employed to generate the flow is composed of 2274 nodes spread along a cuboid of dimensions 80 ds wide and depth and height equal to 10 ds, where ds is the minor axis of the spheroid.
From Table 2.7 we observe that the numerical estimate for the eccentricity is within O(10−3) of the actual eccentricity, although for small α0 the numerics are correct to O(10−4). If ∆s is increased by 25% we observe a 3% increase in the numerical estimate, which over-estimates the actual value by 4%. For ∆s ≈ 0.0330 the estimate for α0 is 0.3332, which is less than 0.4% larger than the actual value of 0.3320. Therefore, the more nodes we have on the spheroid the better the estimate is.
Changes in the rate-of-strain e have very little affect on the accuracy of the numerical estimate.
For e = 2.5 × 10−1, 5 × 10−1, and 7.5 × 10−1 s−1 the numerical estimate for α0 is consistent with the data in Table 2.7 to 5 s.f.; even for e > 7.5 × 10−1 s−1 there is little change in the estimate.
Table 2.7: Analytical and numerical estimates for the cell eccentricity, α0 and the gyrotaxis parameter β. The numerical estimate is based upon a three parameter fit of (2.22) which results in a higher estimate of e than the value e = 5 × 10−2 s−1 initially introduced when creating the flow. The spheroid is composed of 630 nodes distributed around the surface and placed in a shear box with height, width and depth, 10 ds, 80 ds and 10 ds.
Exact Numerical
α0 β α0 β e (s−1)
0.3320 0.0057 0.3350 0.0054 0.055 0.2764 0.0059 0.2803 0.0056 0.055 0.2165 0.0061 0.2198 0.0058 0.055 0.1521 0.0063 0.1553 0.0060 0.055 0.0833 0.0065 0.0856 0.0063 0.055 0.0103 0.0066 0.0109 0.0064 0.055
Although the rate-of-strain does not affect the numerical estimate, the structure of the shear box does have a bearing on the accuracy of the numerical estimate. For fixed α0 = 0.3320 we observe that more nodes on the shear box increases the accuracy of the method. However, when the node spacing is constant and the width and height are changed we observe that making the width of the box too long or too short results in less accurate estimates; a 25% increase in box width sees the estimate rise by 4%, while a box width of 40 ds results in a 2% increase in α0 compared to the estimates obtained in Table 2.7. An increase or decrease in the height by over 40% also causes an increase in the error between the analytical and numerical estimates for the eccentricity.
From our analysis it appears that if the spacing between the nodes is the same there is no great effect on α0 when the aspect ratio of the box is changed. However, if the width is made too small or the height and depth are too large then the shear will not be properly formed in the plane and there may be unwanted boundary effects.
2.8 Discussion
We have detailed the construction of bi-flagellate swimmers, discussing the differences in the various flagellar beats we will employ. The flagellar beats are based on experimental observations and existing models proposed in the literature [32, 59, 96, 101]. The technique of extracting the beat data from the original sources into a Fourier series representation is also detailed, highlighting a process of imaging, discretising and renormalising. The latter processes allow us to have equally spaced nodes along the flagellum. We also outline an approach to obtain approximately equi-distant
nodes on the body. Having equally spaced nodes is important particularly in the case of the flagella where nodes moving tangentially along a flagellum could lead to inaccuracies when looking at the swimming behaviour of cells.
By application of the method of regularised Stokeslets together with a no-slip boundary condi-tion on the cell, and an equilibrium condicondi-tion specifying the behaviour of the forces and torques on the surface of the cell, we formulated a mobility problem. Solution of the mobility problem allows us to determine a cell’s translational and angular velocities. Furthermore, by approximating the cells as self-propelled spheroids we can use the velocity data to obtain estimates for the cell’s effective eccentricity and the time taken to re-orientate due to viscous and external torques. We also detail the passive mechanisms which cause the re-orientation of the cell. The three mechanisms that we consider are due to bottom-heaviness (gravitational torques), shape asymmetry (sedimen-tation torques) and a combination of mass and shape asymmetry. The results can be found in Chapters 3–7 for free-swimming cells in quiescent, shearing and bounded flows. The shear flow is imposed by a shear box, a set of discrete nodes distributed along the surface of a cuboid. The nodes are given a prescribed velocity dictated by the boundary conditions on each face of the box.
To ensure that the mobility problem is implemented correctly we employed it to look at some fluid dynamics problems where a solution is known. These involved the behaviour of a sedimenting sphere in an unbounded fluid and close to a stationary plane boundary. We found that with a good choice of regularisation parameter and a fine discretisation on the surface of the sphere there was good agreement between the numerical solutions and the analytical/asymptotic solutions. Further, we were able to obtain numerical estimates for the eccentricity of a spheroid which were consistent with the exact values. As with the sphere examples the numerical estimate improved for spheroids with more nodes. Also we showed that aspect ratio and distribution of nodes on the shear box had an effect on the numerical result; small widths and/or large height and depths lead to poorer estimates while less nodes implied less accuracy.
The aim of this chapter was to highlight the procedures that will be used in the proceeding chapters. Particularly the methods for estimating the average swimming speeds, rotation rates and trajectories of the cells.