Análisis de posibles implementaciones, componentes o módulos ya existentes

the cell. We consider a spherical body whose radius is equal to the length of the minor axes for the spheroidal body discussed above. The choice of ǫ is based upon the analysis of a translating sphere, see § 3.2, and the number of nodes on the body is approximately equal to that of the spheroid.

We also use the same 6-sided prism structure for the flagella. Hence, we have ∆s_f = 0.0417 and ǫ_f dependent on the beat pattern, see Table 3.6. For the body we have a discretisation size of

∆s_b = 0.0873 and ǫ_b= 0.0310.

Results for the swimming speed against time are shown for all five beat patterns in Fig-ure 3.25(a). The graphs show the velocity at each stage of the beat for a cell with no re-orientation mechanism and θ0 = 0. The velocity-time curves are similar to those obtained for the spheroidal cell, however, there is a drop in magnitude.

0 0.2 0.4 0.6 0.8 1

Figure 3.25: (a) Swimming speed against time for cells with spherical bodies and dual re-orientation mechanisms.

(b) How the sedimentation mechanism affects the change in θ over a single beat.

Estimates for the average swimming speed hUzi, were obtained for cells with spherical body and when θ₀ = 0. With no mechanism or a gravitational torque mechanism we find that hUzi = 0.1011, 0.0385, 0.1314, 0.0864, and 0.0848 d b⁻¹ for the I, F, R, RNL and RNR beats, respectively. This is a reduction of 12 − 19% from the cell with spheroidal body. The estimate for the I beat is close to the values estimated by Jones et al. [59] using a similar beat (0.1 d b⁻¹).

The speed at which the spherical cell sediments is also less than the spheroidal cell, with es-timates for the I, F, R, RNL and RNR predicted at U_sed = 4.1441, 4.1699, 4.1454, 4.1703 and 4.1786 ×10⁻³ d b⁻¹, which is an average reduction from the spheroidal estimates of around 16%.

The behaviour of the horizontal component of the velocity and the estimates for hUzi when swim-ming in the direction of gravity are also smaller than those observed for the spheroidal swimmer.

The change in orientation angle when the cells employ dual or gravitational torques mechanisms are consistent between the beat patterns. However, as Figure 3.25(b) demonstrates the behaviour of the cells when re-orientation is due to sedimentation torques is largely dependent on the beat pattern. For the experimental beat patterns, the R and RN beats, sedimentation torques result in very little rotation midway through the beat, that is when the flagella lie close to the cell body.

Hence, for cells with spherical bodies and natural beat patterns re-orientation due to sedimentation is a less efficient method compared to bottom heaviness. In general the sedimentation torque mechanism appears to have a larger affect on cells with spheroidal bodies than spherical cell bodies.

The results in Figure 3.25(b) suggest that the I and F beats will re-orientate quicker when the cell has a sedimentation torque re-orientation mechanism and this is confirmed when we estimate the re-orientation time B. For the I and F beats B = 9.2 and 9.3 s, respectively, a 3 s increase compared to the RNR beat, which is slowest to re-orientate. Comparisons between the results for the spheroidal cell show that the spherical cells are roughly 3 s slower to re-orientate.

Like the spheroidal case the gravitational torque mechanism lowers the time taken for the cell to re-orient. For the RN beats re-orientation due to gravitational torques is approximately twice as fast compared to when re-orientation is governed by sedimentation torques only. With the gravitational torque mechanism the slowest beats to re-orientate are the I and F beats. Details of these re-orientation times can be found in Table 3.12, which also shows the time taken for cells with dual mechanisms to re-orientate. Regardless of the beat pattern, in the spheroidal results we observed that the individual mechanisms complemented each other and produced a substantial reduction in the time taken for the cell to re-orient. This is also true for the spherical swimmer.

However, spherical swimmers take roughly 1 s longer to re-orientate compared to the bi-flagellates with spheroidal bodies.

Table 3.12: The gyrotactic re-orientation time B for cells with spherical bodies.

Beat B (s)

Grav. Sed. Dual I 6.0912 9.1651 3.6450 F 6.3430 9.2576 3.7286 R 5.5822 10.1827 3.5746 RNL 5.3386 11.2823 3.6130 RNR 5.1728 12.6894 3.6630

Jones [60] obtained slightly larger estimates when he estimated the re-orientation times for cell, which is likely due to the low accuracy of resistive force theory. Furthermore, estimates were only made for cells with gravitational torque as sedimentary forces were deemed to be negligible.

In Figure 3.26 the average velocity field generated by the swimmers is shown, with contour lines highlighting the magnitude of the flow field. Other than the flow at the posterior end of the

cell body there is not much to distinguish between the flows generated by cells with spheroidal bodies and those with spherical bodies. However, there are minor differences like the location of the stagnation points and vortices in relation to the cell’s centre-of-buoyancy; stagnation point and vortices occur 0.1 d closer to and further from the cell, respectively, compared to the spheroidal swimmer. In the far field the behaviour is the same with the magnitude of the average velocity field | hui | decaying, respectively, as r⁻¹ and r⁻² for cells with sedimentation and those without.

0.05

Figure 3.26: Velocity fields averaged over a single flagellar beat for a cell with spherical body. Contours show magnitude of the flow field.

3.5 Discussion

In this chapter we employed the techniques detailed in Chapter 2 to obtain numerical estimates for various properties of bi-flagellate swimming. Five different beat patterns were investigated:

an idealized beat pattern [59], I; a flexible model [32], F; drawings from experiments [96], R; and high-speed photographic observations [101], with either right-symmetric, RNR, or left-symmetric, RNL, flagella. We constructed a two-dimensional model for bi-flagellate swimming and observed

that the average swimming speeds were dependent on the beat patterns: the RN and F beats were within observed experimental ranges, whereas the I and R beats were approximately 13% greater than experimental observations. On the contrary, the three-dimensional results, which were lower in magnitude than the two-dimensional results, were consistent with experimental results regardless of the beat pattern.

Analysis of the swimming speed against time highlighted that the F beat was a poor represen-tation of bi-flagellate locomotion; there was no discernible effective and recovery stroke behaviour.

Furthermore, there is no fluidity to the beat pattern. This was evident in the flow fields generated by the swimmers, where for the F beat we observed lateral and anterior vortices appearing sporad-ically over the course of a beat. In contrast the other beat patterns showed consistent behaviour, with lateral vortices moving along the side of the body, from posterior to anterior end of the cell during the effective stroke and from anterior to posterior end during the recovery stroke. We also found that the two- and three-dimensional representations shared some of the same characteristics, only the magnitude and location of stagnation points in the flow field differed. Hence, the two-dimensional results are fine provided we only wish to study the qualitative behaviour of the cell, but for accurate estimates we need to investigate the behaviour in three-dimensions.

Investigations into the far field behaviour of the cells showed that the magnitude of the flow velocity decays faster with the three-dimensional cell, r⁻² compared to r⁻¹ in the two-dimensional case, although in both instances this alludes to the flow behaving like a stresslet in the far field.

However, we also showed that with the introduction of a re-orientation mechanism due to sedimen-tation torques, or a combination of sedimensedimen-tation and gravisedimen-tational torques, into the bi-flagellate model the far-field behaviour for the three dimensional results was that of a Stokeslet. Further, the flow fields generated by the cells highlighted that at various stages of the flagellar beat large lateral vortices are responsible for the expulsion of fluid out of the flagellar plane. While the exact nature of these eddies are unknown, they could provide a mechanism for the transport of nutrients.

In Chapter 4 we will highlight how these results compare with experimental observations.

Furthermore, we studied the effects that the re-orientation mechanisms had on the time taken for the cells to rotate toward the vertical axis. We compared what affect the three mechanisms had on the five distinct beat patterns and found that sedimentation torques had a greater bearing on the re-orientation times for the I and F beat pattern, whereas the RN and R beats re-orientated quicker

when gravitational torques governed re-orientation. However, when the combined mechanism of gravitational and sedimentation torques was explored we found that rather than competing against one another the individual mechanisms complemented one another. Thus, we observed consistent times for re-orientation over all beat patterns. The complimentary behaviour between mechanisms can be explained by considering the flagellar geometry. An increase in sedimentation torque for a cell swimming at an angle to the vertical relies on a greater average extension of the flagella towards the anterior of the cell, which leads to a larger viscous torque that is to be balanced with the fixed gravitational torque, thus reducing the impact of bottom-heaviness in leading the cell to orient towards the vertical.

The times-scales estimated for the dual mechanism are also similar to those observed exper-imentally suggesting that gravitational and sedimentation torques have equal importance in re-orientation. Results also highlighted that body shape plays a role on the re-orientation times of cells, where cells with spherical bodies were slower to re-orientate than those with spheroidal bodies.

However, body shape appears only to affect the quantitative behaviour rather than the qualitative behaviour. Cells such as Chlamydomonas change their body shape throughout their life-cycle [121], inducing behavioural variation that impacts collective behaviour such as bioconvection.

The results presented here are an improvement on previous models for bi-flagellate swimming.

Not only do we consider a three-dimensional model which is an improvement over the early work by Fauci [32], but our cells have realistic geometry unlike the model proposed by Jones et al. [59].

Rather than employing the low accuracy resistive force theory we employ the method of regularised Stokeslets to give more accurate estimate for the swimming speed, etc.. Furthermore, we consider a combined gravitational and sedimentation torque mechanism for re-orientation, which produces results more in line with experimental observations.

In the next chapter we compare our simulations to recent experimental work, while in subsequent chapters we build upon the basic model to investigate the behaviour of cells in shear flows and the hydrodynamics of interactions between boundaries and cells.

Chapter 4