Descripción de los casos de uso del sistema

As a comparison to our free-swimmer we look at the velocity fields generated by a cell fixed at its centre-of-buoyancy. The properties of the cell and the parameters relating to its discretisation are consistent with those used for the free-swimming cell.

The average velocity fields for the five beat patterns are shown in Figure 3.15. The flow shown is the average flow over a single flagellar beat, while the contours show its average magnitude.

From the velocity field we can see that the fixed cell generates large eddies close to the flagella, similar to those observed with the free-swimmer. However, there are also other vortex like patterns, which are of lower magnitude and caused by the stationary cell body; the flow produced by the beating flagella no longer comes into contact with oncoming fluid produced by the moving body.

Hence, instead of forming eddies close to the body the flow is pushed perpendicular to the cell, then circulated by the motion of the flagella during the next time-step.

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Figure 3.15: The velocity field averaged over a single beat for five distinct beats. Here the cell has been fixed at the geometric centre. The contours show the magnitude of the velocity.

One major difference between the free-swimming cell and the fixed cell can be seen when we

observe what happens in the far field. From Figure 3.16(a) the flow far from a fixed cell can be described by a Stokeslet, whereas Figure 3.16(b) shows that in the far field a free-swimming cell behaves like a two-dimensional stresslet. This is also highlighted in the graphs of the magnitude of the mean flow velocity , | hui |, against r, where r is the distance from the cell’s centre-of-buoyancy:

Figure 3.16(c) is for the fixed cell, and the gradient of the lines are close to zero, Figure 3.16(d) is for the free-swimmer, and the curves for larger r have gradient −1. Both figures highlight the decay of the average velocity field in four directions through the cell with RNR beat; the other beats show similar results. The solid line shows how the velocity decays through the cell’s principal axis toward the anterior end and we observe that for the fixed cell | hui | ≈ ln(r) for large r, whereas for the free-swimmer the drop off in | hui | is r⁻¹. Furthermore, the flow along the cells’ posterior ends and along their minor axes, in both directions, have the same decay as observed along the anterior end.

In the near field, insets of Figures 3.16(c)–(d), the local minima that occur in the curves of

| hui | against r correspond to characteristics of the flow fields. The first minimum for the curve through the anterior end indicates the stagnation point in the flow field, which we can see occurs closer to the fixed cell body than for the free-swimming cell. Moreover, analysis of the lateral decay shows that the vortices occur closer to the fixed cell than to the free swimmer. Along the posterior end for the fixed model we observe a local minimum around one body length from the centre of the cell. This is a consequence of the vortices generated by the flagella interacting with the fluid at the posterior end of the cell. For the free-swimmer the fluid is driven forward by the motion of the cell, which is not the case for a fixed cell. A further distinction between the behaviour of the near field flows for the fixed cell and the free swimmer is that the fixed cell has two stagnation points at the anterior end of the cell rather than one. The first is due to the flow driven by the vortices interacting with the no-slip cell body, whereas the second is a consequence of the vortices interacting with the fluid a few body lengths from the cell. For the free-swimmer the stagnation point occurs due to the advancing cell body interacting with the fluid ahead of the cell.

The two-dimensional results highlighted some important differences in the swimming behaviour for different beat patterns. We found the the R beat was the fastest swimmer, swimming over three times faster than the F beat. However, the estimate for the R beat was larger than recently measured experimental observations. In contrast, the F beat had a mean swimming speeds typical

(a) Fixed cell (b) Free-swimmer

10⁰ 10² 10⁴

10⁻⁶ 10⁻⁴ 10⁻² 10⁰

10² Lateral Left Lateral Right Posterior Anterior

10⁰ 10⁻⁵

10⁰

r d

|hui|db−1

(c) Fixed cell

10⁻² 10⁰ 10² 10⁴

10⁻⁵ 10⁻⁴ 10⁻³ 10⁻² 10⁻¹ 10⁰

Lateral Left Lateral Right Posterior Anterior

10⁰ 10⁻³

10⁻² 10⁻¹ 10⁰

r d

|hui|db−1

(d) Free-swimmer

Figure 3.16: All data is for the two-dimensional cell with RNR beat, however other beat patterns show similar behaviour. (a) The far field effects for a fixed cell resemble that of a Stokeslet. (b) The flow in the far field of a free-swimming cell; flow similar to a Stresslet. (c) A line graph showing the decay of the average velocity field through the geometric centre of a fixed cell. r is the distance from cell centre. (d) The decay of the average velocity field for a free-swimming cell. In both cases the data is shown through four different directions; two in opposite directions along the cell’s minor axis, and two in the opposite directions along the cell’s major axis. The insets show the behaviour of the velocity magnitude in the near field.

of a bi-flagellate, although periods of forward displacement during the recovery stroke where un-characteristic of bi-flagellate locomotion. Like the R beat, the estimate for the swimming speed of the I beat is larger than experimental observations, whereas for the RN beats the swimming speeds were consistent with measured values for C. reinhardtii. A comparison of the flows generated by the flagellar beats showed that despite differences in magnitude the flows were generally the same, except the F beat where the flow fluctuated between time-steps. However, in the far field all the free-swimming cells behaved as two-dimensional stresslets.