The influence of the boundary on swimmers

In this example, we will investigate the influence of boundaries on swimmers. We will performe 10 simulation such that for each simulation, the center sphere of the swimmer is located at \((0,y)\), for \(i=0,2,..., 18\). As \(y\) increases, the swimmer gets closer from the wall. The idea behind performing all these simulations is to see the influence of the boundary on the dynamics of the swimmer and to show that the this influencthe the swimmer close enough to the boundary and see if its motion will be traped by walls depends on the distance between the swimmer and the wall.

The domain geometry for \(y=18\) is shown in the figures below. The mesh is locally refined arround the swimmer.

The case data files are available in Github here [Add the link]

The results obtained by running the ten simulations are given in these figures :

In the two following figures, we show, respectively, the swimmer’s net displacement with respect to \(X\) and \(Y\) of all \(y\)-positions in function of the time.

The two figures below show the net displacement gained at \(T_{final}=20 s\) in function of the swimmer’s position \(y\).

Here, we show the trajectories of the swimmer for each value of \(y\)

From the results above, we can see a trivial fixed point (with respect to \(Y\)-net dispacement) at \(y=0\). There is also another non trivial fixed point at the interval \([10, 12]\).

In the paper of Daddi-Moussa-Ider [Daddi-Moussa-Ider_2018], This non trivial fixed point is located at 0.8 of the channel’s width.

To determine this fixed point for our model, we generate 20 new simulations for \(y = 10, 10.1, 10,2,\cdots ,11.9, 12\).

The figure below shows the \(Y\)-net displacement for all \(y\) values mentioned above.

Note that by the symmetry of the channel domain, another fixes point should be locatd at \(y=-10.9\) (0.22 of the channel’s width).

In this section, we study the influence of the wall on a 3D three sphere swimmer whose arm makes an angle \(\theta\) with the horizon as illustrated below :

where \(g(t)= L+u_1(t)\) and \(h(t)= L+u_2(t)\).

This model has been studied by Zargar and Najafi in their paper [Zargar_2009] and theoritical results were proposed. The functions \(u_1\) and \(u_2\) in Najafi’s paper are taking the fom :

\[ \begin{cases} u_1(t)=u_{10}cos(\omega t)\\ u_2(t)=u_{20}cos(\omega t + \phi) \end{cases} \]

and the swimmer velocity and the angle \(\theta\) are given as follow :

\[ \begin{cases}\displaystyle{ v_x = -\frac{1}{16} \left(\frac{R}{L^{2}} \right)\left[\theta ^{2} + 93 \left(\frac{y}{L}\right) ^2 \right] \mathbf{\Phi},\\ v_y = \frac{3}{16} \left(\frac{R}{L^{2}} \right)\left[ \theta ^{3} - \theta \left(\frac{y}{L}\right) ^2 \right] \mathbf{\Phi},\\ \dot{\theta} = \frac{105}{16}\left(\frac{R}{L^3} \right) \theta ^2 \left(\frac{y}{L} \right) \mathbf{\Phi},} \end{cases} \]

where \(\mathbf{\Phi} = <u_1 \dot{u_2}-u_2 \dot{u}_1>\) denotes the averaging over a complete cycle of deformations.

In our example, the path deformation is not the same as the one proposed in Zagar’s paper. The path deformation we consider in this example is given as follow :

The analytical forms of \(u_1\) and \(u_2\) over one period is given as follow :

\[ u_1(t) =\begin{cases} -\frac{4}{T}\varepsilon t\qquad\qquad \text{if } t\in[0,\frac{T}{4}],\\ -\varepsilon \qquad\qquad\;\;\;\; \text{if } t\in[\frac{T}{4},\frac{T}{2}],\\ \frac{4}{T}\varepsilon t - 3\varepsilon\qquad\;\; \text{if } t\in[\frac{T}{2},\frac{3T}{4}],\\ 0\qquad \qquad\qquad\text{if } t\in[\frac{3T}{4},T] \end{cases} \]

and

\[ u_2(t) =\begin{cases} 0\qquad\qquad\qquad \text{if } t\in[0,\frac{T}{4}],\\ -\frac{4}{T}\varepsilon t + \varepsilon\qquad\; \text{if } t\in[\frac{T}{4},\frac{T}{2}],\\ -\varepsilon \qquad\qquad\;\;\;\; \text{if } t\in[\frac{T}{2},\frac{3T}{4}],\\ \frac{4}{T}\varepsilon t - 4\varepsilon\qquad\;\; \text{if } t\in[\frac{3T}{4},T]. \end{cases} \]

Then,

\[\mathbf{\Phi} = <u_1 \dot{u_2}-u_2 \dot{u}_1> = \frac{8\varepsilon^2}{T}.\]

Hence, the velocity filed and the angle \(\theta\) takes the previous form with :

\[\mathbf{\Phi} = \frac{8\varepsilon^2}{T}.\]

Here are some important points in the Zargar’s paper [zargar_2009] :

Here, we create a three sphere swimmer with an angle \(\theta\) in a rectangular channel of dimensions \([-L,L]\times [-l,l]\). The geometry is illustrated below.

First, we want to check if the swimmer will orient itself parallel to the as the case for [Zargar_2009]. For this, we present here the numerical results obtained using the feelpp fluid toolbox.

The following figures show the center of mass and \(Y\)-displacement for the three spheres.

The \(Y\)-net displacement of each sphere in function of periods is shown in the figure below :