2.1. Swimmer model

This section concerns the modelling of a microswimmer inspired by E. coli. This bacteria has been investigated at length in the literature and a lot of information about its geometry and propulsion mechanism is available. E. coli has a cell length which varies between $1.6$ and $3.9~{\rm\mu}\text{m}$ , the cell width ranges between $0.9$ and $1.7~{\rm\mu}\text{m}$ , with a resulting cell volume from $1.5$ to $4.4~{\rm\mu}\text{m}^{3}$  (Volkmer & Heinemann Reference Volkmer and Heinemann2011) and an average length of flagella of approximately $7~{\rm\mu}\text{m}$  (Lauga et al. Reference Lauga, DiLuzio, Whitesides and Stone2006).

Figure 1.

The model of the flagellated swimmer comprises an ellipsoidal head (cell) and a tubular, helical, rigid, tail (flagellum). The tail rotates about its axis $\boldsymbol{e}_{T}$ of an angle ${\it\phi}(t)$ . The dimensionless semi-axes of the head are $a_{1}$ and $a_{2}$ . The head aspect ratio is kept constant, $AR=a_{1}/a_{2}=2$ , and the equivalent radius (i.e. the radius $\bar{a}$ of a sphere with same volume) is the assumed reference length. The dimensionless tail length is $L=7$ , with a cross-section radius $a_{t}=0.05$ . The dimensionless helix amplitude and wavelength are $A$ and ${\it\lambda}$ , respectively. $\{\boldsymbol{e}_{1},\boldsymbol{e}_{2},\boldsymbol{e}_{3}\}$ are the orthonormal vectors of the frame attached to the swimmer head (body frame). $\boldsymbol{e}_{1}$ is longitudinal and identifies the swimmer orientation with respect to the unit vectors of the fixed frame $\{\boldsymbol{X},\boldsymbol{Y},\boldsymbol{Z}\}$ .

Figure 1 sketches the simplified geometry comprising the ellipsoidal axisymmetric cell and the corkscrew tail. Hereafter, the equivalent radius of the ellipsoidal cell $\bar{a}$ (the radius of the sphere having the same volume) will be used as reference length scale, i.e. the dimensionless cell volume is $V=3V^{\prime }/4{\rm\pi}\bar{a}^{3}=1$ , where the prime identifies dimensional quantities. The cell aspect ratio is $AR=a_{1}/a_{2}=2$ , with $2a_{1}$ and $2a_{2}$ the longitudinal and the transversal (dimensionless) axes. Following Shum et al. (Reference Shum, Gaffney and Smith2010), the tail bundle is modelled as a single helix with radius $a_{T}=0.05$ . The tail rotates around its axis $\boldsymbol{e}_{T}$ (see figure 1). The dimensionless axial length of the helix is $L$ , $A$ denotes the helix amplitude and ${\it\lambda}$ its wavelength. In the present work we selected typical values for the tail length and cell axes, namely $L=7$ , $a_{1}=1.6$ and $a_{2}=0.8$ , while the amplitude $A$ and the wavelength ${\it\lambda}$ of the tail are systematically varied. With the above choices, the swimmer is rescaled into an actual E. coli by assuming $\bar{a}=1~{\rm\mu}\text{m}$ .

Figure 2.

Sketch of the discretized swimmer near a planar surface. The configuration is identified by three parameters, namely, the distance $h$ of the reference point $\boldsymbol{x}_{J}$ from the plane and the pitch angle ${\it\Theta}$ . The third parameter is the tail rotation angle ${\it\phi}$ defined in figure 1.

It is useful to introduce a body reference frame with orthonormal base vectors $\{\boldsymbol{e}_{1},\boldsymbol{e}_{2},\boldsymbol{e}_{3}\}$ where $\boldsymbol{e}_{1}=-\boldsymbol{e}_{T}$ is longitudinal (see figure 1). The subscript $H$ identifies the head and the subscript $T$ refers to the tail, which can rotate with respect to the head about its axis $\boldsymbol{e}_{T}$ . During the tail rotation around $\boldsymbol{e}_{T}$ , each point of the rigid tail describes a circle in the plane normal to $\boldsymbol{e}_{T}$ , with ${\it\phi}$ the corresponding rotation angle (or flagellum phase). The time derivative $\dot{{\it\phi}}(t)$ is the tail rotational velocity ${\it\Omega}_{T}$ . The swimmer position is identified in a fixed reference frame with base $\{\boldsymbol{X},\boldsymbol{Y},\boldsymbol{Z}\}$ by the three coordinates of the cell-to-tail junction point $\boldsymbol{x}_{J}$ , see figure 2. The swimmer translates with dimensionless velocity $\boldsymbol{U}=\boldsymbol{U}^{\prime }/v$ and has angular velocity ${\it\bf\Omega}_{H}={\it\bf\Omega}_{H}^{\prime }\bar{a}/v$ , where $v\simeq 20~{\rm\mu}\text{m}~\text{s}^{-1}$ is a typical swimming velocity taken as a reference quantity. The $\{\boldsymbol{X},\boldsymbol{Y}\}$ coordinate plane is taken to coincide with the planar boundary, either the solid wall or the free surface, see figure 2.

The kinematics of the swimmer is described by seven degrees of freedom, the junction position $\boldsymbol{x}_{J}(t)$ , the flagellum phase ${\it\phi}(t)$ and the three parameters defining the rotation of the head. We stress that the discretely evolved rotation matrix should belong to the proper matrix subspace, namely the $\text{SO}(3)$ subgroup. This is enforced through a description in terms of quaternions with unit norm, see e.g. Diebel (Reference Diebel2006). It follows that

(2.1) $$\begin{eqnarray}\displaystyle & \displaystyle \dot{\boldsymbol{x}}_{J}=\boldsymbol{U}(t), & \displaystyle\end{eqnarray}$$

(2.2) $$\begin{eqnarray}\displaystyle & \displaystyle \dot{{\it\phi}}={\it\Omega}_{T}(t), & \displaystyle\end{eqnarray}$$

(2.3) $$\begin{eqnarray}\displaystyle & \displaystyle \dot{\boldsymbol{q}}={\textstyle \frac{1}{2}}\unicode[STIX]{x1D64E}(\boldsymbol{q}){\it\bf\Omega}_{H}(t), & \displaystyle\end{eqnarray}$$

$\boldsymbol{q}=(q_{0},q_{1},q_{2},q_{3})$

$|\boldsymbol{q}|=1$

${\it\bf\Omega}_{H}(t)$

$\{\boldsymbol{X},\boldsymbol{Y},\boldsymbol{Z}\}$

$\unicode[STIX]{x1D64E}$

(2.4) $$\begin{eqnarray}\unicode[STIX]{x1D64E}(\boldsymbol{q})=\left[\begin{array}{@{}rrr@{}}-q_{1} & -q_{2} & -q_{3}\\ q_{0} & -q_{3} & q_{2}\\ q_{3} & q_{0} & -q_{1}\\ -q_{2} & q_{1} & q_{0}\end{array}\right].\end{eqnarray}$$

Given the quaternion $\boldsymbol{q}(t)$ , the components of the body frame unit vectors can be reconstructed as

(2.5) $$\begin{eqnarray}\displaystyle \left.\begin{array}{@{}c@{}}\boldsymbol{e}_{1}=\left(q_{0}^{2}+q_{1}^{2}-q_{2}^{2}-q_{3}^{2},2(q_{1}q_{2}+q_{0}q_{3}),2(q_{1}q_{3}-q_{0}q_{2})\right),\\ \boldsymbol{e}_{2}=\left(2(q_{1}q_{2}-q_{0}q_{3}),q_{0}^{2}-q_{1}^{2}+q_{2}^{2}-q_{3}^{2},2(q_{2}q_{3}+q_{0}q_{1})\right),\\ \boldsymbol{e}_{3}=\left(2(q_{1}q_{3}+q_{0}q_{2}),2(q_{2}q_{3}-q_{0}q_{1}),q_{0}^{2}-q_{1}^{2}-q_{2}^{2}+q_{3}^{2}\right).\end{array}\right\} & & \displaystyle\end{eqnarray}$$

Time integration of (2.1)–(2.3), allows us to track the swimmer trajectory once $\boldsymbol{U}$ , ${\it\bf\Omega}_{H}$ and ${\it\Omega}_{T}$ are determined by the solution of the hydrodynamical interactions between the swimmer and the fluid, as discussed in the next section. Since the numerical integration error in the equations for the quaternion may affect its norm, the quaternion is normalized to unit length at each time step. It easily shown that the accuracy of the integration scheme is exactly preserved by this procedure.

2.2. Hydrodynamic model

Given the characteristic length scale $\bar{a}$ and swimming velocity $v$ of the micro-organism, the typical Reynolds number in water is $Re={\it\rho}\bar{a}v/{\it\mu}\simeq 2\times 10^{-5}$ where ${\it\mu}$ and ${\it\rho}$ are the water viscosity and density, respectively. Hence the inertial terms are negligible and, for Newtonian fluids, the flow in the domain $\mathscr{D}$ is described by the Stokes equations,

(2.6a ) $$\begin{eqnarray}\displaystyle & \displaystyle \boldsymbol{{\rm\nabla}}\boldsymbol{\cdot }\boldsymbol{u}=0, & \displaystyle\end{eqnarray}$$

(2.6b ) $$\begin{eqnarray}\displaystyle & \displaystyle {\rm\nabla}^{2}\boldsymbol{u}-\boldsymbol{{\rm\nabla}}p=0, & \displaystyle\end{eqnarray}$$

$\boldsymbol{u}=\boldsymbol{u}^{\prime }/v$

$p=p^{\prime }\bar{a}/({\it\mu}v)$

The flow velocity $\boldsymbol{u}$ is forced by the tension $\boldsymbol{f}$ at the swimmer surface acting on the fluid. The swimmer propulsion is due to the internal (constant) torque ${\it\tau}_{M}={\it\tau}_{M}^{\prime }/({\it\mu}v\bar{a}^{2})$ exchanged between head and tail. Being a free body, the total force and torque on the swimmer (head and tail) is zero,

(2.7a ) $$\begin{eqnarray}\displaystyle & \displaystyle \int _{H\cup T}\boldsymbol{f}\,\text{d}S=0, & \displaystyle\end{eqnarray}$$

(2.7b ) $$\begin{eqnarray}\displaystyle & \displaystyle \int _{H\cup T}\boldsymbol{r}\wedge \boldsymbol{f}\,\text{d}S=0, & \displaystyle\end{eqnarray}$$

$\boldsymbol{r}=\boldsymbol{x}-\boldsymbol{x}_{J}$

${\it\tau}_{M}$

$\boldsymbol{f}$

(2.8) $$\begin{eqnarray}\boldsymbol{e}_{T}\boldsymbol{\cdot }\int _{T}\boldsymbol{r}\wedge \boldsymbol{f}\,\text{d}S=-{\it\tau}_{M}.\end{eqnarray}$$

The system (2.6)–(2.8) needs proper boundary conditions at the swimmer surface and external boundaries, i.e. wall or free surface. On the microswimmer surface the no-slip condition yields

(2.9) $$\begin{eqnarray}\displaystyle \boldsymbol{u}(\boldsymbol{x})=\left\{\begin{array}{@{}ll@{}}\displaystyle \boldsymbol{U}+{\it\bf\Omega}_{H}\wedge \boldsymbol{r} & (\boldsymbol{x}\in H),\\ \displaystyle \boldsymbol{U}+({\it\bf\Omega}_{H}+{\it\Omega}_{T}\boldsymbol{e}_{T})\wedge \boldsymbol{r} & (\boldsymbol{x}\in T).\end{array}\right. & & \displaystyle\end{eqnarray}$$

A no-slip boundary condition is also used to model the solid wall. For the free surface, vanishing normal fluid velocity (impermeability) and zero tangential stresses (free-slip condition) are the appropriate prescriptions,

(2.10a ) $$\begin{eqnarray}\displaystyle & \displaystyle \boldsymbol{u}\boldsymbol{\cdot }\boldsymbol{n}=0, & \displaystyle\end{eqnarray}$$

(2.10b ) $$\begin{eqnarray}\displaystyle & \displaystyle \left(\left(\boldsymbol{{\rm\nabla}}\boldsymbol{u}+(\boldsymbol{{\rm\nabla}}\boldsymbol{u})^{\text{T}}\right)\boldsymbol{\cdot }\boldsymbol{n}\right)(\boldsymbol{I}-\boldsymbol{n}\otimes \boldsymbol{n})=0. & \displaystyle\end{eqnarray}$$

2.3. Boundary element method

The solution of the Stokes equations (2.6) can be recast in integral form

(2.11) $$\begin{eqnarray}E(\boldsymbol{x}_{0})\,u_{i}(\boldsymbol{x}_{0})=\int _{\partial \mathscr{D}}\left[u_{j}(\boldsymbol{x})\unicode[STIX]{x1D61B}_{ikj}(\boldsymbol{x},\boldsymbol{x}_{0})n_{k}(\boldsymbol{x})-\unicode[STIX]{x1D60E}_{ij}(\boldsymbol{x},\boldsymbol{x}_{0})f_{j}(\boldsymbol{x})\right]\text{d}S,\end{eqnarray}$$

where $E(\boldsymbol{x}_{0})=1$ for points belonging to the interior of the domain. $\unicode[STIX]{x1D60E}_{ij}(\boldsymbol{x},\boldsymbol{x}_{0})$ is the free-space Green function, i.e. the $i$ th component of the velocity at $\boldsymbol{x}$ induced by a Dirac ${\it\delta}$ -like force at $\boldsymbol{x}_{0}$ acting in direction $j$ , and $\unicode[STIX]{x1D61B}_{ikj}(\boldsymbol{x},\boldsymbol{x}_{0})$ the associated stress tensor (Happel & Brenner Reference Happel and Brenner1983). Equation (2.11) expresses the $i$ th velocity component at the collocation point $\boldsymbol{x}_{0}$ in the fluid domain $\mathscr{D}$ . It requires knowledge of the velocity $u_{j}(\boldsymbol{x})$ and the stresses $f_{j}(\boldsymbol{x})$ at the boundary $\partial \mathscr{D}$ . The integral representation (2.11) can be turned into a boundary integral equation in the limit $\boldsymbol{x}_{0}\rightarrow \partial \mathscr{D}$ . The resulting expression can be written in the same form (2.11) now with $E(\boldsymbol{x}_{0})=1/2$ and the integral understood in the Cauchy principal-value sense. We emphasize that the boundary $\partial \mathscr{D}$ consists of two disjointed parts: the swimmer surface and the planar interface. The integral can be restricted to the swimmer’s surface by exploiting the symmetries associated with the boundary condition at the planar interface. This is tantamount to the use of the appropriate Green’s function, see appendix A, for the free-slip case and Blake (Reference Blake1971) for the no-slip case.

Due to the boundary condition (2.9) the fluid velocity at the swimmer surface is expressed in terms of seven unknowns, namely, the velocity of the junction point $\boldsymbol{U}$ , the angular velocity ${\it\bf\Omega}_{H}$ and the tail rotation velocity ${\it\Omega}_{T}$ , which together with the tension $\boldsymbol{f}$ at the swimmer surface complete the set of unknowns. The system of equations consists of the vector boundary integral equation (2.11), the global force and torque balance (2.7) and the torque balance for the swimmer’s tail (2.8). It can be shown that the solution exists and is unique (Ladyzhenskaya & Silverman Reference Ladyzhenskaya and Silverman1969).

The above system is discretized by means of $N$ curved 6-point elements (typically $N=3518$ total elements, with $N_{H}=512$ panels on the swimmer head), as sketched in figure 2, with piecewise constant shape functions (Pozrikidis Reference Pozrikidis1992). By selecting the centre of each element as a collocation point $\boldsymbol{x}_{0}$ , the boundary integral equation (2.11) with $E(\boldsymbol{x}_{0})=1/2$ is recast in a system of $3N$ scalar equations for the stresses. As will be shown in § 3, the swimmer may approach the interface. In this case some of the panels belonging to the swimmer and to its image may become very close. This potentially spoils the accuracy of the integrals providing the influence coefficients appearing in the discrete equations. Care is taken in the simulations discussed below to have the diameter of the typical panel smaller than the distance between the body and its image, thereby preventing the undesired loss of accuracy. Concerning the solution of the algebraic system, inverting a matrix coming from the discretization of boundary integral equations may require some care, since the algebraic system may be prone to severe ill conditioning. An example is provided by the piecewise constant approximation of a first-kind Fredholm operator (Hsiao & MacCamy Reference Hsiao and MacCamy1973). In the context of the Stokes equations, such an operator arises when solving for the stresses given the velocity. However, the problem of the swimmer addressed in the present paper does not involve a pure first-kind Fredholm operator, since the equations are augmented by the free-body constraints (zero net forces and torques) and by the equation enforcing the internal torque, with the unknowns the stresses on the body plus the rigid body velocities and the relative tail-body rotation rate. The spectral properties of the ensuing matrix differ substantially from those of a pure first-kind Fredholm equation. In fact, the analysis based on the singular value decomposition, (Golub & Van Loan Reference Golub and Van Loan2012), excludes matrix ill conditioning. As a consequence, the complete system of $3N+7$ algebraic equations for the $3N+7$ unknowns collectively denoted ${\bf\chi}$ , $\boldsymbol{A}\boldsymbol{\cdot }{\bf\chi}=\boldsymbol{b}$ , can be solved by standard techniques, such as the Gauss–Jacobi method implemented here in an in-house MPI parallel code. Once the solution ${\bf\chi}$ is obtained, the integration of the kinematic equations (2.1)–(2.3), performed via a third-order low-storage Runge–Kutta method, allows us to track the swimmer trajectory.

3.1. Phase plane analysis and trajectories for a no-slip wall

The interaction between the swimmer and a homogeneous planar surface is completely determined by three parameters: the distance $h$ between the junction and the surface, the flagellum rotation phase angle ${\it\phi}$ and the pitch angle ${\it\Theta}$ between the swimmer longitudinal axis $\boldsymbol{e}_{1}$ and the surface plane, see figures 1 and 2. ${\it\phi}$ is a fast variable, thus, following Shum et al. (Reference Shum, Gaffney and Smith2010), the kinematics of the system can be described in terms of ${\it\phi}$ -averaged quantities, in the following denoted by a circumflex. The kinematics in the reduced $(\hat{{\it\Theta}},{\hat{h}})$ space is ruled by

(3.1) $$\begin{eqnarray}\displaystyle & \displaystyle \dot{\hat{{\it\Theta}}}=\hat{{\it\Omega}}_{2}, & \displaystyle\end{eqnarray}$$

(3.2) $$\begin{eqnarray}\displaystyle & \displaystyle \dot{{\hat{h}}}=\hat{U} _{Z}, & \displaystyle\end{eqnarray}$$

$\hat{{\it\Omega}}_{2}(\hat{{\it\Theta}},{\hat{h}})=\hat{{\it\bf\Omega}}_{H}\boldsymbol{\cdot }\boldsymbol{e}_{2}$

$\hat{U} _{Z}(\hat{{\it\Theta}},{\hat{h}})=\hat{\boldsymbol{U}}\boldsymbol{\cdot }\boldsymbol{Z}$

$\hat{{\it\Omega}}_{2}$

$\hat{U} _{Z}$

Figure 3.

(a) Phase plane $\hat{{\it\Theta}}{-}{\hat{h}}$ for a microswimmer close to a rigid no-slip wall. The curvature radius of the trajectory, (3.3), is provided by the colour map. Due to the presence of the wall the blanked regions are forbidden to the swimmer. Three typical trajectories I, II and III are shown by the solid black lines. Trajectories I and II converge to the stable point. The corresponding trajectory in physical space is shown in (b) where the curvature radius in the $XY$ -plane is colour coded. Results for the free-slip interface are shown in (c,d). All trajectories eventually intersect the planar interface and no stable orbit exists. Note the opposite sign of the curvature radius in comparison with the no-slip case.

The ${\it\phi}$ -averaged trajectory in 3-D space corresponding to I is obtained by integrating the ${\it\phi}$ -averaged version of the kinematic equations (2.1), (2.3) and is reported in figure 3(b). Apparently, after a transient, the swimmer stabilizes on a circular clockwise (CW) orbit which corresponds to the stable point in the reduced space $(\hat{{\it\Theta}},{\hat{h}})$ . Following Shum et al. (Reference Shum, Gaffney and Smith2010), the curvature radius of the stable orbit is given by

(3.3) $$\begin{eqnarray}R=\frac{|\hat{\boldsymbol{U}}(\hat{{\it\bf\Theta}},\hat{\boldsymbol{h}})|}{\hat{{\it\Omega}}_{Z}-\hat{{\it\Omega}}_{X}\tan \hat{{\it\Theta}}},\end{eqnarray}$$

which yields $R=-22.1$ . As expected this result matches within numerical accuracy its direct measure $R=-21.9$ obtained from the 3-D trajectory, see figure 3(b). Note that (3.3) differs in sign from the original equation given in Shum et al. (Reference Shum, Gaffney and Smith2010) due to a different choice of the body reference frame. It is also worth noting that (3.3) provides the exact curvature radius only for the stable orbit. Nevertheless, it also gives a reasonable approximation in the other conditions here explored, as discussed below. In figure 3(a) the colour map refers to $R$ as estimated from (3.3). The colour code in figure 3(b) corresponds to the local curvature radius of the trajectory projection in the $xy$ -plane. This information is also reported along the curve I in the phase plane, 3(a), through the colour of the open circles superimposed to the trajectory $I$ . From the data it is apparent that the colour differences between the circles and the colour map in the background can hardly be appreciated.

3.2. Swimming in presence of a free-slip interface

The phase plane analysis has been performed for a microswimmer moving close to a free-slip interface, as illustrated in figure 3(c,d). The first significant result is the sign of the curvature radius, now positive, i.e. the swimmer exhibits a counter-clockwise (CCW) motion in contrast to what is observed for the no-slip wall, compare (b) and (d) in figure 3. The curvature radius is smaller with respect to the no-slip case at corresponding $(\hat{{\it\Theta}},{\hat{h}})$ . The curvature radius measured in our simulation favourably compares with experimental observation of Di Leonardo et al. (Reference Di Leonardo, Dell Arciprete, Angelani and Iebba2011) where a CCW motion with a curvature radius of $\simeq 10~{\rm\mu}\text{m}$ is described. Indeed, assuming $\bar{a}=1~{\rm\mu}\text{m}$ , appropriate for an E. coli, the typical radius of curvature found from the present numerics, $R\simeq 7$ , gives $R^{\prime }=R\bar{a}\simeq 7~{\rm\mu}\text{m}$ .

The second significant result is that no stable trajectory exists when the swimmer moves close to a free surface. In fact, a swimmer similar to an E. coli moving almost parallel to the free surface (low $\hat{{\it\Theta}}$ ) is always attracted to it. This result confirms the indication of the 2-D model proposed by Crowdy et al. (Reference Crowdy, Lee, Samson, Lauga and Hosoi2011) for undeformable interfaces. This conclusion also partially agrees with the results illustrated by Lopez & Lauga (Reference Lopez and Lauga2014), where the authors, using on a multipole expansion technique, highlighted that the average surface-normal velocity $\hat{U} _{z}$ of the swimmer is always negative for $\hat{{\it\Theta}}=0$ . Our work adds that the surface is no longer able to attract the swimmer if it swims above a certain height $h$ , even with zero tilt angle.

Figure 4.

A wider view of the phase plane is displayed for the microswimmer near the no-slip (a) and the free-slip (b) surface. The range exploited in figure 3 corresponds to the small dashed yellow box. Three regions of the phase plane are identified by colours: a trajectory belonging to the light blue region ends colliding with wall, one in the green region escapes and trajectories in the orange region are attracted to the stable orbit. Note that attraction basin is absent in the free-slip case (b). Unstable equilibrium points are marked by the red circles.

In figure 4 we extend our analysis to a wider range of initial conditions, for both no-slip and free-slip surfaces. In both cases, at high positive $\hat{{\it\Theta}}$ the microswimmer escapes from the surface while, at high negative $\hat{{\it\Theta}}$ , it eventually hits the boundary. On the no-slip wall three cases are possible depending on the initial conditions, namely the swimmer achieves a stable orbit, it escapes from the wall or it collides with the wall. In contrast, for a free-slip surface only two possibilities exist, namely, escaping from the surface or being attracted towards the interface. The maps also highlight the presence of a further equilibrium point $(\hat{{\it\Theta}}_{u},{\hat{h}}_{u})$ where the time derivatives of both ${\hat{h}}$ and $\hat{{\it\Theta}}$ vanish. However, as shown by the behaviour of the streamlines in the neighbourhood of $(\hat{{\it\Theta}}_{u},{\hat{h}}_{u})$ , this point is unstable for both no-slip and free-slip cases, i.e. nearby trajectories escape from the equilibrium point.

In order to extend the analysis, the dynamics of swimmers with different geometrical characteristics was also examined, focusing in particular on the shape of the head and the tail (axial) length. Figure 5 shows the phase plane $\hat{{\it\Theta}}{-}{\hat{h}}$ when the tail length $L\in (3,5,10,15)$ and head aspect ratio $AR=a_{1}/a_{2}\in (1,3,4,5)$ . When changing the aspect ratio, the swimmer does not substantially modify its behaviour. In particular, the orientation as described by the pitch angle $\hat{{\it\Theta}}$ does not qualitatively change and the regions of phase space where the swimmer is attracted to the free surface or where it escapes from it remain very similar. The general trend is an increase of the curvature of the trajectory for given $(\hat{{\it\Theta}},\,{\hat{h}})$ as $AR$ increases. On the other hand, when changing the relative tail-to-head length, the swimmer hits the wall with different pitch angles $\hat{{\it\Theta}}$ . Decreasing the tail length, the region of phase plane where the swimmer escapes becomes larger while the region where it is attracted to the free surface shrinks. The curvature of the trajectory, for given $(\hat{{\it\Theta}},\,{\hat{h}})$ , is found to increase with the tail length.

Figure 5.

Phase plane $\hat{{\it\Theta}}{-}{\hat{h}}$ for a microswimmer close to a free-slip interface for different geometries, namely head aspect ratio $AR=a_{1}/a_{2}\in (1,3,4,5)$ (rows, from bottom to top) and tail length $L\in (3,5,10,15)$ (columns, from left to right). The colour map corresponds to the curvature radius along trajectories (e.g. solid black lines).

3.3. Drift along the trajectory

Typically the longitudinal axis of the microswimmer is misaligned with the local velocity evaluated at the junction point. This behaviour was experimentally observed for E. coli moving close to a liquid–air interface by Di Leonardo et al. (Reference Di Leonardo, Dell Arciprete, Angelani and Iebba2011). In order to compare with experiments we define the drift angle ${\it\alpha}$ as the angle between the $X,Y$ projections of the body frame unit vector $\boldsymbol{e}_{1}$ and the swimmer velocity $\boldsymbol{U}$ . The sign of ${\it\alpha}$ is taken to be positive when the swimmer points outward with respect to the trajectory, see figure 6(a).

In figure 6(b,c), the ${\it\phi}$ -averaged drift angle is reported for both no-slip and free-slip cases. In the free-slip case the head points outside the trajectory and $\hat{{\it\alpha}}$ increases as the swimmer approaches the wall. The spanned range of values, $\hat{{\it\alpha}}\in (10^{\circ },30^{\circ })$ , are in good agreement with the experimental observation (Di Leonardo et al. Reference Di Leonardo, Dell Arciprete, Angelani and Iebba2011). In contrast, in the no-slip case, $\hat{{\it\alpha}}$ is slightly negative, $\hat{{\it\alpha}}\simeq -2^{\circ }$ , meaning that the swimmer is almost aligned with the trajectory of the reference point $\boldsymbol{x}_{J}$ .

Figure 6.

(a,b) Sketch illustrating the drift angle ${\it\alpha}$ . (c,d) The ${\it\phi}$ -averaged drift angle $\hat{{\it\alpha}}$ in the phase plane (colour code), for the no-slip and free-slip interface, respectively. Each inset shows a representative trajectory with superimposed the longitudinal unit vector $\boldsymbol{e}_{1}$ (the colour coding corresponds to the local value of $\hat{{\it\alpha}}$ ).

Here, to complete the discussion, the effect of modifying the tail geometry is briefly addressed. Indeed, among the large number of different parameters defining the flagellated geometry, probably the most uncertain ones concern the tail, specifically the amplitude of the helix, $A$ , and number of turns, $N_{{\it\lambda}}$ . Figure 7 reports the radius of curvature $R$ and the drift angle $\hat{{\it\alpha}}$ for three different tails. The comparison shows that the results discussed so far are generic.

Figure 7.

Different tail geometries. This figure shows the radius of curvature $R$ and the drift angle $\hat{{\it\alpha}}$ when modifying the tail amplitude $A$ or the number of turns $N_{{\it\lambda}}$ . Three cases are reported: $A=0.8$ and $N_{{\it\lambda}}=1$ (a,d,g), $A=0.4$ and $N_{{\it\lambda}}=3$ (b,e,h), $A=0.8$ and $N_{{\it\lambda}}=5$ (c,f,i). The hydrodynamic behaviour is qualitatively the same as observed for the reference configuration in figures 3 and 6 with few changes in the variables values.