2.1. Membrane mechanics

Assume that the membrane thickness is sufficiently small for the membrane to be considered as a two-dimensional hyperelastic material. The membrane surface $S$ is determined by two surface curvilinear coordinates ( $\xi ^1,\xi ^2$ ). The membrane material points in the reference and deformed states are given by $\boldsymbol{X}(\xi ^1,\xi ^2)$ and $\boldsymbol{x}(\boldsymbol{X},t)$ , respectively. The gradient of transformation, denoted by $\unicode{x1D641}$ , is given by (Pozrikidis Reference Pozrikidis2010)

(2.1) \begin{equation} {\rm d}\boldsymbol{x}=\unicode{x1D641} \cdot {\rm d}\boldsymbol{X}. \end{equation}

Then the local deformation of the membrane can be measured by the Green–Lagrange strain tensor

(2.2) \begin{equation} \unicode{x1D640}=\tfrac {1}{2}\big(\unicode{x1D641}^{\kern1pt T} \cdot \unicode{x1D641}-\unicode{x1D644}\big), \end{equation}

where $\unicode{x1D644}$ is the identity tensor. And two invariants of the transformation are given by

(2.3) \begin{equation} I_1=\lambda _1^2+\lambda _2^2-2, \quad I_2=\lambda _1^2\lambda _2^2-1=J_s^2-1, \end{equation}

where $\lambda _1,\lambda _2$ are the principal dilation ratios, and the Jacobian $J_s=\lambda _1 \lambda _2$ expresses the ratio of the deformed to the undeformed surface areas. Assuming that the membrane is an isotropic material, Cauchy tension tensor $\unicode{x1D64F}$ can be related to a strain energy function per unit area of undeformed membrane $\omega _s(I_1,I_2)$ by the equation

(2.4) \begin{equation} \unicode{x1D64F}=\frac {1}{J_s} \unicode{x1D641} \cdot \frac {\partial \omega _s}{\partial \unicode{x1D640}} \cdot \unicode{x1D641}^{\kern1pt T}. \end{equation}

The behaviour of the capsule membrane is described by means of a neo-Hookean (NH) law, the strain energy function of which is given by

(2.5) \begin{equation} \omega _s^{NH}=\frac {G_s}{2} \left (I_1-1+\frac {1}{I_2+1}\right ), \end{equation}

where $G_s$ is the surface elastic shear modulus.

In the case of an infinitely thin membrane, the inertia of the membrane can be considered negligible. Consequently, the motion of the membrane is governed by local mechanical equilibrium:

(2.6) \begin{equation} \boldsymbol{q}_e + \nabla _s \cdot \unicode{x1D64F} = \boldsymbol{0}, \end{equation}

where $\boldsymbol{q}_e$ is the surface load due to membrane in-plane stretch, and $\nabla _s$ is the surface divergence operator. By applying the virtual work principle, the equilibrium equation (2.6) can be written in a weak form (Walter et al. Reference Walter, Salsac, Barthès-Biesel and Tallec2010)

(2.7) \begin{equation} \iint _s \hat{\boldsymbol{v}} \cdot \boldsymbol{q}_e\, {\rm d}S-\iint _s\hat{\boldsymbol{\epsilon }}(\hat{\boldsymbol{v}}):\unicode{x1D64F}\,{\rm d}S=0, \end{equation}

where $\boldsymbol{\hat {v}}$ and $\boldsymbol{\hat {\epsilon }}(\boldsymbol{\hat {v}})= ({1}/{2}) (\nabla _s \boldsymbol{\hat {v}}+\nabla _s \boldsymbol{\hat {v}}^{T})$ are the virtual displacement and virtual deformation tensor, respectively.

The equilibrium equation under consideration pertains to in-plane deformation and does not encompass the bending stiffness of the membrane. This renders the calculations highly unstable for compressive deformations, thus necessitating the use of a weak bending energy in this study. It is assumed that the in-plane deformation load and the bending deformation load can be expressed as a linear sum due to the sufficiently small membrane thickness, and Helfrich’s model (Helfrich Reference Helfrich1973) is adopted for the bending energy:

(2.8) \begin{equation} \omega _b=\frac {E_b}{2} \int \left (2H-c_0\right ){\rm d}S, \end{equation}

where $E_b$ is the bending modulus, $H$ is the mean curvature of the surface, and $c_0$ is spontaneous curvature. Assume that the reference curvature is a flat shape, i.e. $c_0=0$ , and the bending force density $\boldsymbol{q}_b$ is expressed by the first variant of the bending energy (Ou-Yang & Helfrich Reference Ou-Yang and Helfrich1989):

(2.9) \begin{equation} \boldsymbol{q}_b=-2E_b[\varDelta _{s} H +2H(H^2-K)] \boldsymbol{n}, \end{equation}

where $\varDelta _{s}$ is the Laplace–Beltrami operator on the surface, $K$ is the Gaussian curvature, and $\boldsymbol{n}$ is the normal vector on the surface to the outside. Thus the membrane load due to the elastic deformation $\boldsymbol{q}_c$ is determined by $\boldsymbol{q}_c = \boldsymbol{q}_e + \boldsymbol{q}_b$ . The findings of Dupont et al. (Reference Dupont, Tallec, Barthès-Biesel, Vidrascu and Salsac2015) demonstrate that the effect of bending resistance is negligible in comparison to the effect of in-plane stretch. Consequently, the influence of bending resistance on the shape and deformation can be disregarded.

2.2. Damage behaviour

The model proposed by Grandmaison et al. (Reference Grandmaison, Brancherie and Salsac2021) is employed to delineate the damage to a two-dimensional membrane. The damage is modelled on the basis of continuum damage mechanics, with the damage state being defined as an isotropic brittle damage model, and the damage state determined by the history of loading.

We assume that the transformations of the capsule wall correspond to isothermal elastic deformation and damage. The damage variable represents the irreversible growth of microdefects in infinitesimal elements. Here, an overview of the work of Grandmaison et al. (Reference Grandmaison, Brancherie and Salsac2021) will be provided, with the aim of elucidating the brittle damage model.

Assume the presence of irreversible microdefects growing within the membrane in response to large deformations. The deformation is assumed to be isothermal elastic deformation, and the damage is defined by the irreversible growth of microdefects in the infinitesimal element. It is also assumed that the growth of the microdefects in the element is isotropic. Thus the microdefects have no preferential orientation, and the damage variable can be expressed by a scalar function $d$ , which is defined as $d=\delta S_D / \delta S = 1-\delta \tilde {S} / \delta S$ , where $\delta S$ is the total area of the infinitesimal element including the microdefects, and $\delta S_D$ is the maximum intersection of microdefects in $\delta S$ . The function $d$ ranges from 0, for the local undamaged state, to 1, indicating the crack having the size of the element.

By using the principle of strain equivalence (Grandmaison et al. Reference Grandmaison, Brancherie and Salsac2021), tension in the damaged state can be written as

(2.10) \begin{equation} \tilde {\unicode{x1D64F}}= \frac {\delta \tilde { S}}{\delta S} \unicode{x1D64F} = (1-d) \unicode{x1D64F}, \end{equation}

where $\unicode{x1D64F}$ is calculated from an undamaged material. According to classical continuum damage mechanics, strain energy in the damaged state is assumed to be homogeneous, and the strain energy function of a damaged neo-Hookean membrane can be written as

(2.11) \begin{equation} \omega _s(I_1,I_2,d)=(1-d)\omega _s^{NH}(I_1,I_2). \end{equation}

By introducing the damage threshold function (Besson et al. Reference Besson, Cailletaud, Chaboche and Forest2010) and model for quasi-brittle damage developed by (Marigo Reference Marigo1985), the damage variable $d$ can be calculated by

(2.12) \begin{equation} d=\langle \zeta (\omega _s^{{NH}^{MAX}}) \rangle ^+, \end{equation}

where $\langle \, \rangle ^+$ is the Macaulay bracket defined by

(2.13) \begin{equation} \langle x \rangle ^{+}=\begin{cases} x & \mbox{if } x \geqslant 0,\\ 0 & \mbox{otherwise}, \end{cases} \end{equation}

and $\zeta (\omega _s^{{NH}^{MAX}})$ is given by

(2.14) \begin{equation} \left . \begin{array}{ll} \displaystyle \zeta \Big(\omega _s^{{NH}^{MAX}}\Big)=\left(\omega _s^{{NH}^{MAX}}-Y_D\right)/Y_C,\\ \displaystyle \omega _s^{{NH}^{MAX}}(t)=\max _{\tau \leqslant t}\ {\omega _s^{NH}}(t), \end{array}\right \} \end{equation}

where $t$ indicates the time, and $Y_D \geqslant 0$ and $Y_C\gt 0$ represent damage threshold and hardening modulus, respectively.

Grandmaison et al. (Reference Grandmaison, Brancherie and Salsac2021) showed the sheer increase of the damage variable $d$ before rupture, and it is classical in damage mechanics to relax the criterion for rupture to $d=0.9$ or even $d=0.8$ . In this paper, we consider $d=0.9$ as the criterion for membrane rupture.

In this context, a material is said to be in a damaged state if the strain energy $\omega _s$ attains a value greater than the damage threshold $Y_D$ . Conversely, if the strain energy remains below the damage threshold at all times, then the material is designated as undamaged. The categorisation of the damage state is dependent on the final value of $d$ in the results of the simulation. The damage state is divided into three distinct regimes, namely (i) the undamaged state ( $d=0$ ), (ii) the damaged state ( $0 \lt d \lt 0.9$ ), and (iii) the rupture state ( $d \geqslant 0.9$ ). In addition, given that the damage variable $d$ corresponds to the irreversible growth of microdefects in the capsule, it can be concluded that the value of $d$ will be permanently recorded even when deformation of the capsule ceases.

2.3. Swimmer model

The microswimmer is modelled after the Lighthill–Blake squirmer (Lighthill Reference Lighthill1952; Blake Reference Blake1971), a model of a self-propelling sphere by its surface velocity $\boldsymbol{u}^s$ . The squirmer with the unit orientation vector $\boldsymbol{e}$ exhibits axisymmetric and steady surface squirming velocities. The flow field surrounding the squirmer has been derived as an infinite series of eigensolutions of the Stokes equation in axisymmetric spherical polar coordinates ( $r,\psi$ ). The origin of the coordinate system is located at the centre of the squirmer, with $r$ denoting the radial position, and $\psi$ the angle from the unit orientation vector $\boldsymbol{e}$ , as illustrated in figure 1. We follow Ishikawa, Simmonds & Pedley (Reference Ishikawa, Simmonds and Pedley2006) and consider the tangential velocities up to the second mode. The surface velocity is then given by

(2.15) \begin{equation} u_r(r,\psi ) = 0, \quad u_\psi (r,\psi ) = B_1\sin \psi + B_2\sin \psi \cos \psi , \end{equation}

where swimming speed $U_0$ is then given by $U_0=(2B_1)/3$ . The swimming mode and the stresslet strength of the squirmer are controlled by the squirmer parameter $\beta =B_2/B_1$ : $\beta \lt 0$ corresponds to a pusher-type squirmer that has a propulsion apparatus behind the body; $\beta \gt 0$ corresponds to a puller-type squirmer that has a propulsion apparatus in front of the body; $\beta =0$ corresponds to a neutral-type squirmer that has a propulsion apparatus at the centre of the body (see figure 1 b,c,d).

2.4. Flow due to the capsule deformation and the squirmer

In the context of viscous-dominant fluid flow, the flow field is governed by the Stokes equation. The flow field is thus described by a boundary integral equation with the Green’s function. The flow resulting from the deformation of the capsule and the squirmer can be derived as

(2.16) \begin{equation} \boldsymbol{u}(\boldsymbol{x})=-\frac {1}{8\pi \mu } \left [ \int _{S_c}\unicode{x1D645}(\boldsymbol{x},\boldsymbol{y})\cdot \boldsymbol{q}_c(\boldsymbol{y})\,{\rm d}S_c(\boldsymbol{y})+\int _{S_s}\unicode{x1D645}(\boldsymbol{x},\boldsymbol{y})\cdot \boldsymbol{q}_s(\boldsymbol{y})\,{\rm d}S_s(\boldsymbol{y}) \right ], \end{equation}

where $\boldsymbol{q}_s$ is the viscous load exerted on the squirmer, and the subscripts $c$ and $s$ represent the capsule and squirmer surfaces, respectively. The Green’s function of the Stokeslet $\unicode{x1D645}$ is defined by

(2.17) \begin{equation} \unicode{x1D645}(\boldsymbol{x},\boldsymbol{y})=\frac {\unicode{x1D644}}{r}+\frac {1}{r^3}\boldsymbol{r}\otimes \boldsymbol{r}, \end{equation}

where $\boldsymbol{r}=\boldsymbol{x}-\boldsymbol{y}$ , $\boldsymbol{r}=|r|$ and $\unicode{x1D644}$ is the identity tensor.

The determination of the viscous load on the capsule membrane, designated as $\boldsymbol{q}_c$ , can be achieved through the application of membrane mechanics, as elucidated in §§ 2.1 and 2.2, while the load on the squirmer, denoted as $\boldsymbol{q}_s$ , is unknown. From the boundary condition of the rigid motion of squirmer, the following equation holds: $\boldsymbol{u} = \boldsymbol{U} + \boldsymbol{\Omega }\times \boldsymbol{\hat {r}} + \boldsymbol{u}^s$ when $\boldsymbol{x} \in S_s$ , where $\boldsymbol{U}$ is the translational velocity, $\boldsymbol{\Omega }$ is the angular velocity, $\boldsymbol{\hat {r}} = \boldsymbol{x} - \boldsymbol{x}_g$ , and $\boldsymbol{x}_g$ is the centre of the squirmer. In addition, the force-free and torque-free conditions are given by

(2.18) \begin{equation} \int \boldsymbol{q}_s\,{\rm d}S_s = \boldsymbol{0} \end{equation}

and

(2.19) \begin{equation} \int \boldsymbol{q}_s\times \boldsymbol{\hat {r}}\,{\rm d}S_s = \boldsymbol{0}. \end{equation}

We solve for the three unknowns $\boldsymbol{q}_s$ , $\boldsymbol{U}$ and $\boldsymbol{\Omega }$ by coupling (2.16), (2.18) and (2.19). Detailed numerical methods are explained in the next subsection.

2.5. Numerical method

In order to compute the fluid–solid interactions, the finite element–boundary element coupling method is employed for the capsule deformation (Walter et al. Reference Walter, Salsac, Barthès-Biesel and Tallec2010), and a series of linear equations is discretised by the boundary element method for the squirmer motion (Huang et al. Reference Huang, Omori and Ishikawa2020). The capsule membrane is discretised into 20 480 triangular elements with 10 242 nodes, while the squirmer surface is discretised into 1280 triangular elements with 642 nodes.

All material points are tracked in a Lagrangian manner, and the strain energy and Cauchy tension can be calculated explicitly at any time. The equilibrium equation (2.7) is solved with respect to $\boldsymbol{q}_e$ by a finite element method (Walter et al. Reference Walter, Salsac, Barthès-Biesel and Tallec2010). The bending force density $\boldsymbol{q}_b$ is determined from the local mean and Gaussian curvatures according to (2.9), and the load $\boldsymbol{q}_c$ on the capsule membrane is calculated by summing $\boldsymbol{q}_e$ and $\boldsymbol{q}_b$ . Substituting $\boldsymbol{q}_c$ into (2.16), the flow generated by the capsule deformation, i.e. the first term of the equation, is computed by a Gaussian numerical integration scheme (Huang et al. Reference Huang, Omori and Ishikawa2020). To find the surface load on the squirmer $\boldsymbol{q}_s$ , the following simultaneous linear equations are formulated from the boundary integral equation at $\boldsymbol{x} \in S_s$ and the force and torque balance (Huang et al. Reference Huang, Omori and Ishikawa2020):

(2.20) \begin{equation} \begin{bmatrix} \boldsymbol{\mathcal{J}} && \boldsymbol{\mathcal{M}_1} && \boldsymbol{\mathcal{M}_2} \\ \boldsymbol{\mathcal{F}} && \boldsymbol{0} && \boldsymbol{0}\\ \boldsymbol{\mathcal{T}} && \boldsymbol{0} && \boldsymbol{0} \end{bmatrix} \begin{bmatrix} \boldsymbol{q_s}\\ \boldsymbol{U}\\ \boldsymbol{\Omega } \end{bmatrix} = \begin{bmatrix} -\boldsymbol{u^c} + \boldsymbol{u}^s\\ \boldsymbol{0}\\ \boldsymbol{0} \end{bmatrix}, \end{equation}

where the matrix component $\boldsymbol{\mathcal{J}}$ is computed by the second term of (2.16), $\boldsymbol{\mathcal{F}}$ and $\boldsymbol{\mathcal{T}}$ are given by (2.18) and (2.19), $\boldsymbol{\mathcal{M}_1}$ and $\boldsymbol{\mathcal{M}_2}$ are the mobility matrices given by the boundary condition, and the velocity $\boldsymbol{u^c}$ is the flow generated by the capsule deformation. The total matrix size is $(3N+6)\times (3N+6)$ , and the dense matrix system is solved by a lower and upper factorisation technique (Huang et al. Reference Huang, Omori and Ishikawa2020), where $N=642$ is the number of computational points. After solving (2.20), the translational $\boldsymbol{U}$ and rotational $\boldsymbol{\Omega }$ velocities of the squirmer are obtained, and each nodal point on the squirmer surface is updated by the second-order Runge–Kutta method. The flow field is also updated by substituting $\boldsymbol{q}_s$ into (2.16), and new positions of the capsule membrane are given in the same way with the no-slip boundary condition ${\rm d}\boldsymbol{x}/{\rm d}t = \boldsymbol{u}$ .

When the squirmer comes too close to the capsule surface, the solution becomes less accurate and numerical instabilities may arise. To avoid this, we add short-range repulsive forces between the capsule and the squirmer, i.e. providing a force dipole against the nearest neighbour surface of the capsule and the squirmer.

The repulsive force is given by the following equation as an exponentially decaying function, as in previous studies (Brady & Bossis Reference Brady and Bossis1985; Ishikawa et al. Reference Ishikawa, Simmonds and Pedley2006):

(2.21) \begin{equation} \boldsymbol{F}_{rep}=\alpha _1\frac {\alpha _2 \exp(-\alpha _2 r)}{1- \exp(-\alpha _2 r)}(-\boldsymbol{r}), \end{equation}

where $\boldsymbol{r}$ is the vector connecting two nodes, $r=|\boldsymbol{r}|$ , and $\alpha _1$ , $\alpha _2$ are the magnitude of the repulsive force and a parameter related to distance between two nodes. The stress acting on node $i$ is computed by $\boldsymbol{F}^{i}_{rep} / A_i$ , where $A_i$ is the area of the Voronoi cell of node $i$ .

In this study, we choose $\alpha _1=1000$ and $\alpha _2=50$ , corresponding to an effective working distance of the repulsive force of the order of $r/a_s=0.2$ . This value is nearly equal to the mesh size $\Delta x/a_s \sim 0.1$ and can be seen to work only when the swimmer and the capsule are sufficiently close together. When a repulsive force is acting, the force balance of (2.18) changes as follows:

(2.22) \begin{equation} \int \boldsymbol{q}_s\,{\rm d}S_s = \boldsymbol{F}_{rep}. \end{equation}

Another parameter, the capillary number, is defined as $Ca=\unicode{x03BC} U_0/G_s$ , expressing the ratio of viscous force and elastic force. In our simulations, all equations are non-dimensionalised by the free-swimming velocity $U_0$ , the radius of microswimmer $a_s$ , the viscosity $\mu$ , and the elastic modulus $G_s$ . The dimensionless bending rigidity is set to $E_b/G_sa^2_s = 0.01$ , and the size ratio between capsule and squirmer is $a_c/a_s = 4.0$ throughout the study. As for damage behaviour of the capsule, we set $Y_D/ G_s=0.2$ , $Y_C / G_s=2.0$ . A dimensionless time step $\Delta t\,U_0/a_s$ is set to $\Delta t\,U_0/a_s=5.0\times 10^{-4}$ .

3.1. Elastic deformation without damage to membranes

First, the capsule deformation is considered in the low capillary number regime. The swimming mode of the squirmer is set to $\beta = -3$ , corresponding to a pusher-type squirmer. The orientation of the squirmer is set to $\boldsymbol{e} = (1,0,0)$ , and the initial swimming direction is aligned with its orientation. The initial position of the squirmer is set to $(x/a_s,y/a_s,z/a_s) = (0,2,0)$ , which corresponds to the initial incidence angle $\theta = 30^ \circ$ as shown in figure 1(a). The mass centre of the capsule is initially placed at $(0,0,0)$ , and the capillary number is set as $Ca=0.1$ .

Figure 2. Swimming trajectory of a squirmer with different swimming modes in the undamaged regime; each snapshot indicates the time sequence. The incidence angle is set to $\theta = 30^{\circ }$ for all cases. (a) A pusher-type squirmer ( $\beta = -3$ ) case with $Ca = 0.1$ . (b) A neutral-type squirmer ( $\beta =0$ ) case with $Ca = 0.07$ . (c) A puller-type squirmer ( $\beta = 3$ ) case with $Ca = 0.09$ . A pusher-type squirmer circles stably near the capsule membrane, while a neutral-type or puller-type squirmer swims perpendicularly towards the membrane, reaching equilibrium at $tU_0/a_s \geqslant 15$ . Blue arrows corresponds to the orientation vector $\boldsymbol{e}$ .

As illustrated in figure 2(a), the temporal development of the swimmer’s trajectory is demonstrated.

Subsequent to the initiation of the calculation, the swimming direction of the squirmer is modified by hydrodynamic interactions with the membrane, resulting in the adoption of a circular trajectory along the capsule membrane. In the low $Ca$ regime, the strain energy does not reach the threshold for brittle damage due to the small deformation of the membrane. Consequently, deformation of the membrane invariably occurs within the elastic region, without compromising the integrity of the membrane.

The same tendency can be observed for the puller-type and neutral-type squirmers, but their swimming trajectories are different from those of the pusher-type squirmer (cf. figure 2 b,c). In the cases of neutral-type and puller-type squirmers ( $\beta = 0,3$ ), the squirmer swims towards the membrane and finally comes to rest in a position of equilibrium.

As demonstrated in figure 2(c), a dimple is also visible at the upper part of the capsule. The observed asymmetric distribution can be attributed to the initial conditions. Specifically, the initial position of the swimmer is near the top of the capsule, and the dimple manifests at the top of the capsule. Conversely, if the swimmer is initially positioned at the bottom, then the dimple will also appear at the bottom.

The underlying reason for these trajectory differences is elucidated in Ishikawa (Reference Ishikawa2019) by lubrication theory, where a pusher-type squirmer tends to escape from a free surface, while a puller-type squirmer tends to be trapped by a free surface. It is important to note that the findings of the present study demonstrate discrepancies with those of previous research. For instance, while the literature suggests that a neutral-type squirmer will typically escape from a free surface, our study demonstrates that it swims towards the capsule membrane. This deviation can be attributed to the deformation of the capsule membrane, which, upon being deformed around the squirmer, cancels out the rotational torque generated on the opposite side. Consequently, the torque generated in front of the squirmer is insufficient to alter its direction, resulting in the neutral-type squirmer swimming towards the capsule surface.

3.2. Effect of capillary number; three different states of membrane damage

The subsequent investigation focuses on the effect of capillary number. A pusher-type squirmer ( $\beta =-3$ ) is confined within the capsule, and the initial conditions are the same as in figure 2; we observed three distinct states of membrane damage, which varied according to the capillary number.

For low $Ca$ , e.g. $Ca = 0.1$ , the deformation of the capsule is small, thereby preserving the membrane’s undamaged state. Consequently, the maximum damage variable on the surface $d_{max}$ remains at zero, and the squirmer swims along the membrane surface (cf. figure 3 a,b).

With an augmented capillary number, $Ca=0.11$ , deformation of the capsule increases, and the strain energy reaches the threshold for the onset of brittle damage. The damage variable $d_{max}$ then gradually increases but reaches a stable state, while the squirmer creates a ring-like damage area on the membrane surface (cf. figure 3 c). And this state is defined as the damaged state.

With an even larger capillary number, $Ca = 0.12$ , the membrane damage no longer remains in the stable regime, but increases, eventually leading to membrane rupture forming a scratch-like rupture area diagonally ahead of its swimming direction (cf. figure 3 d). Accordingly, the damage variable $d_{max}$ increases quickly and reaches its maximum in a short time, as shown in figure 3(a).

Figure 3. Different states of the capsule confining a pusher-type squirmer ( $\beta = -3$ ): ruptured state ( $Ca = 0.12$ ), damaged state ( $Ca = 0.11$ ) and undamaged state ( $Ca = 0.1$ ). (a) Damage evolution of the capsule with different $Ca$ . When $Ca = 0.12$ , the capsule is ruptured at $tU_0/a_s=32.5$ , whereas the damage to the membrane remains small below $Ca \leqslant 0.11$ . Snapshots of (b) an undamaged state ( $Ca = 0.1,\ tU_0/a_s = 50$ ), (c) a damaged state ( $Ca = 0.11,\ tU_0/a_s = 150$ ), and (d) a ruptured state ( $Ca = 0.12,\ tU_0/a_s = 32.5$ ). The initial incidence angle is set to $\theta = 30^{\circ }$ , and the colour band indicates the damage variable on the membrane.

3.3. Membrane rupture in different swimming modes

The squirmer with different $\beta$ is placed inside the capsule, and it is found find that the critical capillary number for membrane rupture depends on $\beta$ .

Figure 4. Rupture of the capsule by squirmers with different swimming modes: (a) a pusher-type squirmer ( $\beta = -3$ ), (b) a neutral-type squirmer ( $\beta = 0$ ), and (c) a puller-type squirmer ( $\beta = 3$ ). The time of the rupture is indicated by the broken line in the graph; $tU_0/a_s = 32.5$ with a pusher-type squirmer, $tU_0/a_s = 9.6$ with a neutral-type squirmer, and $tU_0/a_s = 9.1$ with a puller-type squirmer. The upper panels are viewed from the $(x,y)$ -plane, while the middle panels are viewed from the $(y,z)$ -plane. The initial incidence angle is set to $\theta = 30^{\circ }$ .

In the case of a pusher-type squirmer ( $\beta = -3$ ), the critical capillary number for membrane rupture, $Ca_0$ , is in the range $0.11 \lt Ca_0 \lt 0.12$ when the initial incidence angle is $\theta = 30^ \circ$ . As previously outlined, the pusher-type squirmer swims along the membrane, and membrane rupture occurs near the oblique front of the swimming direction (cf. figures 3(d) and 4(a), and supplementary movie 1).

In the case of a puller-type squirmer ( $\beta =3$ ) and a neutral-type squirmer ( $\beta =0$ ), the damage area is similar in appearance, forming a small damage patch in front of the squirmer, and breaking the membrane from the centre of the patch (see figure 4(b,c) and supplementary movies 2 and 3). The time to membrane rupture for the puller-type and neutral-type squirmers is $tU_0/a_s \approx 10$ , which is faster than for the pusher-type squirmer ( $tU_0/a_s = 32.5$ ). Despite the similarity in the motions of the squirmer and the formations of membrane damage for the puller-type and neutral-type squirmers, their critical capillary numbers $Ca_0$ are different, estimated to be $0.12 \lt Ca_0 \lt 0.13$ for a puller-type squirmer, and $0.1 \lt Ca_0 \lt 0.11$ for a neutral-type squirmer.

Here, we compare the present results with the study by Grandmaison et al. (Reference Grandmaison, Brancherie and Salsac2021), who investigated capsule rupture in simple shear flow.

Previous studies have defined the capillary number as $Ca_{pre}=\unicode{x03BC} \gamma a_c/G_s$ , where $\gamma$ corresponds to the shear rate. However, due to the difference in capsule radius ( $a_c / a^{pre}_c = 4$ ), it is necessary to divide $Ca_{pre}$ by 4, converting to the one based on the capsule radius for a meaningful comparison with our $Ca_0$ . When they used $Y_D / G_s = 0.2$ and $Y_C / G_s = 2.0$ , the capsule is damaged with $Ca_{pre}/4 \approx 0.1$ , and broken with $Ca_{pre}/4 \approx 0.18$ . On the other hand, when we choose the same $Y_D$ and $Y_C$ in the present setting, the capsule can be damaged with $Ca_0 \approx 0.1$ , and broken with $Ca_0 \approx 0.13$ . These results indicate that capsules can be broken more easily when subjected to the action of an internal swimmer than when subjected to a simple shear flow.

3.4. Phase diagram of membrane damage

The effect of capillary number $Ca$ and swimming mode $\beta$ on the membrane damage is summarised in figure 5. The initial angle of incidence is set to $\theta = 30^ \circ$ for all cases, and the effect of $\theta$ is explained in the next subsection.

Figure 5. Phase diagram of damage development in $\beta$ and $Ca$ space. The initial incidence angle is set to $\theta = 30^{\circ }$ .

When $Ca$ is less than 0.05, the membrane deformation is sufficiently small that the strain energy does not reach the threshold for onset of brittle damage for all $\beta$ . In regimes where $Ca$ is greater than 0.05, the damage state depends on the swimming mode $\beta$ . In the case of a strong pusher-type squirmer, i.e. in the $\beta \lt -2$ regime, the squirmer tends to swim along with the membrane surface, and the undamaged regime expands with $|\beta |$ . In this regime, the critical capillary number for membrane rupture $Ca_0$ is close to the $Ca$ value at which brittle damage begins, and the undamaged and rupture states are close together. However, in the case of a weak pusher-type squirmer ( $-2\leqslant \beta \lt -1$ ), the squirmer tends to swim towards the membrane surface, and the membrane is susceptible to damage from the recirculating flow in front of the squirmer (cf. figure 1 b), with $Ca_0$ reaching a minimum value at $\beta = -2$ , where the swimming direction undergoes a transition from being aligned with the surface to being oriented against it. In regimes where $\beta \geqslant -1$ , in particular where $\beta \geqslant 1$ , the undamaged regime tends to widen as $\beta$ increases. In the $-1 \lt \beta \lt 1$ regime, the flow generated by the squirmer is relatively smooth and weak, and all squirmers in this regime tend to swim towards the membrane surface, thus the critical capillary numbers become the same. Conversely, in the $\beta \gt 1$ regime, the flow in front of the squirmer tends to draw the capsule membrane close to the squirmer, thereby cancelling out a part of the deformation towards outside, and thus expanding the undamaged region as $\beta$ increases.

3.5. Effect of incidence angle

Figure 6. Effect of the initial incidence angle $\theta$ : capsule enclosing (a) a pusher-type squirmer ( $\beta = -3$ ), (b) a neutral-type squirmer ( $\beta = 0$ ), and (c) a puller-type squirmer ( $\beta = 3$ ).

In the following investigation, the influence of the incidence angle $\theta$ is examined by varying the initial position of the squirmer $y$ (see figure 1 a). The swimming mode $\beta$ is set to $\beta =-3,0,3$ , and the results are shown in figure 6.

When $\beta =-3$ , the critical capillary number increases with the incidence angle (see figure 6 a), indicating that the capsule becomes harder to damage. When the angle of incidence is large, the initial swimming direction of the squirmer is almost tangential to the capsule surface, and it swims stably along the surface (see supplementary movie 5). Conversely, when the angle of incidence is small, the initial swimming direction is almost perpendicular to the capsule surface, resulting in severe deformation of the capsule (see supplementary movie 4). This makes the capsule more susceptible to damage.

When $\beta =0$ and 3, the critical capillary number is found to be unaffected by the incidence angle (figure 1 b,c), indicating that the occurrence of brittle damage is independent of the initial condition. In the case of a puller-type squirmer or a neutral-type squirmer, the squirmer tends to swim perpendicular to the capsule surface. Therefore, in the present study, the direction of swimming near the membrane is determined regardless of the initial conditions, and as a result, $\theta$ dependence cannot be observed (see supplementary movies 6 and 7).

4.1. Breaking a harder capsule with a static magnetic field

First, drug release is assumed, then an external magnetic field is applied to rupture the elastically deforming capsule membrane. A pusher-type squirmer ( $\beta = -3$ ) is used, and a static magnetic field is considered for the external magnetic field, $\boldsymbol{b}=(\sqrt {3},1,0)/2$ .

The capillary number is set to $Ca=0.07$ to correspond to a brittle damage-free regime with stable elastic deformation except for initial defects with initial incidence angle less than $10^ \circ$ .

The magnetic torque activates at the dimensionless time $tU_0/a_s =15.0$ , and its amplitude is set to $A_m/\unicode{x03BC} a_s^2U_0 = 30$ . Before the activation of the magnetic torque, the squirmer swims along with the surface. However, after the activation of the magnetic field, the squirmer’s movement is directed towards the direction of the magnetic field $\boldsymbol{b}$ due to the magnetic torque, as shown in figure 7 and supplementary movie 8. The second incident angle is smaller, and is accompanied by deformation of the rupture regime. This results in the eventual rupture of the membrane.

We then investigate the effect of $A_m$ on the critical capillary number. As demonstrated in figure 7(a), the critical capillary number undergoes a substantial decrease due to the influence of the external magnetic torque $A_m$ . The findings suggest that the rupture of the membrane can be controlled by the squirmer through the application of a sufficiently large external torque.

Figure 7. Rupture of the capsule with a pusher-type squirmer inside controlled ( $\beta = -3$ ) by a constant external magnetic torque. The normalised magnetic field vector is set to $\boldsymbol{b} = (\sqrt {3},1,0)/2$ , and the initial incidence angle is set to $\theta = 30^{\circ }$ . (a) Effect of the amplitude of external magnetic torque $A_m$ . (b) Behaviour of the pusher-type squirmer until the membrane becomes ruptured. Magnetic torque is applied at $tU_0/a_s = 15$ (indicated by the cross) to control the swimming direction of the squirmer. By applying the magnetic torque with amplitude $A_m/\unicode{x03BC} a_s^2U_0 = 30$ , the stable circular swimming of the inside squirmer changes to a one-directional swimming, which allows the squirmer to break the capsule ( $Ca$ is set to 0.07).

4.2. Preserving a capsule by a rotational magnetic field

Figure 8. Rupture of the capsule with a pusher-type squirmer inside controlled by a rotating magnetic field; $\beta = -3$ , $Ca = 0.15$ and $\theta = 30^{\circ }$ for all cases. (a) Phase diagram in $A_m$ and $\omega _m$ space. (b–g) Effects of the angular velocity on the swimming of the squirmer: $\omega _ma_s/U_0 = 0.2, 1.0,2.0, 2.2,5.0,50.0$ , respectively, while the amplitude is set to $A_m/\unicode{x03BC} a_s^2U_0 = 40$ for all cases. The black line indicates the swimming trajectory of the squirmer, and the blue and the red arrows correspond to the swimming direction $\boldsymbol{e}$ and the normalised magnetic field vector $\boldsymbol{b}$ , respectively.

Next, the potential of a rotating magnetic field in preventing the rupture of a softer capsule is explored. A pusher-type squirmer ( $\beta = -3$ ) is employed as the internal swimmer. Under these conditions, the capsule remains in the ruptured regime in the absence of magnetic torque. The rotating magnetic field is defined as $\boldsymbol{b}=(\cos \omega _m t,-\sin \omega _m t,0)$ , where $t$ is time. The magnitude of the torque is set to $A_m/\unicode{x03BC} a_s^2U_0 = 40$ , and the angular velocity $\omega _m$ is parametrised (see figure 8).

When the angular velocity is sufficiently small, e.g. $\omega _ma_s/U_0 = 0.2$ in figure 8(b), the magnetic field changes too slowly to make the swimmer move away from the membrane. This condition is analogous to the absence of a magnetic field, and the capsule reaches a rupture state (see supplementary movie 9). When the angular velocity of the magnetic field is comparable to the swimming speed ( $0.2 \lt \omega _ma_s/U_0 \leqslant 1.5$ ), the squirmer traces a circular orbit of its radius, and the interaction with the membrane also becomes weak. Therefore, the capsule membrane is preserved (cf. figure 8(c) and supplementary movie 10). However, when the angular velocity is approximately $1.5 \lt \omega _ma_s/U_0 \leqslant 2$ , the swimming path becomes unstable, and the increased frequency of near-field interactions between the capsule and the squirmer tends to increase the susceptibility to rupture (cf. figure 8(d) and supplementary movie 11). Then in the regime with larger $\omega _m$ ( $2 \lt \omega _ma_s/U_0 \leqslant 10$ ), the angular velocity of the magnetic field gradually becomes dominant in relation to the angular velocity of swimming ( $\omega _m \gt |\boldsymbol{\Omega }|$ ), and the swimming trajectory again approaches a smooth circular path (cf. figure 8(e,f ) and supplementary movies 12 and 13). However, with extremely large angular velocities, e.g. $\omega _ma_s/U_0=50$ in figure 8(g), the magnetic field undergoes such rapid changes that the swimmer is unable to maintain its course. Consequently, the motion of the squirmer reverts to a state of sufficiently small angular velocity, resulting in the membrane reaching a rupture state (see supplementary movie 14).

The phase diagram of the membrane damage due to torque magnitude $A_m$ and angular velocity $\omega _s$ is summarised in figure 8(a). Evidence suggests that there is indeed a damage-free region, indicating the potential for utilising the rotating magnetic field as a soft capsule transport technique.