Finally, we want to lay out how to extend our iterative method, primarily to actively swimming capsules[35]. One possible swimming mechanism[51], for example in squirmer type swimmer models [36, 37, 52], is a finite velocity field at the capsule surface in its rest frame. Our solution technique remains still applicable as long as there is no net velocity normal to the surface.
Figure 79: The flow field around three
types of squirmers – from left to right α=0 (neutral), α= −1(puller), α=1(pusher) – and a passive particle for comparison. The left half of each diagram depicts the flow in the resting frame of the immersed particle, whereas the flow in the lab frame is shown in the right half.
The squirmer model for a spherical capsule corresponds to a decomposi- tion of the slip velocity into Legendre polynomials. We restrict ourselves to the first two modes and write the tangential slip velocity as
us∝ sin(s0) +α/2 sin(2s0).
The first mode corresponds to a “force” velocity field (resembling a dipole field in standard multipole expansions) and the latter to a “force dipole” field (a quadrupole contribution). For α>0there is a flow towards the middle, the thrust comes from the bow of the capsule, this case is therefore commonly referred to as apusher. The case α<0is referred to as apulleras the flow is directed away from the middle and the thrust comes from the stern of the capsule. The velocity a rigid sphere assumes due to the finite slip velocity is independent from α and depends only on the omitted prefactor168.
168Generally, the resulting velocity
of a rigid sphere due to a given slip velocity distribution is given by the
average slip velocity. This follows[53] from
an application of Lorentz’ reciprocal theorem[30]. For a general body of revolution we can analogously relate the translational velocity u∞ and the
slip velocity us if we know the surface
forces ef for a rigid, passive translation of this body by R Cds r sin ψ ef · u∞ = −RCds r sin ψ ef · us or u∞R Cds r sin ψ efz=RCds r sin ψpesus.
Figure 80: Generatrices of a neutrally
squirming (gsincreases from left to right)
for ˜EB=0.001. The deformation is domi-
nated by the tangential stress.
The justification of a slip velocity of any kind is that this allows us to incorporate the effect of some underlying propulsion mechanism that we do not model explicitly. A large variety of propulsion mechanisms have been studied. However, we are interested in deformable swimmers and in general quantifying the change of the effective slip velocity with the deformation would require a profound knowledge of the underlying mechanism. Usually this would increase the complexity of the problem by coupling to a third system of differential equations for the drive, for example a heat equation for thermophoretic particles[54] or a diffusion equation for diffusiophoretic particles[55, 56]. One rather generic possibility would be a propulsion by cilia (biological or artificial) on the surface, we would then assume that there
is a given density of cilia of a given strength on any surface element of the undeformed capsule. As the cilia number is fixed and the cilia itself are im- motile this constitutes a model where we are able to give the slip velocity for the deformed capsule as well. We call the previously omitted proportionality factor gs and can then give the slip velocity on the surface of the capsule
at any state of deformation us(s0) = √gs
λsλϕ
(sin(s0) +α/2 sin(2s0)). By
including the local area dilatation λsλϕthe power density (and hence the
total propulsion power) is conserved P ∝R
dAu2s(s) ∝ R
dA0λsλϕu2s(s0).
As stated before the initial force-velocity relation (for a spherical resting shape) is only dependent on gs and one finds[37] that the stationary velocity
is given by u = 2/3gs. On the technical level the additional slip velocity
adds little overhead and some results are shown in Fig. 80. The slip velocity enters on the level of the boundary conditions, that is the left hand side of eq. (56). As it is more efficient to adapt the velocity of the capsule u, while computing the deformation of the capsule169, we want to separate the flow
169This is a technical subtlety. At any
stage of the iterative process (except for the initial iteration) we use the flow correspond- ing to the shape of the previous iteration in the solution of the shape equations. How- ever, we cannot deduce the absence of a net flow on the new shape based upon the previous shape, thus violating global force balance, which severely afflicts the ability to find a solution of the shape equations. Our approach to this is to adapt the veloc- ity in the solution of the shape equations (using a convex combination of old and new stresses, as stated before). The alter- native would be to compute u based upon the old shape at the level of the solution of the Stokes equation. For the finally found stationary solution these approaches coin- cide.
and the stresses into the contributions from the motion of the capsule (∝ u) and the contributions from the propulsion (∝ gs). We are free to do this since
the Stokes equation is linear. We therefore solve two versions of eq. (57), one with the same boundary condition as for sedimentation and one with the slip velocity as boundary condition on the capsule surface. A variation to the fixed “squirming pattern” considered before would be to search for the optimalslip velocity distribution. As we separated the two hydrodynamics and the capsule deformability, we can directly170 adapt the approach for
170However, this renders the remark of side-
note 169 mute, as we now do compute the force and velocity based upon the old shape, but as stated before this is a minor techni- cality.
rigid bodies of Ref. 63.
A more direct approach to swimming would be to consider a direct de- formation of the capsule. To avoid subtleties[57] let us assume that the deformation is given by a deformation velocity uDdescribing the time evolu-
tion of each surface segment. The resulting motion of the entire capsule leads to a velocity boundary condition u+uD(s) on the surface of the capsule.
The correct value of the swimming velocity is fixed by the absence of a net force and therefore easily computed as we can separate the two contributions (translation, deformation) due to the linearity of the Stokes equation171.
171The deformation pattern has to be cyclic
to sustain prolonged locomotion (and rea- sonably be called swimming), but must not be reciprocal[58] as the absence of a direct time dependence in the Stokes equation leads to an invariance under (nonlinear) reparametrisation of time (including time reversal). This is commonly referred to as thescallop theorem.
However, the very nature of this locomotion procedure rules out the existence of asteady state and thus the incorporation of elasticity would have to be dynamic in nature and thus out of the scope of the method presented here.
Aside from actively swimming capsules there are also a number of in- teresting problems with externally controlled flow fields172, which one can
172In light of the nature of the developed
method, we only consider arrangements that lead to a non-trivial hydrodynamic stress. For example, we could also com- pute the deformation of a capsule inside a spinning drop apparatus, but this only amounts for adding a pressure contribution
p∝ sin(ψ)r3er.
employ to probe elastic properties of the capsule. For example, we could prescribe an extensional flow[31] uext = −εrer+2ε(z−zcm). This is the
most general form of an axisymmetric extensional flow, the factor two guar- antees compatibility with incompressibility. We are generally interested in stationary solutions, but this flow field has broken symmetry along the (axial) direction of motion, hence we can only consider a capsule that is localised (with vanishing centre of mass velocity) around the centre of mass. We already took this into account by centring the flow around the centre of mass of the capsule zcm. Stationary shapes in this flow are not only axisymmetric,
but also have a fore-aft symmetry, as this is needed to establish the absence of a net force. An experimentally important setup is a capsule inside a (rigid) spherical tube, that is in a Poiseuille flow[25]. However, the inclusion of additional walls alters the boundary conditions at infinity, hence we can no longer use the free propagator to find an exact solution173.
173Sadly, the only representation[31, 59]
of the propagator for Poiseuille boundary conditions known to the author in Fourier space is exceptionally badly fit for a nu- merical approach. However, we can try to find an approximate solution in a way that has been shown to work for the deforma- tion of vesicles[60]: we ignore the influence of the walls and reduce the problem to a prescribed parabolic velocity profile
upoi=gp(1− (r/d)2)ez.
Here, gp quantifies the strength of the
Poiseuille flow (given by the pressure gra- dient along the tube without the capsule in it) and d is the radius of the “phantom tube”. A different approach with more ef- fort, but better control of the error would be to consider[61] the motion in a finite tube. After separating the problem into a capsule with free boundary conditions and a tube with given velocity on its surface, we can solve both subproblems using our standard boundary integral formalism.