The hydrodynamics of slender swimmers near deformable interfaces

We study the coupled hydrodynamics between a motile slender microswimmer and a deformable interface that separates two Newtonian fluid regions. From the disturbance field generated by the swimming motion, we quantitatively characterize the interface deformation and the manner in which the coupling modifies the microswimmer translation itself. We treat the role of the swimmer type (pushers and pullers), size and model an interface that can deform due to both surface tension and bending elasticity. Our analysis reveals a strong dependence of the hydrodynamics on the swimmer orientation and position. Given the viscosities of the two fluid media, the interface properties and the swimmer type, a swimmer can either migrate towards or away from the interface depending on its configurations. When the swimmer is oriented parallel to the interface, a pusher-type swimmer is repelled from the interface at short times if it is swimming in the more viscous fluid. At long times however, pushers are always attracted to the interface, and pullers are always repelled from it. On the other hand, swimmers oriented orthogonal to the interface exhibit a migration pattern opposite to the parallel swimmers. In consequence, a host of complex migration trajectories emerge for swimmers arbitrarily oriented to the interface. We find that confining a swimmer between a rigid boundary and a deformable interface results in regimes of attraction towards both surfaces depending on the swimmer location in the channel, irrespective viscosity ratio. The differing migration patterns are most prominent in a region of order the swimmer size from the interface, where the slender swimmer model yields a better approximation to the coupled hydrodynamics.

Analytical studies that have investigated the hydrodynamics of microswimmers and passive particles near deformable interfaces, have treated the bodies as singularities (point forces, force-dipoles, quadropoles, etc.) [12,19,42,44,45], two-dimensional swimming sheets [14], or spherical particles/squirmers [17,47]. Each of these unique approaches can explore observations in the regime of their applicability. For example, Dias and Powers [14] showed that a two-dimensional sheet swimming near a deformable interface can generate a pumping velocity that can be either attractive or repulsive to the interface depending on the viscosity ratio of the two fluid regions. Moreover, they found that an increase in the bending stiffness resulted in enhanced swimmer translation but reduced the fluid pumping. Daddi-Moussa-Ider et al. [19] considered higher order terms of a multipole analysis for determining the effect of the coupled hydrodynamics on the translational and rotational motion of a microswimmer near a deformable interface. By considering an interface which exhibits resistance to both bending and shear, they established that the swimmer translation was enhanced due to interface bending and suppressed due to shear resistance. On the other hand Shaik and Ardekani [17] modeled the swimmer as a sphere with a prescribed surface slip velocity (the spherical squirmer approach [48]) and quantitatively characterized both the interface deformation and the swimmer translation FIG. 1: A schematic representation of a slender swimmer translating near a deformable interface with speed V s along its director vector p. The swimmer is characterized by an aspect ratio κ defined as the ratio of its total length L to its lateral extent W . The circles along the axial length of the fore-aft symmetric swimmer represent a line distribution of stokeslets characterizing the head and tail for pushers. The disturbance flow field generated by the swimming motion deforms the interface, and u z is the interface deformation (solid gray line) relative to its initially flat undeformed (dotted line) state z 0 .

A. Equations for the fluid velocity and the boundary condition
The equations governing the two fluid regions are the Stokes equations and the continuity equations which characterize the disturbance flow field v and the pressure field P , and are: Here, the subscript α ∈ [1,2] represents the two fluid regions, η is the fluid viscosity, δ(x) is the Dirac-delta function [58], L is the total swimmer length, V s is the swimming speed, and s refers to the distance along the slender swimmer axial coordinate, whose center-of-mass is at position x s and is oriented along the direction p. The orientation vector p is characterized by a polar angle θ relative to the vertical and azimuthal angle φ in the plane of the undeformed interface. The kernel in the integral on the right-hand side of Eq. (1a), f α , represents the forcing due to the slender swimmer and characterizes a line distribution of Stokeslets along the axial coordinate of the swimmer. For a fore-and aft-symmetric swimmer, which disturbs in the surrounding fluid medium as it moves, one may write f α as [51,52]: where κ refers to the slender swimmer aspect ratio, and is defined as the ratio of the total swimmer length to its lateral extent, as shown in Fig. 1. The above form of f α describes two swimming mechanisms, namely, pushers (rear propelled) and pullers (fore propelled). In the far-field both swimming mechanisms exhibit a disturbance flow field akin to a point-force dipole with opposing dipole strength [59,60]. This opposing character is encoded in the parameter D = −1 (+1) in Eq. (2) which describes the specific nature of the force-dipole swimming mechanism, that is extensile for pushers (contractile for pullers). Typically, pusher-type bacteria such as E.coli, B.subtilis are fairly slender with κ ∼ 10 when the swimmer length is measured based on the cell body and the flagellar bundle length combined [31,51,61,62]. Equation (1) must be solved subject to appropriate boundary conditions at the interface. We consider an impenetrable no-slip interface of infinitesimal thickness within the continuum framework [41]. This translates to the following velocity boundary conditions: v 1 · n = v 2 · n and v 1 · (I − nn) = v 2 · (I − nn), where n = ∇F/|∇F | is the unit normal to the interface F = z − z 0 − u z [41], and I is the identity tensor. The z-component of the interface deformation relative to its undeformed planar state is u z (see Fig. 1). As described in Sec. I, the interface can deform when there is a flow in the fluid medium surrounding it, since it has a finite surface tension and can to bend elastically. Therefore, the normal component of the stress undergoes a jump across the interface: σ zz is the stress tensor, γ the isotropic surface tension and δF bend represents the stress jump due to bending. Here, we consider linear elastic bending, and use the Helfrich model [63] to relate the interface deformation to the bending stress. This model has been used extensively in the literature for analyzing swimming sheets, Stokeslets and other higher-order singularities near deformable interfaces [see 14,19,44,45,64, and also where Eqs. (6a) and (6c) correspond to Eq. (1a), and Eqs. (6b) and (6d) to Eq. (1b) for α ∈ [1,2]. Note that, for brevity, we retain the original notation of the field variables in the non-dimensional Eqs. (6a)-(6d). Now, the leading order flow field generated by a slender-body is O(ln κ) −1 [51,53,59,65], where we emphasize that the aspect ratio κ 1. As the disturbance field v in the Stokes equations (6a) is linear in the forcing, it will also scale as O(ln κ) −1 , and is therefore asymptotically small. Hence, we can consider small interface deformations without having to invoke the smallness of the ratio of viscous fluid stresses to the deformation driving stresses on the interface (surface tension and bending). In other words, owing to the weak flow field of a slender-body, the slender swimmer model enables simplifications of the boundary conditions.
For small deformations, we can linearize the expressions for the unit normal and the curvature at the interface as: n ≈ 1 z and ∇ r · n ≈ −∆ r u z , where 1 z is the r z -component of the unit normal [42,44]. Within the linear response framework the height function, u z , depends only upon the r x and r y coordinates, and an analytical expression for the stress jump due to bending using the Helfrich elastic energy function dimensionally is: δF bend = κ β ∆ 2 u z , where ∆ represents the Laplacian in the r x − r y plane and κ β the bending modulus of the interface [44]. Thus, the non-dimensional form of the boundary conditions specified above Eq. (3) are: , and (7d) where λ = η 2 /η 1 is the viscosity ratio. We are studying an interfacial deformation process and to non-dimensionalize the kinematic condition Eq. (3), we can choose between the scale for the elastic bending and surface tension, corresponding to t c1 = η 1 L 3 /κ β and t c2 = η 1 L/γ, respectively. For ultra-soft interfaces, the surface tension can be γ ∼ O(10) −5 N/m or smaller [54,56,66], and for example the bending rigidity of vesicles is κ β ∼ O(10) −19 J [54,55,67,68]. In a fluid medium with a viscosity approximately that of water, η 1 ∼ O(10) −3 Ns/m 2 , and a swimmer of . Independent of the time scale chosen, the relative importance of the surface tension and bending stress on the right-hand side of Eq. (7e) can then be characterized by the ratio of the two scales, namely γL 2 /κ β . Depending on the stiffness of the interface, for typical values of the time scales, γL 2 /κ β can vary from O(1) to γL 2 /κ β 1. Therefore, we choose t c1 as a time scale noting that choosing t c2 is equally plausible. In the literature, γL 2 /κ β is also treated as an elasto-capillary length [69,70], to characterize the length scales over which surface tension or elasticity dominate. The dimensionless kinematic boundary condition Eq. (3) becomes where p z = cos θ, and the right-hand side of Eq. (8) applies at the location of the deformed interface u z . In principle, there is a third time scale in the problem if we account for the intrinsic reorientation time scale of a microswimmer. For instance, t c3 = τ or t c3 = D −1 r , where τ is the mean run duration of a run-and-tumble particle (RTP) [32,49,50,[71][72][73] and D r the rotary diffusivity coefficient of an active Brownian particle (ABP) [31,37,71,73,74]. For simplicity, we neglect this third scale and treat only straight-swimmers, that is, the class of swimmers that do not tend to intrinsically reorient. In Sec. V, we briefly discuss the relevance of the swimmer reorientation time and how it compares to the response scale of the interface deformation.
For small deformations, one can use the method of domain perturbation [41,64] to represent this term at the location of the planar undeformed interface z 0 instead. Namely, v z | uz , can be perturbed about r z0 as: v z | uz ≈ v z | rz 0 + u z (∂v z /∂r z )| rz 0 + O(u z ) 2 + . . . , and therefore considering terms up to O(u z ) in the velocity, one may rewrite Eq. (8) as: Note that the second and third nonlinear terms on the left-hand side of Eq. (9) involve products of the field variable (u z , v) and their associated gradients. These terms are usually neglected if the forcing is sufficiently far away from the interface so that terms involving their products are asymptotically small in the far-field [42,44]. Given that we consider swimmers at O(1) distances from the interface, these terms can not be neglected a-priori. However, as noted below Eq. 6, the disturbance field v is O(ln κ) −1 . Therefore, if we express the interface deformation as a perturbation series in (ln κ) −1 , u z = u z1 × (ln κ) −1 + u z2 × (ln κ) −2 + . . ., the second and third terms on the left-hand side of Eq. (9) will be O(ln κ) −2 , and hence asymptotically smaller than the disturbance field v. A similar argument, owing to the asymptotically weak flow field of a slender swimmer, has recently been used to treat the tracer diffusivity and the velocity variance in a suspension of interacting slender swimmers in bulk [52]. Thus, to a leading order in (ln κ) −1 , one can simplify Eq. (9) as: for a slender-body near a deformable interface. We can solve the above system of linear equations (6a)-(6d) and the boundary conditions (7a)-(7e) in Fourier space [42,44,58], and we define the two-dimensional Fourier transform of any variable A(r x , r y ) as: . Determining the Fourier transformed disturbance fieldv z is straightforward, as shown in Appendix A, B for swimmers oriented parallel and perpendicular to the interface, respectively. However, here we show the Fourier transformed kinematic boundary condition Eq. (10): which is solved in the next section for different swimmer configurations relative to the interface. For swimmers translating parallel to the interface p z = 0, in which case the right-hand side of Eq. (11) is only the Fourier transformed disturbance velocity at the undeformed interface. This approximation holds as long as the ratio of the viscous to bending stress: η 1 V s L 2 /κ β remains O(1). However, if η 1 V s L 2 /κ β 1, the nonlinear boundary condition given by Eq. (9) would have to be used. In Appendix C, we validate the above approximation for the interface deformation by comparing u z derived from solving Eq. (9) with that from Eq. (10), for the specific case of swimmers oriented orthogonal to the interface over a range of η 1 V s L 2 /κ β . In the following sections, we fix the aspect ratio κ = 10 and the ratio of the viscous to bending stresses η 1 V s L 2 /κ β = 1.

III. SWIMMERS NEAR A SINGLE INTERFACE
In general, simultaneously solving Eq. (5) and Eq. (11) results in an inherently unsteady problem, with the coupling involving a dependence on both the distance of the swimmer to the interface as well as its orientation. Therefore, in what follows, we first discuss the results for a swimmer oriented parallel to the interface, then for a swimmer orthogonal to the interface, and finally for a swimmer with an arbitrary orientation. In Sec. IV we consider a swimmer confined between a rigid boundary and a deformable interface.

A. Microswimmers swimming parallel to the interface
To understand the coupled hydrodynamics, it is useful to refer to the equation for the interface deformation and for the swimmer motion for swimmers oriented parallel to the interface. For the former, we use the expression for v 1z | rz 0 from Eq. (A15) in Appendix A 3 and obtain the following equation for the time evolution ofû z : where Γ ≡ γL 2 /κ β characterizes the relative importance of surface tension to bending stress. Note that the interface deformation is linked to the swimmers instantaneous configuration via r z0 and p, both of which can evolve in time.
Here,ṗ = 0, as is also true for a generic force-dipole swimmer near a planar or weakly deforming boundary [13,19,75]. However, the same is not true for V T , since the z-component of the translation is responsible for the coupling with the interface deformation in Eq. (12) via r z0 . We only focus on V T z , which is, where we have used the definition of the inverse Fourier transform: A(r x , r y ) = dkÂ exp(2πik · r ); k = (k x , k y ) and r = (r x , r y ). Thus, characterizing the coupled hydrodynamics due to a swimmer translating parallel to the interface reduces to simultaneously solving Eqs. (12) and (13) to obtainû z and r z0 , and then the disturbance flow field given by Eq. (A12) and Eq. (A15) in Appendix A 3. We now interpret the nature of the deformation influenced by the disturbance field generated from the swimmer. In Fig. 2a we plot the three-dimensional contour of the interface deformation due to a pusher in fluid region 1 for the parameters λ = 0.5 and Γ = 0.1. As expected, for pushers (or pullers) oriented parallel to the interface, the deformation is not radially symmetric [17]. The qualitative nature of the deformation remains the same for all values of λ (not shown), with the magnitude of the deformation decreasing as λ increases, as expected from Eq. (12), noting that for λ 1, u z ∝ 1/λ. spanning from r x ∈ [−1/2, 1/2] signifies the slender swimmer. The colorbar given on the right-hand side of (b) applies to both figures. The plot is for Γ = 0.1, λ = 0.5 at t ≈ 2, κ = 10 and η 1 V s L 2 /κ β = 1. Legends under • refer to u z obtained from solving the pair of equations (12), (13), and those under from solving the approximate equations (15), (16). The (black) dotted line in the insets is the far-field approximation from Eq. (14).
To characterize the interface response with Γ, the deformation is plotted as a function of the radial distance parallel and orthogonal to the swimmer axis in Fig. 3 for λ = 0.5 and Γ = 0.1, 1, 5. It is clear that increasing the importance of surface tension, decreases the magnitude of the deformation. In the far-field, however, at any finite time, the deformation is independent of Γ and scales as O(r ) −3 , as shown in the insets of Fig. 3. In fact, for any small but finite time, one can obtain an analytical expression for the interface deformation assuming r z0 to be nearly constant; this constraint of a constant r z0 is reasonable up to an O(1) change in time, as will be seen below while interpreting the swimmer translation. The resulting expression for u z is: highlighting independence from Γ, and the O(r ) −3 scaling in the far-field. The approximate form of u z given by Eq. (14) is shown in the insets of Fig. 3, and it agrees well with the numerically determined values. Now, the interface deformation is also a function of the distance of a swimmer from the interface: At distances of order the swimmer size, one needs to numerically solve Eqs. (12) and (13) simultaneously, whereas simplifications can be made when the swimmer is farther away. We rescale the wavevector in the interface deformation, Eq. (12), as k ≡ kr z0 and time ast = t/r z0 , which is valid in the regimer z0 ≡ |r z0 (0)| 1. Seeking simplifications appropriate fork,t ∼ O(1) [76] yields the following reduced equation for the interface deformation: The simplified expression for the vertical component of the swimmer translation is Eq. (15) highlights two important qualitative changes in the interface deformation for swimmers that are far from the interface. First, the interface deformation is controlled by surface tension alone, as the term proportional to bending is O(r z0 ) −2 smaller than Γ on the left-hand side of Eq. (12), and hence is absent from the leading order approximation. Second, the above rescalings of k and t recover the original dependence of the rescaled variables on r z0 , as in Eq. (12). This implies that, for r z0 1, the time evolution slows down in proportion to r z0 . In other words, swimmers at an O(1) distance from the interface influence an O(1) growth of the interface deformation at O(1) times. However, a swimmer that is farther away (r z0 1) will influence an O(1) growth of the interface deformation only when t ∼ O(r z0 ). In the insets of Fig. 3 we plot the interface deformation determined from simultaneously solving Eqs. (15) and (16). The agreement of both equations is excellent for Γ = 5 for r ∼ O(1), but the deviation of the approximate solution is large for Γ = 0.1. This is expected as Eq. (15) relies on Γ being larger than the bending term, and hence, forr z0 = 1 it relies solely on the largeness of Γ. At large r , the agreement is good for the same reason underlying Eqs. (15) and (16), but now for r 1 instead of r z0 1. For larger z0 (not shown), the approximations in Eqs. (15) and (16) agree well even for Γ < O(1). We note that the far-field approximation corresponds to modeling the swimmer as a point force-dipole, and the disagreement at small to moderate values of Γ, particularly for r z0 ∼ O(1) is indicative of the dipolar swimmer overestimating the interface deformation.
A result arising from the neglect of the bending term in Eq. (15) is a subtle qualitative change in the deformation behavior when bending dominates (Γ 1) versus when surface tension dominates (Γ 1). In particular, following a zero-crossing at an O(1) distance away from the swimmer, when Γ 1, the deformation monotonically transitions to an O r −3 far-field character, whereas for Γ up to O(1), the deformation exhibits an intermediate scaling before transitioning to the far-field decay. As shown in the insets of Fig. 3, there is a characteristic maxima in |u z | for Γ = 0.1 (see the red curve corresponding to the solution of Eqs. (12) and (13)). These features are absent in the approximate Eqs. (15) and (16), and are most evident in Fig. 3a (see orange and dark green curves in the inset). This qualitative difference emerges since the bending mode, which dominates in this regime, depends on higher order derivatives in the curvature as specified by the stress boundary condition, Eq. (7e). In Fig. 4, we plot the vertical translation velocity V T z of a pusher oriented parallel to the interface for three specific viscosity ratios λ = 0.5, 1 and 1.5. We note two interesting features of the vertical migration. First, as shown in Fig. 4a, when λ < 1, V T z is positive for short times and negative for long times. That is, a pusher is repelled from the interface at short times, while at long times it is eventually attracted to the interface. Second, the crossover time from repulsion to attraction is shorter for swimmers closer to the interface. For r z0 ∼ O(1), the swimmer undergoes a change in its vertical translational velocity at a t transition < O(1), whereas, when r z0 1, t transition ∼ O(r z0 ); here, t transition is the time taken for V T z to vanish. For instance, when r z0 ∈ [1, 2], t transition < 0.5 and when r z0 ≥ 4, t transition > 1. This is unlike the case of λ ≥ 1 as shown in Fig. 4b and c, where the pusher is monotonically attracted to the interface. Pullers (not shown) behave in the exact opposite manner, that is, for λ < 1 they are attracted to the interface at short times and repelled at long times, with the crossover time for attraction to repulsion having the same character; for λ ≥ 1 pullers are monotonically repelled from the interface.
In Fig. 5, we plot the relative swimmer trajectories of pushers as a function of time for swimmers starting from different initial locations r z0 (0), for λ = 0.5, 1 and 1.5 as well. Owing to the repulsion (V T z > 0) at short times when λ < 1 as highlighted in Fig. 4a, the relative swimmer trajectory: r z0 (t) − r z0 (0) ≥ 0 in Fig. 5a. Moreover, as expected from the discussion surrounding Fig. 4, the time it takes for the swimmer to return to its initial location depends sensitively upon its initial distance from the interface. For instance, when r z0 (0) = 1, the return time is about t ∼ 0.25, whereas, when r z0 (0) = 5, return time is t ∼ 3.7. Such a non-monotonic vertical swimmer translation for λ < 1 stands in contrast to what has been observed for swimmers near rigid boundaries [4,13,30,32,50], where the pushers are, hydrodynamically, only attracted to the interface. Even for model swimmers near deformable interfaces, the swimmer migration has been reported to behave monotonically for a given λ [14,17]. For instance, Dias and Powers [14] noted that the average pumping velocity between a swimming sheet and the interface was of negative for λ < 1 and positive for λ > 1. Shaik and Ardekani [17] calculated the vertical translational velocity of a spherical squirmer over a range of λ, and did not find a qualitative change in the swimmer migration across λ = 1 (see Figs. 2 and 8 therein). The first of the above observations regarding the non-monotonic swimmer translational for λ < 1 can be explained from a closer inspection of V T z given by Eq. (13). The translational velocity has two principal contributions: (1) the instantaneous contribution from Stokes flow that is proportional to (1 − λ)/(1 + λ), (2) the time dependent term associated with the interface deformationû z . In fact, at t = 0, V T z (red circles in Fig. 4) is precisely the instantaneous contribution before the interface deforms. The time dependent term remains negative at all times t > 0, and its magnitude gradually increases as the interface deformation grows. However, the instantaneous term changes sign depending on whether λ is greater than or less than unity, although its magnitude remains nearly constant. For pushers, when λ < 1 (> 1), the instantaneous term is positive (negative), and it vanishes when λ = 1. Its magnitude remains nearly constant because it changes only via the hydrodynamic interaction-induced change in the swimmer vertical motion r z0 (t), which does not change appreciably over O(1) times implying that ∆r z0 (t) ≡ r z0 (t) − r z0 (0) r z0 (0). Hence, among the two fluid regions, if a pusher is present in the more viscous fluid (λ < 1), it starts off being repelled at short times due to the dominance of the repulsive instantaneous term. At t = t transition , the growing time dependent deformation term exactly matches the nearly constant instantaneous Stokes term, whence V T z vanishes, and this is characterized by the peak in the relative swimmer trajectory in Fig. 5 (left panel). For t > t transition the swimmer experiences an attraction to the interface owing to the dominance of the time dependent term. This explains the repulsion of pushers from the interface at short times and attraction at long times only when it is in the more viscous fluid.
To explain the second observation of the time dependence of the swimmer translation as a function of its distance from the interface, we note that the time dependence in the problem originates from the kinematic boundary condition of the interface deformation given by Eq. (12). The swimmer translation velocity V T z (and thus, r z0 ) obeys the same time dependence as u z due to the linear relationship between the two field variables in Eq. (13), independent of λ. Therefore, such a delay in the time to return back to its original position when λ < 1 is expected following the arguments explained in the context of the interface deformation. That is, when r z0 ∼ O(1), the interface deformation, and hence, V T z evolves on O(1) time units. However, when r z0 1, the interface deformation evolves on a time scale of O(r z0 ). Therefore, as shown in Fig. 5 for λ = 0.5, the farther the pusher (puller) is from the interface, the longer it will sense a repulsion (an attraction), albeit with reduced intensity.
To show the relative importance of surface tension to bending stress on the swimmer motion, in Fig. 6a we plot the relative swimmer translation as a function of time for different values of Γ(≡ γL 2 /κ β ). Clearly, increasing the surface tension results in a more rapid translation, with a dramatic reduction in the time spent in the repulsive state. For instance, as Γ is increased from 0.1 to 10, the return time decreases from t ≈ 0.28 to t ≈ 0.06. Moreover, as evident from the inset of Fig. 6a, the swimmer translation velocity approaches a quasi-steady state when Γ 1. Note the emphasis on 'quasi-steady state', as the problem is inherently an unsteady one, even though V T z appears to be almost steady for Γ = 10 following the initial transient. This is because as the swimmer is moving closer to the interface, it will continue to accelerate, given that the interface undergoes a stronger deformation from swimmers closer to it. In turn, the translation velocity will continue to grow. Nevertheless, this acceleration remains small after sustaining an initial growth in the deformation, as the driving force is still small (see the scales along the ordinate of Fig. 6a).
This analysis shows that the swimmer translation must be analyzed in terms of both distance to the interface and interface properties. That is, both r z0 and Γ have comparable effects on the swimmer translation when these quantities are large. Note, however, that owing to the exponential damping of u z and V T z with respect to r z0 even an O(1) change in the distance of the swimmer to the interface results in a large reduction of the swimmer translational velocity. Therefore, a swimmer near an interface with relatively low surface tension can exhibit similar translational dynamics to that of a swimmer that is slightly farther away from the interface with higher surface tension. An example of such a trajectory is highlighted in Fig. 6b, where the initial locations of the swimmers are displaced by 0.5 units of r z0 , but Γ is more than doubled. For an O(1) change in time, the trajectories in two scenario are similar.

B. Microswimmers swimming orthogonal to the interface
We now consider swimmers oriented orthogonal to the interface (p z = ±1 and p x = p y = 0), since the results are easier to interpret relative to swimmers with arbitrary orientation. Here too, the rate of rotationṗ = 0, and therefore characterizing the swimmer motion requires simultaneously solving for the interface deformation and the vertical translation.
The equation governing the interface deformation is obtained using the expression forv 1z | rz 0 from Eq. (B7) (Appendix B) in Eq. (11), and is: The expression for the vertical component of the swimmer translation velocity V T z is: Here, the interface deformation and the swimmer translation exhibit a radial symmetry in the plane of the interface, which enables a convenient analytical formulation of the instantaneous part of V T z as given in Eq. (18). Such a radial symmetry is specific to swimmers orthogonal to the interface, and is a consequence of the linearity of the problem. Note that we have not incorporated the self-swimming term in V T z , which would involve addition of a constant factor p z to Eq. 18. Therefore, the discussion here pertains to shakers -swimmers that disturb the fluid with a flow field V s / ln κ but do not self-propel [77].
We now analyze the interface deformation created by the swimming motion. In Fig. 7, we plot the interface deformation due to a pusher as a function of the radial distance along the plane of the interface, r , for Γ = 0.1, 1 and 10. As defined previously, r = 0 is the swimmer location on the interface. For large r the deformation u z is independent of Γ, similar to that of swimmers parallel to the interface (see the collapse in the inset of Fig. 7 for r 1). This far-field scaling is O r −3 , and can be identified by solving Eq. (17) analytically, applying the approach used to obtain Eq. (14) in Sec. III A, yielding: The above approximation also holds for r z0 1 at r = O(1), and is plotted in the inset of Fig. 7b. However, the interface deformation for this swimmer configuration exhibits an opposite trend to that of swimmers oriented parallel to the interface; namely, for a pusher the deformation occurs away from the swimmer (into fluid region 2) at short separations, whereas, its positive for r 1, as also noted previously in Lee et al. [10] for point force-dipoles and Shaik and Ardekani [17] for spherical squirmers. This is expected as the disturbance field generated by a pusher exits along the ends of its long-side and is thereby pushing onto the interface when it is oriented orthogonal to it. Given that the interface deformation also contributes to the evolution of the vertical translation, this opposing trend will have important consequences for the nature of the migration of swimmers oriented orthogonal to the interface, as will be discussed below.  (20), (21). The 'Far-field' in the inset of (b) refers to u z from Eq. (19) for Γ = 1.
The radial symmetry in the plane of the interface enables simplifications to both u z and V T z for swimmers positioned far away from the interface (r z0 O(1)). Following the discussion in Sec. III A for swimmers oriented parallel to the interface, we rescale the wavevectork = kr z0 and timet = t/r z0 in Eq. 17 to find with the corresponding expression for vertical component of the translation velocity becoming In the insets of Fig. 7, we plot |u z | using both the full equations (17) and (18), and the approximate equations (20) and (21). Similar to the discussion in Sec. III A, for r z0 (0) = 1 in Fig. 7a, the agreement of both pair of equations is excellent for Γ = 10, but the deviation of the approximate solution is large for Γ = 1 and 0.1. For r z0 = 5 however, this holds for Γ < O(1), as seen in the inset of Fig. 7b. In Fig. 8, we plot the time trace of the vertical component of the swimmer translation velocity V T z of a pusher at different initial swimmer distances from the interface r z0 (0), for λ = 0.5, 1, and 1.5. Similar to swimmers parallel to the interface, for λ < 1, both repulsive and attractive regimes are exhibited by V T z as a function of time. Here, the deformation term is repulsive owing to the opposite character of the interface deformation, as discussed below Eq. (18). The instantaneous term in V T z is attractive (repulsive) when λ < 1 (λ > 1). Therefore, when λ < 1, for short times, V T z is attractive (negative), and is repulsive (positive) at long times. In turn, the relative swimmer trajectory r z0 (t) − r z0 (0) shown in Fig. 9 is the qualitative mirror image about r z0 (t) − r z0 (0) = 0, to that of a swimmer parallel to the interface in Fig. 5.
The transition time from attraction to repulsion, t transition , exhibits a character similar to that of swimmers parallel to the interface, and is expected from the similar scaling's of Eq. (15) and Eq. (20). Namely, for r z0 ∼ O(1), t transition ∼ O(1), whereas for large r z0 , t transition ∼ O(r z0 ). However, the magnitude of V T z is larger for swimmers oriented orthogonal to the interface compared to swimmers oriented parallel to the interface. At r z0 (0) = 1, for a swimmer parallel to the interface V T z changes from +0.5 × 10 −3 to about −10 −3 in a time interval t ∈ [0, 4], as can be seen in Fig. 4a (see the dotted trace of the first vertical line). In contrast, for swimmers oriented orthogonal to the interface, V T z transitions from about −10 −3 to about +2.5 × 10 −3 in the same time interval, as shown in Fig. 8a. Since the velocity at t = 0, is that prescribed by the instantaneous component, and that at large times is dominated by the deformation, such character shows that both terms of V T z are larger for swimmers oriented orthogonal to the interface. Therefore, in a given time interval, the relative swimmer vertical translation of an orthogonally oriented swimmer as shown in Fig. 9 is larger than that of a swimmer parallel to the interface (see Fig. 5). In other words, the coupled hydrodynamics more strongly repels a pusher oriented orthogonal to the interface than it attracts a pusher oriented parallel to the interface.
The relative importance of surface tension to bending stress on the swimmer motion (not shown), is similar to that of swimmers parallel to the interface, as discussed in Sec. III A and as is evident in Eqs. (17) and (18). Increasing  the surface tension results in a more rapid translation, and for λ < 1 this implies a reduction in the time spent being attracted to the interface. Thus, this monotonic response in Γ implies that swimmers farther away from an interface with larger Γ can exhibit migration similar to those closer to an interface with lower Γ. This is qualitatively similar in character to Fig. 6b, but again of an opposite trend.

C. Microswimmers swimming arbitrarily oriented to the interface
We now generalize the coupled hydrodynamics to arbitrary swimmer orientations. As explained in Appendix B, the derivation procedure for any non-parallel swimmer orientation remains the same, and the changes to the fluid equations due to a non-trivial in-plane orientation component are discussed in Appendix D. Unlike the previous cases, here the vertical component of the rotation rate is non-zero;ṗ z = 0. For brevity we omit the large equations forû z , V T z andṗ z , the girth of latter two associated with the contributions from all three orientation components. In the following, we only focus on the role of the coupled hydrodynamics on the swimmer vertical translation and rotation. We study how the change in the swimmer orientation affects the nature of the swimmer migration, and the final swimmer orientation given an arbitrary initial orientation. As in Sec. III B, here too we do not account for the self-swimming term in V T z , which is instantaneously shifted by constant factor p z . As in the case of swimmer translation, the rotation rate has two parts; one from the time dependent interface deformation and the other from the instantaneous Stokes flow field. However, because the time dependent interface deformation term inṗ z is smaller than the instantaneous term, the rotation rate is dominated by the latter. This quasi-steady behavior allows us to construct a phase portrait in orientation space, as shown in Fig. 10a [78]. For pushers, only trajectories in the first and third quadrants are admissible and hence any initial orientation p z = 0 approaches p z = ±1 depending on whether its directed away from (p z → 1 or θ → 0) or towards the interface (p z → −1 or θ → 180 • ). Now, from the analysis of swimmers parallel to the interface in Sec. III A, we know thaṫ p z = 0, for p z = 0. Therefore, p z = 0 is an unstable fixed point for pusher, and hence a stable fixed point for pullers. Similar rotational preferences have been reported by Shaik and Ardekani [17] for spherical squirmers near a deformable interface.
In Fig. 10b the time dependence of the vertical component of the swimmer translation velocity V T z is shown for pushers starting from different initial orientations θ(0). When a swimmer is nearly aligned with the interface (θ(0) = 89.5 • ), V T z decreases until t ∼ O(1), owing to the dominant contribution from the parallel configuration, as anticipated from Sec. III A. On the other hand, V T z grows at large times, owing to the dominant contribution of the orthogonal configuration as θ(t) → 0. Therefore, the translation velocity V T z of swimmers nearly aligned with the interface goes through a minimum, which is negative for nearly parallel swimmers (see curves of θ(0) = 85 • and 89.5 • ) and positive for those that start at a smaller initial orientation, say θ(0) 75 • and 45 • . At θ(0) 45 • , V T z only increases in time with a θ(0) dependent t = 0 intercept. Therefore, the early time trajectories are qualitatively and quantitatively dependent upon θ(0) but all of the curves collapse at late times.
Owing to the sensitive dependence of V T z (t) on θ(0) at short times, the swimmer migration has a rich set of trajectories up to times of O(1). In Fig. 11a, we plot the relative vertical swimmer trajectories r z0 (t) − r z0 (0) for the same values of θ(0). The inset shows short time oscillations for θ(0) = 85 • and 89.5 • , the latter of which briefly becomes weakly attractive, whereas the curvature gradually changes as θ(0) decreases. At long times the slopes asymptote for all θ(0) and the overall displacement increases as θ(0) decreases.
In Fig. 11b, we plot the relative vertical swimmer trajectory r z0 (t) − r z0 (0), for three values of the viscosity ratio λ = 0.5, 1 and 1.5 for a pusher that starts off initially almost parallel to the interface (θ = 85 • ). Although the three  trajectories are qualitatively similar, here a swimmer exhibits regimes of attraction and repulsion for λ ≥ 1. The underlying principle for such a response can be explained as follows. For swimmers that start off nearly aligned with the interface, at short times V T z is positive for λ < 1, zero for λ = 1 (no contribution from instantaneous component), and negative for λ > 1. As time evolves the swimmer rotates to an orthogonal configuration and V T z increases, independent of λ. In turn, V T z (t = 0) of a pusher nearly aligned to the interface positive, zero or negative if λ < 1, λ = 1 or λ ≥ 1 respectively. Therefore, for t < O(1), a swimmer will always have an attractive component for λ ≥ 1, and r z0 (t) − r z0 (0) < 0.
We now understand that hydrodynamic interactions with a deformable interface lead to pushers reorienting into an orthogonal configuration, pullers reorient into a parallel configuration. Therefore, unlike their response to a rigid boundary, both swimmer types seek a configuration that drives them away from the interface. As shown in Fig. 12, pushers and pullers that start at the same location will eventually segregate spatially, independent of their initial orientation but to a degree that increases as θ(0) decreases. Note that p = 0 is a stable configuration for pushers only the absence of an imposed flow, which may be present in many microswimmer settings, as we discuss in Sec. V.

IV. MICROSWIMMERS CONFINED BETWEEN A RIGID BOUNDARY AND AN INTERFACE -SWIMMING PARALLEL TO BOUNDARIES
Here, we generalize our approach to treat the coupled hydrodynamics of a swimmer confined between a rigid boundary and an underlying deformable interface (Fig. 13). We note that in this configuration the rotation rateṗ = 0, similar to a swimmer oriented parallel to a single deformable interface (see Sec. III A), or to a single rigid boundary [4]. Therefore, the hydrodynamics is characterized by the interface deformation u z and the vertical component of the translation velocity V T z . It is convenient to choose a reference frame moving with the swimmer but shifted to the plane of the undeformed interface, and hence the definition of r z0 used here is the opposite of that used for a swimmer near a single interface in Sec. III (see the direction of arrows in Figs. 1 and 13).
The equation for the Fourier transformed interface deformationû z is and the expression for V T z is where, E in both Eqs. (22) and (23) is We note that the additional H-dependent terms in Eq. (22) for the interface deformation are at least exponentially smaller by a factor exp(−2πk(2H − r z0 ). Unless H → r z0 (extreme confinement), the contributions from these terms remains small, apart from there being an H-dependent amplitude, as the terms with the smaller exponents are proportional to 1 − λ. Hence, varying λ about unity does not yield a qualitatively different deformation than that of a swimmer parallel a single interface. Therefore, in the following we focus only on the change in the swimmer translation effected by the coupled hydrodynamics between the swimmer and the boundaries. In Fig. 14a we plot the relative swimmer trajectory r z0 (t) − r z0 (0) as a function of the distance between the boundaries H, for r z0 (0) = 1 and λ = 0.5. In the limit H 1, we recover the trajectory corresponding to that of a swimmer in the vicinity of a single interface as discussed in Sec. III A. Upon increased confinement by reducing H and increasing confinement, the residence time of the swimmer in the repulsive state increases. When the swimmer begins at the midpoint, it always moves towards the rigid boundary. This is also shown in the inset of Fig. 14a, where for H = 2, V T z does not undergo a zero-crossing in finite time. Physically, a rigid boundary provides an infinite resistance to both bending and shear, and thus facilitates a stronger attraction relative to an interface that only supports finite bending [19]. In other words, the flow field induced by the rigid surface dominates that of the deformable interface.
In Fig. 14b we plot the relative swimmer trajectory r z0 (t) − r z0 (0) for three different viscosity ratios λ = 0.5, 1, 1.5, with the distance between the interface and the rigid boundary fixed to H = 2.3. Interestingly, unlike a swimmer in the vicinity of a standalone interface, the swimmer experiences a transient attraction and repulsion for all values of λ. In Sec. III A we explained that for λ < 1 the swimmer translation had both repulsive and attractive component FIG. 13: A sectional schematic representation of a slender swimmer translating with speed V s along its director vector p in a channel with a deformable interface below and a rigid boundary on top. The distance between the plane of the undeformed interface and the rigid boundary is H. The circles along the axial length of the fore-aft symmetric swimmer represent a line distribution of stokeslets characterizing the head and tail for pushers. The disturbance flow field generated by the swimming motion deforms the interface, and u z is the interface deformation (solid gray line) relative to its initially flat undeformed (dotted line) state z 0 . 3. The insets contains the corresponding vertical component of the translation velocity V T z ≡ dr z0 /dt. The "repulsion" and "attraction" are to be interpreted relative to the interface. In both figures, the initial swimmer distance to the interface is |r z0 | = 1 and the ratio of surface tension to bending stress is Γ = 1. The attraction and repulsion are specified relative to the deformable interface.
due to the difference in the instantaneous contribution, dominant at short times, and the interface deformation dependent contributions of V T z , dominant at long times (see Eq. (13)). Here, however, the rigid boundary creates repulsion, through an H-dependent positive shift of the instantaneous component of V T z , the largest contribution being proportional to (1 + λ) 2 , and thus positive for any λ. As the confinement increases, the contribution of this term increases, and eventually for H 2r z0 , the swimmer only moves towards the rigid boundary. Although not shown, the dependence on the ratio of surface tension to bending stress, Γ, remains the same as that for a swimmer near a single deformable interface, and seen in Eqs. (22) and (23).

V. SUMMARY AND CONCLUSION
We have investigated the coupled hydrodynamics between finite sized orientable swimmers and a deformable interface. By treating the swimmers as slender bodies, we have gone beyond a far-field picture, solving the hydrodynamics appropriate at O(1) swimmer lengths from the interface. Moreover, we predict the hydrodynamics for a robust range of the ratio of the viscous stress to bending stress, η 1 V s L 2 /κ β , and the capillary number, η 1 V s /γ. Our analysis reveals that the swimmer orientation plays a crucial role in the response of both the swimmer and deformable interface. Because parallel and perpendicular orientations exhibit differing overall trajectories, when swimmers take an arbitrary orientations we find a rich dynamical behavior. Importantly, given that an arbitrarily oriented pusher (puller) preferentially rotates to a perpendicular (parallel) orientation, the migration pattern for both swimmer types remains the same. However, given the difference in the migration speed in the two orientations, a pusher and puller that start from the same location with any orientation will spatially segregate.
When confined to swimming between a rigid and a deformable boundary, pushers closer to the latter interface experience repulsion that is extended to time scales longer than the elastic response time. Such confinement, between rigid and soft boundaries, provide new controls on swimmer migration. For example, the different rotational preferences near deformable versus rigid boundaries provides one such confinement-dependent control of swimmers.
In general, the dynamics of swimmer translation is extremely sensitive to the distance from the interface. Even an O(1) increase in this distance results in a large slowdown in the migration towards or away from the interface. The interfacial properties play a crucial role in migration, second only to the distance from the interface. Therefore, it is possible that two scenarios in which swimmers have different distances from the interface display similar migration patterns. Owing to the strong sensitivity of swimmer migration to distance from the interface, even a small change in the latter would require an O(1) change in the interface properties to obtain a nearly matching migration pattern. This is particularly relevant from a practical standpoint, because interfacial properties are not necessarily spatially homogeneous. In consequence, swimmers in a suspension at different distances from the interface can, in principle, redistribute in a spatially similar manner.
We find that the viscosity ratio plays a controlling role in whether a swimmer is attracted or repelled from the interface. Indeed, viscosity variations may be dictate the migration of planktonic biota in marine ecology, by modifying both swimming and the nutrient dispersion [79][80][81]. On the other hand, whereas interfacial deformation is clearly controlled by the relative importance of bending elasticity and surface tension, it also depends on the swimmer orientation and distance from the interface. However, independent of the swimmer orientation, the interfacial deformation in the interfacial plane extends well beyond distances of order the swimmer size L. Moreover, the far-field scaling of the deformation in the interfacial plane remains O r −3 for swimmers oriented parallel or perpendicular to the interface.
Even in the dilute limit, such far-field scaling can have important consequences in analyzing the deformation due to a suspension of microswimmers near deformable interfaces. This is because inclusion of the collective contribution from the swimmers upon the interface involves an integral over the domain volume, which decays as O r −3 , implying the potential for long-ranged radial contributions.
It is important to note that time scales short relative to the elastic response time, t c1 = η 1 L 3 /κ β , are still usually very long compared to the time scales of swimming. For typical soft interfaces, the bending modulus κ β ≈ 10 −19 J [55,57], so if one considers a viscosity approximately that of water, η 1 ≈ 10 −3 N s/m 2 , and swimmers of length L ≈ 10 µm (cell body+flagellar bundle combined), then t c1 ≈ 10 s. In general, microscopic swimmers also respond to external cues, such as an externally imposed flow [26-28, 31, 34, 37, 72, 82] or some form of chemical actuation [83][84][85][86][87][88][89][90], that can influence their migration on such long time scales. Typically, even in the absence of such an external driving force, the reorientations take seconds, as most microscopic swimmers have an inherent mechanism allowing them to change their orientation [49,71,72,91]. We note that when λ < 1, a swimmer aligned with the interface might only sense either a repulsion (pusher) or an attraction (puller), before reorienting itself. Therefore, the nature of the short time dynamics shown in Fig. 5 implies the possibility of pushers migrating away from the interface on average. A similar argument can be made for other swimmer orientations.
Although we have considered the motion of a single swimmer near a deformable interface and confined between a deformable interface and a rigid boundary, one can readily extend this study to a dilute suspension of non-interacting swimmers. This requires a more careful consideration of the single-swimmer statistics developed here, as the expressions for the instantaneous field variables depend on the swimmer configuration. Moreover, given the complex dynamics involved, deriving a position-orientation space swimmer distribution function is challenging. Finally, a host of new dynamical processes will be revealed when extending this approach beyond the linear regime.
Now, the right-hand side of Eq. (A3b) is proportional to δ(r z ), and hence, we seek the Greens function of the differential equation. From Appendix A 2, we have two boundary conditions forv αt , namely, the no-slip condition given by Eq . (A6b) and the continuity of tangential stress given by Eq. (A7b), and four unknown constants A 1 -A 4 to determine. Therefore, we seek two conditions to uniquely determine the constants. To extract additional boundary conditions, we use the continuity properties of the Greens function; (a) about the location of forcing (r z = 0), and (b) the first derivative with respect to r z exhibiting a finite-jump discontinuity about r z = 0 [42,92]. These additional conditions are:v Using Eqs. (A6)b, (A7b) and (A11), the four constants are uniquely determined, and we obtain the following expressions for the transverse velocity components in the two fluid region: Next, we consider the normal velocity components in Fourier spacev αz . A general solution for the 4 th -order differential equations Eqs. (A4) and (A5a) can be written as:v αz = (B + Cr z ) exp(±2πkr z ) [42,44,92]. Again, noting that the disturbance flow field must decay in the far-field, we write the solutions as: v 1z = (B 1 + B 2 r z ) exp(−2πkr z ); r z > 0 (B 3 + B 4 r z ) exp(2πkr z ) + (B 5 + B 6 r z ) exp(−2πkr z ); r z0 < r z < 0, and (A13a) v 2z = (B 7 + B 8 r z ) exp(2πkr z ); r z < r z0 .
In Appendix A 2, we have four boundary conditions for the normal velocity components, namely, the impenetrability of the velocity across the interface Eq. (A6a), the no-slip velocity at the interface Eq. (A6c), the continuity of tangential stress Eq. (A7a) and the normal stress jump Eq. (A9). Therefore, we require four relations to uniquely determine the eight unknown constants B 1 − B 8 , and we follow the protocol outlined above forv αt , using the Greens function properties. In this case, however, the forcing on the right-hand side of Eq. (A4) is proportional to δ (r z ). Therefore, the additional boundary conditions are the continuity ofv 1z , its first and third derivative with respect r z at r z = 0, whereas the second derivative undergoes a jump discontinuity [42,44,92]. Note that if the third derivative exhibited a finite jump discontinuity, then a δ(r z ) would appear on the right-hand side of Eq. (A4). The four additional boundary conditions arev allowingv αz to be uniquely determined as: ; r z > r z0 , and (A15a) ; r z < r z0 .
We now use the continuity equations (A3d) and (A5c) to determine the longitudinal velocity componentsv αl . Note that the longitudinal component of the fluid velocity field is independent of the interface deformation at r z = r z0 .
where only the terms involving F 1 in Eq. (B5) are to be integrated over. This yieldŝ Again, usingv αz from Eq. (B7) in the continuity equations (A3d) and (A5c) readily yield the longitudinal velocity componentsv αl .
the gradient of the normal velocity components to get the following expressions for v x , v y , v z and ∂v z /∂r z at r z0 : The radial symmetry of v x and v y allows us to solve Eq. (C1) in cylindrical polar coordinates (r ,θ,ẑ). Because v θ = 0, there is a dimensional reduction in the polar coordinate system; r 2 = r 2 x + r 2 y and θ = arctan(r y /r x ). Thus, we solve: where v r = r x v x /r + r y v y /r. Exploiting radial symmetry, we calculate ∂u z /∂r as an inverse Hankel transform ofû z with the Bessel function J 1 (2πkr ). Here, we refer to the interface deformation obtained from solving the approximate kinematic condition Eq. (17) as u z−approx .
In the discussion preceeding Eq. (10), we argued that the approximation to the kinematic boundary condition will hold when the ratio of the viscous stress to the bending stress η 1 V s L 2 /κ β was O(1) or smaller. As this ratio becomes larger, one would expect the largeness in η 1 V s L 2 /κ β to dominate over the weak perturbation of the inverse logarithm of the slender swimmer aspect ratio κ, and hence necessitate the inclusion of the neglected nonlinear terms. In Fig. 15 we compare u z obtained from Eq. (C6) with u z−approx (both solved in along with Eq. (18) for V T z ), for η 1 V s L 2 /κ β = 1, 10 to assess the relative importance of the nonlinear terms on the left-hand side of Eq. (C6). We focus only on equations specific to shakers, and thus p z = 0 on the right-hand side of both Eqs. (17) and (C6). It is clear that even for an initial distance of the swimmer from the interface of r z0 (0) = 1, u z−approx provides a very good approximation to the full solution obtained from Eq. (C6). In the inset, a linear plot is shown for η 1 V s L 2 /κ β = 10, showing that the deviation between the two is small, with a maximum in the region of r = 0 having a relative error of about 0.24. Although not shown here, u z−approx approaches the u z obtained form Eq. (C6) as r z0 increases. This validates the boundary approximation for a slender swimmer even at O(1) distances from the interface and for values of η 1 V s L 2 /κ β ∼ O(10).

Appendix D: Swimmers arbitrarily oriented to the interface -Two-dimensional Fourier transform summary
To solve for a swimmer arbitrarily oriented to the interface, we follow the same approach to that of swimmers oriented orthogonal to the interface. Here, the equations forv z andv t are similar to that for swimmers orthogonal to the interface, as given by Eq. (B2) in Appendix B. For an arbitrarily oriented slender swimmer, we have the following Appendix E: Swimmers confined between a deformable interface and a rigid boundary -Two-dimensional Fourier transform summary Here, we briefly describe the formulation for a swimmer confined between a rigid boundary and an initially undeformed interface, specifically when its oriented parallel to the boundaries. We follow the development in Appendix A, and outline only the additional boundary conditions that emerge owing to the additional rigid boundary. For this swimmer orientation only the z-component of the velocity is relevant, given thatṗ = 0 as mentioned in Sec. IV. We therefore focus solely on the additional boundary conditions forv αz , where α ∈ [1, 2] for the two fluid regions. For a swimmer of arbitrary orientation, however, the rotation rate would be non-trivial and it would be essential to derive the transverse and longitudinal velocity components,v αt andv αl .
The equations forv αz are solved subject to the appropriate boundary conditions; the impenetrability and the no-slip conditions at the deformable interface and the rigid boundary, the continuity of tangential stress and the normal-stress jump across the deformable interface. The boundary conditions at the deformable interface are as in Appendix A 2, namely Eqs. (A6a), (A6c), (A7a) and (A9). The additional Fourier transformed velocity boundary conditions owing to the rigid boundary are:v A general solution for the 4 th -order differential equations Eqs. (A4) and (A5a), noting that the disturbance flow field must decay in the far-field, iŝ v 1z = (B 1 + B 2 r z ) exp(−2πkr z ) + (B 3 + B 4 r z ) exp(−2πkr z ); H > r z > r z0 (B 5 + B 6 r z ) exp(2πkr z ) + (B 7 + B 8 r z ) exp(−2πkr z ); 0 < r z < r z0 , and (E2a) v 2z = (B 9 + B 10 r z ) exp(2πkr z ); r z < 0.
We use the six boundary conditions mentioned above, in addition to the four conditions emerging from the properties of the Green's function, which remain the same as in Eq. (A14), to obtain the constants B 1 − B 10 . We omit for brevity the cumbersome expressions forv αz .