Mobility tensor of a sphere moving on a superhydrophobic wall: application to particle separation

Abstract

The paper addresses the hydrodynamic behavior of a sphere close to a micropatterned superhydrophobic surface described in terms of alternated no-slip and perfect-slip stripes. Physically, the perfect-slip stripes model the parallel grooves where a large gas cushion forms between fluid and solid wall, giving rise to slippage at the gas–liquid interface. The potential of the boundary element method in dealing with mixed no-slip/perfect-slip boundary conditions is exploited to systematically calculate the mobility tensor for different particle-to-wall relative positions and for different particle radii. The particle hydrodynamics is characterized by a nontrivial mobility field which presents a distinct near-wall behavior where the wall patterning directly affects the particle motion. In the far field, the effects of the wall pattern can be accurately represented via an effective description in terms of a homogeneous wall with a suitably defined apparent slippage. The trajectory of the sphere under the action of an external force is also described in some detail. A “resonant” regime is found when the frequency of the transversal component of the force matches a characteristic crossing frequency imposed by the wall pattern. It is found that under resonance, the particle undergoes a mean transversal drift. Since the resonance condition depends on the particle radius, the effect can in principle be used to conceive devices for particle sorting based on superhydrophobic surfaces.

Appendix: Boundary integral formulation

In this section, the application of the boundary element method (BEM) to the present geometrical configuration is shortly described (see, e.g., Pozrikidis (1992); Kim and Karrila (2005) for further details).

Due to linearity, the constant coefficient Stokes problem, Eq. (2), can be recast into a boundary integral representation formula

$$\begin{aligned} E({\boldsymbol \xi}) u_j({\boldsymbol \xi}) &= \frac{1}{8 \pi} \int\nolimits_{\partial B} t_i({\bf x}) G_{ij}({\bf x},{\boldsymbol \xi}) {\hbox{d}}S_{B} \\ &\quad-\frac{1}{8 \pi } \int\limits_{W_{NS}} t_i({\bf x}) G_{ij}({\bf x},{\boldsymbol \xi}) dS_{NS} \\ &\quad-\frac{1}{8 \pi } \int\limits_{ W_{PS}} t_i({\bf x}) G_{ij}({\bf x},{\boldsymbol \xi}) {\hbox{d}}S_{PS} \\ &\quad- \frac{1}{8\pi} \int\limits_{\partial B} u_i({\bf x}) {\mathcal T}_{ijk} ({\bf x},{\boldsymbol \xi}) n_k({\bf x}) {\hbox{d}}S_{B} \\ &\quad- \frac{1}{8\pi} \int\limits_{W_{NS}} u_i({\bf x}) {\mathcal T}_{ijk} ({\bf x},{\boldsymbol \xi}) n_k({\bf x}) {\hbox{d}}S_{NS} \\ &\quad- \frac{1}{8\pi} \int\limits_{W_{PS}} u_i({\bf x}) {\mathcal T}_{ijk} ({\bf x},{\boldsymbol \xi}) n_k({\bf x}) {\hbox{d}}S_{PS}. \end{aligned}$$

(26)

In Eq. (3) the boundary $\partial \varOmega$ of the fluid domain is explicitly decomposed into three parts: the particle surface ∂B, the no-slip stripes on the patterned wall, collectively denoted W _NS , and the complementary part of the wall with the perfect-slip stripes W _PS . The contributions arising from the portion of the boundary at infinity (not included in the present formulation where the fluid is assumed to be at rest) can be easily incorporated under suitable assumption on the asymptotic behavior of the field. The effects of body forces such as gravity and electric fields can be accounted for by a convolution integral extended to the fluid domain between the force and the free-space Green’s tensor (see below). In representation (26), t _i and $u_i, i=1,\ldots,3$, are the Cartesian components of the surface stress and velocity, respectively. The free-space Green’s function (the so-called steady Stokeslet) is defined as

$$G_{ij}({\bf r})=\left(\frac{\delta_{ij}}{r} + \frac{r_i r_j}{r^3}\right) ,$$

(27)

and the associated stress tensor is

$${\mathcal T}_{ijk}({\bf r}) =-6 \frac{r_i r_j r_k }{r^5}.$$

(28)

In the above expression, G _ij (r) provides the contribution to the jth velocity component at ${\boldsymbol \xi}$ due to a concentrated force acting in the ith direction at x. The associated Green’s stress tensor, as always, should be contracted with the outward unit normal n _k (x) to the boundary $\partial \varOmega$ in order to provide the effect on the jth velocity component at ${\boldsymbol \xi}$ of the ith boundary velocity at x. The vector r is defined as ${\bf r}={\bf x}-{\boldsymbol \xi}$ with $r=\sqrt{r_k r_k}$ its modulus. In Eq. (3) $E({\boldsymbol \xi})=1$ when ${\boldsymbol \xi}\in\varOmega$ and $E({\boldsymbol \xi})=1/2$ for ${\boldsymbol \xi}\in\partial\varOmega$ (the existence of a regular tangent plane is assumed throughout). When ${\boldsymbol \xi}\in\partial\varOmega$, representation (3) becomes a boundary integral equation where the unknowns can either be the three stress vector components t _i , the three velocity components u _i , or a combination thereof, depending on the boundary conditions assigned on the specific portion of boundary. This approach allows us to discretize only the boundary surfaces of the flow domain instead of considering the entire volume. This results in: (1) a substantial reduction in the number of unknowns; (2) the possibility to easily specify different kinds of boundary condition on different surface patches; and (3) the simple update of the geometry when dealing with time-dependent configurations. Once the boundary $\partial\varOmega$ is discretized into panels, Eq. (3) is recast into an algebraic linear system whose solution can be achieved by standard linear algebra packages. In simple cases, part of the boundary can be accounted for by symmetry, as it happens for a flat homogeneous wall.

In this paper, given the generality of the boundary condition to be used at the wall (either patterned perfect/no-slip stripes or effective slip Navier-like boundary conditions), the complete formulation of the boundary integral problem based on the free-space Green’s function has been retained, with the use of the wall Green’s function demanded of providing reference results for accuracy tests.

Finally, a few more words may be useful concerning the specific boundary conditions used in the paper. On the perfect-slip boundary patches, W _PS , the normal velocity vanishes u _⊥ = 0 due to impermeability, while the tangential velocity u _∥ (two Cartesian components) is unknown. Moreover, the tangential stress t _∥ vanishes by perfect slip such that the stress is aligned to the normal, $t_i=-\varPhi({\bf x}) n_i$, with $\varPhi$ representing a further scalar unknown. On the no-slip surfaces, W _NS and ∂B, velocities are completely assigned while stresses are unknown. Concerning the partial-slip condition used as an effective model of the stripe pattern, zero normal velocity u _⊥ = 0 at the wall is implied, while tangential velocities and stresses are coupled by the Navier condition (Lauga et al. 2007),

$${\bf u}_{\parallel} = {\boldsymbol \ell}_{\bf s} {\bf n} \cdot ( {\boldsymbol \nabla}{{\bf u}} + ({\boldsymbol \nabla}{{\bf u}})^T) \cdot ({\bf 1}-{\bf n} \otimes {\bf n}) .$$

(29)

Here ${\boldsymbol \ell}_{\bf s}$ is a 2 × 2 symmetric (Kamrin et al. 2010) tensor describing the directionally dependent slip length. In the present case, this tensor is diagonalized when expressed in stripe-parallel and stripe-normal Cartesian coordinates. In this case the two diagonal entries are different as a consequence of the orientation of the stripe pattern. For a flat wall, Eq. (29) is rewritten as

$${\bf u}_{\parallel}={\boldsymbol \ell}_{\bf s} \cdot {\bf t_{\parallel}} ,$$

(30)

that is the vectorial form of Eq. (7), the two nonzero components of ${\boldsymbol \ell}_{\bf s}$ being reported in Eq. 8 (Philip 1972; Ng and Wang 2010). The boundary integral Eq. (3) supplemented with Eq. (30) provides a closed system that after inversion determines all the unknowns involved in the problem.