2.1. Colloidal-probe AFM set-up

A schematic of the experimental system is shown in figure 1(a). A borosilicate sphere (MOSci Corporation, radius $R=27 \pm 0.5 \ \mathrm {\mu } {\rm m}$) is glued (Epoxy glue, Araldite) to the end of an AFM cantilever (SNL-10, Brukerprobes), and located near a planar mica surface. The cantilever stiffness $k_{c}=0.68\pm 0.05 \ {\rm N}\ {\rm m}^{-1}$ is calibrated using the drainage method proposed by Craig & Neto (Reference Craig and Neto2001). The experiments were performed using an AFM (Bruker, Dimension3100) in three different liquids, i.e. water, dodecane and silicone oil, whose densities and dynamic viscosities are $1000$ kg m$^{-3}$ and 1 mPa s, $750$ kg m$^{-3}$ and 1.34 mPa s, and $930$ kg m$^{-3}$ and 9.3 mPa s, respectively, at room temperature. The sphere–wall distance $D$ was controlled by an integrated stage step motor. Each separation distance was adjusted by displacing the cantilever vertically using the step motor with precision in position <0.1 $\mathrm {\mu }$m. The zero distance was determined as the point where the deflection signal changed sharply due to the solid–solid contact of the probe with the wall. The cantilever's deflection produced by thermal and hydrodynamic forces acting on the probe was acquired directly using an analogue to digital (A/D) acquisition card (PCI-4462, NI, USA) with sampling frequency 200 kHz. The amplitude of the thermal fluctuation of the probe remained smaller than $\sim$1 nm, in all the experiments.

Figure 1.

Schematics of the system. A borosilicate sphere with radius $R$ is glued at the end of an AFM cantilever, and fluctuates thermally within a viscous liquid and near a mica substrate, with a distance $D$ between the sphere and the substrate. The sphere and mica surfaces are denoted as $\mathcal {S}_0$ and $\mathcal {S}_{w}$, respectively.

2.2. Confined thermal dynamics

The time-dependent position of the spherical probe is denoted $Z(t)$. For a given sphere–wall distance $D$, the probe dynamics can be modelled by a forced harmonic oscillator (see Jai, Cohen-Bouhacina & Maali Reference Jai, Cohen-Bouhacina and Maali2007; Kiracofe & Raman Reference Kiracofe and Raman2011; Maali & Boisgard Reference Maali and Boisgard2013), as

(2.1)\begin{equation} m_\infty \ddot{Z}+ \gamma_\infty \dot{Z} + k_{c} Z = F_{th} + F_{int}, \end{equation}

where $m_\infty$ is the effective mass of the probe in the bulk, and $\gamma _\infty$ is the bulk damping coefficient. These two coefficients correspond to the free dynamics of the probe far from the surface, and can thus be obtained by measuring the resonance properties of the AFM probe in the far field, as shown below. Besides the elastic restoring force by the cantilever of stiffness $k_{c}$, and in the absence of conservative forces (e.g. van der Waals or electrostatic forces), the two main forces acting on the sphere along the $z$ direction are the random thermal force $F_{th}$ and the hydrodynamic interaction force with the wall $F_{int}$. The latter corresponds to the deviation of the hydrodynamic drag with respect to the bulk drag force and depends on the sphere–wall distance.

Taking the Fourier transform of (2.1), we find

(2.2)\begin{equation} - m_\infty \omega^2 \tilde{Z} + {\rm i}\omega \gamma_\infty \tilde{Z} + k_{c} \tilde{Z} = \tilde{F}_{th} + \tilde{F}_{int}, \end{equation}

where $\tilde {f}(\omega ) = ({1}/{2{\rm \pi} })\int _{-\infty }^{\infty } \mathrm {d}t\,f(t) \,{\rm e}^{-{\rm i} \omega t}$ is the Fourier transform of the function $f(t)$. The real and imaginary parts of $\tilde {F}_{int}$ correspond to an inertial force and a dissipative force, respectively, that can be recast into

(2.3)\begin{equation} \tilde{F}_{int} = m_{int} \omega^2 \tilde{Z} - {\rm i}\omega\gamma_{int} \tilde{Z}, \end{equation}

where $m_{int}$ and $\gamma _{int}$ are the wall-induced variations of the effective mass and dissipation coefficient. For the sake of simplicity, we neglect in the following the possible frequency dependencies of $m_{int}$ and $\gamma _{int}$. With this main assumption, and injecting (2.3) into (2.2), the probe's motion follows a thermally forced harmonic oscillator dynamics with spring constant $k_{c}$, effective damping coefficient $\gamma \equiv \gamma _\infty + \gamma _{int}$, and effective mass $m \equiv m_\infty + m_{int}$. For the latter problem, one can then derive the one-sided power spectral density $S(\omega ) \equiv 2 \langle \vert \tilde {Z}(\omega ) \vert ^2 \rangle$, as

(2.4)\begin{equation} S(\omega) = \frac{2 \langle \vert F_{th} \vert ^2 \rangle / (m^{2}\omega_0^4)}{\left[1-\left(\dfrac{\omega}{\omega_0}\right)^2 \right]^2+\left(\dfrac{\omega}{\omega_0 Q} \right)^2} = \frac{2k_{B}T / ({\rm \pi} Q m \omega_0^3)}{\left[1-\left(\dfrac{\omega}{\omega_0} \right)^2 \right]^2+\left(\dfrac{\omega}{\omega_0 Q} \right)^2} , \end{equation}

where $\langle \cdot \rangle$ denotes the ensemble average, $k_{B}T$ is the thermal energy, $\omega _0 = \sqrt {k_{c}/m}$ is the resonance angular frequency, and $Q=m\omega _0/\gamma$ is the quality factor. The second equality in (2.4) is obtained by using the correlator of the noise $\langle F_{th}(t)\,F_{th}(t') \rangle = 2 \gamma k_{B}T \delta _{D}(t-t')$, where we assumed a white noise through the Dirac distribution $\delta _{D}$ , and where we invoked the fluctuation–dissipation theorem to set the amplitude of the noise. The experimental power spectral densities are fitted by the function (Honig et al. Reference Honig, Sader, Mulvaney and Ducker2010; Bowles, Honig & Ducker Reference Bowles, Honig and Ducker2011)

(2.5)\begin{equation} S(\omega) = \frac{c_1}{\left[1-\left(\dfrac{\omega}{\omega_0} \right)^2 \right]^2+\left(\dfrac{\omega}{\omega_0 Q} \right)^2} + c_2, \end{equation}

where $\omega _0$ and $Q$ are the key adjustable parameters indicating the position and width of the resonance, and $c_1$ and $c_2$ are unimportant extra parameters allowing us to accommodate for potential spurious experimental offset and/or prefactor.

2.3. Power spectral density

Figure 2 displays the power spectral densities for probes immersed in dodecane or silicone oil (water was employed as well, but the similar results are not shown here), and for a variety of sphere–wall distances. A well-defined peak can be observed for each spectrum, indicating the fundamental resonance. The resonance properties are well described by the damped harmonic oscillator model above. The largest sphere–wall distance ($D=100\ \mathrm {\mu }{\rm m}$) corresponds to nearly four times the sphere radius, so that the hydrodynamic interactions between the probe and the wall can be neglected. At such distances, the bulk resonance frequency $\omega _0^\infty = \sqrt {k_{c}/m_\infty }$ and bulk quality factor $Q_\infty = m_\infty \omega _0^\infty /\gamma _\infty$ are extracted from the fitting procedure, giving respective values $7070\pm 5$ Hz and $3.3\pm 0.1$ in dodecane, and $5320\pm 5$ Hz and $1.3\pm 0.1$ in silicone oil. In the more viscous fluid (silicone oil), the resonance is broader since the dissipation is larger, as expected. Also, in both liquids, we observe that the resonance is broader as the sphere gets closer to the hard wall, which indicates that the near-wall dissipation is larger as compared to the bulk situation, as expected too. Besides, and interestingly, the bulk resonance frequency changes significantly from silicone oil to water ($\approx$20 % difference in $\omega _0^\infty$, which means $\approx$50 % difference in effective mass), although the liquid density is similar (less than 10 % difference). Hence the effective mass depends on the viscosity of the ambient fluid in a non-trivial way. Moreover, the resonance frequency depends on the sphere–wall distance.

Figure 2.

Experimental power spectral densities in arbitrary units (a.u.), for the colloidal probe's vertical position, in (a) dodecane and (b) silicone oil, for various sphere–wall distances as indicated. The curves are shifted vertically for clarity. The solid lines show the best fits to the damped harmonic oscillator model, using (2.5).

To be quantitative, the fitted values of the resonance frequency $\omega _0$ and the quality factor $Q$ are shown in figure 3 as functions of the normalized separation distance $D/R$, for the three liquids studied. Intriguingly, we observe an increase of the resonance frequency in silicone oil near the wall as compared to the bulk resonance frequency (figure 3a), and a corresponding decrease in dodecane (figure 3c) and water (figure 3e). We point out that the sphere–wall distances in the present experiments are large enough ($D>0.5\ \mathrm {\mu } {\rm m}$) so that molecular interactions (e.g. electrostatic or van der Waals forces) can be neglected safely. Therefore, the changes in resonance frequency observed here should result only from hydrodynamic contributions. The next section aims to model this intricate behaviour.

Figure 3.

Resonance frequency $\omega _0/(2{\rm \pi} )$ (blue dots) and quality factor $Q$ (orange dots) of the normal mode of the colloidal AFM probe as a function of dimensionless sphere–wall distance $D/R$. The dashed lines represent the resonance frequencies and quality factors calculated by (4.3) and (4.4), respectively, without adjustable parameter. (a,b) Results for silicone oil ($\eta = 9.3$ mPa s, $\rho = 930$ kg m$^{-3}$), with squared Womersley number $Wo^2 =2.4$. (c,d) Results for dodecane ($\eta = 1.34$ mPa s, $\rho = 750$ kg m$^{-3}$), with $Wo^2 =18.1$. (e,f) Results for water ($\eta = 1$ mPa s, $\rho = 1000$ kg m$^{-3}$), with $Wo^2 =31.7$.

3.1. Governing equations

We aim here to calculate the hydrodynamic force exerted on an immersed sphere moving normally near a rigid, flat and immobile wall. The hydrodynamic pressure field is denoted $p$. The fluid velocity field $\boldsymbol {v}$ satisfies the incompressible Navier–Stokes equations. In order to identify the relevant terms in these, we non-dimensionalize them. Rescaling times by $\omega ^{-1}$, lengths by $R$, velocities by $A\omega$, and pressures by $\eta A\omega /R$, and adding bars for all the dimensionless variables, the dimensionless Navier–Stokes equations read

(3.1)\begin{equation} Wo^2\,\partial_{\bar{t}} \bar{\boldsymbol{v}} +Re\,\bar{\boldsymbol{v}}\boldsymbol{\cdot}\bar{\boldsymbol{\nabla}}\bar{\boldsymbol{v}}={-}\bar{\boldsymbol{\nabla}} \bar{p} + \bar{\boldsymbol{\nabla}}^2 \bar{\boldsymbol{v}}, \quad \bar{\boldsymbol{\nabla}}\boldsymbol{\cdot} \bar{\boldsymbol{v}} = 0 , \end{equation}

where $Re=\rho RA\omega /\eta$ is the Reynolds number, and $Wo^2 = R^2 \omega / \nu$ is the squared Womersley number. The amplitude of thermal oscillations in the experiments is nanometric, which implies a relatively small Reynolds number for all accessible frequencies. Therefore, we can neglect the convective term of the incompressible Navier–Stokes equations. Nonetheless, the typical resonance frequency is in the kHz range, such that the squared Womersley number $Wo^2 = R^2 \omega / \nu$ is in the 1–50 range. As a consequence, we expect unsteady inertial effects to be important. The fluid velocity field thus satisfies the unsteady incompressible Stokes equations. Putting back dimensions, the latter reads

(3.2)\begin{equation} \rho\,\partial_t \boldsymbol{v} ={-}\boldsymbol{\nabla} p + \eta\,\nabla^2 \boldsymbol{v}, \quad \boldsymbol{\nabla}\boldsymbol{\cdot} \boldsymbol{v} = 0 . \end{equation}

Without loss of generality, the sphere's position is supposed to oscillate normally to the substrate at frequency $\omega$ and with amplitude $A$, which correspond to a given Fourier mode of the full fluctuation spectrum. Applying the Fourier transform to the unsteady incompressible Stokes equations, we get

(3.3)\begin{equation} {\rm i}\rho\omega \tilde{\boldsymbol{v}} ={-}\boldsymbol{\nabla} \tilde{p} + \eta\,\nabla^2 \tilde{\boldsymbol{v}},\quad \boldsymbol{\nabla}\boldsymbol{\cdot} \boldsymbol{\tilde{v}} = 0. \end{equation}

A no-slip condition is assumed at both the wall and the sphere surfaces, denoted by $\mathcal {S}_{w}$ and $\mathcal {S}_0$, respectively (see figure 1b), leading to the following boundary conditions for the fluid velocity field:

(3.4a,b)\begin{equation} \tilde{\boldsymbol{v}}(\boldsymbol{r} \in \mathcal{S}_0) = {\rm i}\omega A \boldsymbol{e}_z, \quad \tilde{\boldsymbol{v}}(\boldsymbol{r} \in \mathcal{S}_{w}) = \boldsymbol{0} , \end{equation}

with $\boldsymbol {e}_z$ the unit vector in the $z$ direction. Here, we suppose that the sphere position is fixed at its average position during one oscillation, as the oscillation amplitude is assumed to be small. The hydrodynamic drag force applied on the sphere is given by

(3.5)\begin{equation} \tilde{\boldsymbol{F}} = \int_{\mathcal{S}_0} \boldsymbol{n} \boldsymbol{\cdot} \tilde{\boldsymbol{\sigma}}\, \mathrm{d}\mathcal{S}_0, \end{equation}

where $\tilde {\boldsymbol {\sigma }} = -\tilde {p} \boldsymbol {I} + \eta [\boldsymbol {\nabla }\tilde {\boldsymbol {v}} + (\boldsymbol {\nabla }\tilde {\boldsymbol {v}})^{\rm T}]$ is the fluid stress tensor, and $\boldsymbol {n}$ denotes the unit vector normal to $\mathcal {S}_0$ oriented towards the fluid. To the best of our knowledge, there is no closed-form solution of the problem, in contrast with the steady case (see Brenner Reference Brenner1961). Indeed, the stream function follows the Helmholtz equation (resp. the Laplace equation) associated with the unsteady (resp. steady) Stokes equation. The steady Stokes equation can be solved exactly, using the spectral decomposition of the Laplacian operator in the bispherical coordinate system. Nevertheless, the same methodology cannot be applied to the Helmholtz equation, hence to the unsteady Stokes problem.

By symmetry, the drag force is directed along the $z$ direction, i.e. $\tilde {\boldsymbol {F}}=\tilde {F}_z\boldsymbol {e}_z$. Using dimensional analysis, and assuming that the oscillation amplitude $A$ is much smaller than $D$, one can show that the drag force $\tilde {F}_z$ normalized by the bulk Stokes reference $-6{\rm i}{\rm \pi} \eta R A \omega$ to form the dimensionless drag force $\tilde {f}_z=\tilde {F}_z/(-6{\rm i}{\rm \pi} \eta R A \omega )$, depends on only two dimensionless parameters: (i) the Womersley number $Wo$, and (ii) the sphere–wall distance relative to the sphere radius $D/R$. As a consequence, the dimensionless hydrodynamic interaction force (see § 2.2 and (2.3)) reads

(3.6)\begin{equation} \frac{\tilde{F}_{int}}{6{\rm i}{\rm \pi}\eta R A \omega} = \tilde{f}_z(D/R \to \infty, Wo) - \tilde{f}_z(D/R, Wo)= \frac{(m_{int}\omega^2-{\rm i}\omega \gamma_{int})\tilde{Z}}{6{\rm i}{\rm \pi}\eta R A \omega}. \end{equation}

Although there is no general analytical solution of (3.3) with the boundary conditions (3.4a,b), the hydrodynamic drag force has known asymptotic expressions in certain limits, some of which are given in the next two subsections.

3.2. Large-distance regime

In the infinite-distance limit, the force expression reduces to the BBO equation (see (1.1)) for a sphere in an unbounded space, which gives in Fourier space

(3.7)\begin{equation} \tilde{F}_z ={-}6{\rm i}{\rm \pi}\eta R A\omega \left(1 + \sqrt{-{\rm i}}\,Wo - \frac{{\rm i}\,Wo^2}{9}\right), \quad \text{for} \ D/R \to \infty. \end{equation}

The last term of (3.7) corresponds to an inertial force of added mass $2{\rm \pi} \rho R^3/3$, and the $\sqrt {-{\rm i}}\,Wo$ term corresponds to the Basset force, with $\sqrt {-{\rm i}} = (1-i)/\sqrt {2}$. The large-distance asymptotic correction to the added-mass contribution due to a rigid wall has been computed using the potential flow theory, and gives $2{\rm \pi} \rho R^3 \{1 + 3R^3/[8(R+D)^3]\}/3$ (see Lamb Reference Lamb1932). By using a boundary integral formulation of the unsteady incompressible Stokes equations, Fouxon & Leshansky (Reference Fouxon and Leshansky2018) have generalized the latter result by including the Basset force, to obtain the large-distance asymptotic drag force that reads

(3.8)\begin{equation} \tilde{F}_z ={-}6{\rm i}{\rm \pi}\eta R A\omega\left(1 + \sqrt{-{\rm i}}\,Wo - \frac{{\rm i}\,Wo^2}{9} + B\,\frac{R^3}{(D+R)^3}\right), \quad \text{for} \ D/R \gg 1, \end{equation}

where the numerical prefactor $B$ depends on $Wo$:

(3.9)\begin{equation} B = \frac{1}{4}\left( 1+\sqrt{-{\rm i}}\,Wo-\frac{{\rm i}\,Wo^2 }{3} \right) \left[\frac{1}{3}+\frac{3{\rm i}}{2\,Wo^2 } \left(1+\sqrt{-{\rm i}}\,Wo -\frac{{\rm i}\,Wo^2 }{9} \right) \right]. \end{equation}

3.3. Small-distance regime

In the limit of small sphere–wall distance, which is of importance for colloidal-probe experiments, the drag force is usually dominated by viscous effects. The out-of-phase component of the force can be described by lubrication theory (see Batchelor Reference Batchelor1967), in which the main contribution to the drag comes from the confined region between the sphere and the wall, which leads to the expression

(3.10)\begin{equation} \tilde{F}_z ={-}\frac{6{\rm i}{\rm \pi}\eta R^2 A\omega}{D}. \end{equation}

We stress that the in-phase correction to the latter is still unknown in the lubricated limit. It would be interesting to perform asymptotic-matching calculations on the unsteady Stokes equations (see Cox & Brenner Reference Cox and Brenner1967) to obtain a self-consistent expression of the effective added-mass in this limit.

3.4. Low-Womersley-number regime

As pointed out by Fouxon & Leshansky (Reference Fouxon and Leshansky2018), in the small-frequency limit, which corresponds to a small Womersley number, the drag force can be expressed in terms of known integrals by using the Lorentz reciprocal theorem (see Masoud & Stone Reference Masoud and Stone2019; Fouxon et al. Reference Fouxon, Rubinstein, Weinstein and Leshansky2020). We provide here an alternative derivation of this result.

We introduce the model steady problem of a sphere moving normally to a surface in a viscous fluid, which corresponds to the problem of § 3.1, at zero frequency, i.e.

(3.11a,b)\begin{equation} \boldsymbol{\nabla} \boldsymbol{\cdot} \hat{\boldsymbol{\sigma}} = \boldsymbol{0} , \quad \boldsymbol{\nabla} \boldsymbol{\cdot} \hat{\boldsymbol{v}} = 0 , \end{equation}

with the same boundary conditions

(3.12a,b)\begin{equation} \hat{\boldsymbol{v}}(\boldsymbol{r} \in \mathcal{S}_0) = {\rm i}\omega A\boldsymbol{e}_z , \quad \hat{\boldsymbol{v}}(\boldsymbol{r} \in \mathcal{S}_w) = \boldsymbol{0}, \end{equation}

where $\hat {\boldsymbol {\sigma }}$ and $\hat {\boldsymbol {v}}$ are the fluid stress and velocity fields of the model problem, respectively. Integrating the Lorentz identity $\boldsymbol {\nabla } \boldsymbol {\cdot } (\tilde {\boldsymbol {\sigma }}\boldsymbol {\cdot } \hat {\boldsymbol {v}} - \hat {\boldsymbol {\sigma }}\boldsymbol {\cdot } \tilde {\boldsymbol {v}}) = {\rm i}\omega \rho \tilde {\boldsymbol {v}} \boldsymbol {\cdot } \hat {\boldsymbol {v}}$ on the total fluid volume, we obtain

(3.13)\begin{equation} ({\rm i}\omega A \boldsymbol{e}_z) \boldsymbol{\cdot} \left[\int_{\mathcal{S}_0} \hat{\boldsymbol{\sigma}} \boldsymbol{\cdot} \boldsymbol{n}\, \mathrm{d}\mathcal{S}_0 - \int_{\mathcal{S}_0} \tilde{\boldsymbol{\sigma}} \boldsymbol{\cdot} \boldsymbol{n}\, \mathrm{d}\mathcal{S}_0\right] = {\rm i}\omega\rho \int_\mathcal{V} \tilde{\boldsymbol{v}} \boldsymbol{\cdot} \hat{\boldsymbol{v}} \, \mathrm{d}\mathcal{V}, \end{equation}

where the divergence theorem has been used. Recalling (3.5), we get

(3.14)\begin{equation} \tilde{F}_z = \hat{F}_z -\frac{\rho}{A} \int_\mathcal{V} \tilde{\boldsymbol{v}} \boldsymbol{\cdot} \hat{\boldsymbol{v}} \, \mathrm{d}\mathcal{V}. \end{equation}

The force $\hat {F_z}$ and velocity field $\hat {\boldsymbol {v}}$ of the model problem correspond to those derived analytically by Brenner (Reference Brenner1961) and Maude (Reference Maude1961) using a modal decomposition. The force of the model problem thus reads

(3.15)\begin{align} &\dfrac{\hat{F}_z}{6{\rm i}{\rm \pi}\eta R \omega A} = \dfrac{4}{3} \sinh(\alpha) \sum_{n=1}^\infty \dfrac{n(n+1)}{(2n-1)(2n+3)} \nonumber\\ &\quad \times \left\{1-\dfrac{2\sinh[(2n+1)\alpha] + (2n+1)\sinh (2\alpha)}{t[2\sinh((n+\frac{1}{2})\alpha)]^2 - [(2n+1)\sinh(\alpha)]^2} \right\}, \end{align}

with $\cosh (\alpha ) = 1 + D/R$. Nevertheless, the unsteady velocity field $\tilde {\boldsymbol {\boldsymbol {v}}}$ in (3.14) is still unknown, so the drag force $\tilde {F}_z$ cannot be found exactly.

Analytical progress can be made in the low-$Wo$ regime, where the unsteady velocity field can be approximated by the steady solution with $\mathcal {O}(Wo^2)$ corrections, as $\tilde {\boldsymbol {\boldsymbol {v}}} = \hat {\boldsymbol {v}} [1 + \mathcal {O}(Wo^2)]$. In this limit, at leading order in inertial contributions, the drag force reduces to

(3.16)\begin{equation} \tilde{F}_z = \hat{F}_z -\frac{\rho}{A} \int_\mathcal{V} \hat{\boldsymbol{v}}^2 \, \mathrm{d}\mathcal{V}. \end{equation}

The volume integral in (3.16) can then be evaluated numerically using the model velocity field provided by Brenner (Reference Brenner1961). The volume integral in (3.16) always converges as the velocity field $\hat {\boldsymbol {v}}$ decays exponentially at large radius.

3.5. Finite-element method

We complement the previous asymptotic expressions of the drag force with full numerical solutions. Using the open-source finite-element library Nutils (see van Zwieten, van Zwieten & Hoitinga Reference van Zwieten, van Zwieten and Hoitinga2022), we solve (3.3). The axisymmetric velocity and pressure fields are defined on a $320 \times 320$ element mesh, spaced uniformly on a rectangular domain $[0\leq \tau \leq \alpha,\; 0\leq \sigma \leq {\rm \pi}]$. We then use the bipolar coordinate transform

(3.17a,b)\begin{equation} r = a\,\frac{\sin(\sigma)}{ \cosh(\tau) - \cos(\sigma)}, \quad z = a\, \frac{\sinh(\tau) }{ \cosh(\tau) - \cos(\sigma)}, \end{equation}

with $a = R\sinh \alpha$. The resulting mesh, when axisymmetry is considered, spans the entire domain where $r>0$ and $z>0$, with the exception of a circular region corresponding to the sphere, as shown in figure 4. On the symmetry axis ($r=0$), the flow in the radial direction is constrained and the vertical flow is required to be shear-free. At the wall surface ($z=0$), the velocity field is set to zero. Finally, on the surface of the sphere, the radial and vertical velocity components are set to zero and unity (imaginary part) respectively, following (3.4a,b). From the calculated velocity and pressure fields, the total force exerted on the particle can be computed directly using (3.5). Typical flow fields are shown in figures 4(b,c).

Figure 4.

(a) Typical mesh used in the finite-element method. (b) Streamlines of the in-phase flow field, obtained numerically. (c) Streamlines of the out-of-phase flow field, obtained numerically. The squared Womersley number is set to $Wo^2 = 10$, such that $\delta \approx 0.31 R$. The red dashed lines indicate a sphere of radius $R+\delta$.