2. Capillary-lubrication theory

We consider a sphere of radius $a$ translating with a prescribed time-dependent horizontal velocity $\boldsymbol {u}=u(t)\boldsymbol {e}_x$ near a fluid interface, as shown in figure 1. The interface is characterized by its surface tension $\sigma$, and separates two incompressible Newtonian viscous liquids with dynamic shear viscosities $\eta$ and $\eta _{sl}$, as well as densities $\rho$ and $\rho _{sl}$ (with $\rho <\rho _{sl}$). The acceleration due to gravity is denoted by $g$. The gap profile between the sphere and the undeformed fluid interface is denoted by $h(r,t)$, which depends on the horizontal radial coordinate $r$ and time $t$. The $x$-direction oriented along $\boldsymbol {e}_x$ corresponds to the horizontal angular coordinate $\theta =0$. The time-dependent distance between the bottom of the sphere's surface and the undeformed interface is denoted by $d(t)$, with $d>0$. Its temporal derivative $\dot {d}(t)$ denotes the vertical velocity of the sphere. We focus on the case where the bottom layer is extremely viscous compared with the top layer, i.e. $\eta _{sl}\gg \eta$, and is infinitely thick compared with the gap between the sphere and the undeformed interface. These conditions allow us to assume that the tangential velocity at the fluid interface vanishes because the shear stress from the top layer is insufficient to generate a flow in the lower layer.

Figure 1.

Schematic of the system. A sphere of radius $a$ immersed in a viscous fluid of viscosity $\eta$ and density $\rho$ moves near a fluid interface. The undeformed gap profile is noted $h(r,t)$, with $r$ the horizontal radial coordinate and $t$ the time. The origin of coordinates is located at the undeformed fluid interface ($z = 0$) in line with the centre of mass of the sphere ($r = 0$). The interface separates the top fluid from a secondary fluid of viscosity $\eta _{sl}$, with $\eta _{sl}\gg \eta$, and density $\rho _{sl}$ at the bottom, i.e. $\rho _{sl}>\rho$. The sphere has prescribed horizontal velocity $u$ and vertical velocity $\dot {d}$, where $d=h(0,t)$ denotes the instantaneous distance between the sphere bottom and the undeformed fluid interface. The interface deflection field is denoted as $\delta (r,t)$, the acceleration due to gravity is denoted as $g$ and the surface tension is denoted as $\sigma$.

We neglect fluid inertia and assume that the typical gap between the sphere and the interface is much smaller than the typical horizontal length scale, allowing us to invoke lubrication theory (Reynolds Reference Reynolds1886; Oron, Davis & Bankoff Reference Oron, Davis and Bankoff1997). Introducing the excess pressure field $p$, in the top layer, with respect to the hydrostatic contribution, and the horizontal velocity field $\boldsymbol {v}$ of the fluid in the gap, the incompressible Stokes equations thus read at leading lubrication order

(2.1)$$\begin{gather} \frac{\partial p}{\partial z} = 0, \end{gather}$$

(2.2)$$\begin{gather}\boldsymbol{\nabla} p= \eta\frac{\partial^2 \boldsymbol{v}}{\partial z^2}, \end{gather}$$

where $\boldsymbol {\nabla }$ denotes the gradient in the horizontal plane and where $z$ is the vertical coordinate. In the limit of a small gap, the thickness profile of the fluid between the sphere and the undeformed fluid interface can be approximated by its parabolic expansion, leading to

(2.3)\begin{equation} h(r,t) \simeq d(t)+\frac{r^{2}}{2 a}.\end{equation}

The no-slip boundary condition is imposed at both the sphere's surface and the fluid interface. Hence, in the frame of the moving sphere, at $z = h$, one has

(2.4)\begin{equation} \boldsymbol{v} = \boldsymbol{0}, \end{equation}

and at the interface, i.e. $z = \delta$, one has

(2.5)\begin{equation} \boldsymbol{v} ={-}\boldsymbol{u} ={-}u\boldsymbol{e}_x. \end{equation}

Given the boundary conditions above, we can easily write the horizontal velocity profile in the gap between the sphere and the interface as

(2.6)\begin{equation} \boldsymbol{v} = \frac{\boldsymbol{\nabla} p}{2\eta}(z-h)(z-\delta)+\boldsymbol{u}\frac{z-h}{h-\delta}.\end{equation}

The conservation of the fluid volume in the gap allows for the derivation of the Reynolds thin-film equation for the system, which reads

(2.7)\begin{equation} \frac{\partial}{\partial t}(h-\delta) = \boldsymbol{\nabla}\boldsymbol{\cdot} \left[\frac{\boldsymbol{\nabla} p}{12\eta}(h-\delta)^3+\frac{\boldsymbol{u}}{2}(h-\delta)\right].\end{equation}

The deflection of the fluid interface is controlled by the Laplace pressure jump. Thus, at $z = \delta$, one has

(2.8)\begin{equation} p \simeq\sigma \nabla^2\delta+g\delta(\rho-\rho_{sl}), \end{equation}

where we have assumed a small deformation of the interface and linearized the curvature.

Let us now non-dimensionalize the equations through

(2.9) \begin{equation} \left.\begin{gathered} d(t)= d^*D(T),\quad h(r,t) = d^*H(R,T),\quad \boldsymbol{r}=l\boldsymbol{R},\quad z =d^*Z,\\ t = \frac{l}{c}T,\quad \boldsymbol{v}(\boldsymbol{r},z,t)=c\boldsymbol{V}(\boldsymbol{R},Z,T),\quad \boldsymbol{u}(t) =cU(T)\boldsymbol{e}_x,\\ p(\boldsymbol{r},t) =\frac{\eta c l}{{d^*}^2}P(\boldsymbol{R},T),\quad \delta(\boldsymbol{r},t) = d^*\varDelta(\boldsymbol{R},T), \end{gathered}\right\} \end{equation}

where $c$ and $d^*$ represent characteristic in-plane velocity and gap-thickness scales, respectively, with $l = \sqrt {2ad^*}$ denoting the characteristic hydrodynamic radius.

At this point, we discuss the assumption of an infinitely viscous bottom layer and the resulting vanishing of the velocity, which has been previously addressed in the literature (Yiantsios & Davis Reference Yiantsios and Davis1990). Since the bottom fluid layer is considered to be much thicker than the upper one, the flow generated in the former evolves over a much larger length scale, typically given by $a$. As a consequence, by adapting the thin-bilayer-film normal stress balance (see (A4) in Bertin et al. Reference Bertin, Lee, Salez, Raphaël and Dalnoki-Veress2021) to the current case of a thick bottom layer, we find that the dimensionless normal viscous stress at the fluid interface has a $(\eta _{sl}/\eta )(d^*/l)^4$ prefactor. We assume this numerical prefactor to be very small in our study, due to the vanishing of $d^*/l$, despite the large viscosity ratio – thus the non-inclusion of bottom-layer stress terms in (2.8).

In dimensionless terms, the undeformed thickness profile and Reynolds equation become

(2.10)\begin{equation} H(R,T) \simeq D(T)+R^{2}, \end{equation}

and

(2.11)\begin{equation} \frac{\partial}{\partial T}(H-\varDelta) = \boldsymbol{\nabla}\boldsymbol{\cdot} \left[\frac{\boldsymbol{\nabla} P}{12}(H-\varDelta)^3+\frac{\boldsymbol{U}}{2}(H-\varDelta)\right].\end{equation}

Besides, the deflection field $\varDelta$ is related to the excess pressure field $P$ by the dimensionless version of the Laplace equation, which reads

(2.12)\begin{equation} \nabla^2\varDelta-{Bo}\varDelta=\kappa P, \end{equation}

where ${Bo} = (l/l_{c})^2$ denotes the Bond number, $l_{c} = \sqrt {\sigma /[g(\rho _{sl}-\rho )]}$ denotes the capillary length and where we have introduced the capillary compliance of the interface, denoted by

(2.13)\begin{equation} \kappa = \frac{\eta cl^3}{\sigma {d^*}^3}. \end{equation}

3. Perturbation analysis

As in previous studies regarding soft lubrication (Sekimoto & Leibler Reference Sekimoto and Leibler1993; Skotheim & Mahadevan Reference Skotheim and Mahadevan2005; Urzay et al. Reference Urzay, Llewellyn Smith and Glover2007; Snoeijer et al. Reference Snoeijer, Eggers and Venner2013; Salez & Mahadevan Reference Salez and Mahadevan2015; Essink et al. Reference Essink, Pandey, Karpitschka, Venner and Snoeijer2021; Bertin et al. Reference Bertin, Amarouchene, Raphael and Salez2022), we perform an asymptotic expansion of the unknown fields at first order in dimensionless capillary compliance, $\kappa$, as

(3.1)$$\begin{gather} \varDelta \simeq \kappa \varDelta_{1}+O(\kappa^2), \end{gather}$$

(3.2)$$\begin{gather}P \simeq P_{0}+\kappa P_{1}+O(\kappa^2), \end{gather}$$

where $\kappa ^i\varDelta _i$ is the $i$th-order contribution to the deflection of the interface, and $\kappa ^iP_i$ is the $i$th-order contribution to the excess pressure field.

3.1. Zeroth-order pressure

At zeroth order in $\kappa$, (2.11) reduces to

(3.3)\begin{equation} \dot{D} = \boldsymbol{\nabla}\boldsymbol{\cdot} \left(\frac{\boldsymbol{\nabla} P_{0}}{12}H^3+\boldsymbol{U}\frac{H}{2}\right). \end{equation}

This equation is identical to the one for a perfectly rigid, flat and no-slip boundary. Since the zeroth-order pressure field results from linear terms in velocity, we decompose it azimuthally as

(3.4)\begin{equation} P_0(\boldsymbol{R},T) = P_{00}(R,T)+P_{01}(R,T)\cos\theta, \end{equation}

with $R$ and $\theta$ the horizontal polar coordinates of $\boldsymbol {R}$. Assuming a vanishing pressure field at large $R$, and that it must remain finite and single valued at $R=0$, the equations can be solved to give the zeroth-order components of the excess pressure field as (O'Neill & Stewartson Reference O'Neill and Stewartson1967)

(3.5)$$\begin{gather} P_{00} ={-}\frac{3\dot{D}}{2(D+R^2)^2}, \end{gather}$$

(3.6)$$\begin{gather}P_{01} = \frac{6UR}{5(D+R^2)^2}. \end{gather}$$

3.2. Interface deflection

We now decompose the first-order deflection of the interface azimuthally as

(3.7)\begin{equation} \varDelta_1(\boldsymbol{R},T) = \varDelta_{10}(R,T)+\varDelta_{11}(R,T)\cos\theta. \end{equation}

Writing (2.12) at first order in $\kappa$, one has

(3.8)\begin{equation} \frac{1}{R}\frac{\partial}{\partial R}\left(R\frac{\partial \varDelta_{10}}{\partial R}\right)- {Bo}\varDelta_{10}=P_{00}, \end{equation}

and

(3.9)\begin{equation} \frac{1}{R}\frac{\partial}{\partial R}\left(R\frac{\partial \varDelta_{11}}{\partial R}\right)- \frac{\varDelta_{11}}{R^2}-{Bo}\varDelta_{11}=P_{01}, \end{equation}

with the boundary conditions $\varDelta _{1i}\rightarrow 0$ as $R\rightarrow \infty$, and finite and single-valued $\varDelta _{1i}$ at $R=0$. The solutions of the above equations can be written in the most general form as

(3.10)\begin{align} \varDelta_{10}(R) &={-}I_0(\sqrt{Bo}R)\int_R^\infty K_0(\sqrt{Bo}\xi)\xi P_{00}(\xi)\,\textrm{d}\kern0.7pt \xi \nonumber\\ &\quad -K_0(\sqrt{Bo}R)\int_0^R I_0(\sqrt{Bo}\xi)\xi P_{00}(\xi)\,\textrm{d}\kern0.7pt \xi, \end{align}

(3.11)\begin{align} \varDelta_{11}(R) &={-}I_1(\sqrt{Bo}R)\int_R^\infty K_1(\sqrt{Bo}\xi)\xi P_{01}(\xi)\,\textrm{d}\kern0.7pt \xi \nonumber\\ &\quad -K_1(\sqrt{Bo}R)\int_0^R I_1(\sqrt{Bo}\xi)\xi P_{01}(\xi)\,\textrm{d}\kern0.7pt \xi, \end{align}

where $I_j$ and $K_j$ denote the $j$th-order modified Bessel functions of the first and second kinds, respectively.

To understand the parametric influence of the Bond number ${Bo}$, we explore the behaviours of the interface deflection for both small and large ${Bo}$ values. For vanishing ${Bo}$, the anisotropic deflection component $\varDelta _{11}$ reaches a limiting behaviour given by the following expression:

(3.12)\begin{equation} \varDelta_{11} \simeq{-}\frac{3U}{10}\frac{\ln\left(1+\dfrac{R^2}{D}\right)}{R}. \end{equation}

In contrast, the isotropic deflection component $\varDelta _{10}$ does not show any limiting behaviour at vanishing ${Bo}$, and a reduction in ${Bo}$ leads to an unbounded increase in $\varDelta _{10}$. To understand the behaviour of $\varDelta _{10}$ as ${Bo}\rightarrow 0$, we take an asymptotic approach previously used in problems relating to capillary deformations (James Reference James1974; Lo Reference Lo1983; Dupré de Baubigny et al. Reference Dupré de Baubigny, Benzaquen, Fabié, Delmas, Aimé, Legros and Ondarçuhu2015). The vanishing of ${Bo}$ allows for a scale separation in the radial coordinate $R$ into: (i) an inner region controlled by surface tension; and (ii) an outer region, where gravity is present. In the inner region ($R\ll 1/\sqrt {Bo}$), the interface deformation denoted by $\varDelta _{10}^{i}$ satisfies

(3.13)\begin{equation} \frac{1}{R}\frac{\partial}{\partial R}\left(R\frac{\partial \varDelta_{10}^{i}}{\partial R}\right)=P_{00}. \end{equation}

The general solution of the above equation reads

(3.14)\begin{equation} \varDelta_{10}^{i} = \mathcal{A}-\frac{3\dot{D}}{8D}\ln(D+R^2), \end{equation}

where $\mathcal {A}$ is an integration constant. The far-field behaviour of the inner solution reads

(3.15)\begin{equation} \varDelta_{10}^{i} \sim \mathcal{A}-\frac{3\dot{D}}{4D}\ln(R). \end{equation}

In the outer region ($R \gg 1/\sqrt {Bo}$), gravity matters but there is no lubrication pressure, which leads to the following governing equation for the interface deformation denoted by $\varDelta _{10}^{o}$:

(3.16)\begin{equation} \frac{1}{R}\frac{\partial}{\partial R}\left(R\frac{\partial \varDelta_{10}^{o}}{\partial R}\right)- {Bo}\varDelta_{10}^{o}=0. \end{equation}

The latter equation is solved, along with the condition that the deflection vanishes at infinity, leading to

(3.17)\begin{equation} \varDelta_{10}^{o} = \mathcal{B}K_0(\sqrt{Bo}R), \end{equation}

where $\mathcal {B}$ is an integration constant. The near-field behaviour of the outer solution reads

(3.18)\begin{equation} \varDelta_{10}^{o} \sim{-}\mathcal{B}\left(\gamma-\ln 2+\frac{\ln {Bo}}{2}+\ln R\right), \end{equation}

where $\gamma$ is Euler's constant. Matching (3.15) with (3.18) leads to

(3.19)$$\begin{gather} \mathcal{A} ={-}\frac{3\dot{D}}{4D}\left(\gamma-\ln 2+\frac{\ln {Bo}}{2}\right), \end{gather}$$

(3.20)$$\begin{gather}\mathcal{B} = \frac{3\dot{D}}{4D}. \end{gather}$$

Substituting these constants in (3.14) and (3.17), we find the matched asymptotes of the interface deflection at small ${Bo}$. The interface deflection can then be approximated by the matched cross-over expression

(3.21)\begin{equation} \varDelta_{10}|_{{Bo}\rightarrow0} \approx \frac{3\dot{D}}{4D} \left[K_0(\sqrt{Bo}R)+\frac12\ln\left(\frac{R^2}{D+R^2}\right)\right].\end{equation}

The interface deflection is shown in figure 2 at a fixed low value of ${Bo}$. The cross-over expression at small ${Bo}$ described above matches the exact one calculated using (3.10), with improving precision as ${Bo}$ reduces. Figure 2 also shows the inner and outer solutions, which diverge in the far and near fields, respectively.

Figure 2.

Isotropic component $\varDelta _{10}$ of the amplitude of the first-order interface deflection as a function of the radial coordinate $R$ (solid black line), as calculated from (3.10), for ${Bo} = 0.01$, $D = 1$ and $\dot {D} = 1$. For comparison, the inner solution (red solid line), the outer solution (blue solid line) and the matched cross-over expression (symbols), from (3.14), (3.17) and (3.21), respectively, are also shown.

The other interesting limit arises when ${Bo}\rightarrow \infty$, leading to the curvature-related terms in (3.8)–(3.9) dropping out, giving us

(3.22)$$\begin{gather} \varDelta_{10} ={-}\frac{P_{00}}{Bo} = \frac{3\dot{D}}{2{Bo}(D+R^2)^2}, \end{gather}$$

(3.23)$$\begin{gather}\varDelta_{11} ={-}\frac{P_{01}}{Bo} ={-}\frac{6UR}{5{Bo}(D+R^2)^2}. \end{gather}$$

As a consequence, the deflection is directly proportional to the pressure applied, with $\kappa /{Bo} = \eta c l/[g{d^*}^3(\rho _{sl}-\rho )]$ as an effective compliance. From the latter, we see that in the limit of large ${Bo}$ the surface tension no longer controls the deformation. This large-${Bo}$ response is akin to the Winkler response for thin compressible elastic materials, which has been studied previously (Skotheim & Mahadevan Reference Skotheim and Mahadevan2005; Urzay et al. Reference Urzay, Llewellyn Smith and Glover2007; Salez & Mahadevan Reference Salez and Mahadevan2015; Chandler & Vella Reference Chandler and Vella2020; Bertin et al. Reference Bertin, Amarouchene, Raphael and Salez2022).

Apart from these two extremes, further exploration of the influence of ${Bo}$ can be done using (3.10) and (3.11). Figure 3 shows the isotropic and anisotropic profiles of the amplitude of the first-order interface deflection, for different values of ${Bo}$. We see that capillarity leads to a long-ranged interface deflection whose range and magnitude both decrease with increasing ${Bo}$.

Figure 3.

Isotropic component $\varDelta _{10}$ (a) and anisotropic component $\varDelta _{11}$ (b) of the amplitude of the first-order interface deflection as a function of the radial coordinate $R$, as calculated from (3.10) and (3.11), for $D = 1$, various ${Bo}$ as indicated, and for either $\dot {D} = 1$ (a) or $U=1$ (b). The black solid line denotes the limiting profile for ${Bo} = 0$ in the anisotropic case, as given in (3.12).

3.3. First-order pressure

At first order in $\kappa$, the pressure field involves in particular the squared velocity, and can thus be decomposed as

(3.24)\begin{align} P_1(\boldsymbol{R},T,{Bo},D) = P_{10}(R,T,{Bo},D)+P_{11}(R,T,{Bo},D)\cos \theta+P_{12}(R,T,{Bo},D)\cos2\theta. \end{align}

The governing equations for the components $P_{1i}$ of the first-order magnitude $P_1$ of the excess pressure field can be derived by considering (2.11) at first order in $\kappa$. Since $P_{12}$ is not needed to compute the forces and torques, we restrict ourselves to the two following equations:

(3.25)\begin{align} \frac{1}{R}\frac{\partial}{\partial R}\left(RH^3\frac{\partial P_{10}}{\partial R}\right) &= \frac{1}{R}\frac{\partial}{\partial R}\left[3RH^2\left(\varDelta_{10} \frac{\partial P_{00}}{\partial R}+\frac{\varDelta_{11}}{2} \frac{\partial P_{01}}{\partial R}\right)\right] \nonumber\\ &\quad +3U\frac{\partial\varDelta_{11}}{\partial R} +3U \frac{\varDelta_{11}}{R}-12\frac{\partial \varDelta_{10}}{\partial T}, \end{align}

(3.26)\begin{align} \frac{1}{R}\frac{\partial}{\partial R}\left(RH^3\frac{\partial P_{11}}{\partial R}\right)- \frac{H^3}{R^2}P_{11} &= \frac{1}{R}\frac{\partial}{\partial R}\left[3RH^2\left(\varDelta_{10} \frac{\partial P_{01}}{\partial R}+\varDelta_{11}\frac{\partial P_{00}}{\partial R}\right)\right] \nonumber\\ &\quad -\frac{3H^2\varDelta_{10}P_{01}}{R^2}+6U\frac{\partial\varDelta_{10}}{\partial R}-12\frac{\partial \varDelta_{11}}{\partial T}. \end{align}

Using the linearity of the equations above, the components of the pressure field can be expressed as follows:

(3.27)$$\begin{gather} P_{10} = \frac{U^2}{D^2}\phi_{U^2}+\frac{\dot{D}^2}{D^3}\phi_{\dot{D}^2}+ \frac{\ddot{D}}{D^2}\phi_{\ddot{D}}, \end{gather}$$

(3.28)$$\begin{gather}P_{11} = \frac{\dot{U}}{D}\phi_{\dot{U}}+\frac{U\dot{D}}{D^2}\phi_{U\dot{D}}, \end{gather}$$

where the $\phi _k$ represent the auxiliary functions for the corresponding second-order forcing parameters $k$, such as $U^2$ etc., which all vary with distance $R$ and depend upon the values of $D$ and ${Bo}$. These functions can then be evaluated by numerically solving (3.25)–(3.26), together with the boundary conditions that the pressure vanishes in the far field and remains finite and single valued at the origin. The results are shown in figures 4 and 5. For all components, as ${Bo}$ increases, the magnitudes of the auxiliary functions decrease due to the decreasing interface deflection. Interestingly, even though the deflection studied above is long ranged, the pressure field decays quite rapidly.

Figure 4.

Auxiliary functions $\phi _k$ (see (3.27)) of the isotropic component $P_{10}$ of the first-order magnitude $P_1$ of the excess pressure field, as functions of the radial coordinate $R$, for various Bond numbers ${Bo}$, as obtained by numerically solving (3.25) with $D = 1$.

Figure 5.

Auxiliary functions $\phi _k$ (see (3.28)) of the anisotropic component $P_{11}$ of the first-order magnitude $P_1$ of the excess pressure field, as functions of the radial coordinate $R$, for various Bond numbers ${Bo}$, as obtained by numerically solving (3.26) with $D = 1$.

4. Capillary-lubrication forces and torques

Since the zeroth-order forces and torques acting on the sphere are identical to the known ones for the motion near a rigid, flat and no-slip boundary, we focus here on the first-order forces and torques acting on the sphere, and resulting from the interface deflection. These can be calculated from the stress tensor $\boldsymbol {\varSigma }$, that reads in the lubrication approximation: $\boldsymbol {\varSigma } \simeq -p\boldsymbol {I}+\eta \boldsymbol {e}_z\partial _z\boldsymbol {v}$, where $\boldsymbol {I}$ denotes the identity tensor. In dimensional units, the first-order vertical force acting on the sphere can be evaluated as

(4.1)\begin{equation} f_{z,1} \simeq \frac{2^{5/2}{\rm \pi}\kappa\eta ca^{3/2}}{{d^*}^{1/2}}\int_0^{\infty} P_{10}(R) R\,\textrm{d}R, \end{equation}

which can be decomposed using the auxiliary functions calculated before as

(4.2)\begin{equation} f_{z,1} \simeq \alpha_{U^2}\frac{\eta^2 u^2 a^3}{\sigma d^2}-\alpha_{\dot{D}^2} \frac{\eta^2 \dot{d}^2 a^4}{\sigma d^3}+\alpha_{\ddot{D}}\frac{\eta^2 \ddot{d} a^4}{\sigma d^3}, \end{equation}

where the $\alpha _k$ (with $k$ indicating here the forcing source, such as $U^2$) are the prefactors of the respective scalings. These prefactors are plotted in figure 6 as functions of ${Bo}$. An important difference arises between the various prefactors at small ${Bo}$. Indeed the prefactor $\alpha _{U^2}$ reaches a plateau, whereas $\alpha _{\ddot {D}}$ and $\alpha _{\dot {D}^2}$ increase logarithmically with decreasing ${Bo}$. These asymptotic behaviours at small ${Bo}$ can be calculated using the Lorentz reciprocal theorem (Daddi-Moussa-Ider et al. Reference Daddi-Moussa-Ider, Rallabandi, Gekle and Stone2018; Rallabandi et al. Reference Rallabandi, Oppenheimer, Zion and Stone2018; Masoud & Stone Reference Masoud and Stone2019), by invoking as well (3.12) and (3.21), as detailed in Appendix A and summarized in table 1. They are in agreement with the numerical solutions, as shown in figure 6. Besides, as ${Bo}$ increases, all the prefactors decrease, and this decrease becomes inversely proportional to ${Bo}$ in the large-${Bo}$ limit, highlighting the transition to the Winkler-like regime. The corresponding asymptotic expressions have been calculated previously (Bertin et al. Reference Bertin, Amarouchene, Raphael and Salez2022), are summarized in table 1, and are in agreement with the numerical solutions, as shown in figure 6.

Figure 6.

Prefactors $\alpha _k$ of the scalings of the vertical-force terms, defined in (4.2), as functions of Bond number ${Bo}$. The solid and dashed lines correspond to the small- and large-${Bo}$ behaviours (see table 1), respectively.

Table 1.

Asymptotic behaviours of the scaling prefactors $\alpha _k$ and $\beta _k$ at small (see Appendix A) and large (see Bertin et al. Reference Bertin, Amarouchene, Raphael and Salez2022) Bond numbers ${Bo}$, where $\gamma$ denotes Euler's constant.

Similarly, the horizontal force acting on the sphere is given by the expression (Bertin et al. Reference Bertin, Amarouchene, Raphael and Salez2022)

(4.3)\begin{align} f_{x,1} &\simeq 2{\rm \pi}\eta c a\kappa\int_0^\infty \left[{-}2RP_{11}-\frac{H}{2}\left(\partial_R P_{11}+\frac{P_{11}}{R}\right)+\frac{\varDelta_{11}}{2}\partial_R P_{00} \right. \nonumber\\ &\quad \left. +\frac{\varDelta_{10}}{2}\left(\partial_RP_{01}+\frac{P_{01}}{R}\right)-2 \frac{U\varDelta_{10}}{H^2}\right]R\,\textrm{d}R, \end{align}

which can be decomposed into

(4.4)\begin{equation} f_{x,1} \simeq{-}\beta_{U\dot{D}}\frac{\eta^2 u \dot{d}a^3}{\sigma d^2}+\beta_{\dot{U}} \frac{\eta^2 \dot{u}a^3}{\sigma d},\end{equation}

where the $\beta _k$ (with $k$ indicating here the forcing source, such as $\dot {U}$) are the prefactors of the respective scalings. These prefactors are plotted in figure 7 as functions of ${Bo}$. An important difference arises once again between the two prefactors at small ${Bo}$. Indeed, the prefactor $\beta _{\dot {U}}$ reaches a plateau, whereas $\beta _{U\dot {D}}$ increases logarithmically with decreasing ${Bo}$. These asymptotic behaviours at small ${Bo}$ can be once again calculated using the Lorentz reciprocal theorem, as detailed in Appendix A and summarized in table 1. They are in agreement with the numerical solutions, as shown in figure 7. Besides, as ${Bo}$ increases, all the prefactors decrease, and this decrease becomes inversely proportional to ${Bo}$ in the large-${Bo}$ limit, highlighting once again the transition to the Winkler-like regime. The corresponding asymptotic expressions have been calculated previously (Bertin et al. Reference Bertin, Amarouchene, Raphael and Salez2022), are summarized in table 1 and are in agreement with the numerical solutions, as shown in figure 7.

Figure 7.

Prefactors $\beta _k$ of the scalings of the horizontal-force terms, defined in (4.4), as functions of Bond number ${Bo}$. The solid and dashed lines correspond to the small- and large-${Bo}$ behaviours (see table 1), respectively.

As shown in previous studies (Salez & Mahadevan Reference Salez and Mahadevan2015; Bertin et al. Reference Bertin, Amarouchene, Raphael and Salez2022), the contributions to the first-order torque felt by the sphere along the $y$-axis have the same numerical prefactors as those for the first-order horizontal force, with the inclusion of a supplementary length-scale factor $a$. Hence, the first-order torque exerted on the sphere is given by

(4.5)\begin{equation} \tau_{y,1} \simeq \beta_{U\dot{D}}\frac{\eta^2 u \dot{d}a^4}{\sigma d^2}-\beta_{\dot{U}} \frac{\eta^2 \dot{u}a^4}{\sigma d}.\end{equation}

We conclude this whole section with an important remark. The cross-over from the capillary-dominated to the Winkler-like regime occurs at ${Bo} \sim 1$. This occurs when the hydrodynamic radius is comparable to the capillary length. Since typical capillary lengths are of the order of $\sim$1 mm, and given the lubrication conditions, such a cross-over can only be felt with spheres of millimetric radii and above.