4.1.1. Thin film contact region (outer solution)

In the outer region ($0< Z < 1$) the term multiplying $\lambda ^{1/2}$ can be neglected in (4.1), so the film thickness $H(Z)$ can be found by solving the algebraic equation

(4.4)\begin{equation} \frac{H - H^*}{H^3} = S(Z). \end{equation}

If we approximate $\varDelta$ by a parabola (the ‘parabolic approximation’), and since $\varDelta (0) = \varDelta (1) = 0$, as well as the maximum thickness being normalized to unity, we obtain the universal shape

(4.5)\begin{equation} \varDelta = 4Z(1-Z) . \end{equation}

As a result, the dimensionless slopes at the entry become $S_0=4$, and $S_1=-4$ at the exit.

Observe that $H = H^*$ when $S = 0$; thus, $H^*$ is identified as the film thickness at the maximum radial extent of the intruder. Solutions to the cubic equation (4.4) are of the form $H(Z) = f(S(Z); H^*)$, illustrated in figure 3(a). Roots of the cubic equation have to be chosen such that $H(Z)$ remains continuous, with $H=H^*$ at the maximum, and physically meaningful solutions can only be found for a finite range of combinations of $S(Z)$ and $H^*$.

Figure 3.

(a) The solution $H(Z)$ of (4.4) for the generic parabolic intruder shape (4.5); $H=H^*$ for the maximum of $\varDelta$, at $Z = 1/2$. (b) The local film thickness $H$ as a function of $H^*$ for $S = S_0 > 0$ (red) and $S = S_1 < 0$ (black), as constrained by the outer solution (4.4). The dashed line $H = H^*$ corresponds to $S = 0$. The space of possible solutions sweeps from the red curve to the black curve across the intruder as the slope changes from $S_0$ to $S_1$. Continuity with the asymptote $H (S \rightarrow 0) \rightarrow H^*$ implies that the upper branch (crosses) is unphysical. Thus, the outer solution restricts $H \leq 1/\sqrt {3 S_0}$ and $H^* < 2/(3\sqrt {3 S_0})$.

To illustrate this point, it is convenient to think of $H^*$ as the dependent variable; rearranging (4.4) gives

(4.6)\begin{equation} H^* = H (1 - H^2 S), \end{equation}

where we recall that $H$ and $S$ are functions of $Z$, but $H^*$ is a constant that characterizes the entire solution. As shown in figure 3(b), (4.6) describes two types of curve depending on the sign of $S$. At the maximum, the two solutions have to be joined together continuously. For regions where $S < 0$, $H^*$ grows monotonically with $H$, while for any value $S > 0$, $H^*$ is only positive for $H$ in the range $0 < H \leq 1/{\sqrt {S}}$, attaining a maximum value of $2/(3\sqrt {3S})$ when $H = 1/\sqrt {3 S}$. Because $H^*$ is a constant, the largest permissible value of $H^*$ is set by the maximum positive value of the intruder slope $S(Z)$ over the dry contact region. If the intruder shape is convex, this maximum slope occurs at the entrance $Z=0$, and we obtain the upper bound (see figure 3)

(4.7)\begin{equation} H^* \leq \frac{2}{3 \sqrt{3 S_0}}, \quad \text{where}\ S_0 \equiv S(Z = 0). \end{equation}

This condition must be satisfied for (4.6) to admit solutions throughout the ‘outer’ region $0< Z<1$.

Furthermore, since the local thickness $H(Z)$ must approach $H^*$ as $S(Z) \rightarrow 0$, only the solution branch $H < 1/\sqrt {3 S}$ is physically meaningful (figure 3b), leading to the upper bound

(4.8)\begin{equation} H(Z) \leq \frac{1}{\sqrt{3 S_0}}. \end{equation}

Thus, for convex intruder shapes the slope at the entry region $S_0$ limits the space of possible solutions. This conclusion does not involve any detailed analysis of the entry region, but is rather a consequence of the outer problem. We now show that the structure of the entry region picks a unique value of $H^*$.

4.1.2. Boundary layers at entrance and exit (inner solutions)

Within the contact region, the film thickness is of order $h_s=\lambda ^{1/2}\delta _m$ as noted in § 2, while the typical length scale is $\ell$, and thus gradients in the film thickness are small in the limit of small $\lambda$. However, at the edges of the contact region, located at $Z=0$ and $Z=1$, the film profile has to adapt to the shape of a tube of constant radius, and axial derivatives in (4.1) can no longer be neglected. In the scaling of (4.1), since $S$ and $H$ are of order one, balancing the first and the last term we find that boundary layers are of dimensionless size $\lambda ^{1/2}$. Since both the film thickness and the width of the boundary layers scale like $\lambda ^{1/2}$, the lubrication approximation is not automatically satisfied for small $\lambda$. However, within these thin boundary layers, $\mathrm {d} h/\mathrm {d} z = O(\delta _m/\ell )$, so the lubrication approximation remains valid for a slender intruder.

We first analyse the boundary layer near the entrance to the film $(Z = 0)$ which, as we will show, uniquely determines the dimensionless flux $H^*$ when combined with (4.4). This feature is analogous to many other similar problems, such as the Bretherton problem for the motion of long bubbles in tubes (Bretherton Reference Bretherton1961), or the motion of an elastic object translating next to an elastic wall (Snoeijer et al. Reference Snoeijer, Eggers and Venner2013). The scaling of the film thickness remains unchanged from the outer problem, so we avoid introducing a new ‘inner’ variable for $H$. Rescaling $Z = \lambda ^{1/2} \zeta$, we obtain the similarity equation

(4.9)\begin{equation} \frac{\mathrm{d} H}{\mathrm{d} \zeta} - \frac{\left(H - H^*\right)}{H^3} + S_0 = 0 + O(\lambda^{1/2}), \end{equation}

since the slope $S_0 = S(0)$ is approximately constant across the boundary layer.

This equation has to satisfy matching conditions for $\zeta \rightarrow \pm \infty$: towards the film ($\zeta \rightarrow \infty$), where the film thickness is varying slowly, $H$ has to match the film thickness of the outer solution, and thus $H(\zeta \rightarrow \infty ) = H_0 = f(S_0;H^*)$ (cf. Figure 4a). Towards the tube of constant radius, the film thickness increases linearly with the slope of the intruder: $H'(\zeta \rightarrow -\infty ) = -S_0$ (figure 4a). This latter condition is satisfied automatically as $H$ increases towards $\zeta \rightarrow -\infty$, since the second term of (4.9) becomes smaller.

Figure 4.

The self-similar profiles (a) at the entry to the film, located at $Z=0$, and (b) at the back of the film, located at $Z=1$, indicated as solid curves. Towards the left and right in panels (a,b), respectively, the profiles asymptote to the undeformed tube, represented by the dashed lines. The dimensionless slopes are those of the parabolic approximation (4.5).

We now show that the condition as $\zeta \rightarrow \infty$ selects a unique solution to (4.9): substituting $H(\zeta ) \approx H_0 + G(\zeta )$ into (4.9) and linearizing in $G$ we find

(4.10)\begin{equation} \frac{\mathrm{d} G}{\mathrm{d} \zeta} = \frac{1}{H_0}\left(\frac{1}{H_0^2} -3 S_0\right) G \implies G(\zeta) \propto \exp\left\{\frac{1}{H_0} \left(\frac{1}{H_0^2} -3 S_0\right) \zeta\right\}. \end{equation}

For the perturbation $G(\zeta )$ to remain small as $\zeta \gg 1$ requires that $H_0 \geq 1/\sqrt {3 S_0}.$ This condition is consistent with the previously obtained upper bound (4.8) only when

(4.11)\begin{equation} H_0 = \frac{1}{\sqrt{3 S_0}}, \end{equation}

which sets the film thickness at the entrance. Using (4.4) we obtain

(4.12)\begin{equation} H^* = \frac{2}{3 \sqrt{3 S_0}}, \end{equation}

which shows that the boundary layer solution selects the largest possible flux $H^*$ that permits solutions of the outer problem (4.4). Separating variables, and using (4.12), (4.9) can be solved as

(4.13a)\begin{align} \zeta &={-}S_0^{{-}3/2}\int^{S_0^{1/2}H}\frac{G^3\, \mathrm{d} G} {\left(G + 2/\sqrt{3}\right) \left(G - 1/\sqrt{3}\right)^2} \end{align}

(4.13b)\begin{align} &={-}\frac{H}{S_0} + \frac{1}{9 S_0^{2}\left(H - H_0\right)} + \frac{8 }{9\sqrt{3} S_0^{3/2}} \ln\left(\frac{H+2H_0}{H-H_0}\right), \end{align}

which contains $S_0$ as the sole parameter. The constant of integration has been chosen such that $H = -S_0\zeta + O(1/\zeta )$ for $\zeta \rightarrow -\infty$, which is the profile of the undeformed tube. This function is plotted in figure 4(a). Clearly, (4.13a) has the required linear growth for $\zeta \rightarrow -\infty$ (first term on the right-hand side). On the other hand, as $H$ approaches $H_0$ for $\zeta \rightarrow \infty$, the second term on the right-hand side of (4.13a) dominates, and the profile approaches its limiting thickness as a power law $H \sim H_0 + 1/({9 S_0^2\zeta })$.

A boundary layer of size $\lambda ^{1/2}$ also exists near the exit $Z= 1$, where we denote the slope as $S_1 = S(Z=1) < 0$. However, $H^*$ is now fixed, having been selected at the entrance as usual (see also Bretherton (Reference Bretherton1961) and Snoeijer et al. (Reference Snoeijer, Eggers and Venner2013)). Introducing the similarity variable $\xi = (Z - 1)/\lambda ^{1/2}$ at the exit, (4.9) remains the same (with the substitution $S_0 \mapsto S_1$), but the matching conditions are reversed: $H(\xi \rightarrow -\infty ) = H_1 = f(S_1;H^*)$ toward a film on the left, and $H'(\xi \rightarrow \infty ) = S_1$ toward the tube on the right. Linearizing around a constant film thickness $H_1$ as before, we find

(4.14)\begin{equation} H(\zeta) = H_1 + A \exp\left\{ \frac{1}{H_1} \left(\frac{1}{H_1^2} - 3 S_1 \right) \xi\right\} , \quad \text{as}\ \xi \rightarrow - \infty. \end{equation}

Now the prefactor in the exponential is always positive, so for $\xi \rightarrow -\infty$ the solution converges exponentially onto the correct boundary condition (the outer problem), as seen in figure 4(b), regardless of $H^*$. The solution near the exit is given by

(4.15)\begin{equation} \xi = \int^{H}\frac{G^3 \,\mathrm{d} G}{G-H^*-S_1 G^3} , \end{equation}

subject to $H = -S_1\zeta + O(1/\zeta )$ for $\zeta \rightarrow \infty$, which we calculate numerically (figure 4b). The result now depends on the intruder shape at both the entrance and the exit through the values of $S_0$ and $S_1$ separately. As before, numerical solutions of (4.1) near the exit collapse onto the asymptotic solution (4.15) for small $\lambda$. It will turn out that the entry region remains nearly unchanged in the fully coupled problem (3.12), while the exit region is modified quantitatively by membrane tension.

4.1.3. Force on the intruder and tension in the tube

The force on the intruder is obtained by evaluating the integral in (4.3). This calculation simplifies for $\lambda ^{1/2} \ll 1$ since its main contribution comes from the thin film ‘contact’ region $0< Z<1$, with the contribution from boundary layers being $O(\lambda ^{1/2})$ smaller. Thus, we can approximate

(4.16)\begin{equation} \mathcal{F} = \lambda^{1/2}\int_{0}^{1} \frac{4 H - 3 H^*}{6 H^2} \,\mathrm{d} Z + O(\lambda). \end{equation}

Within the limits of the above integral, $H(Z)$ is obtained by solving the cubic equation (4.4), where $S(Z)$ is given by the shape of the intruder. Note that at low speeds contributions to the force only come from the contact region, where the lubrication approximation is valid even without assuming a slender intruder.

Equation (4.16) must be integrated numerically in general, though for the parabolic approximation (4.5) (so $S_0 = 4$) the integral can be evaluated analytically by transforming to $H$ as the integration variable, to give $\mathcal {F} \simeq 0.873 \lambda ^{1/2}$. For a more general shape, a convenient approximation for $\mathcal {F}$ is obtained by observing that near the middle of the thin film (where $S$ is small), $H \approx H^*$ (the profile $H(Z)$ is approximated by the dashed line in figure 3a). With this approximation, the integral (4.16) becomes trivial, and using (4.12), results in

(4.17)\begin{equation} \mathcal{F} \simeq \frac{\sqrt{3}}{4}(\lambda S_0)^{1/2} \ \text{for}\ \lambda^{1/2} \ll 1. \end{equation}

For the parabolic intruder, this approximation yields $\mathcal {F} \simeq 0.866 \lambda ^{1/2}$, which is remarkably close to the exact result ($\approx 0.8\,\%$ error). This surprisingly simple expression (4.17) predicts that the dimensionless force depends solely on the slope of the intruder surface at the entrance region. The dependence of $\mathcal {F}$ on local slope $S_0$ at the entry despite shear stresses being significant over the entire contact region ($0 < Z < 1$) is due to the fact that the entrance selects the fluid flux $H^* \propto S_0^{-1/2}$ through the film, see (4.12). Later we will show that the simple approximation (4.17) is robust to changes in intruder shape as long as it remains slender.

4.2.1. Symmetric intruders

We first focus on the generic slender axisymmetric intruder ($\delta _m/\ell \ll 1$), which is also fore-and-aft symmetric (e.g. a prolate ellipsoid) and can be approximated by the quadratic function (4.5) with $S_0 = \varDelta '(0) = 4$. For an ellipsoid of axial half-width $R_z$ (the half-width in the radial direction is $a + \delta _m$), the dry contact length is $\ell \simeq 2 \sqrt {2R_z^2 \delta _m/a}$ in the limit $\delta _m \ll a$.

Figure 5(a) shows numerical results of lubrication theory (4.1) for the normalized radial displacement of the tube $U(Z) = u_r(z)/\delta _m = \varDelta (Z) + \lambda ^{1/2} H(Z)$ for this shape. The tube deformation relaxes over a length scale that increases with increasing $\lambda$ corresponding, for example, to faster translation speeds or softer tubes. As $\lambda \rightarrow 0$, the solution adopts a boundary-layer structure with a dimensional axial width $\lambda ^{1/2} \ell$, as expected from § 4.1. Solutions for the film near the entrance $(Z = 0)$ and exit $(Z = 1)$ regions are shown as functions of the rescaled boundary layer coordinates in figure 5(b) and figure 5(c), respectively. Numerical solutions are found to collapse onto the theoretical predictions (4.13a) and (4.15) for the entry and exit regions for values of $\lambda \lesssim 1$. The qualitative structure of the entry and exit layers remains unchanged even when $\lambda \gtrsim 1$.

Figure 5.

The film shape for an intruder with parabolic shape (4.5) within the hoop stress approximation. The object translates leftward relative to the tube. (a) Numerical results from solving (4.1) for the normalized radial displacement $U(Z)$ of the tube. The tube deformation relaxes over a length scale that increases with $\lambda$. (b,c) Profiles of film thickness $H$ in the boundary layers near the entry region (b) and exit region (c), where numerical solutions (symbols) and theory (solid lines) are shown. Observe that the film thickness in the dry contact region $0< Z<1$ is relatively insensitive to $\lambda$.

We now test the dependence of the solution on $\lambda$ more systematically, and over a very wide range. We will see below that even in the limit $\lambda \gg 1$ lubrication theory can be used to calculate the force. As seen in figure 6(a), the dimensionless fluid film thickness $H^*$ (thickness at the maximum of the intruder), is relatively insensitive to $\lambda$, even over several orders of magnitude. The asymptotic result $H^* = 1/{(3 \sqrt {3})}$ from (4.12) for $\lambda \ll 1$ (note that $S_0 = \varDelta '(0) = 4$ for the chosen intruder shape) is indicated by the horizontal line.

Figure 6.

Numerical results from solving (4.1). (a) Rescaled flux $H^*$ versus $\lambda$ for the generic symmetric intruder (4.5), showing the asymptotic behaviour $H^* = 1/{(3 \sqrt {3})}$ at small $\lambda$ (solid line) and the scaling behaviour $H^* \propto \lambda ^{-1/10}$ at large $\lambda$. (b) Rescaled force $\mathcal {F}$ as a function of $\lambda$, showing the small-$\lambda$ asymptotic results (4.17) (solid line) and the scaling laws $\mathcal {F} \propto \lambda ^{2/3}$ and $\mathcal {F} \propto \lambda ^{4/5}$ for intermediate and large $\lambda$, respectively. (c) Dimensionless tension $\mathcal {T}(Z)$ for different values of $\lambda$. The limiting value for $Z \to \infty$ is the dimensionless force $\mathcal {F}$. The dashed line is $\varDelta ^2/2$ and corresponds to the tension over the intruder in the dry limit $\lambda \to 0$.

When $\lambda \gg 1$, both the axial length scale and the film thickness become much greater than the geometric scales imposed by the intruder, a trend which is already apparent in figure 5(a). At large axial distances from a paraboloidal intruder the surface slope grows as $S(Z) = \varDelta '(Z) \sim 4Z$. Requiring a balance of all three terms of (4.1) far away from the intruder reveals a film thickness $H = O(\lambda ^{-1/10})$ with variations over a characteristic axial scale $Z = O(\lambda ^{1/5})$ (i.e. $h = O(\delta _m \lambda ^{2/5})$, $z = O(\ell \lambda ^{1/5})$). This scaling law for $H$ is reflected in the behaviour of $H^*$ for $\lambda \gtrsim 10^2$, indicated in figure 6(a). Note that in this limit $\mathrm {d} h/\mathrm {d} {z} = O(\lambda ^{1/5}\delta _m/ \ell )$, which is assumed small in the theory, so the intruder must be sufficiently slender for the thin-film and the hoop-stress approximations to remain valid at large $\lambda$. Note also that the paraboloidal approximation of the surface shape is valid over axial length scales $Z \lesssim R_z/\ell$, so the above scaling relations for large $\lambda$ are valid if $\lambda ^{1/5} \lesssim R_z/\ell$.

Using (4.17), the dimensionless force on the generic symmetric intruder is approximately

(4.18)\begin{equation} \mathcal{F} \simeq \left(3 \lambda/4\right)^{1/2} \quad \text{or} \quad F \simeq 6 {\rm \pi}\left( \mu V E b\right)^{1/2} (2R_z^2 \delta_m^3/a)^{1/4} , \quad \text{for}\ \lambda \ll 1. \end{equation}

Figure 6(b) shows numerical results (symbols) for the dimensionless force on the intruder, with the prediction of (4.18) indicated as a solid line, showing good agreement for $\lambda \lesssim 0.1$.

At intermediate $\lambda \gtrsim 1$, the axial length is still set by the intruder geometry, while the film thickness becomes comparable to $\delta _{m}$ (recall that the characteristic film thickness is $h_s = \lambda ^{1/2} \delta _m$). A balance between the first two terms in (4.1) with $Z = O(1)$ determines $H = O(\lambda ^{-1/6})$. Then, the force integral (4.3) yields the scaling estimate

(4.19)\begin{equation} \mathcal{F} \propto \lambda^{2/3} \quad \text{or} \quad F \propto (E b \delta_m R_z^2)^{1/3} (\mu V)^{2/3} , \quad \text{for}\ \lambda \gtrsim 1. \end{equation}

This approximate scaling law at intermediate $\lambda$ is indicated in figure 6(b), and accurately represents the numerical results in the range $0.1 \lesssim \lambda \lesssim 10$.

For $\lambda \gg 1$, the characteristic scales $Z = O(\lambda ^{1/2})$ and $H = O(\lambda ^{-1/10})$, obtained earlier, yield the scaling law

(4.20)\begin{equation} \mathcal{F} \propto \lambda^{4/5} \quad \text{or} \quad F \propto (\mu V)^{4/5} (E b R_z^4 a)^{1/5}, \quad \text{for}\ \lambda \gg 1, \end{equation}

when substituted into (3.13). This prediction is consistent with numerical calculations for $\lambda \gtrsim 10^2$, as indicated in figure 6(b). Since the force grows sublinearly in (4.20), this is subdominant to the contribution (3.14) coming from the pipe, in the limit of large $\lambda$. Whether the contribution to the force (3.14) from the Poiseuille flow can in fact be neglected has to be checked in the individual cases. The forgoing results are summarized in table 1.

Table 1.

Summary of scaling laws in different ranges of $\lambda$ (see (4.1)) for a generic fore-and-aft-symmetric intruder with axial curvature $1/R_z$ and a maximum radius $a + \delta _{m}$. The shape of such an intruder is $\varDelta (Z) = 4 Z (1- Z)$. The results are valid for $\delta _{m} \ll \ell$, where $\ell = 2\sqrt {2 \delta _{m} R_z^2/a\,}$. Prefactors for $\lambda \ll 1$ are provided by the asymptotic theory.

The membrane tension profiles $\mathcal {T}(Z)$, obtained by integrating (4.2) subject to the stress-free condition at the left-hand end ($\mathcal {T}_{-\infty } = 0$), are shown in figure 6(c) for a few different $\lambda$. In the laboratory frame, the intruder moves leftward and the tube is fixed at the right-hand end, so the tension approaches zero on the left-hand end of the tube (which is stress free) and approaches a finite value $T_{\infty }$ on the right-hand end. This is made clear in the limit $\lambda ^{1/2} \ll 1$, where $\mathcal {T}(Z<0) \sim \mathcal {T}_{-\infty } = 0$ due to the small $U$ and large $H$. Over the contact region, where $U(Z) \approx \varDelta (Z)$, we find $\mathcal {T} = \varDelta ^2(Z)/2 + O(\lambda ^{1/2})$, which is the only part of the tension that survives in the static dry limit. To the right, however, the tension approaches the drag force on the intruder, having ‘picked up’ the shear stress over the thin-film contact region, so $\mathcal {T}(Z > 0) \sim \mathcal {T}_{\infty } = \mathcal {F} \simeq ({\sqrt {3}}/{4})(\lambda S_0)^{1/2}$. The transition between $T \sim \varDelta ^2/2$ over the film and $T \sim T_{\infty }$ on the right of the object occurs through the boundary layer at the film's exit. This distribution of tension will remain qualitatively intact for the full membrane theory (3.12). Observe that the shear stress, which is the derivative of the tension, is restricted to $0< Z<1$ for $\lambda ^{1/2}$, consistent with the approximation (4.16).

The necessary condition for the thin-film approximation $(\mathrm {d} h/\mathrm {d} z \ll 1)$ to apply is that the intruder is slender: $\delta _m \ll \ell$ for moderate $\lambda$, and the slightly stricter condition $\lambda ^{1/5} \delta _m \ll \ell$ for $\lambda \gtrsim 10^2$. For small $\lambda$, $\mathrm {d} h/\mathrm {d} z$ scales as $\lambda ^{1/2}\delta _m/\ell$ in the outer region and as $\delta _m/\ell$ in the entry and exit regions. This means that the present thin-film analysis of the outer problem for $\lambda ^{1/2} \ll 1$ remains valid even for non-slender intruders even though this approximation is inadequate in the entry and exit regions. Because the force is dominated by the outer problem (see § 4.1.3), we speculate that $\mathcal {F} \propto \lambda ^{1/2}$ may persist for non-slender intruders at small $\lambda$, although the modified flow in the entry region may select a different $H^*$ and, therefore, a different prefactor for $\mathcal {F}$. We note the subinear power-law dependence of the force on the velocity is qualitatively similar to the experimental results of Tani et al. (Reference Tani, Cambau, Bico and Reyssat2017).