2.1. The equation for the free surface

To derive the equation governing the height of the free surface of the fluid, $z = h(x)$ , we apply boundary conditions on the solid and free surfaces:

(2.4a) \begin{align} \text {no slip or penetration on substrate} \quad & (u,w) = (1,0) & \text {on} &\quad z = 0, \end{align}

(2.4b) \begin{align} \text {no slip or penetration on plate} \quad & (u,w) = (0,0) & \text {on} &\quad z = H(x), \end{align}

(2.4c) \begin{align} \text {normal stress balance on free surface} \quad & p = -\delta ^3 h_{xx} & \text {on} &\quad z = h(x), \end{align}

(2.4d) \begin{align} \text {tangential stress balance on free surface} \quad & u_z = 0 & \text {on} &\quad z = h(x), \end{align}

where $\delta ^3 = \epsilon ^3 \gamma /(\mu U) = 1/{Ca}$ is the non-dimensional inverse capillary number previously introduced in (1.2), and subscripts denote partial derivatives. Note that due to the thinness of the film, (2.4c ) involves a linearised expression for the curvature of the free surface.

Classical lubrication (i.e. thin-film) theory now allows us to derive the Reynolds equation for the pressure gradient,

(2.5) \begin{equation} p_x = \frac {3(h - h_{\infty })}{h^3}, \end{equation}

where $h \to h_{\infty }$ as $x \to \infty$ . Substituting this expression for $p_x$ into the normal stress condition (2.4c ) yields the Landau–Levich equation

(2.6) \begin{equation} \delta ^3 h_{xxx} = \frac {3(h_{\infty }-h)}{h^3}, \end{equation}

which governs the height of the free surface.

2.2. The equation for the elastic plate

Equation (2.6) for the height of the free surface must be coupled to an equation governing the height of the elastic plate, $z = H(x)$ , which we now derive. First, let us consider the dimensional force per unit area exerted by the fluid on the plate, $\tilde {\boldsymbol {F}} = (\tilde {F}_{\tilde {x}}, \tilde {F}_{\tilde {z}})$ . As in § 2.1, we make the lubrication approximation and assume that the plate is nearly horizontal. Then the leading order in $\epsilon$ horizontal and vertical forces on the plate are given by

(2.7) \begin{equation} \tilde {F}_{\tilde {x}} = -\tilde {p} \frac {\mathrm {d}\tilde {H}}{\mathrm {d}\tilde {x}} + \mu \frac {\partial \tilde {u}}{\partial \tilde {z}} \quad \text {and} \quad \tilde {F}_{\tilde {z}} = \tilde {p}. \end{equation}

Let the dimensional tension in the plate be $\tilde {T}=\tilde {T}(\tilde {x})$ . By balancing the horizontal forces we have ${\operatorname {d\!}{}}\tilde {T}/{\operatorname {d\!}{}}\tilde {x} = \tilde {F}_{\tilde {x}}$ . Consideration of the surface tension at the end of the plate suggests a re-scaling of $\tilde {T} = \gamma T$ , so using (2.3), we then have

(2.8) \begin{equation} \frac {\mathrm {d}T}{\mathrm {d}x} = \frac {\epsilon ^2}{\delta ^3} \left ( -p\frac {\mathrm {d}H}{\mathrm {d}x} + \frac {\partial u}{\partial z} \right ). \end{equation}

We assume that the aspect ratio is such that $\epsilon ^2 \ll \delta ^3 = 1/{Ca}$ . Under this assumption, the tension is constant at leading order in $\epsilon$ , thus the balance with surface tension at the end of the plate implies that $T \equiv 1$ .

We let $\tilde {N}=\tilde {N}(\tilde {x})$ be the dimensional transverse shear force and let $\tilde {M} = \tilde {M}(\tilde {x})$ be the dimensional moment. The normal force and moment balances on the elastic plate are given, respectively, by

(2.9a) \begin{align} \frac {\mathrm {d}\tilde {N}}{\mathrm {d}\tilde {x}} + \tilde {T} \frac {\mathrm {d}^2\tilde {H}}{\mathrm {d}\tilde {x}^2} + \tilde {p} &= 0, \end{align}

(2.9b) \begin{align} \frac {\mathrm {d}\tilde {M}}{\mathrm {d}\tilde {x}} = \tilde {N} \end{align}

(see, for example, Howell et al. Reference Howell, Kozyreff and Ockendon2009). Furthermore, for small displacements, we assume the constitutive relation

(2.10) \begin{equation} \tilde {M} = -B \frac {\mathrm {d}^2\tilde {H}}{\mathrm {d}\tilde {x}^2}, \end{equation}

where $B = EI$ is the bending stiffness, in which $E$ is the elastic modulus and $I$ is the area moment of inertia. Recalling that $T \equiv 1$ , we now differentiate (2.4c ) with respect to $\tilde {x}$ , and use (2.3), the Reynolds equation (2.5), the constitutive relation (2.10), and $T \equiv 1$ to obtain the equation governing the height of the plate, namely

(2.11) \begin{equation} \mathcal {B}^5 \frac {\mathrm {d}^5 H}{\mathrm {d}x^5} - \delta ^3 \frac {\mathrm {d}^3 H}{\mathrm {d}x^3} = \mathcal {R}(H; h_{\infty }), \end{equation}

where, for later convenience, we have defined the function

(2.12) \begin{equation} \mathcal {R}(H; h_{\infty }) = \frac {6(H - 2h_{\infty })}{H^3}, \end{equation}

and $\mathcal {B}^5 = \epsilon ^3 B/(\mu U L^2)$ is the non-dimensional elasticity number previously introduced in (1.4).

Turning now to the boundary conditions at the pinned ( $x=0$ ) and free ( $x=1$ ) ends of the plate, we recall that from the constitutive relation (2.10), the moment is proportional to the second derivative of $H$ , hence

(2.13a) \begin{align} H(0) & = 1, \end{align}

(2.13b) \begin{align} H_{xx}(0) = 0 & = H_{xx}(1). \end{align}

In light of the fact that the free surface exerts a tangential force due to surface tension at the downstream end of the plate, $x = 1$ , the zero moment condition (2.13b ) is not obvious. We will discuss this condition further in § 2.3.

Examining figure 2(a), which shows quantities defined near the downstream end of the plate, we see that a balance of shear stress at the end of the plate, $x = 1$ , implies the dimensional balance

(2.14) \begin{equation} \tilde {N}(L) + \tilde {T} \big[ \tilde {H}_{\tilde {x}}(L) - \tilde {h}_{\tilde {x}} \big] = 0, \end{equation}

so using (2.3), (2.9b ) and (2.10) gives the shear boundary condition

(2.15) \begin{equation} \mathcal {B}^5 H_{xxx}(1) - \delta ^3 \left [ H_x(1) - h_x(1) \right ] = 0. \end{equation}

Figure 2. (a) Quantities defined near the downstream end of the plate. (b) An illustration of the moment arm due to surface tension forces.

Without loss of generality, we take $\tilde {p}_{{a}} \equiv 0$ . Then the upstream fluid pressure must match the prescribed pressure of the reservoir, $p_0 \equiv p(0)$ , and similarly, the pressure of the fluid at the downstream end of the plate must match the capillary pressure $-\delta ^3 h_{xx}(1)$ . Since the pressure underneath the plate is given by (2.9a ), or alternatively the integral of (2.11), the non-dimensional pressure under the plate is given by

(2.16) \begin{equation} p=\mathcal {B}^5 H_{xxxx} - \delta ^3 H_{xx}. \end{equation}

Then applying the boundary conditions gives

(2.17a) \begin{align} \mathcal {B}^5 H_{xxxx}(0) &= p_0, \end{align}

(2.17b) \begin{align} \mathcal {B}^5 H_{xxxx}(1) &= -\delta ^3 h_{xx}(1). \end{align}

Finally, we require additional details of the far-field behaviour, $h \to h_{\infty }$ as $x \to \infty$ . We linearise about the uniform downstream height by writing $h = h_{\infty } + \overline {h}$ , where $\overline {h} \ll h_{\infty }$ . The resulting modes of the linear equation for $h_{\infty }$ can be obtained using a standard WKBJ (Liouville–Green) analysis (see, for example, Tuck & Schwartz Reference Tuck and Schwartz1990) to yield

(2.18) \begin{equation} \overline {h} \sim C_1 \exp \left ( -\frac {3^{1/3} x}{\delta h_{\infty }} \right ) + C_2 \exp \left ( \frac {3^{1/3}\mathrm {e}^{ \pi i/3} x}{\delta h_{\infty }} \right ) + C_3 \exp \left ( \frac {3^{1/3}\mathrm {e}^{-\pi i/3} x}{\delta h_{\infty }} \right ) \end{equation}

as $x \to \infty$ , where $C_1$ is real and $C_2$ and $C_3$ are complex conjugates. The two exponentially growing modes, which represent capillary waves, are ruled out on physical grounds, so that $C_2 = 0$ and $C_3 = 0$ , leaving only $C_1$ to be determined. Thus we see that, as is typical for problems of this type, the downstream condition $h \to h_{\infty }$ effectively provides two boundary conditions for the third-order Landau–Levich equation (2.6). This completes our derivation of the governing equations and boundary conditions.

2.3. Summary and remarks on the non-trivial nature of the mathematical formulation

Together, we may consider the system of equations and boundary conditions for the pinned elastic problem to correspond to a ‘ninth-order’ problem; the relevant expressions are given in table 1. In particular, the system consists of the third-order Landau–Levich equation for $h(x)$ , the fifth-order beam equation for $H(x)$ , and an unknown eigenvalue corresponding to the far-field film height $h_{\infty }$ . In total, this system requires nine boundary conditions. These nine boundary conditions consist of: [1, 2, 3] fixed height, zero moment, and prescribed pressure at $x = 0$ ; [4, 5, 6, 7] continuity of height, zero moment, continuity of shear stress and pressure at $x = 1$ ; and [8, 9] far-field conditions as $x \to \infty$ .

Table 1. A summary of the ‘ninth-order’ system of equations and boundary conditions for the pinned elastic problem analysed in the present work, comprising a third-order equation for $h$ , a fifth-order equation for $H$ , and an unknown ‘eigenvalue’ $h_\infty$ .

Although we have chosen to derive the governing equations and boundary conditions using local force and moment balances, we note that it is also possible to use a variational approach. Using the small displacement and lubrication approximations, we see that the equilibrium configuration of the plate should extremise the dimensional energy:

(2.19) \begin{align} \mathcal {E} &= \int _0^L \bigg \{ \underbrace {\tfrac {1}{2} \mathcal {B} \tilde {H}_{\tilde {x}\tilde {x}}^2 }_{\mathrm{bending\,energy}}- \underbrace {\tilde {p}\tilde {H}}_{\mathrm{pressure\,energy}} + \underbrace {\tilde {T} \left [ \sqrt {1 + \tilde {H}_{\tilde {x}}^2} - 1\right ]}_{\mathrm{tension\,energy}} \, \bigg \} {\operatorname {d\!}{}} \tilde {x} \nonumber\\ &\quad + \int _L^\infty \biggl \{ \underbrace { \gamma \left [ \sqrt {1 + \tilde {h}_{\tilde {x}}^2} - 1 \right ]}_{\mathrm{surface\,tension\,energy}} - \underbrace {\tilde {p}\tilde {h}}_{\mathrm{pressure\,energy}} \, \biggr \} {\operatorname {d\!}{}} \tilde {x}. \end{align}

It can then be verified that the equations and boundary conditions given in table 1 can be recovered if we require that the first variation of $\mathcal {E}$ given by (2.19) is zero.

We make some remarks about the non-trivial nature of the mathematical formulation just presented. In the statement of the zero moment condition (2.13b ), we indicated that it is not entirely obvious that this condition guarantees that the plate is in rotational equilibrium. Although the conditions emerge naturally from application of the variational approach described above, we can directly verify that the dimensional moment about $\tilde {x} = 0$ is zero, i.e. $\tilde {M}_0 =0$ . This moment has a pressure contribution and a contribution from surface tension forces, and is given by

(2.20) \begin{equation} \tilde {M}_0 = \gamma d + \int _0^L \tilde {x}\, \tilde {p}(\tilde {x}) \, {\operatorname {d\!}{}}{\tilde {x}}, \end{equation}

where $d$ is the length of the moment arm due to the surface tension forces (i.e. the perpendicular distance from $(0, \tilde {H}_0)$ to the line parallel to the tension force), as shown in figure 2(b). For small displacements of the plate, this distance is given by

(2.21) \begin{equation} d \sim L \left [ \tilde {h}_{\tilde {x}}(L) + \frac {\tilde {H}_0 - \tilde {H}(L)}{L} \right ]. \end{equation}

We may now substitute the dimensional version of the pressure under the plate given by (2.16) into (2.20), integrate by parts, and simplify using the boundary conditions at $\tilde {x} = 0$ and $\tilde {x}=L$ , to confirm that $\tilde {M}_0 = 0$ .

4.1. The limit of a rigid plate, $\mathcal {B} \to \infty$

In the limit of a rigid plate, $\mathcal {B} \to \infty$ , we assume that there are no boundary layers (notably no boundary layers at $x = 0$ and/or $x = 1$ ), and expand the free surface and plate heights and the far-field film height as

(4.1) \begin{equation} h(x) = \sum _{n=0}^\infty \frac {h_n(x)}{\mathcal {B}^{5n}}, \quad H(x) = \sum _{n=0}^\infty \frac {H_n(x)}{\mathcal {B}^{5n}} \quad \text {and} \quad h_{\infty } = \sum _{n=0}^\infty \frac {h_{\infty n}}{\mathcal {B}^{5n}}. \end{equation}

At leading order, the beam equation (2.11), together with the boundary conditions $H_0(0) = 1$ and $H_{0xx}(0) = 0$ , yields a solution for $H_0=H_0(x)$ corresponding to a straight (but not, in general, horizontal) rigid plate, namely

(4.2) \begin{equation} H_0 = 1 + \alpha x, \end{equation}

where the constant $\alpha$ is the unknown tilt angle. What is noteworthy about this solution is that in order to obtain the values of $\alpha$ and $h_{\infty }$ , the moment and force balances must be applied at $\mathcal {O}(1/\mathcal {B}^5)$ . This occurs because while for a rigid plate, the curvature (and all higher-order derivatives) of $H$ are identically zero, for an elastic plate, these higher-order derivatives are non-zero and are essential in order to satisfy the boundary conditions.

At $\mathcal {O}(1/\mathcal {B}^5)$ , (2.11) yields

(4.3) \begin{equation} H_{1xxxxx} - \delta ^3 H_{0xxx} = \frac {6}{H_0^3} (H_0 - 2h_{\infty 0}). \end{equation}

Integrating (4.3) from $x = 0$ to $x = 1$ , and using the zero moment (2.13b ) and pressure (2.17) boundary conditions yields the leading-order shear condition

(4.4) \begin{equation} F_z^{(0)} = p_0 + \delta ^3\, h_{xx}(1) + \left [6\, I_2(1) - 12h_{\infty 0}\, I_3(1)\right ] = 0, \end{equation}

where we have introduced the notation

(4.5) \begin{equation} I_k(x) \equiv \int _0^x \frac {1}{H_0^k(x')} \, {\operatorname {d\!}{}}{x'}. \end{equation}

In fact, the shear condition (4.4) was obtained by Trinh et al. (Reference Trinh, Wilson and Stone2014) for a rigid plate (their (2.15)), but with our leading-order $H_0$ replaced by their full expression for $H(x)$ . Thus (4.4) is a leading-order shear condition, $F_z \sim F_z^{(0)}$ , valid in the limit $\mathcal {B} \to \infty$ . Since there are two remaining unknowns, namely $\alpha$ and $h_{\infty 0}$ , to be determined at $\mathcal {O}(1)$ , we expect to complement (4.4) with an additional equation expressing a moment balance. Integrating (4.3) three times and using the zero moment conditions (2.13b ) yields

(4.6) \begin{align} &H_{1xx}(x) - H_{1xxxx}(0)\left [\frac {x^2}{2} - x\right ] - x\, H_{1xxx}(1) + \delta ^3 \left [ H_{0x}(1) - \left \{H_0(x) - H_0(0)\right \}\right ] \nonumber\\ &\qquad \qquad = \int _0^x \int _1^{x_2} [6\,I_2(x_1) - 12 h_{\infty 0}\, I_3(x_1)] \, {\operatorname {d\!}{}}{x_1}{\operatorname {d\!}{}}{x_2}. \end{align}

Setting $x = 1$ in (4.6), reversing the order of integration, and applying the boundary conditions (2.15), (2.17) and the expression for the height of a rigid plate (4.2), we obtain a leading-order zero moment condition,

(4.7) \begin{equation} M_0^{(0)} = \frac {p_0}{2} + \delta ^3 \left [h_{0x}(1) - \alpha \right ] + \int _0^1 [6x\, I_2(x_1) - 12 h_{\infty 0} x_1\, I_3(x_1)] \, {\operatorname {d\!}{}}{x_1} = 0, \end{equation}

which coincides with the corresponding expression obtained by Trinh et al. (Reference Trinh, Wilson and Stone2014) (their (3.5)). Our (4.7), however, is a leading-order moment condition $M_0 \sim M_0^{(0)}$ , valid in the limit $\mathcal {B} \to \infty$ .

In summary, we have demonstrated that the behaviour in the limit of a rigid plate $\mathcal {B} \to \infty$ is nearly the same as that when elasticity is entirely ignored. However, when elasticity is included and $\mathcal {B}$ is large but finite, the plate is not quite flat, but is gently curved with $\mathcal {O}(1/\mathcal {B}^{5})$ curvature. The determination of the final two unknowns requires the solution of the third-order Landau–Levich equation (2.6), subject to two far-field conditions ([8, 9] given in table 1), the continuity condition $h_0(1) = H_0(1) = 1 + \alpha$ , the shear condition (4.4), and the moment condition (4.7). Although this can be done numerically, asymptotic formulae in the limits $\delta \to 0$ and $\delta \to \infty$ were derived by Trinh et al. (Reference Trinh, Wilson and Stone2014). For instance, in the limit $\delta \to \infty$ ,

(4.8) \begin{equation} \alpha \sim \frac {p_0 - 6}{3^{1/3} \delta ^2} \quad \text {and} \quad h_{\infty 0} \sim 1 + \frac {p_0 - 6}{3^{2/3}\delta }, \end{equation}

while the limit of $\delta \to 0$ is more complicated because of the existence of possibly more than one solution due to the effects of the prescribed pressure, $p_0$ .

In figure 4, we present the far-field film height $h_{\infty }$ as a function of $\delta$ for values of $\mathcal {B}$ ranging from $\mathcal {B} = 0.1$ to $\mathcal {B} = 1$ . In particular, figure 4 shows that by $\mathcal {B} = 1$ , the far-field film height is nearly identical to the leading-order asymptotic solution in the limit $\mathcal {B} \to \infty$ , obtained from numerical solutions of the problem for a rigid plate.

Figure 4. The far-field film height $h_{\infty }$ plotted as a function of $\delta$ for various values of $\mathcal {B}$ in the case $p_0 = 0$ . From top to bottom, the curves correspond to $\mathcal {B} = 0.1$ , 0.2, 0.5 and 1.0. The leading-order asymptotic solution in the limit of a rigid plate $\mathcal {B} \to \infty$ described in § 4.1 is shown with the dashed line.

4.2. The limit of a very flexible plate, $\mathcal {B} \to 0$

We now study the limit of a very flexible plate, $\mathcal {B} \to 0$ . In fact, the asymptotic solution in this limit turns out to be very useful because the numerical solution of the governing boundary-value problem becomes increasingly (mathematically) stiff as the plate becomes increasing (physically) flexible. We expand the (outer) solution as

(4.9) \begin{equation} H(x) \sim H_{{outer}}(x) = \sum _{n=0}^\infty \mathcal {B}^n H_n, \end{equation}

with similar expansions for $h(x) \sim h_0(x)$ and $h_{\infty } \sim h_{\infty 0}$ . From (2.11), the equation for the leading-order (outer) plate height is

(4.10) \begin{equation} -\delta ^3 H_{0xxx} \sim \mathcal {R}(H_0; h_{\infty }), \end{equation}

so we need to provide four boundary conditions at $x = 0$ and $x = 1$ .

We find a rather complicated boundary-layer structure in this singular limit. We will begin by considering the simpler case of zero prescribed pressure $p_0 = 0$ , and then return to the case of general values of $p_0$ in § 4.2.5. When $p_0 = 0$ , numerically calculated solutions of the problem for small values of $\mathcal {B}$ indicate that there is no boundary layer near $x = 0$ , thus we can apply the conditions $H(0) = 1$ and $H_{xx}(0) = 0$ there. The remaining three conditions must be selected from amongst the four conditions [3, 5, 6, 7] given in table 1, and the challenge is to determine which ones. As it turns out, the outer solution, valid away from $x = 1$ , is coupled to inner solutions near $x = 1$ via three nested boundary layers.

The full beam equation involves contributions from the elasticity, surface tension and pressure. We write

(4.11) \begin{equation} \underbrace {\mathcal {B}^5 H_{xxxxx}}_{\unicode {x2460}} - \underbrace {\delta ^3 H_{xxxx}}_{\unicode {x2461}} = \underbrace {\mathcal {R}(H; h_\infty )}_{\unicode {x2462}}, \end{equation}

and define the following regions and asymptotic balances.

(4.12a) \begin{align} & \text{Outer} \!\qquad\qquad x - 1 = \mathcal {O}(1) \qquad\qquad\qquad\qquad\qquad \text{where}\ \unicode {x2461}\,\sim \,\unicode {x2462},\end{align}

(4.12b) \begin{align} &\text{Region I} \!\!\quad\qquad x - 1 = \mathcal {O}(\mathcal {B}^{\alpha }) \qquad \text{for} \quad \alpha \lt 5/2 \qquad \text{where}\ \unicode {x2460}\,\ll \,\unicode {x2461},\end{align}

(4.12c) \begin{align} & \text{Region II} \!\!\!\!\quad\qquad x - 1 = \mathcal {O}(\mathcal {B}^{5/2})\qquad\qquad\qquad\qquad\quad\! \text{where}\ \unicode {x2460}\,\sim \,\unicode {x2461},\end{align}

(4.12d) \begin{align} & \text{Region III} \qquad x - 1 = \mathcal {O}(\mathcal {B}^{\alpha }) \qquad \text{for} \quad \alpha \gt 5/2 \qquad \text{where}\ \unicode {x2460}\,\gg \,\unicode {x2461}.\end{align}

In the outer region, surface tension balances the contribution from pressure forces. Moving towards the inner region near $x=1$ , first the outermost boundary layer (region I) in which surface tension dominates elasticity is encountered, then the intermediate boundary layer (region II) in which surface tension and elasticity balance, and finally the innermost boundary layer (region III) in which elasticity dominates.

Figure 5. The curvature of the plate $H_{xx}$ plotted as a function of $x$ for various values of $\mathcal {B}$ in the case $\delta = 1$ and $p_0 = 1$ . From top to bottom on the right, the curves correspond to $\mathcal {B} = 0.05$ , 0.1, 0.2, 0.3 and 0.4. The asymptotic solution in region II given by (4.23b ) is shown with the dashed line. Plot (b) is an enlargement of plot (a) near $x=1$ .

Before we present the analysis of this inner region, we display in figure 5 typical profiles of the curvature of the plate, $H_{xx}(x)$ , calculated from the numerical solutions of the full system in the case $p_0=1$ . The reason for examining $H_{xx}$ (rather than $H$ or $H_x$ ) is that the boundary layers appear only in the second and higher derivatives of the leading-order terms. Crucially, the numerical solutions indicate that

(4.13) \begin{equation} \text {$H(x)$, $H_x(x)$ and $H_{xx}(x)$ are all bounded and non-zero as $\mathcal {B} \to 0$} \end{equation}

for fixed values of $\delta$ and fixed $x \in (0,1)$ .

Based on figure 5, we see that although the zero moment condition (2.13b ) requires $H_{xx}(1) = 0$ , the leading-order solution in the outer region has a curvature that approaches a non-zero value as $x \to 1$ . Our principal task is to determine this effective value of the curvature, shown with the dashed line in figure 5, which provides a boundary condition on the outer solution.

4.2.1. Region II: elasticity and surface tension balance

We begin in region II, where elasticity $\unicode {x2460}$ balances surface tension $\unicode {x2461}$ . Solving the leading-order equation $\mathcal {B}^5 H_{xxxxx} \sim \delta ^3 H_{xxx}$ gives

(4.14) \begin{equation} H^{\text {II}}_{xxx} \sim A \exp \left [-\frac {\delta ^{3/2}}{\mathcal {B}^{5/2}} (1-x)\right ] + \hat {A} \exp \left [\frac {\delta ^{3/2}}{\mathcal {B}^{5/2}} (1-x)\right ], \end{equation}

where $A$ and $\hat {A}$ are constants (possibly dependent on $\mathcal {B}$ and $\delta$ ). We assume that $\hat {A} = 0$ , as this suppresses the otherwise unmatchable exponential growth. Integrating (4.14) once yields

(4.15) \begin{equation} H^{\text {II}}_{xx} \sim \frac {A\mathcal {B}^{5/2}}{\delta ^{3/2}} \exp \left [-\frac {\delta ^{3/2}}{\mathcal {B}^{5/2}} (1-x)\right ] + C, \end{equation}

where $C$ is constant. However, note from the assumption (4.13) that the curvature is bounded and non-zero, so $A = \mathcal {O}(\mathcal {B}^{-5/2})$ and $C = \mathcal {O}(1)$ . We therefore re-scale the constants $A$ and $C$ and write (4.15) as

(4.16) \begin{equation} H^{\text {II}}_{xx} \sim \frac {\tilde {A}}{\delta ^{3/2}} \exp \left [-\frac {\delta ^{3/2}}{\mathcal {B}^{5/2}} (1-x)\right ] + \tilde {C}, \end{equation}

where $\tilde {A}$ and $\tilde {C}$ are $\mathcal {O}(1)$ .

In region III, we assume the scaling $1 - x = \mathcal {B}^\alpha s$ for $\alpha \gt 5/2$ and $s = \mathcal {O}(1)$ . Thus if we take $x$ from region II to region III, then within the exponential argument, $(1-x)/\mathcal {B}^{5/2} = \mathcal {B}^{-(5/2-\alpha )} s$ , and this quantity must be small. Consequently, we may expand

(4.17) \begin{equation} H^{\text { II } \to \text { III}}_{xx} \sim \frac {\tilde {A}}{\delta ^{3/2}} - \frac {\tilde {A}(1-x)}{\mathcal {B}^{5/2}} + \mathcal {O}\left (\frac {(1-x)^2}{\mathcal {B}^{10/2}}\right ) + \tilde {C}. \end{equation}

From the zero moment condition (2.13b ), we require $H_{xx}(1) = 0$ . We make the assumption (which can be verified a posteriori) that this applies even for the inner limit of the region II solution, thus

(4.18) \begin{equation} \tilde {C} = -\frac {\tilde {A}}{\delta ^{3/2}}. \end{equation}

Proceeding now in the other direction, and studying the transition from region II to region I, we find that

(4.19) \begin{equation} H^{\text { II } \to \text { I}}_{xx} \sim -\frac {\tilde {A}}{\delta ^{3/2}} + \, \text {exponentially small terms in} \, \frac {\delta ^{3/2}(1-x)}{\mathcal {B}^{5/2}}, \end{equation}

once we have made the substitution (4.18). Thus the curvature is effectively constant to all algebraic orders, and it is precisely this constant, namely $\tilde {C} = -\tilde {A}/\delta ^{3/2}$ , that is shown with the dashed line in figure 5.

4.2.2. Region I: surface tension dominates elasticity and pressure

In region I, we have $\delta ^3 H_{xxx} \ll 1$ , which gives $H^{\text {I}}_{xx} \sim \text {constant}$ . This solution must match (4.19), so the constant must be equal to (4.18), thus

(4.20) \begin{equation} H^{\text {II}}_{xx} = -\frac {\tilde {A}}{\delta ^{3/2}}. \end{equation}

4.2.3. Region III: elasticity dominates surface tension and pressure

Let us consider the solution in the innermost region III, where elasticity $\unicode {x2460}$ dominates both surface tension $\unicode {x2461}$ and pressure $\unicode {x2462}$ . In this case, we have $H_{xxxxx} \ll 1$ , so integrating three times yields

(4.21) \begin{equation} H^{\text {III}}_{xx} \sim \frac {C_A}{2} (1-x)^2 + C_B(1-x) + C_C, \end{equation}

for new constants $C_A$ , $C_B$ and $C_C$ . From the zero moment condition (2.13b ), $H_{xx}(1) = 0$ , thus $C_C = 0$ . Based on the $H^{\text { II } \to \text { III}}$ limit of (4.17), we argue that the solution in region III cannot diverge quadratically as $x$ moves away from $x = 1$ , and consequently $C_A = 0$ . Then we match (4.17) and (4.21) to obtain

(4.22) \begin{equation} C_B = -\frac {\tilde {A}}{\mathcal {B}^{5/2}}. \end{equation}

4.2.4. Summary and final matching

At this point, we have the following solutions:

(4.23a) \begin{align} H_{xx}^{\text {I}} &\sim -\frac {\tilde {A}}{\delta ^{3/2}}, \end{align}

(4.23b) \begin{align} H_{xx}^{\text {II}} &\sim \frac {\tilde {A}}{\delta ^{3/2}} \exp \left [-\frac {\delta ^{3/2}(1-x)}{\mathcal {B}^{5/2}} \right ] -\frac {\tilde {A}}{\delta ^{3/2}}, \end{align}

(4.23c) \begin{align} H_{xx}^{\text {III}} &\sim -\frac {\tilde {A}(1-x)}{\mathcal {B}^{5/2}}. \end{align}

We now apply the remaining boundary conditions. Since $\mathcal {B}^5 H_{xxx}(1) = \mathcal {O}(\mathcal {B}^{5/2})$ according to the solution in region III (or region II), the shear condition (2.15) reduces to $H_x(1) \sim h_x(1)$ at leading order. Thus in the limit $\mathcal {B} \to 0$ the plate and free surface contact tangentially. The pressure condition (2.17b ) applied to the inner limit of the solution in region II (4.23b ) indicates that

(4.24) \begin{equation} \mathcal {B}^5 H_{xxxx}(1) \sim \mathcal {B}^5 \frac {\tilde {A}}{\delta ^{3/2}} \left (\frac {\delta ^{3/2}}{\mathcal {B}^{5/2}}\right )^2 = \tilde {A} \delta ^{3/2} = -\delta ^3\, h_{xx}(1), \end{equation}

thus we have determined the final constant to be

(4.25) \begin{equation} \tilde {A} = -\delta ^{3/2}\, h_{xx}(1). \end{equation}

Substitution of this value of $\tilde {A}$ into (4.23a ) gives the effective value of the curvature to be applied as a boundary condition on the outer solution, namely $H_{xx}(1) = h_{xx}(1)$ .

In summary, in order to determine the leading-order outer solution in the limit $\mathcal {B} \to 0$ , we solve the outer boundary-value problem

(4.26a) \begin{align} -\delta ^3 H_{xxx} = \mathcal {R}(H; h_{\infty }), \end{align}

(4.26b) \begin{align} H(0) = 1, \quad H_{xx}(0) = 0, \end{align}

(4.26c) \begin{align} H(1) = h(1), \quad H_x(1) = h_x(1), \quad H_{xx}(1) = h_{xx}(1), \end{align}

in the following way. First, values of $h_{\infty }$ and $H(1)$ are used as initial guesses. The Landau–Levich equation (2.6) is solved starting at a large value of $x = x_{\text {max}} \gg 1$ , with $h$ close to its far-field value, and stopping once $h = H(1)$ . At this point, we solve the third-order boundary-value problem (4.26a ) subject to three of the five conditions (4.26b ) and (4.26c ). The shooting algorithm is then repeated to converge to the correct values of $h_{\infty }$ and $H(1)$ in order to satisfy the remaining two conditions.

It is a curious fact that the effective boundary conditions on the outer problem, namely (4.26), impose the constraint that the elastic plate, $H(x)$ , and the free surface, $h(x)$ , are continuous in their heights, derivatives and curvatures. As a consequence, the plate and the film behave very similarly to a fluid with an uncovered free surface. However, we should recall that the equation for $H(x)$ imposes a zero-slip condition on the free surface, whereas the velocity profiles for $h(x)$ will exhibit slip at the free surface. Additionally, the interpretation of the elastic plate as continuously attaching to the fluid is true only of the outer solution, as is shown clearly in figure 5. In particular, there is a rapid variation in the curvature of the elastic plate from its non-zero effective value (proportional to the curvature $h_{xx}(x)$ ) to the zero value required for the zero moment condition.

4.2.5. Modification for $p_0 \neq 0$

Now that we have understood the nested boundary-layer structure of the inner region, we see that the introduction of regions I and III is not strictly necessary in order to derive the effective boundary conditions on the outer problem. This is because the solution in the intermediate region (region II) contains the information about all three boundary-layer regions, and in fact, the boundary conditions exactly at $x = 1$ can be applied directly to the solution in region II. The crucial quality that allows for this is that the terms that change balance between regions I, II and III are exponential in nature.

With the introduction of $p_0 \neq 0$ , another boundary layer (or, more accurately, three new boundary layers) must be introduced in a new inner region near $x = 0$ . Using the same naming convention for the three regions, the solution in the intermediate region, where the dominant balance is $\mathcal {B}^5 H_{xxxxx} \sim \delta ^3 H_{xxx}$ , has the curvature

(4.27) \begin{equation} H_{xx}^{\text {II, left}} \sim \frac {a_1}{\delta ^{3/2}} \exp \left (-\frac {\delta ^{3/2}}{\mathcal {B}^{5/2}} x\right ) + a_2, \end{equation}

where, as in § 4.2.1, we have scaled the constants $a_1$ and $a_2$ so that the curvature remains non-zero as in the limit $\mathcal {B} \to 0$ . Next, matching with the innermost solution, II $\to$ III requires that $H_{xx}(0) = 0$ for the hinged plate. Thus $a_2 = -a_1/\delta ^{3/2}$ . The last step is to apply the pressure condition (2.17a ), requiring $\mathcal {B}^5 H_{xxxx}(0) = p_0$ ; this gives $a_1 = p_0/\delta ^{3/2}$ , so the final intermediate solution is

(4.28) \begin{equation} H_{xx}^{\text {II},\ {\rm left}} \sim \frac {p_0}{\delta ^3} \left [\exp \left (-\frac {\delta ^{3/2}}{\mathcal {B}^{5/2}}\, x\right ) - 1\right ], \end{equation}

and we observe that the effective curvature to apply to the outer solution is $-p_0/\delta ^3$ .

This completely determines the problem, and we are left with the effective outer problem:

(4.29a) \begin{gather} -\delta ^3 H_{xxx} = \mathcal {R}(H; h_{\infty }), \end{gather}

(4.29b) \begin{gather} H(0) = 1, \quad H_{xx}(0) = -\frac {p_0}{\delta ^3}, \end{gather}

(4.29c) \begin{gather} H(1) = h(1), \quad H_x(1) = h_x(1), \quad H_{xx}(1) = h_{xx}(1), \end{gather}

which is solved similarly to the problem (4.26). Numerical solutions are shown in figure 6, which shows that the outer curvature $H_{xx}$ tends to (different) non-zero values at the ends of the interval.

Figure 6. The curvature of the plate $H_{xx}$ plotted as a function of $x$ for various values of $\mathcal {B}$ in the case $\delta = 1$ and $p_0 = 1$ . From top to bottom on the right, the curves correspond to $\mathcal {B} = 0.1$ , 0.2, 0.3 and 0.4.

5.1. The limit of slow substrate motion and/or strong surface tension, $\delta \to \infty$

In the limit $\delta \to \infty$ , it can be verified that both the height of the plate $H(x)$ and the height of the free surface $h(x)$ uniformly tend to unity. We then expand

(5.1) \begin{equation} H(x) = 1 + \sum _{n=1}^\infty \frac {H_n}{\delta ^n}. \end{equation}

As in the corresponding analysis for the bending stiffness $\mathcal {B}$ described in § 4, we are interested in developing a uniformly valid solution for the curvature of the plate $H_{xx}$ , which is expected to be composed of boundary-layer solutions valid in inner regions near the ends of the plate, $H_{xx}^{{\rm left}}$ and $H_{xx}^{{\rm right}}$ , and an outer solution valid elsewhere, $H_{xx}^{{\rm outer}}$ . Also as in the previous analysis, if the prescribed pressure is zero, then there is no boundary layer at the left-hand end of the plate.

First, we consider the general case $p_0 \neq 0$ . In the outer region $H \sim 1$ and $h_{\infty } \sim 1$ , and assuming a dominant balance between the surface tension $\unicode {x2461}$ and pressure $\unicode {x2462}$ , we find that $\delta ^3 H_{xxx} \sim 6$ and hence

(5.2) \begin{equation} H^{{\rm outer}}_{xx} \sim \dfrac {1}{\delta ^3} (6x - d) \end{equation}

for constant $d$ . Notice that, unlike in § 4.2, we do not assume that the curvature is $\mathcal {O}(1)$ in this limit.

In the left-hand boundary layer near $x=0$ , we seek a balance between elasticity $\unicode {x2460}$ and surface tension $\unicode {x2461}$ in (4.11), thus the boundary layer is of width $x = \mathcal {O}(\delta ^{-3/2})$ . Solving $\mathcal {B}^5 H_{xxxxx} \sim \delta ^3 H_{xxx}$ , and using the zero moment condition $H_{xx}(0) = 0$ , we obtain

(5.3) \begin{equation} H_{xx}^{{\rm left}} \sim -\dfrac {D}{\delta ^3} \left [ 1 - \exp \left ( -\dfrac {\delta ^{3/2} x}{\mathcal {B}^{5/2}} \right ) \right ] \end{equation}

for a constant $D$ . In order for this curvature to match the outer curvature (5.2), we take $\delta ^{3/2}x \to \infty$ and thus require $D = d$ . Moreover, imposing the pressure condition (2.17a ) yields $D = p_0$ .

Figure 7. Plots of the re-scaled curvature of the plate $\delta ^3 H_{xx}$ as a function of $x$ for various values of $\delta$ in the case $\mathcal {B} = 0.5$ and $p_0 = 1$ . From top to bottom on the right in (a), the curves correspond to $\delta = 500$ , 100, 10 and 1. The asymptotic solution given by (5.5) for $\delta =500$ is shown with the dashed line. Plot (b) is an enlargement of plot (a) near $x=1$ , including only the curves for $\delta =500$ for clarity.

It remains to determine the curvature in the right-hand boundary layer near $x=1$ . Seeking a balance between elasticity $\unicode {x2460}$ and surface tension $\unicode {x2461}$ in (4.11) yields a boundary layer of width $1 - x = \mathcal {O}(\delta ^{-3/2})$ . Imposing the condition $H_{xx}(1) = 0$ yields

(5.4) \begin{equation} H_{xx}^{{\rm right}} \sim \dfrac {E}{\delta ^3}\left [ 1 - \exp \left (-\dfrac {\delta ^{3/2}(1-x)}{\mathcal {B}^{5/2}}\right )\right ], \end{equation}

where the constant $E$ is determined to be $E = 6 - d = 6 - p_0$ once $H_{xx}^{{\rm right}}$ is matched to $H_{xx}^{{\rm outer}}$ . Thus we conclude that

(5.5) \begin{equation} H_{xx} \sim \begin{cases} -\dfrac {p_0}{\delta ^3} \left [ 1 - \exp \left (-\dfrac {\delta ^{3/2}x}{\mathcal {B}^{5/2}}\right )\right ] & \text {for $x = \mathcal {O}(\delta ^{3/2})$}, \\ \dfrac {1}{\delta ^3} (6x - p_0) & \text {for $x \in (0, 1)$}, \\ \dfrac {6-p_0}{\delta ^3} \left [ 1 - \exp \left (-\dfrac {\delta ^{3/2}(1-x)}{\mathcal {B}^{5/2}}\right )\right ] & \text {for $x = 1 - \mathcal {O}(\delta ^{-3/2})$}. \end{cases} \end{equation}

Figure 7 shows the re-scaled curvature $\delta ^3 H_{xx}$ for $\delta = 1, 10, 100, 500$ , together with the nearly visually indistinguishable asymptotic solution given by (5.5) for $\delta =500$ . Note that we plotted $\delta ^3 H_{xx}$ rather than $H_{xx}$ in order to remove the algebraic dependence on $\delta$ .

With the curvature of the plate determined, we are now free to proceed similarly to the study of a rigid plate by Trinh et al. (Reference Trinh, Wilson and Stone2014) and investigate the Landau–Levich equation (2.6) in the limit $\delta \to \infty$ . In this limit, the deviation from $h \sim 1$ for finite $x$ is exponentially small in $\delta$ , so we seek a re-scaling near the the right-hand edge of the plate at $x = 1$ . Setting $x = 1 + \delta X$ , we substitute $h = 1 + h_1(X)/\delta + \mathcal {O}(1/\delta ^2)$ and $h_{\infty } = 1 + h_{\infty 1}/\delta + \mathcal {O}(1/\delta ^2)$ into the re-scaled Landau–Levich equation given by $h_{XXX} = 3(h_{\infty } - h)/h^3$ . Solving the equation at $\mathcal {O}(1/\delta )$ , we set the two integration constants to zero to remove the exponentially growing modes (see [8,9] in table 1), and match the heights of the free surface and the plate, $h_1(X = 0) = H_1(x = 1)$ , where $H_1$ is from the series expansion (5.1). Written in terms of $x$ , this yields

(5.6) \begin{equation} h_1 = h_{\infty 1} + \left [ H_1(1) - h_{\infty 1}\right ] \exp \left (-\frac {3^{1/3} (x-1)}{\delta }\right ). \end{equation}

From (5.5), we observe that the $\mathcal {O}(1/\delta )$ contribution to the plate is identically zero, $H_1 \equiv 0$ , thus the free-surface correction in (5.6) depends only on $h_{\infty 1}$ . Consequently, (5.6) yields an expression for the leading-order free-surface curvature near the edge of the plate given by $h_{xx}(1) \sim - 3^{2/3} h_{\infty 1}/\delta ^3$ . The normal load at the edge of the elastic plate is $\mathcal {B}^5 H_{xxxxx}(1) \sim (p_0 - 6)/\delta ^3$ , which follows from (5.5). Finally, the fluid pressure and the load on the plate are related through (2.17b ). Solving for the far-field film height gives

(5.7) \begin{equation} h_{\infty } = 1 + \frac {p_0 - 6}{3^{2/3}\delta }, \end{equation}

which is verified in figure 8.

Figure 8. The far-field film height $h_{\infty }$ plotted as a function of $\delta$ in the case $p_0 = 1$ . From top to bottom, the curves correspond to $\mathcal {B} = 0.2$ , 0.5 and 1. The two-term asymptotic approximation in the limit $\delta \to \infty$ given by (5.7) is shown with the dashed line.

The above results, and, in particular, the separation of the curvature $H_{xx}$ into the three regions in (5.5), correspond to the situation in which there is a non-zero prescribed pressure $p_0 \neq 0$ . If $p_0 = 0$ , then the asymptotic structure is simpler and the outer solution with $H_{xx} \sim 6x/\delta ^3$ applies all the way to $x = 0$ and satisfies the necessary zero moment condition. The limit $p_0 \to 0$ is regular: the above analysis, now consisting of only a single boundary layer near $x = 1$ , is otherwise unaltered, and the results are consistent with setting $p_0 = 0$ in the previously derived formulae.

5.2. The limit of fast substrate motion and/or weak surface tension, $\delta \to 0$

In the limit $\delta \to 0$ , the plate uniformly tends to unit height, $H \sim 1$ , while the downstream height of the fluid tends to $h \sim h_{\infty } \sim 1/2$ , except near the plate, where the free surface rises rapidly to match the edge of the plate. In terms of the analysis for the elastic plate, the principal difference between the limit $\delta \to 0$ , the limit $\delta \to \infty$ of §. 5.1, and the limit $\mathcal {B} \to 0$ of § 4.2 is that for the case here, the $\delta \to 0$ limit is regular and there are no boundary layers. We thus expand as usual as $H = 1 + \delta H_1 + \mathcal {O}(\delta ^2)$ and $h_{\infty } = 1/2 + \delta h_{\infty 1} + \mathcal {O}(\delta ^2)$ . At $\mathcal {O}(\delta )$ in (2.11), we find $\mathcal {B}^5 H_{1xxxxx} = 6(H_1 - 2h_{\infty 1})$ , which has the general solution

(5.8a) \begin{equation} H_1(x; \, h_{\infty 1}) = 2h_{\infty 1} + \sum _{j=0}^4 C_j \exp \left ( \frac {6^{1/5} \mathrm {e}^{2\pi \mathrm {i} k/5} x}{\mathcal {B}}\right ), \end{equation}

with constants $C_j$ for $j=0,\ldots ,4$ . The values of the six unknowns $C_0$ , …, $C_4$ and $h_{\infty 1}$ can be determined by imposing the six boundary conditions

(5.8b) \begin{align} H_1(0) = 0, \quad H_{1xx}(0) = 0, \quad H_{1xxx}(0) = \frac {p_0}{\mathcal {B}^5}, \end{align}

(5.8c) \begin{align} H_{1xx}(1) = 0, \quad H_{1xxx}(1)= 0, \quad H_{1xxxx}(1) = -\frac {h_{0xx}(1)}{\mathcal {B}^5}. \end{align}

The only unknown in the above set of boundary conditions is the value of $h_{0xx}(1)$ . This value can be found by numerically solving the Landau–Levich equation (2.6), starting at a large value of $x$ and stopping once $h(1) \sim h_0(1) = 1$ is reached. Based on the results of Trinh et al. (Reference Trinh, Wilson and Stone2014), this value is approximately

(5.9) \begin{equation} h_{0xx}(1) \sim \frac {1.7639}{\delta ^2}. \end{equation}

Figure 9. (a) The far-field height correction $h_{\infty 1}$ defined by $h_\infty = 1/2 + \delta h_{\infty 1} + \mathcal {O}(\delta ^2)$ , plotted as function of $\mathcal {B}$ . From top to bottom, the curves correspond to $p_0 = 0.1$ , 0 and $-0.1$ . (b) The full numerical solutions for $h_\infty$ in the case $p_0 = 0$ for $\delta = 0.1$ and $\delta = 0.01$ (shown with the solid lines) and the two-term asymptotic solution given by $1/2 + \delta h_{\infty 1}$ (shown with the dashed line) plotted as functions of $\mathcal {B}$ . The pair of solid and dashed curves with the larger deviation from $h_{\infty }=0.5$ corresponds to $\delta =0.1$ , and the pair of solid and dashed curves with the smaller deviation (which are nearly visually indistinguishable) corresponds to $\delta =0.01$ .

In figure 9(a), we display the values of $h_{\infty 1}$ for $p_0 = 0.1$ , 0 and $-0.1$ , calculated using the above approach (i.e. solving (5.8)). In figure 9(b), we compare the two-term asymptotic solution $1/2 + \delta h_{\infty 1}$ with the full numerical solution for $\delta = 0.1$ and $\delta = 0.01$ when $p_0 = 0$ . The figure confirms that there is good agreement between the numerical and asymptotic results; by $\delta = 0.01$ the curves are nearly visually indistinguishable.

There is, in fact, an additional distinguished limit when both $\mathcal {B} \to 0$ and $\delta \to 0$ simultaneously, which produces the increasingly rapid oscillations visible in figure 9. In the analysis leading to the asymptotic solution (5.8), $\mathcal {B}$ was assumed to be fixed while $\delta \to 0$ . However, we note that the solutions that correspond to (afterwards) setting $\mathcal {B} \to 0$ have $H_1(1) \to -\infty$ , and are thus inadmissible as soon as $H_1(1)$ exceeds $\mathcal {O}(1/\delta )$ in magnitude. Examples of solutions in this limit are given in figure 10. We observe that as $\mathcal {B} \to 0$ and $\delta \to 0$ , $H$ exhibits a series of maxima and minima in $x$ , for which the primary maximum moves closer to the boundary at $x = 1$ in the double singular limits. The behaviour of these ripples seems to parallel that observed in the analyses of Wilson & Jones (Reference Wilson and Jones1983), Snoeijer et al. (Reference Snoeijer, Ziegler, Andreotti, Fermigier and Eggers2008), Benilov et al. (Reference Benilov, Chapman, McLeod, Ockendon and Zubkov2010) and McKinlay et al. (Reference McKinlay, Wray and Wilson2023); we leave this particular special limit as an open problem.

Figure 10. The height of the plate $H(x)$ plotted as a function of $x$ for various values of $\mathcal {B}$ in the case $p_0 = 0$ and $\delta = \mathcal {B}$ . From left to right, the curves correspond to $\delta = \mathcal {B} = 0.0735$ , 0.0652, 0.0573 and 0.0500. There are further minima and maxima at smaller values of $x$ not visible on the scale of the diagram.