2.1 Formulation of the dimensional problem

We consider an infinite one-dimensional substrate, on which viscous liquid is injected uniformly through the surface in the region $|\hat{x}|\leqslant L$ at a constant speed $W$ . We let $\hat{x}$ measure the distance along the substrate and $\hat{z}$ measure the distance perpendicular to the substrate, and we show a schematic of the geometry under consideration in figure 1. We expect the reduced Reynolds number to be very small for flows of practical interest in purification devices, and so, we ignore the effect of inertia. Further, assuming that gravity can also be neglected, the flow is surface-tension driven and governed by Stokes’ equations

(2.1) $$\begin{eqnarray}\displaystyle & -\unicode[STIX]{x1D735}\hat{p}+\unicode[STIX]{x1D707}\unicode[STIX]{x1D6FB}^{2}\hat{\boldsymbol{u}}=0, & \displaystyle\end{eqnarray}$$

(2.2) $$\begin{eqnarray}\displaystyle & \unicode[STIX]{x1D735}\boldsymbol{\cdot }\hat{\boldsymbol{u}}=0, & \displaystyle\end{eqnarray}$$

where $\hat{\boldsymbol{u}}=(\hat{u} ,{\hat{w}})$ is the fluid velocity, $\hat{p}$ is the pressure and $\unicode[STIX]{x1D707}$ is the viscosity of the liquid. We have the following no-slip, kinematic and stress boundary conditions,

(2.3) $$\begin{eqnarray}\displaystyle & \hat{u} =0\quad \text{on }\hat{z}=0, & \displaystyle\end{eqnarray}$$

(2.4) $$\begin{eqnarray}\displaystyle & {\hat{w}}=W\unicode[STIX]{x1D6E9}(L-|\hat{x}|)\quad \text{on }\hat{z}=0, & \displaystyle\end{eqnarray}$$

(2.5) $$\begin{eqnarray}\displaystyle & \displaystyle {\hat{w}}={\displaystyle \frac{\unicode[STIX]{x2202}{\hat{h}}}{\unicode[STIX]{x2202}\hat{t}}}+\hat{u} {\displaystyle \frac{\unicode[STIX]{x2202}{\hat{h}}}{\unicode[STIX]{x2202}\hat{x}}}\quad \text{on }\hat{z}={\hat{h}}(\hat{x},\hat{t}), & \displaystyle\end{eqnarray}$$

(2.6) $$\begin{eqnarray}\displaystyle & \displaystyle \hat{\boldsymbol{n}}\boldsymbol{\cdot }\hat{\unicode[STIX]{x1D64F}}\boldsymbol{\cdot }\hat{\boldsymbol{n}}=\unicode[STIX]{x1D6FE}(-\unicode[STIX]{x1D735}\boldsymbol{\cdot }\hat{\boldsymbol{n}})=\unicode[STIX]{x1D6FE}{\displaystyle \frac{\unicode[STIX]{x2202}^{2}{\hat{h}}}{\unicode[STIX]{x2202}\hat{x}^{2}}}\left(1+\left({\displaystyle \frac{\unicode[STIX]{x2202}{\hat{h}}}{\unicode[STIX]{x2202}\hat{x}}}\right)^{2}\right)^{-3/2}\quad \text{on }\hat{z}={\hat{h}}(\hat{x},\hat{t}), & \displaystyle\end{eqnarray}$$

(2.7) $$\begin{eqnarray}\displaystyle & \hat{\boldsymbol{n}}\boldsymbol{\cdot }\hat{\unicode[STIX]{x1D64F}}\boldsymbol{\cdot }\hat{\boldsymbol{t}}=0\quad \text{on }\hat{z}={\hat{h}}(\hat{x},\hat{t}), & \displaystyle\end{eqnarray}$$

where $\unicode[STIX]{x1D6E9}(\hat{x})$ is the Heaviside step function, $\hat{\boldsymbol{t}}$ and $\hat{\boldsymbol{n}}$ are the unit tangent and outward-pointing normal vectors to the free surface $\hat{z}={\hat{h}}(\hat{x},\hat{t})$ , defined by

(2.8a,b ) $$\begin{eqnarray}\hat{\boldsymbol{t}}=\left(1+\left({\displaystyle \frac{\unicode[STIX]{x2202}{\hat{h}}}{\unicode[STIX]{x2202}\hat{x}}}\right)^{2}\right)^{-1/2}\left(1,{\displaystyle \frac{\unicode[STIX]{x2202}{\hat{h}}}{\unicode[STIX]{x2202}\hat{x}}}\right),\quad \hat{\boldsymbol{n}}=\left(1+\left({\displaystyle \frac{\unicode[STIX]{x2202}{\hat{h}}}{\unicode[STIX]{x2202}\hat{x}}}\right)^{2}\right)^{-1/2}\left(-{\displaystyle \frac{\unicode[STIX]{x2202}{\hat{h}}}{\unicode[STIX]{x2202}\hat{x}}},1\right),\end{eqnarray}$$

$\unicode[STIX]{x1D6FE}$ is the (constant) surface tension and $\hat{\unicode[STIX]{x1D64F}}$ is the stress tensor, defined by

(2.9) $$\begin{eqnarray}\hat{\unicode[STIX]{x1D64F}}=\left(\begin{array}{@{}cc@{}}-\hat{p}+2\unicode[STIX]{x1D707}{\displaystyle \frac{\unicode[STIX]{x2202}\hat{u} }{\unicode[STIX]{x2202}\hat{x}}} & \unicode[STIX]{x1D707}\left({\displaystyle \frac{\unicode[STIX]{x2202}\hat{u} }{\unicode[STIX]{x2202}\hat{z}}}+{\displaystyle \frac{\unicode[STIX]{x2202}{\hat{w}}}{\unicode[STIX]{x2202}\hat{x}}}\right)\\ \unicode[STIX]{x1D707}\left({\displaystyle \frac{\unicode[STIX]{x2202}\hat{u} }{\unicode[STIX]{x2202}\hat{z}}}+{\displaystyle \frac{\unicode[STIX]{x2202}{\hat{w}}}{\unicode[STIX]{x2202}\hat{x}}}\right) & -\hat{p}+2\unicode[STIX]{x1D707}{\displaystyle \frac{\unicode[STIX]{x2202}{\hat{w}}}{\unicode[STIX]{x2202}\hat{z}}}\end{array}\right),\end{eqnarray}$$

and evaluated on the free surface.

Figure 2.

A schematic diagram of the thin-film problem on half-space with injection and a precursor layer of thickness ${\hat{h}}_{\infty }$ .

At the moving front, $\hat{x}=\hat{a}(\hat{t})$ , say, we could impose zero-thickness and no-flux conditions along with a slip law along the whole of the substrate. For late time, as in the non-injection case we expect a quasi-static profile for the film thickness, where the relationship between the dynamic contact angle, $\unicode[STIX]{x1D703}_{d}$ , and the speed of the contact line, $\dot{\hat{a}}$ (where a dot $(\dot{\,})$ denotes differentiation with respect to time), is given by a version of Tanner’s law. For small contact angles, this can be written as

(2.10) $$\begin{eqnarray}{\displaystyle \frac{\unicode[STIX]{x2202}{\hat{h}}}{\unicode[STIX]{x2202}\hat{x}}}=\hat{k}\dot{\hat{a}}^{1/3},\end{eqnarray}$$

for some constant of proportionality $\hat{k}$ , depending on the system under consideration (Tanner Reference Tanner1979; Myers Reference Myers1998).

However, instead we choose to regularize the problem using a precursor-film regularization. In this regularization, we assume that the whole substrate is initially covered in a thin layer of liquid of thickness ${\hat{h}}_{\infty }$ (see figure 2). Thus, we assume that

(2.11) $$\begin{eqnarray}{\hat{h}}\rightarrow {\hat{h}}_{\infty }\quad \text{as }\hat{x}\rightarrow \infty ,\end{eqnarray}$$

and

(2.12) $$\begin{eqnarray}{\hat{h}}={\hat{h}}_{\infty }\quad \text{at }\hat{t}=0.\end{eqnarray}$$

This regularization is sufficient to close the problem (King & Bowen Reference King and Bowen2001), it resolves the well-known stress singularity that occurs at the moving contact line in film-spreading problems (Huh & Scriven Reference Huh and Scriven1971), and we will find that this recovers a version of Tanner’s law describing the motion of the apparent contact line, $\hat{a}$ . The position of the apparent contact line, which lies in a narrow region where the film thickness adjusts to the precursor thickness, is sometimes defined as the point where the curvature of the free surface attains a local maximum (see, for example, Ajaev Reference Ajaev2012; Carlson, Mandre & Mahadevan Reference Carlson, Mandre and Mahadevan2015). However, any point in the narrow transition region can serve as a definition for the contact line. For concreteness, in this analysis, as well as the numerical solutions, we define the position of the contact line to be the smallest value of $\hat{x}$ at which ${\hat{h}}$ equals the precursor-layer thickness, ${\hat{h}}_{\infty }$ (see figure 2).

Our final lateral boundary conditions are to assume symmetry at $\hat{x}=0$ , i.e.

(2.13) $$\begin{eqnarray}\hat{u} ={\displaystyle \frac{\unicode[STIX]{x2202}\hat{p}}{\unicode[STIX]{x2202}\hat{x}}}={\displaystyle \frac{\unicode[STIX]{x2202}{\hat{h}}}{\unicode[STIX]{x2202}\hat{x}}}={\displaystyle \frac{\unicode[STIX]{x2202}^{3}{\hat{h}}}{\unicode[STIX]{x2202}\hat{x}^{3}}}=0\quad \text{at }\hat{x}=0.\end{eqnarray}$$

From now on, we assume $x\geqslant 0$ and will drop the modulus sign on the right-hand side of (2.4).

2.2 Dimensionless model

We non-dimensionalize (2.1)–(2.7) using

(2.14) $$\begin{eqnarray}\left.\begin{array}{@{}c@{}}(\hat{x},\hat{a})=L(x,a),\quad (\hat{z},{\hat{h}})=(\unicode[STIX]{x1D707}WL^{4}/\unicode[STIX]{x1D6FE})^{1/4}(z,h),\\ \hat{u} =(\unicode[STIX]{x1D6FE}W^{3}/\unicode[STIX]{x1D707})^{1/4}u,\quad {\hat{w}}=Ww,\quad \hat{t}=(\unicode[STIX]{x1D707}L^{4}/\unicode[STIX]{x1D6FE}W^{3})^{1/4}t,\quad \hat{p}=(\unicode[STIX]{x1D707}W\unicode[STIX]{x1D6FE}^{3}/L^{4})^{1/4}p.\end{array}\right\}\end{eqnarray}$$

We note that we have chosen the intrinsic scale for ${\hat{h}}$ that emerges as a result of non-dimensionalizing (2.6) and balancing viscous forces in the normal direction with surface-tension forces. We assume that the aspect ratio in the problem $\unicode[STIX]{x1D716}=\left(\unicode[STIX]{x1D707}W/\unicode[STIX]{x1D6FE}\right)^{1/4}\ll 1$ , and expand the dependent variables in asymptotic series in powers of $\unicode[STIX]{x1D716}^{2}$ . This gives the following leading-order dimensionless problem

(2.15) $$\begin{eqnarray}\displaystyle & \displaystyle -{\displaystyle \frac{\unicode[STIX]{x2202}p}{\unicode[STIX]{x2202}x}}+{\displaystyle \frac{\unicode[STIX]{x2202}^{2}u}{\unicode[STIX]{x2202}z^{2}}}=0, & \displaystyle\end{eqnarray}$$

(2.16) $$\begin{eqnarray}\displaystyle & \displaystyle -{\displaystyle \frac{\unicode[STIX]{x2202}p}{\unicode[STIX]{x2202}z}}=0, & \displaystyle\end{eqnarray}$$

(2.17) $$\begin{eqnarray}\displaystyle & \displaystyle {\displaystyle \frac{\unicode[STIX]{x2202}u}{\unicode[STIX]{x2202}x}}+{\displaystyle \frac{\unicode[STIX]{x2202}w}{\unicode[STIX]{x2202}z}}=0, & \displaystyle\end{eqnarray}$$

together with

(2.18) $$\begin{eqnarray}\displaystyle & u=0\quad \text{on }z=0, & \displaystyle\end{eqnarray}$$

(2.19) $$\begin{eqnarray}\displaystyle & w=\unicode[STIX]{x1D6E9}(L-x)\quad \text{on }z=0, & \displaystyle\end{eqnarray}$$

(2.20) $$\begin{eqnarray}\displaystyle & \displaystyle w={\displaystyle \frac{\unicode[STIX]{x2202}h}{\unicode[STIX]{x2202}t}}+u{\displaystyle \frac{\unicode[STIX]{x2202}h}{\unicode[STIX]{x2202}x}}\quad \text{on }z=h(x,t), & \displaystyle\end{eqnarray}$$

(2.21) $$\begin{eqnarray}\displaystyle & \displaystyle p=-{\displaystyle \frac{\unicode[STIX]{x2202}^{2}h}{\unicode[STIX]{x2202}x^{2}}}\quad \text{on }z=h(x,t), & \displaystyle\end{eqnarray}$$

(2.22) $$\begin{eqnarray}\displaystyle & \displaystyle {\displaystyle \frac{\unicode[STIX]{x2202}u}{\unicode[STIX]{x2202}z}}=0\quad \text{on }z=h(x,t). & \displaystyle\end{eqnarray}$$

Integrating (2.17), and using (2.15)–(2.16) together with (2.18)–(2.22), we find that $h$ satisfies the lubrication equation

(2.23) $$\begin{eqnarray}{\displaystyle \frac{\unicode[STIX]{x2202}h}{\unicode[STIX]{x2202}t}}+\frac{1}{3}{\displaystyle \frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}x}}\left(h^{3}{\displaystyle \frac{\unicode[STIX]{x2202}^{3}h}{\unicode[STIX]{x2202}x^{3}}}\right)=\unicode[STIX]{x1D6E9}(1-x).\end{eqnarray}$$

The dimensionless boundary and initial conditions become

(2.24) $$\begin{eqnarray}\displaystyle & \displaystyle {\displaystyle \frac{\unicode[STIX]{x2202}h}{\unicode[STIX]{x2202}x}}={\displaystyle \frac{\unicode[STIX]{x2202}^{3}h}{\unicode[STIX]{x2202}x^{3}}}=0\quad \text{at }x=0, & \displaystyle\end{eqnarray}$$

(2.25) $$\begin{eqnarray}\displaystyle & h\rightarrow \unicode[STIX]{x1D6FF}\quad \text{as }x\rightarrow \infty , & \displaystyle\end{eqnarray}$$

(2.26) $$\begin{eqnarray}\displaystyle & h=\unicode[STIX]{x1D6FF}\quad \text{at }t=0, & \displaystyle\end{eqnarray}$$

where $\unicode[STIX]{x1D6FF}={\hat{h}}_{\infty }/(\unicode[STIX]{x1D716}L)\ll 1$ is the dimensionless precursor-film thickness and the only remaining parameter in the system. Although we have only three boundary conditions for our fourth-order equation, the precursor-layer regularization (2.25) is enough to ensure a unique solution (Myers Reference Myers1998; King & Bowen Reference King and Bowen2001).

Integrating (2.23) with respect to both $x$ and $t$ and applying (2.24) and (2.26), we obtain

(2.27) $$\begin{eqnarray}\int _{0}^{\infty }(h(x,t)-\unicode[STIX]{x1D6FF})\,\text{d}x=t.\end{eqnarray}$$

Equation (2.27) is an alternative expression for global conservation of mass, but will be convenient to use in the following analysis.

In the next section, we will explore the behaviour of the film thickness for early, $O(1)$ , and late time, which will enable us to make substantial analytical progress with the problem.

3.1 Early-time behaviour

To examine the early-time behaviour of $h$ , we rescale the thickness $h$ with the initial thickness, $\unicode[STIX]{x1D6FF}$ , and scale time to balance the film growth with the liquid supply, i.e. we set

(3.1a,b ) $$\begin{eqnarray}h=\unicode[STIX]{x1D6FF}\tilde{h},\quad t=\unicode[STIX]{x1D6FF}\tilde{t}.\end{eqnarray}$$

We pose the asymptotic expansions

(3.2) $$\begin{eqnarray}\displaystyle & \tilde{h}(x,\tilde{t})\sim \tilde{h}_{0}(x,\tilde{t})+\unicode[STIX]{x1D6FF}\tilde{h}_{1}(x,\tilde{t})+\cdots \quad \text{as }\unicode[STIX]{x1D6FF}\rightarrow 0, & \displaystyle\end{eqnarray}$$

(3.3) $$\begin{eqnarray}\displaystyle & a(\tilde{t})\sim 1+\unicode[STIX]{x1D6FF}a_{1}(\tilde{t})+\cdots \quad \text{as }\unicode[STIX]{x1D6FF}\rightarrow 0, & \displaystyle\end{eqnarray}$$

and assume, following King & Bowen (Reference King and Bowen2001), that on this time scale the apparent contact line moves only an $O(\unicode[STIX]{x1D6FF})$ distance from its original location at the end of the injection region. Thus, to leading order we have

(3.4) $$\begin{eqnarray}{\displaystyle \frac{\unicode[STIX]{x2202}\tilde{h}_{0}}{\unicode[STIX]{x2202}\tilde{t}}}=\unicode[STIX]{x1D6E9}(1-x),\end{eqnarray}$$

with initial condition

(3.5) $$\begin{eqnarray}\tilde{h}_{0}=1\quad \text{at }\tilde{t}=0.\end{eqnarray}$$

The solution, in terms of the original variables, is

(3.6) $$\begin{eqnarray}h_{0}(x,t)=\unicode[STIX]{x1D6FF}+t\unicode[STIX]{x1D6E9}(1-x).\end{eqnarray}$$

We immediately note that, since this solution is spatially uniform on either side of $x=1$ , the boundary conditions (2.24) are automatically satisfied. The global conservation of mass (2.27) also holds. However, there is a discontinuity in our solution at $x=1$ and, to determine the smooth transition between the two solutions given in (3.6), we need to consider a boundary layer near $x=1$ . Specifically, we rescale

(3.7) $$\begin{eqnarray}x=1+\unicode[STIX]{x1D6FF}\tilde{x}.\end{eqnarray}$$

In this boundary layer, we obtain the following problem

(3.8) $$\begin{eqnarray}{\displaystyle \frac{\unicode[STIX]{x2202}\tilde{h}}{\unicode[STIX]{x2202}\tilde{t}}}+\frac{1}{3}{\displaystyle \frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}\tilde{x}}}\left(\tilde{h}^{3}{\displaystyle \frac{\unicode[STIX]{x2202}^{3}\tilde{h}}{\unicode[STIX]{x2202}\tilde{x}^{3}}}\right)=\unicode[STIX]{x1D6E9}(-\tilde{x}),\end{eqnarray}$$

subject to

(3.9) $$\begin{eqnarray}\displaystyle & \tilde{h}\rightarrow 1\quad \text{as }\tilde{x}\rightarrow \infty , & \displaystyle\end{eqnarray}$$

(3.10) $$\begin{eqnarray}\displaystyle & \tilde{h}\rightarrow 1+\tilde{t}\quad \text{as }\tilde{x}\rightarrow -\infty , & \displaystyle\end{eqnarray}$$

(3.11) $$\begin{eqnarray}\displaystyle & \tilde{h}=1\quad \text{at }\tilde{t}=0. & \displaystyle\end{eqnarray}$$

We will solve this boundary-layer problem numerically in § 4.

Before we focus on the behaviour of the thickness of the liquid film $h(x,t)$ for late time when we expect $O(1)$ movement of the contact line, we look at the $O(1)$ -time problem and discuss the behaviour of the apparent contact line.

3.2 The $O(1)$ -time behaviour

For $O(1)$ time, all terms in (2.23) come into play, and analytical progress is difficult. However, we will carefully analyse the motion of the apparent contact line with the purpose of exploiting these results during the late-time analysis. The scalings that we obtain in this section are identical to those used in the no-injection case considered in King & Bowen (Reference King and Bowen2001). They find that the apparent contact line is stationary for $O(1)$ time, which is supported by our numerical solutions found in § 4. We will focus on the next-order correction to this leading-order behaviour. Following the work of King & Bowen (Reference King and Bowen2001), we note that the leading-order (in $\unicode[STIX]{x1D6FF}$ ) solution for $h$ in the injection region has the following behaviour as $x\rightarrow 1^{-}$

(3.12) $$\begin{eqnarray}h_{0}\sim A(t)(1-x)\quad \text{as }x\rightarrow 1^{-},\end{eqnarray}$$

where $A(t)>0$ is some function that can, in theory, be determined from solving the leading-order problem

(3.13) $$\begin{eqnarray}{\displaystyle \frac{\unicode[STIX]{x2202}h_{0}}{\unicode[STIX]{x2202}t}}+\frac{1}{3}{\displaystyle \frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}x}}\left(h_{0}^{3}{\displaystyle \frac{\unicode[STIX]{x2202}^{3}h_{0}}{\unicode[STIX]{x2202}x^{3}}}\right)=1,\end{eqnarray}$$

together with

(3.14) $$\begin{eqnarray}\displaystyle & \displaystyle {\displaystyle \frac{\unicode[STIX]{x2202}h_{0}}{\unicode[STIX]{x2202}x}}={\displaystyle \frac{\unicode[STIX]{x2202}^{3}h_{0}}{\unicode[STIX]{x2202}x^{3}}}=0\quad \text{at }x=0, & \displaystyle\end{eqnarray}$$

(3.15) $$\begin{eqnarray}\displaystyle & h_{0}=0\quad \text{at }x=1, & \displaystyle\end{eqnarray}$$

(3.16) $$\begin{eqnarray}\displaystyle & h_{0}=0\quad \text{at }t=0, & \displaystyle\end{eqnarray}$$

and conservation of mass

(3.17) $$\begin{eqnarray}\int _{0}^{1}h_{0}(x,t)\,\text{d}x=t.\end{eqnarray}$$

In a transition region at the contact line near $x=1$ , we write

(3.18a,b ) $$\begin{eqnarray}x=1+\unicode[STIX]{x1D713}c(t)+\unicode[STIX]{x1D711}d(t)\unicode[STIX]{x1D702},\quad h=\unicode[STIX]{x1D6FF}H,\end{eqnarray}$$

where we take $d={\dot{c}}^{-1/3}$ for convenience in subsequent manipulations, and $\unicode[STIX]{x1D713}=\unicode[STIX]{x1D713}(\unicode[STIX]{x1D6FF})$ and $\unicode[STIX]{x1D711}=\unicode[STIX]{x1D711}(\unicode[STIX]{x1D6FF})$ are anticipated to be small parameters, since the contact line is expected to be close to $x=1$ . Noting that the spatial and time derivatives transform to

(3.19a,b ) $$\begin{eqnarray}{\displaystyle \frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}t}}\rightarrow {\displaystyle \frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}t}}-\left(\frac{\unicode[STIX]{x1D713}{\dot{c}}}{\unicode[STIX]{x1D711}d}+\frac{\unicode[STIX]{x1D702}{\dot{d}}}{d}\right){\displaystyle \frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}\unicode[STIX]{x1D702}}},\quad {\displaystyle \frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}x}}\rightarrow \frac{1}{\unicode[STIX]{x1D711}d}{\displaystyle \frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}\unicode[STIX]{x1D702}}},\end{eqnarray}$$

after rearranging, equation (2.23) becomes

(3.20) $$\begin{eqnarray}\unicode[STIX]{x1D6FF}d^{4}{\displaystyle \frac{\unicode[STIX]{x2202}H}{\unicode[STIX]{x2202}\bar{t}}}-\left(\frac{\unicode[STIX]{x1D6FF}\unicode[STIX]{x1D713}}{\unicode[STIX]{x1D711}}+\unicode[STIX]{x1D702}\unicode[STIX]{x1D6FF}{\dot{d}}d^{3}\right){\displaystyle \frac{\unicode[STIX]{x2202}H}{\unicode[STIX]{x2202}\unicode[STIX]{x1D702}}}+\frac{\unicode[STIX]{x1D6FF}^{4}}{3\unicode[STIX]{x1D711}^{4}}{\displaystyle \frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}\unicode[STIX]{x1D702}}}\left(H^{3}{\displaystyle \frac{\unicode[STIX]{x2202}^{3}H}{\unicode[STIX]{x2202}\unicode[STIX]{x1D702}^{3}}}\right)=d^{4}\unicode[STIX]{x1D6E9}\left(-\unicode[STIX]{x1D713}c-\unicode[STIX]{x1D711}d\unicode[STIX]{x1D702}\right).\end{eqnarray}$$

In order to obtain a non-trivial dominant balance in (3.20), we balance the second and fourth terms, which requires

(3.21a,b ) $$\begin{eqnarray}\unicode[STIX]{x1D713}=\frac{\unicode[STIX]{x1D6FF}^{3}}{\unicode[STIX]{x1D711}^{3}},\quad \unicode[STIX]{x1D711}\ll \unicode[STIX]{x1D713}\quad \text{as }\unicode[STIX]{x1D6FF}\rightarrow 0.\end{eqnarray}$$

The second condition tells us that for $O(1)$ time, in the limit $\unicode[STIX]{x1D6FF}\rightarrow 0$ , the apparent contact line moves by an amount that is much greater in extent than the width of the transition region. Thus, we may ignore injection in the transition region, and the corresponding equation for $H$ becomes

(3.22) $$\begin{eqnarray}-{\displaystyle \frac{\unicode[STIX]{x2202}H}{\unicode[STIX]{x2202}\unicode[STIX]{x1D702}}}+\frac{1}{3}{\displaystyle \frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}\unicode[STIX]{x1D702}}}\left(H^{3}{\displaystyle \frac{\unicode[STIX]{x2202}^{3}H}{\unicode[STIX]{x2202}\unicode[STIX]{x1D702}^{3}}}\right)=0.\end{eqnarray}$$

Analysing the behaviour of (3.22) as $\unicode[STIX]{x1D702}\rightarrow -\infty$ , matching it to the outer solution (3.12), as has been done in King & Bowen (Reference King and Bowen2001), and using (3.21), we obtain the necessary scalings for $\unicode[STIX]{x1D713}$ and $\unicode[STIX]{x1D711}$ , namely

(3.23a,b ) $$\begin{eqnarray}\unicode[STIX]{x1D713}=\frac{1}{\log (1/\unicode[STIX]{x1D6FF})},\quad \unicode[STIX]{x1D711}=\unicode[STIX]{x1D6FF}\log ^{1/3}(1/\unicode[STIX]{x1D6FF}).\end{eqnarray}$$

We immediately note that (3.21) hold true. These scalings will be further verified by the numerical solutions presented in § 4. Thus, we conclude that for $O(1)$ time, the contact line is indeed stationary to leading order in $\unicode[STIX]{x1D6FF}$ , and moves a distance $O(1/\log (1/\unicode[STIX]{x1D6FF}))$ at next order.

3.3 Late-time behaviour

To explore the late-time behaviour of the system, we will rescale both time and the film thickness appropriately to arrive at a limit in which the surface-tension term in (2.23) is the only dominant term at leading order in the relevant asymptotic parameter (which will be determined as part of the analysis). This leading-order problem constitutes the solution in an outer region, the region to the left of the moving apparent contact line (see figure 3). At this stage, though, the position of the apparent contact line is an unknown function of time. We will solve for the film thickness in the transition region near the apparent contact line and match the solutions from the outer, transition and precursor-layer regions to obtain the full profile of the film thickness and the leading-order solution for the location of the moving boundary $a(t)$ . We will also see at the end of this section, and in § 5, that the late-time behaviour of our model is a generalization of the classical droplet-spreading problem when we allow the volume of liquid to be time dependent. However, unlike the classical droplet-spreading problem, we will obtain unbounded growth of the film, which is entirely due to the finite-width injection region. In § 5 we will consider some generalizations to our injection-spreading problem, and we will see that unbounded growth is not always guaranteed and depends on the functional form of the injection rate.

Figure 3.

A schematic diagram of the outer, transition and precursor-layer regions, and the position of the apparent contact line. The film thicknesses are not drawn to scale.

3.3.1 The outer region

Unlike the spreading-droplet scenario considered in King & Bowen (Reference King and Bowen2001), here we anticipate that the injection results in a film thickness that is much larger than $\unicode[STIX]{x1D6FF}$ for late time. Thus, we scale

(3.24a,b ) $$\begin{eqnarray}t=\frac{\bar{t}}{\unicode[STIX]{x1D706}},\quad h=\frac{\bar{h}}{\unicode[STIX]{x1D706}},\end{eqnarray}$$

where $\unicode[STIX]{x1D706}=\unicode[STIX]{x1D706}(\unicode[STIX]{x1D6FF})\ll 1$ is to be determined. We note that we do not scale $x$ , because we anticipate that $a$ will move by an $O(1)$ amount on this time scale. After rearranging, equation (2.23) becomes

(3.25) $$\begin{eqnarray}\unicode[STIX]{x1D706}^{4}{\displaystyle \frac{\unicode[STIX]{x2202}\bar{h}}{\unicode[STIX]{x2202}\bar{t}}}+\frac{1}{3}{\displaystyle \frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}x}}\left(\bar{h}^{3}{\displaystyle \frac{\unicode[STIX]{x2202}^{3}\bar{h}}{\unicode[STIX]{x2202}x^{3}}}\right)=\unicode[STIX]{x1D706}^{4}\unicode[STIX]{x1D6E9}(1-x).\end{eqnarray}$$

The surface-tension term in (3.25) is dominant to leading order in $\unicode[STIX]{x1D706}^{4}$ . We assume $\unicode[STIX]{x1D6FF}\ll \unicode[STIX]{x1D706}^{4}\ll 1$ as $\unicode[STIX]{x1D6FF}\rightarrow 0$ , and will verify this assertion a posteriori. Expanding the dependent variables as

(3.26) $$\begin{eqnarray}\displaystyle & \bar{h}(x,\bar{t})\sim \bar{h}_{0}(x,\bar{t})+\unicode[STIX]{x1D706}^{4}\bar{h}_{1}(x,\bar{t})+\cdots \quad \text{as }\unicode[STIX]{x1D706}\rightarrow 0, & \displaystyle\end{eqnarray}$$

(3.27) $$\begin{eqnarray}\displaystyle & a(\bar{t})\sim a_{0}(\bar{t})+\unicode[STIX]{x1D706}^{4}a_{1}(\bar{t})+\cdots \quad \text{as }\unicode[STIX]{x1D706}\rightarrow 0, & \displaystyle\end{eqnarray}$$

the leading-order problem becomes

(3.28) $$\begin{eqnarray}{\displaystyle \frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}x}}\left(\bar{h}_{0}^{3}{\displaystyle \frac{\unicode[STIX]{x2202}^{3}\bar{h}_{0}}{\unicode[STIX]{x2202}x^{3}}}\right)=0,\end{eqnarray}$$

with

(3.29) $$\begin{eqnarray}\displaystyle & \displaystyle {\displaystyle \frac{\unicode[STIX]{x2202}\bar{h}_{0}}{\unicode[STIX]{x2202}x}}={\displaystyle \frac{\unicode[STIX]{x2202}^{3}\bar{h}_{0}}{\unicode[STIX]{x2202}x^{3}}}=0\quad \text{at }x=0, & \displaystyle\end{eqnarray}$$

(3.30) $$\begin{eqnarray}\displaystyle & \bar{h}_{0}=0\quad \text{at }x=a_{0}(\bar{t}). & \displaystyle\end{eqnarray}$$

The leading-order version of the conservation-of-mass equation (2.27) becomes

(3.31) $$\begin{eqnarray}\bar{t}=\int _{0}^{a_{0}(\bar{t})}\bar{h}_{0}(x,\bar{t})\,\text{d}x.\end{eqnarray}$$

Equation (3.28) with conditions (3.29)–(3.31) has a parabolic solution, namely,

(3.32) $$\begin{eqnarray}\bar{h}_{0}(x,\bar{t})=\frac{3\bar{t}(a_{0}^{2}-x^{2})}{2a_{0}^{3}},\end{eqnarray}$$

where $a_{0}(\bar{t})$ is unknown. In order to determine $a_{0}(\bar{t})$ , we will explore the solution in the transition region near the moving boundary and then match to (3.32).

3.3.2 The transition region

In the transition region near the moving front, we write

(3.33a-c ) $$\begin{eqnarray}t=\frac{\bar{t}}{\unicode[STIX]{x1D706}},\quad x=a_{0}(\bar{t})+\unicode[STIX]{x1D708}s(\bar{t})\unicode[STIX]{x1D709},\quad h=\unicode[STIX]{x1D6FF}H,\end{eqnarray}$$

where $\unicode[STIX]{x1D708}\ll 1$ , and we set $s(\bar{t})={\dot{a}}_{0}^{-1/3}$ for convenience in subsequent manipulations. With this change of variables, the spatial and time derivatives transform to

(3.34a,b ) $$\begin{eqnarray}{\displaystyle \frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}\bar{t}}}\rightarrow {\displaystyle \frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}\bar{t}}}-\left(\frac{{\dot{a}}_{0}}{\unicode[STIX]{x1D708}s}+\frac{\unicode[STIX]{x1D709}{\dot{s}}}{s}\right){\displaystyle \frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}\unicode[STIX]{x1D709}}},\quad {\displaystyle \frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}x}}\rightarrow \frac{1}{\unicode[STIX]{x1D708}s}{\displaystyle \frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}\unicode[STIX]{x1D709}}}.\end{eqnarray}$$

Furthermore, from § 3.2, we know that $a$ moves $O(1/\log (1/\unicode[STIX]{x1D6FF}))$ away from the end of the injection region for $O(1)$ time. Therefore, assuming that $\unicode[STIX]{x1D708}\ll 1/\log (1/\unicode[STIX]{x1D6FF})$ as $\unicode[STIX]{x1D6FF}\rightarrow 0$ (to be verified a posteriori), the transition region will lie outside the injection region for late time, and (2.23) becomes

(3.35) $$\begin{eqnarray}\unicode[STIX]{x1D706}\left(s^{4}{\displaystyle \frac{\unicode[STIX]{x2202}H}{\unicode[STIX]{x2202}\bar{t}}}-\left(\frac{1}{\unicode[STIX]{x1D708}}+\unicode[STIX]{x1D709}{\dot{s}}s^{3}\right){\displaystyle \frac{\unicode[STIX]{x2202}H}{\unicode[STIX]{x2202}\unicode[STIX]{x1D709}}}\right)+\frac{\unicode[STIX]{x1D6FF}^{3}}{3\unicode[STIX]{x1D708}^{4}}{\displaystyle \frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}\unicode[STIX]{x1D709}}}\left(H^{3}{\displaystyle \frac{\unicode[STIX]{x2202}^{3}H}{\unicode[STIX]{x2202}\unicode[STIX]{x1D709}^{3}}}\right)=0.\end{eqnarray}$$

In order to obtain a non-trivial dominant balance in (3.35), we balance the second and fourth terms, which requires that

(3.36) $$\begin{eqnarray}\unicode[STIX]{x1D708}=\frac{\unicode[STIX]{x1D6FF}}{\unicode[STIX]{x1D706}^{1/3}}.\end{eqnarray}$$

After rearranging, equation (3.35) then becomes

(3.37) $$\begin{eqnarray}\unicode[STIX]{x1D708}\left(s^{4}{\displaystyle \frac{\unicode[STIX]{x2202}H}{\unicode[STIX]{x2202}\bar{t}}}-\unicode[STIX]{x1D709}{\dot{s}}s^{3}{\displaystyle \frac{\unicode[STIX]{x2202}H}{\unicode[STIX]{x2202}\unicode[STIX]{x1D709}}}\right)-{\displaystyle \frac{\unicode[STIX]{x2202}H}{\unicode[STIX]{x2202}\unicode[STIX]{x1D709}}}+\frac{1}{3}{\displaystyle \frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}\unicode[STIX]{x1D709}}}\left(H^{3}{\displaystyle \frac{\unicode[STIX]{x2202}^{3}H}{\unicode[STIX]{x2202}\unicode[STIX]{x1D709}^{3}}}\right)=0.\end{eqnarray}$$

We expand the dependent variables in powers of $\unicode[STIX]{x1D708}=\unicode[STIX]{x1D708}(\unicode[STIX]{x1D6FF})$ , assuming $\unicode[STIX]{x1D6FF}\ll \unicode[STIX]{x1D708}\ll 1$ as $\unicode[STIX]{x1D6FF}\rightarrow 0$ , which we will also check a posteriori, and, thus, we write

(3.38) $$\begin{eqnarray}H(\unicode[STIX]{x1D709},\bar{t})\sim H_{0}(\unicode[STIX]{x1D709},\bar{t})+\unicode[STIX]{x1D708}H_{1}(\unicode[STIX]{x1D709},\bar{t})+\cdots \quad \text{as }\unicode[STIX]{x1D708}\rightarrow 0.\end{eqnarray}$$

Substituting (3.38) into (3.37), retaining leading-order terms, integrating once, and applying the matching condition with the precursor layer

(3.39) $$\begin{eqnarray}H_{0}\rightarrow 1\quad \text{as }\unicode[STIX]{x1D709}\rightarrow \infty ,\end{eqnarray}$$

we find that $H_{0}$ satisfies

(3.40) $$\begin{eqnarray}-H_{0}+\frac{1}{3}H_{0}^{3}{\displaystyle \frac{\unicode[STIX]{x2202}^{3}H_{0}}{\unicode[STIX]{x2202}\unicode[STIX]{x1D709}^{3}}}=-1.\end{eqnarray}$$

To match with the outer solution, we need to understand the behaviour of the transition-region solution as $\unicode[STIX]{x1D709}\rightarrow -\infty$ . We assume the solution is of the form $H_{0}\sim B(-\unicode[STIX]{x1D709})^{m}\log ^{n}(-\unicode[STIX]{x1D709}),$ as $\unicode[STIX]{x1D709}\rightarrow -\infty$ , for some values of $B,m$ and $n$ . Choosing $m=1$ , $n=1/3$ and $B=9^{1/3}$ gives the leading-order balance as $\unicode[STIX]{x1D709}\rightarrow -\infty$ between the two terms on the left-hand side of (3.40). Thus, our second matching condition is

(3.41) $$\begin{eqnarray}H_{0}\sim 9^{1/3}(-\unicode[STIX]{x1D709})\log ^{1/3}(-\unicode[STIX]{x1D709})\quad \text{as }\unicode[STIX]{x1D709}\rightarrow -\infty .\end{eqnarray}$$

We note that (3.40) has been solved by Duffy & Wilson (Reference Duffy and Wilson1997) for the case when the right-hand side is zero. Furthermore, they also obtain the asymptotic behaviour (3.41) (modulo the pre-factor), which confirms our result, since as $\unicode[STIX]{x1D709}\rightarrow -\infty$ , $H_{0}\rightarrow \infty$ and the right-hand side of (3.40), which is constant, does not appear in the dominant balance. We solve for $H_{0}$ using a shooting method. Specifically, we expand $H_{0}$ using

(3.42) $$\begin{eqnarray}H_{0}\sim 1+\tilde{H}_{0}\quad \text{as }\unicode[STIX]{x1D709}\rightarrow \infty ,\end{eqnarray}$$

where $\tilde{H}_{0}\ll 1$ . The equation for $\tilde{H}_{0}$ is

(3.43) $$\begin{eqnarray}\frac{1}{3}{\displaystyle \frac{\unicode[STIX]{x2202}^{3}\tilde{H}_{0}}{\unicode[STIX]{x2202}\unicode[STIX]{x1D709}^{3}}}-\tilde{H}_{0}=0,\end{eqnarray}$$

which has a solution that is a combination of exponentials. Retaining only decaying solutions and appropriately choosing the origin, we may write the solution as

(3.44) $$\begin{eqnarray}H_{0}\sim 1+\tilde{A}\exp (-3^{1/3}\unicode[STIX]{x1D709}/2)\cos (3^{5/6}\unicode[STIX]{x1D709}/2)\quad \text{as }\unicode[STIX]{x1D709}\rightarrow \infty .\end{eqnarray}$$

We then solve (3.40) by using $H_{0},H_{0}^{\prime },H_{0}^{\prime \prime }$ given by (3.44) evaluated at some large value of $\unicode[STIX]{x1D709}$ (we used $\unicode[STIX]{x1D709}=10$ ); $\tilde{A}$ is our shooting parameter. We note that we can restrict the interval for possible values for $\tilde{A}$ to, for example, $[1,\exp (2\unicode[STIX]{x03C0}/\sqrt{3})]$ (McEwan & Taylor Reference McEwan and Taylor1966; Tuck & Schwartz Reference Tuck and Schwartz1990). We vary $\tilde{A}$ in order to match the behaviour as $\unicode[STIX]{x1D709}\rightarrow -\infty$ specified by (3.41). As in Tuck & Schwartz (Reference Tuck and Schwartz1990), we use the fact that $H_{0}H_{0}^{\prime }H_{0}^{\prime \prime }\rightarrow -3$ as $\unicode[STIX]{x1D709}\rightarrow -\infty$ to provide us with an accurate indicator of how close we are to the required solution.

3.3.3 Matching

We first match the transition region with the outer region. We write (3.41) in outer variables as

(3.45) $$\begin{eqnarray}\displaystyle \bar{h}_{0} & {\sim} & \displaystyle \frac{9^{1/3}\unicode[STIX]{x1D706}^{4/3}(a_{0}-x)\log ^{1/3}(\unicode[STIX]{x1D706}^{1/3}(a_{0}-x)/(\unicode[STIX]{x1D6FF}s))}{s}\nonumber\\ \displaystyle & {\sim} & \displaystyle \frac{9^{1/3}\unicode[STIX]{x1D706}^{4/3}(a_{0}-x)\log ^{1/3}(1/\unicode[STIX]{x1D6FF})}{s}\quad \text{as }\unicode[STIX]{x1D6FF}\rightarrow 0.\end{eqnarray}$$

The leading-order term in the outer solution (3.32) is

(3.46) $$\begin{eqnarray}\bar{h}_{0}\sim \frac{3\bar{t}(a_{0}-x)}{a_{0}^{2}}\quad \text{as }x\rightarrow a_{0}^{-}.\end{eqnarray}$$

This suggests that, to match (3.45) and (3.46), the appropriate scaling for $\unicode[STIX]{x1D706}$ should be

(3.47) $$\begin{eqnarray}\unicode[STIX]{x1D706}=\frac{1}{\log ^{1/4}(1/\unicode[STIX]{x1D6FF})}.\end{eqnarray}$$

Thus, equation (3.36) gives

(3.48) $$\begin{eqnarray}\unicode[STIX]{x1D708}=\unicode[STIX]{x1D6FF}\log ^{1/12}(1/\unicode[STIX]{x1D6FF}).\end{eqnarray}$$

We note that this confirms the hypotheses that $\unicode[STIX]{x1D6FF}\ll \unicode[STIX]{x1D708}\ll \unicode[STIX]{x1D706}^{4}\ll \unicode[STIX]{x1D706}\ll 1$ , as $\unicode[STIX]{x1D6FF}\rightarrow 0$ . Logarithmic scales also appear in similar problems described in Lacey (Reference Lacey1982), Hocking (Reference Hocking1983) and King & Bowen (Reference King and Bowen2001), for example. However, for the classical problem of a spreading droplet, the relevant scale is $1/\log (1/\unicode[STIX]{x1D6FF})$ (King & Bowen Reference King and Bowen2001), which is asymptotically smaller (in the limit $\unicode[STIX]{x1D6FF}\rightarrow 0$ ) than our result (3.47) in the injection case. Matching (3.45) with (3.46), we obtain

(3.49) $$\begin{eqnarray}{\dot{a}}_{0}=\frac{3\bar{t\,}^{3}}{a_{0}^{6}},\end{eqnarray}$$

which can be solved to determine $a_{0}(\bar{t})$ . We also remark that (3.49) is a version of Tanner’s law, since it follows that, at the contact line,

(3.50) $$\begin{eqnarray}{\displaystyle \frac{\unicode[STIX]{x2202}\bar{h}_{0}}{\unicode[STIX]{x2202}x}}=-(9{\dot{a}}_{0})^{1/3}.\end{eqnarray}$$

We match the late-time behaviour for the contact line with the behaviour for $O(1)$ time, which results in imposing the initial condition $a_{0}\rightarrow 1$ as $t\rightarrow 0$ . Thus, equation (3.49) can be solved to yield

(3.51) $$\begin{eqnarray}a_{0}(\bar{t})=\left(1+\frac{21\bar{t\,}^{4}}{4}\right)^{1/7},\end{eqnarray}$$

and so,

(3.52) $$\begin{eqnarray}\bar{h}_{0}(0,\bar{t})=\frac{3\bar{t}}{2a_{0}}=\frac{3\bar{t}}{2}\left(1+\frac{21\bar{t\,}^{4}}{4}\right)^{-1/7}.\end{eqnarray}$$

In terms of the original variables, $h_{0}=\bar{h}_{0}/\unicode[STIX]{x1D706}$ and $t=\bar{t}/\unicode[STIX]{x1D706}$ ,

(3.53a,b ) $$\begin{eqnarray}a_{0}(t)=\left(1+\frac{21\unicode[STIX]{x1D706}^{4}t^{4}}{4}\right)^{1/7},\quad h_{0}(0,t)=\frac{3t}{2}\left(1+\frac{21\unicode[STIX]{x1D706}^{4}t^{4}}{4}\right)^{-1/7}\quad \text{for }t=O(1/\unicode[STIX]{x1D706}).\quad\end{eqnarray}$$

We observe that, for late time, the apparent contact line moves much faster ( ${\sim}t^{4/7}$ ) than the classical non-injection case ( ${\sim}t^{1/7}$ ) considered in King & Bowen (Reference King and Bowen2001). We also note that, through $\unicode[STIX]{x1D706}$ , the leading-order results (3.53) have a dependence on the precursor thickness, $\unicode[STIX]{x1D6FF}$ . In physical systems where precursor layers are not observed, $\unicode[STIX]{x1D6FF}$ can be treated as an adjustable parameter to be determined from experimental data.

As mentioned at the beginning of § 3, we can generalize the results from the classical droplet-spreading problem (see, for example, Voinov Reference Voinov1976; Cox Reference Cox1986; Bonn et al. Reference Bonn, Eggers, Indekeu, Meunier and Rolley2009) to the case of a time-dependent injection rate using the Cox–Voinov law (Bonn et al. Reference Bonn, Eggers, Indekeu, Meunier and Rolley2009) to obtain the full late-time ( $t\gg 1/\unicode[STIX]{x1D706}$ ) behaviour of the film spreading (see appendix C for details).

We further note that we can also obtain the power-law dependence on time in (3.53) using scaling arguments. The matching condition (3.49) can also simply be deduced from a comparison of the slopes of $h$ in the inner and outer regions obtained from (3.45) and (3.46), respectively. Scaling $x\sim a$ , the slope of the outer solution ${\sim}h/a\sim t/a^{2}$ from (3.32), while the inner region has height ${\sim}\unicode[STIX]{x1D6FF}$ and length ${\sim}\unicode[STIX]{x1D6FF}{\dot{a}}^{-1/3}$ from (3.33) (with $\unicode[STIX]{x1D706}=1$ ) and, thus, has slope $t/a^{2}\sim {\dot{a}}^{1/3}$ confirming the scalings $a\sim t^{4/7}$ and $h\sim t^{3/7}$ . Furthermore, if we look for a similarity solution to the problem for late time (essentially, away from the injection region) of the form $t^{\unicode[STIX]{x1D6FC}}f(x/t^{\unicode[STIX]{x1D6FD}})$ , for some exponents $\unicode[STIX]{x1D6FC}$ and $\unicode[STIX]{x1D6FD}$ , we can recover $\unicode[STIX]{x1D6FC}=3/7$ and $\unicode[STIX]{x1D6FD}=4/7$ and the parabolic profile for the outer solution. This, as we saw, is confirmed by the late-time asymptotic analysis. Thus, for late time the aspect ratio given by $h_{0}(0,t)/a_{0}(t)\sim t^{-1/7}$ , so remains small for all time.

5.1 Power-law injection

We first consider the case in which $\hat{f}$ has a power-law dependence in time, i.e. $\hat{f}=\hat{t}^{k}{\hat{g}}(\hat{x})$ , where ${\hat{g}}$ is some integrable function on $[0,1]$ , and $k$ is some real exponent whose range we will specify later on. We expect that $k<0$ for physically sensible models, such as a chemical reaction whose rate decreases with time. However, we will also study the case $k>0$ . We note that, for $-1<k<0$ , the injection rate is singular at $t=0$ , but the volume of liquid (which is the time integral of the flux) is still finite. The case $k\leqslant -1$ corresponds to an infinite volume of liquid injected at $t=0$ , which we exclude on physical grounds. For simplicity, in the following analysis we assume that ${\hat{g}}=W$ is constant, even though more general spatial variance could be incorporated using the same approach. Thus, the boundary condition on the surface $\hat{z}=0$ becomes

(5.2) $$\begin{eqnarray}{\hat{w}}=W\hat{t}^{k}\unicode[STIX]{x1D6E9}(L-\hat{x})\quad \text{on }\hat{z}=0.\end{eqnarray}$$

In this case, we require a different non-dimensionalization and we use

(5.3) $$\begin{eqnarray}\left.\begin{array}{@{}c@{}}(\hat{x},\hat{a})=L(x,a),\quad (\hat{z},{\hat{h}})=(\unicode[STIX]{x1D707}W^{1/(k+1)}L^{4}/\unicode[STIX]{x1D6FE})^{(k+1)/(3k+4)}(z,h),\\ \hat{u} =(\unicode[STIX]{x1D6FE}W^{3}L^{3k}/\unicode[STIX]{x1D707})^{1/(3k+4)}u,\quad {\hat{w}}=(\unicode[STIX]{x1D707}W^{4/k}L^{4}/\unicode[STIX]{x1D6FE})^{k/(3k+4)}w,\\ \hat{t}=(\unicode[STIX]{x1D707}L^{4}/\unicode[STIX]{x1D6FE}W^{3})^{1/(3k+4)}t,\quad \hat{p}=(\unicode[STIX]{x1D707}^{k+1}\unicode[STIX]{x1D6FE}^{2k+3}W/L^{2k+4})^{1/(3k+4)}p.\end{array}\right\}\end{eqnarray}$$

We assume a long and thin geometry, i.e. that the aspect ratio

(5.4) $$\begin{eqnarray}\unicode[STIX]{x1D716}=(\unicode[STIX]{x1D707}W^{1/(k+1)}L^{k/k+1}/\unicode[STIX]{x1D6FE})^{(k+1)/(3k+4)}\ll 1,\end{eqnarray}$$

and that we have a thin precursor film, i.e. $\unicode[STIX]{x1D6FF}=h_{\infty }/(\unicode[STIX]{x1D716}L)\ll 1$ . Following the same steps as in § 2.2, we manipulate the dimensionless versions of (2.1)–(2.3), (2.5)–(2.7) and (5.2) and we arrive at the evolution equation for $h$ , namely

(5.5) $$\begin{eqnarray}{\displaystyle \frac{\unicode[STIX]{x2202}h}{\unicode[STIX]{x2202}t}}+\frac{1}{3}{\displaystyle \frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}x}}\left(h^{3}{\displaystyle \frac{\unicode[STIX]{x2202}^{3}h}{\unicode[STIX]{x2202}x^{3}}}\right)=t^{k}\unicode[STIX]{x1D6E9}(1-x),\end{eqnarray}$$

together with the boundary conditions given by (2.24)–(2.26).

For early time, we use a scaling similar to (3.1), but take account of the exponent $k$ in the injection function by setting

(5.6a,b ) $$\begin{eqnarray}h=\unicode[STIX]{x1D6FF}\tilde{h},\quad t=\unicode[STIX]{x1D6FF}^{1/(k+1)}\tilde{t},\end{eqnarray}$$

where, here, we restrict ourselves to considering $k>-1$ so that the scalings are valid for early time. After asymptotically expanding $h$ and $a$ in powers of $\unicode[STIX]{x1D6FF}$ as before, we arrive at a dominant balance between the first and third term in (5.5). Using the same notation as in § 3, we obtain

(5.7) $$\begin{eqnarray}\tilde{h}_{0}(x,\tilde{t})=1+\displaystyle \frac{\tilde{t}^{k+1}}{k+1}\unicode[STIX]{x1D6E9}(1-x).\end{eqnarray}$$

As in § 3.1, it is possible to obtain a problem for the boundary layer near $x=1$ , analogous to (3.8)–(3.11), which smoothly joins the two solutions given in (5.7), but we do not pursue that here. Instead, we point out that we can consider two distinct temporal regimes in which more analytical progress can be made (see appendix B for the details of this analysis).

For late time, we scale

(5.8a,b ) $$\begin{eqnarray}h=\frac{\bar{h}}{\unicode[STIX]{x1D706}_{k}},\quad t=\frac{\bar{t}}{\unicode[STIX]{x1D706}_{k}^{1/(k+1)}},\end{eqnarray}$$

where we assume $\unicode[STIX]{x1D706}_{k}\ll 1$ . Repeating the analysis in § 3.3, we find the appropriate choice of $\unicode[STIX]{x1D706}_{k}$ is (cf. (3.47))

(5.9) $$\begin{eqnarray}\unicode[STIX]{x1D706}_{k}=\frac{1}{\log ^{(k+1)/(3k+4)}(1/\unicode[STIX]{x1D6FF})},\end{eqnarray}$$

which is indeed small for $k>-1$ . Therefore, we obtain the following leading-order results for the position of the apparent contact line and the maximum film thickness

(5.10) $$\begin{eqnarray}\displaystyle & \displaystyle a_{0}(\bar{t})=\left(1+\frac{21\bar{t}^{\,3k+4}}{(k+1)^{3}(3k+4)}\right)^{1/7}, & \displaystyle\end{eqnarray}$$

(5.11) $$\begin{eqnarray}\displaystyle & \displaystyle \bar{h}_{0}(0,\bar{t})=\frac{3\bar{t}^{\,k+1}}{2(k+1)a_{0}}=\frac{3\bar{t}^{\,k+1}}{2(k+1)}\left(1+\frac{21\bar{t}^{\,3k+4}}{(k+1)^{3}(3k+4)}\right)^{-1/7}. & \displaystyle\end{eqnarray}$$

In an analogous way to § 3, we can use the Cox–Voinov law to obtain the full late-time behaviour of the film spreading when $t\gg 1/\unicode[STIX]{x1D706}_{k}^{1/(k+1)}$ .

We note that, in the case when $k=0$ , we recover the results (3.51) and (3.52). Furthermore, when $k=-3/4$ , $\bar{h}_{0}$ tends to a constant, i.e. injection is just enough to balance the surface-tension flow. Consequently, for $-1<k<-3/4$ , injection can no longer support a growing or constant film thickness, and, thus, the film thickness decreases at large time. For very large time, we expect capillarity to force the film to approach the thickness of the precursor layer again, and, thus, using the scaling from (5.11), $h\sim t^{(4k+3)/7}$ gives $t=\unicode[STIX]{x1D6FF}^{7/(4k+3)}t^{\star }$ as the required scaling for $t$ . We note that the same scaling can be derived from the balance $h-\unicode[STIX]{x1D6FF}\sim t^{k+1}/x,x\sim (\unicode[STIX]{x1D6FF}^{3}t)^{1/4}$ between the growth and the surface-tension term in (5.5). We do not pursue this situation further.

In figure 7, we show log–log plots of the numerical solution and our asymptotic predictions for the maximum film thickness at the origin and the position of the apparent contact line as functions of time for different values of $k$ . We see a good agreement between the numerical solution and the asymptotic results for early and late time.

Additionally, we observe that the aspect ratio, given by $h_{0}(0,t)/a_{0}(t)\sim t^{(k-1)/7}$ , grows in time if $k>1$ . In this case, for late time, the thin-film hypothesis will eventually break down.

Figure 7.

Log–log plots of the numerical solution to (5.5) with (2.24)–(2.26) (solid blue) for $h(0,t)$ (a,c,e) and $a(t)$ (b,d,f) for (a,b) $k=1/2$ , (c,d)  $k=-1/2$ , (e,f) $k=-3/4$ . The asymptotic predictions for early time, equation (5.7) (dotted green), and late time, equations (5.10)–(5.11) (dashed red), are also shown. In all figures, $\unicode[STIX]{x1D6FF}=10^{-3}$ .

Finally, we note that the results (5.10)–(5.11) can be used in a simple inverse problem, in which we can determine the exponent of the injection rate, assuming it has a power-law dependence, by observing the spreading of the film front or the evolution of its maximum thickness at the origin for late time. Thus, if the film front is observed to spread like $t^{\unicode[STIX]{x1D70E}}$ for some exponent $\unicode[STIX]{x1D70E}$ , then the exponent of the injection rate is given by $k=(7\unicode[STIX]{x1D70E}-4)/3$ . For example, if experimental data reveal a linear spread of the film front, then the liquid is also injected linearly in time.

5.2 Point-source injection

We now consider the case when liquid is injected at the origin, i.e. a point-source injection with $\hat{f}=Q\unicode[STIX]{x1D6FF}(\hat{x})$ in (5.1), where $Q$ is a given liquid flux. We note that, due to the definition of the delta function, $Q=2WL$ in terms of the notation in § 3. The boundary condition on $\hat{z}=0$ then becomes

(5.12) $$\begin{eqnarray}{\hat{w}}=Q\unicode[STIX]{x1D6FF}(\hat{x})\quad \text{on }\hat{z}=0.\end{eqnarray}$$

We non-dimensionalize using

(5.13) $$\begin{eqnarray}\left.\begin{array}{@{}c@{}}(\hat{x},\hat{a})=(\unicode[STIX]{x1D6FE}Q^{3}\unicode[STIX]{x1D70F}^{4}/\unicode[STIX]{x1D707})^{1/7}(x,a),\quad (\hat{z},{\hat{h}})=(\unicode[STIX]{x1D707}Q^{4}\unicode[STIX]{x1D70F}^{3}/\unicode[STIX]{x1D6FE})^{1/7}(z,h),\\ \hat{u} =(\unicode[STIX]{x1D6FE}Q^{3}/\unicode[STIX]{x1D707}\unicode[STIX]{x1D70F}^{3})^{1/7}u,\quad {\hat{w}}=(\unicode[STIX]{x1D707}Q^{4}/\unicode[STIX]{x1D6FE}\unicode[STIX]{x1D70F}^{4})^{1/7}w,\quad \hat{t}=\unicode[STIX]{x1D70F}t,\quad \hat{p}=(\unicode[STIX]{x1D707}^{3}\unicode[STIX]{x1D6FE}^{4}/Q^{2}\unicode[STIX]{x1D70F}^{5})^{1/7}p,\end{array}\right\}\end{eqnarray}$$

where we can choose the time scale $\unicode[STIX]{x1D70F}=(h_{\infty }^{7}\unicode[STIX]{x1D6FE}/\unicode[STIX]{x1D707}Q^{4})^{1/3}$ so that the dimensionless precursor-layer thickness $\unicode[STIX]{x1D6FF}=h_{\infty }(\unicode[STIX]{x1D707}Q^{4}\unicode[STIX]{x1D70F}^{3}/\unicode[STIX]{x1D6FE})^{-1/7}=1$ and we remove all dimensionless parameters from the model. However, we are interested in the situation where the film thickness is much larger than the precursor thickness and a long time scale over which the aspect ratio $\unicode[STIX]{x1D716}=\left(\unicode[STIX]{x1D707}^{2}Q/\unicode[STIX]{x1D6FE}^{2}\unicode[STIX]{x1D70F}\right)^{1/7}\ll 1$ so that the thin-film hypothesis is valid. Thus, we rescale our problem using

(5.14a-c ) $$\begin{eqnarray}t\rightarrow Tt,\quad (x,a)\rightarrow T^{4/7}(x,a),\quad (z,h)\rightarrow T^{3/7}(z,h)\end{eqnarray}$$

for some $T\gg 1$ . The evolution equation for $h$ becomes

(5.15) $$\begin{eqnarray}{\displaystyle \frac{\unicode[STIX]{x2202}h}{\unicode[STIX]{x2202}t}}+\frac{1}{3}{\displaystyle \frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}x}}\left(h^{3}{\displaystyle \frac{\unicode[STIX]{x2202}^{3}h}{\unicode[STIX]{x2202}x^{3}}}\right)=\unicode[STIX]{x1D6FF}(x),\end{eqnarray}$$

together with (2.24)–(2.26), in which the dimensionless precursor thickness now appears as $\unicode[STIX]{x1D6FF}=1/T^{3/7}\ll 1$ , and where we have used the fact that $\unicode[STIX]{x1D6FF}(\unicode[STIX]{x1D705}x)=\unicode[STIX]{x1D6FF}(x)/\unicode[STIX]{x1D705}$ for any $\unicode[STIX]{x1D705}>0$ . We note that, the thin-film hypothesis will hold as long as the new aspect ratio $\unicode[STIX]{x1D716}=\left(\unicode[STIX]{x1D707}^{2}Q/\unicode[STIX]{x1D6FE}^{2}\unicode[STIX]{x1D70F}T\right)^{1/7}\ll 1$ , which is true for sufficiently large $T$ . The appropriate limit of the late-time analysis presented in § 3.3 holds in this case too, since the width of the slot is asymptotically small for late time, and so, can be described by a delta function. We, thus, recover the power-law dependencies

(5.16) $$\begin{eqnarray}\displaystyle & \displaystyle a_{0}(\bar{t})=\left({\textstyle \frac{21}{32}}\right)^{1/7}\bar{t\,}^{4/7}, & \displaystyle\end{eqnarray}$$

(5.17) $$\begin{eqnarray}\displaystyle & \displaystyle \bar{h}_{0}(0,\bar{t})={\textstyle \frac{3}{4}}\left({\textstyle \frac{32}{21}}\right)^{1/7}\bar{t\,}^{3/7}, & \displaystyle\end{eqnarray}$$

where, again, we can use the Cox–Voinov law, as mentioned in § 3, to obtain the full late-time behaviour. However, we note that the numerical pre-factors are not the same as those in (3.51) and (3.52), since the strength of the source in this case is half that considered in § 3.3.

Figure 8.

Log–log plots of the numerical solution to (5.19) with (5.20) and (2.25)–(2.26) (solid blue), the numerical solutions to (5.18) with (2.24)–(2.26) with $L_{s}=1$ (orange), $L_{s}=0.1$ (green) and $L_{s}=0.01$ (purple) for (a) $h(0,t)$ and (b) $a(t)$ , compared with the asymptotic predictions for late time (3.53) (dashed red). Here, $\unicode[STIX]{x1D6FF}=10^{-3}$ .

In figure 8, we show log–log plots of the maximum film thickness and the position of the apparent contact line as functions of time for the case of point-source injection. To simulate the behaviour of the delta function, we have used two approaches. First, we approximate a point-source injection using the original numerical solution with a finite injection slot of small width $\hat{L}_{s}=(\unicode[STIX]{x1D6FE}Q^{3}\unicode[STIX]{x1D70F}^{4}/\unicode[STIX]{x1D707})^{1/7}L_{s}$ , keeping the total flux of liquid constant by increasing the injection rate, i.e. replacing (5.15) with

(5.18) $$\begin{eqnarray}{\displaystyle \frac{\unicode[STIX]{x2202}h}{\unicode[STIX]{x2202}t}}+\frac{1}{3}{\displaystyle \frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}x}}\left(h^{3}{\displaystyle \frac{\unicode[STIX]{x2202}^{3}h}{\unicode[STIX]{x2202}x^{3}}}\right)=\frac{\unicode[STIX]{x1D6E9}(L_{s}-x)}{2L_{s}},\end{eqnarray}$$

where $L_{s}\ll 1$ . Second, we prescribe the flux at the origin by solving the problem

(5.19) $$\begin{eqnarray}{\displaystyle \frac{\unicode[STIX]{x2202}h}{\unicode[STIX]{x2202}t}}+\frac{1}{3}{\displaystyle \frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}x}}\left(h^{3}{\displaystyle \frac{\unicode[STIX]{x2202}^{3}h}{\unicode[STIX]{x2202}x^{3}}}\right)=0,\end{eqnarray}$$

subject to (2.25), (2.26), and

(5.20a,b ) $$\begin{eqnarray}{\displaystyle \frac{\unicode[STIX]{x2202}h}{\unicode[STIX]{x2202}x}}=0,\quad \frac{1}{3}h^{3}{\displaystyle \frac{\unicode[STIX]{x2202}^{3}h}{\unicode[STIX]{x2202}x^{3}}}=\frac{1}{2}\quad \text{at }x=0.\end{eqnarray}$$

We also plot the asymptotic results (5.16) and (5.17) across the entire $t$ range (despite them only being formally valid for $t\gg 1$ ). We see a very good agreement between the numerical results obtained using the second approach and the asymptotic results for a wider range of time than anticipated. The asymptotic results provide a good estimate at early time, because, in this case, the slot has been shrunk to a point. For early time, we observe a slight discrepancy between the numerical and the asymptotic results, which can be explained by the finite precursor film, which is present everywhere on the substrate at $t=0$ and is not reflected by our ‘late-time’ results. We also see that, as time increases, the numerical solution from the first approach converges to the numerical solution from the second approach (as anticipated) and to the asymptotic predictions (5.16) and (5.17), and the time to convergence decreases as $L_{s}$ decreases. To illustrate this, we consider the early-time behaviour. As before, we determine the linear growth for early time by balancing the first and third terms in (5.18), which gives

(5.21) $$\begin{eqnarray}h=\unicode[STIX]{x1D6FF}+\frac{t}{L_{s}}\unicode[STIX]{x1D6E9}(L_{s}-x).\end{eqnarray}$$

We obtain the range of validity of the early-time solution (5.21) by requiring that the ‘discontinuous’ step that is initially at $x=L_{s}$ has not spread to the origin. By considering (5.21), we see that, if $t\ll \unicode[STIX]{x1D6FF}L_{s}$ , then $h\sim \unicode[STIX]{x1D6FF}$ , and the validity condition, found using a scaling argument on (5.19), reads

(5.22) $$\begin{eqnarray}(\unicode[STIX]{x1D6FF}^{3}t)^{1/4}\ll L_{s}.\end{eqnarray}$$

If $\unicode[STIX]{x1D6FF}L_{s}\ll t\ll 1$ , then $h\sim t/L_{s}$ from (5.21), and the validity condition is instead

(5.23) $$\begin{eqnarray}(h^{3}t)^{1/4}\ll L_{s}.\end{eqnarray}$$

The two conditions (5.22) and (5.23) can be combined into a single range of validity, $t\ll \min \{L_{s}^{4}/\unicode[STIX]{x1D6FF}^{3},L_{s}^{7/4}\}$ . We note that when $L_{s}=1$ , the validity condition becomes $t\ll 1$ , as seen in § 3.1.