3.1.1. Statement of outer problem

Away from the evaporating interface (an O(1) macro-lengthscale distance away), we (re)-introduce microscale variables $\xi=X/\epsilon$ and $\eta=Z/\epsilon$ (these move with the macroscale interface, hence differ from the static coordinates x and y that we had previously) and work on the timescale $t=T/\epsilon$ . We introduce a microscale cell containing a periodic section of the microstructure, denoted by $\omega_c$ , and illustrated in Figure 3 (left), with fluid-occupied region $\omega_f(t)$ , and solid inclusion $\omega_s(t)$ . We treat the dependent variables $\xi,\eta,t$ , and Z and T as independent variables and so (2.9a)–(2.9c) in the outer region become

(3.2a) \begin{align} \epsilon\left(\nabla_\xi+\epsilon{\textbf{e}}_2\frac{\partial}{\partial Z}\right)\cdot{\textbf{u}}&=0, \end{align}

(3.2b) \begin{align} \epsilon\left(\nabla^2_\xi+\epsilon\left(\nabla_\xi\cdot{\textbf{e}}_2\frac{\partial}{\partial Z}+{\textbf{e}}_2\frac{\partial}{\partial Z}\cdot\nabla_\xi\right)+\epsilon^2\frac{\partial^2}{\partial Z^2}\right){\textbf{u}}&=\left(\nabla_\xi+\epsilon{\textbf{e}}_2\frac{\partial}{\partial Z}\right)p,\end{align}

(3.2c) \begin{align} \epsilon\delta\left(\rho_t+\epsilon\rho_T-H_T\left(\rho_\eta+\epsilon\rho_Z\right)\right)+\left(\nabla_\xi+\epsilon{\textbf{e}}_2\frac{\partial}{\partial Z}\right)&\cdot\left(\epsilon\nu\rho{\textbf{u}}-\left(\nabla_\xi+\epsilon{\textbf{e}}_2\frac{\partial}{\partial Z}\right)\rho\right)=0.\end{align}

The vapour density $\rho$ , gas velocity ${\textbf{u}}$ and pressure p in this outer region are all functions of $\xi,$ $\eta,$ t, Z, and T. On the boundary of the cell $\omega_c$ , we assume that the dependent variables are periodic, with period 1, in $\xi,\eta$ . We note that the geometry of the microscale cell changes over time, since the solid inclusions move up through the cell at speed $H_T$ . Since the microstructure is spatially periodic, this means that the cell geometry is also periodic in t, with period $1/H_T$ . We also therefore assume that the dependent variables are periodic in t, with period $1/H_T$ . On the surface $\partial\omega_s(t)$ of the solid material (2.9d) becomes

(3.2d) \begin{align} {\textbf{u}}={\textbf{0}}, \end{align}

(3.2e) \begin{align} \left(\nabla_\xi+\epsilon{\textbf{e}}_2\frac{\partial}{\partial Z}\right)\rho\cdot{\textbf{n}}_s=0, \end{align}

where ${\textbf{n}}_s(t)$ is the unit normal to the solid surface. In (3.2), we have introduced the notation:

(3.3) \begin{equation} \nabla_\xi={\textbf{e}}_1\frac{\partial}{\partial \xi}+{\textbf{e}}_2\frac{\partial}{\partial \eta},\end{equation}

for the microscale gradient operator, where ${\textbf{e}}_1$ and ${\textbf{e}}_2$ are the unit vectors in the X or $\xi$ , and Z or $\eta$ directions, respectively (we use the same notation for unit vectors in both coordinate systems).

Figure 3.

Schematics of the microscale cells regions. Left: the cell $\omega_f(t)=[-1/2,1/2]^2\setminus \omega_s(t)$ for the outer and intermediate regions. Right: the inner cell $\omega_{{\mathcal{H}}}$ formed of the region between $z={{\mathcal{H}}}(x,t)$ and $\Gamma$ , $x\in[-1/2,1/2]$ , excluding the solid structure $\omega_s(t)$ . In the analysis we consider the limit as the height of $\Gamma$ increases, so that $\omega_{{\mathcal{H}}}$ becomes semi-infinite in z. The solid inclusions $\omega_s(t)$ (shown as grey circles) move up through both cells at speed $H_T$ , so that the cells are periodic in time t with period $1/H_T$ .

3.1.2. Statement of the inner problem

As in [Reference Luckins, Breward, Griffiths and Wilmott16], we will find that there must be a boundary layer near the gas–liquid interface in which oscillations due to the motion of the interface are felt at leading order. In our moving coordinate system, the macroscale interface is at $Z=0$ . We look for a boundary layer of the macroscale problem by changing to the microscale variables $Z=z/\epsilon$ , $X=x/\epsilon$ centred on the interface. We define the interface to be located at $z={\mathcal{H}}$ on the microscale, where $\mathcal{H}(x,t)$ is the microscale variation of the evaporating interface about its mean position $z=0$ . We assume the problem is periodic in x and t, and so study the inner region on the microscale ‘cell’ region, $\omega_{\mathcal{H}}(t)$ , illustrated on the right of Figure 3. This cell is the region $x\in[-1/2,1/2]$ between $z={{\mathcal{H}}}(x,t)$ and some curve $\Gamma$ , excluding the solid structure $\omega_s(t)$ . However, we cannot assume periodicity in z. Instead, we will match with the outer region in the ${\textbf{e}}_2$ direction by sending the height z of the curve $\Gamma$ to $\infty$ so that the inner cell approaches a semi-infinite region. We call this region a ‘cell’ by analogy with the cell region $\omega_c(t)$ for the outer problem. We will homogenise along the macroscale interface in the x direction and in time t.

In this inner region, we use an overbar for the dependent variables $\bar{\rho}$ , $\bar{p}$ and $\bar{{\textbf{u}}}$ , to differentiate from the variables in the outer region. These inner region variables are functions of x, z, t and T (via the value of $H_T$ ) but are independent of Z. We also use the gradient operator:

(3.4) \begin{equation} \nabla_x={\textbf{e}}_1\frac{\partial}{\partial x}+{\textbf{e}}_2\frac{\partial}{\partial z}.\end{equation}

The equations (2.9a)–(2.9c) in $\omega_{{\mathcal{H}}}(t)$ are

(3.5a) \begin{align} \epsilon\nabla_x\cdot\bar{{\textbf{u}}}&=0, \end{align}

(3.5b) \begin{align} \epsilon\nabla_x^2\bar{{\textbf{u}}}&=\nabla_x \bar{p}, \end{align}

(3.5c) \begin{align} \epsilon\delta(\bar{\rho}_t+\epsilon\bar{\rho}_T-H_T\bar{\rho}_z)+\nabla_x\cdot\left(\epsilon\nu\bar{\rho}\bar{{\textbf{u}}}-\nabla_x\bar{\rho}\right)&=0.\end{align}

On the solid surface $\partial\omega_s(t)$ , (2.9d) becomes

(3.5d) \begin{align} \bar{{\textbf{u}}}={\textbf{0}}, \end{align}

(3.5e) \begin{align} \nabla_x\bar{\rho}\cdot{\textbf{n}}_s=0,\end{align}

and (2.9e)–(2.9g) become, on $z={{\mathcal{H}}}(x,t)$ ,

(3.5f) \begin{align} \bar{{\textbf{u}}}\cdot{\textbf{m}}&=-(1-\delta)\frac{H_T+{{\mathcal{H}}}_t}{\sqrt{1+{{\mathcal{H}}}_x^2}},\end{align}

(3.5g) \begin{align} \nabla_x\bar{\rho}\cdot{\textbf{m}}&=\epsilon(1-\nu\bar{\rho})\frac{H_T+{{\mathcal{H}}}_t}{\sqrt{1+{{\mathcal{H}}}_x^2}}, \end{align}

(3.5h) \begin{align} \bar{u}+\bar{v}{{\mathcal{H}}}_x&=0.\end{align}

The chemistry interface condition, (2.9h), applies at the microscale interface $z={{\mathcal{H}}}$ . Although we will not use this condition until Section 4, it reads

(3.6) \begin{equation} 1-\bar{\rho}=-\alpha \left(\frac{H_T+{{\mathcal{H}}}_t}{\sqrt{1+{{\mathcal{H}}}_x^2}}\right) \qquad \text{ on }z={{\mathcal{H}}}(x,t).\end{equation}

3.2.1. Outer region

The homogenisation of Stokes flow to Darcy’s law has been performed in, for instance, [Reference Dalwadi, Griffiths and Bruna6, Reference Keller14]. We follow the analysis presented in [Reference Dalwadi, Griffiths and Bruna6] closely, but note that, in our evaporation setting, the velocity and the pressure gradients in general all vary in time t as well as in space. We will therefore need to average the dependent variables in time as well, although this does not significantly change the analysis or results of the homogenisation.

We expand the dependent variables ${\textbf{u}}$ and p in powers of $\epsilon$ , using superscript notation to denote the order, for example, $ p=p^{(0)}+\epsilon p^{(1)}+\dots$ . At O(1) we find that (3.2b) becomes

(3.7) \begin{equation} \nabla_\xi p^{(0)}={\textbf{0}},\end{equation}

so that $p^{(0)}$ is independent of the microscale spatial variables $\xi$ and $\eta$ . At $O(\epsilon)$ we find that (3.2a)–(3.2b) become

(3.8a) \begin{align} \nabla_\xi\cdot{\textbf{u}}^{(0)}&=0, \end{align}

(3.8b) \begin{align} \nabla_\xi^2{\textbf{u}}^{(0)}&=p^{(0)}_Z{\textbf{e}}_2+\nabla_\xi p^{(1)}, \end{align}

with the boundary condition ${\textbf{u}}^{(0)}={\textbf{0}}$ on $\partial\omega_s(t)$ . As in [Reference Dalwadi, Griffiths and Bruna6], the problem is linear and so has solution of the form:

(3.9) \begin{align} {\textbf{u}}^{(0)}=- {\textbf{K}}(\xi,\eta,t)p^{(0)}_Z{\textbf{e}}_2, \qquad p^{(1)}=-{\boldsymbol\Pi}(\xi,\eta,t)\cdot {\textbf{e}}_2 p^{(0)}_Z+\pi_1(t,T),\end{align}

where ${\textbf{K}}$ is a tensor, ${\boldsymbol\Pi}$ a vector, and $\pi_1$ is a scalar. Substituting these forms into (3.8) we obtain a ‘cell problem’ for ${\textbf{K}}$ and ${\boldsymbol\Pi}$ , namely

(3.10a) \begin{align} \nabla_\xi\cdot{\textbf{K}}&={\textbf{0}}, \end{align}

(3.10b) \begin{align} \nabla_\xi^2{\textbf{K}}&=\nabla_\xi{\boldsymbol\Pi}-{\textbf{I}}, \end{align}

where ${\textbf{I}}$ is the identity matrix, along with the boundary condition ${\textbf{K}}={\textbf{0}}$ on $\partial\omega_s(t)$ , and the assumption of periodicity of ${\textbf{K}}$ and ${\boldsymbol\Pi}$ in $\xi$ , $\eta$ and t. We note that ${\textbf{K}}$ and ${\boldsymbol\Pi}$ depend on the short timescale t, since the position of the solid inclusion $\omega_s(t)$ depends on t.

We define the Darcy velocity to be the leading-order velocity, averaged over the cell, and over the micro-timescale:

(3.11) \begin{equation} {\textbf{q}}\;:\!=\; H_T\int_{t=0}^{1/H_T}\iint_{\omega_f(t)}{\textbf{u}}^{(0)}\,\mathrm{d}\xi\mathrm{d}\eta\,\mathrm{d}t,\end{equation}

where the average over the cell has been reduced to an integral over the fluid region since there is no flow in the solid region and the cell has unit area. Since we are assuming a one-dimensional macroscale flow with no X dependence, we note that ${\textbf{q}}$ is unidirectional, ${\textbf{q}}=q{\textbf{e}}_2$ .

While $p^{(0)}$ is independent of $\xi$ and $\eta$ , it is not necessarily independent of t, and so we define the time-averaged pressure to be

(3.12) \begin{equation} P(Z,T)\;:\!=\;H_T\int_{t=0}^{1/H_T}p^{(0)}\,\mathrm{d}t.\end{equation}

Substituting the definition (3.9) of ${\textbf{u}}^{(0)}$ into (3.11), we obtain Darcy’s law:

(3.13) \begin{equation} q=-H_T\int_{t=0}^{1/H_T}\iint_{\omega_f(t)}p^{(0)}_Z{\textbf{K}}_{22}\,\mathrm{d}\xi\mathrm{d}\eta\,\mathrm{d}t=-kP_Z,\end{equation}

using the definition of the average pressure P, and where we have defined the average permeability (for the 1D flow) to be

(3.14) \begin{equation} k=\iint_{\omega_f(t)}{\textbf{K}}_{22}\,\mathrm{d}\xi\mathrm{d}\eta.\end{equation}

The equations (3.10) are quasi-steady; the dependence of ${\textbf{K}}$ and ${\boldsymbol\Pi}$ on t is through the moving solid inclusions only. Due to the periodicity of ${\textbf{K}}$ in both time t and over the cell $\omega_c(t)$ , we see that the permeability, k, given by (3.14), must be independent of t.

We have found the form of Darcy’s law but are yet to derive an expression of overall conservation of mass. To do this we now look at the $O(\epsilon^2)$ problem, where we find that (3.2a) becomes

(3.15) \begin{equation} \nabla_\xi\cdot{\textbf{u}}^{(1)}+{\textbf{e}}_2\frac{\partial}{\partial Z}\cdot{\textbf{u}}^{(0)}=0.\end{equation}

We also require periodicity of ${\textbf{u}}^{(1)}$ and impose ${\textbf{u}}^{(1)}={\textbf{0}}$ on the solid surface $\partial \omega_s(t)$ . Integrating (3.15) over the fluid-occupied region of the cell, $\omega_f$ , using the divergence theorem, periodicity and boundary condition for ${\textbf{u}}^{(1)}$ , we find that

(3.16) \begin{align} 0&=\iint_{\omega_f(t)}{\textbf{e}}_2\frac{\partial}{\partial Z}\cdot{\textbf{u}}^{(0)}\,\mathrm{d}\xi\mathrm{d}\eta={\textbf{e}}_2\frac{\partial}{\partial Z}\cdot\left(\iint_{\omega_f(t)}{\textbf{u}}^{(0)}\,\mathrm{d}\xi\mathrm{d}\eta\right).\end{align}

Averaging over t (and using the fact that $H_T$ is independent of Z) gives the macroscale expression of conservation of mass:

(3.17) \begin{equation} \frac{\partial q}{\partial Z}=0.\end{equation}

This completes the analysis for the flow problem in the outer region: we have derived Darcy’s law (3.13) and the average conservation of mass equation (3.17).

3.2.2. Inner region

We now study the boundary layer near to the macroscale evaporating interface in order to derive the boundary conditions for the averaged flow problem in the outer region. We analyse the model (3.5a)–(3.5b) with the boundary conditions (3.5d), (3.5f) and (3.5h).

At leading order, we find, in $\omega_{{{\mathcal{H}}}^{(0)}}$ (which is bounded by the curve $z={{\mathcal{H}}}^{(0)}(x,t)$ , rather than $z={{\mathcal{H}}}(x,t)$ ),

(3.18a) \begin{align} \nabla_x\cdot\bar{{\textbf{u}}}^{(0)}&=0, \end{align}

(3.18b) \begin{align} \nabla_x\bar{p}^{(0)}&={\textbf{0}}, \end{align}

along with the boundary conditions, on $z={{\mathcal{H}}}^{(0)}(x,t)$ ,

(3.18c) \begin{align} \bar{{\textbf{u}}}^{(0)}\cdot{\textbf{m}}^{(0)}&=-(1-\delta)\frac{H_T+{{\mathcal{H}}}^{(0)}_t}{\sqrt{1+\left({{\mathcal{H}}}^{(0)}_x\right)^2}}, \end{align}

(3.18d) \begin{align} \bar{u}^{(0)}+\bar{v}^{(0)}{{\mathcal{H}}}^{(0)}_x&=0, \end{align}

where ${\textbf{m}}^{(0)}=\big(1/\sqrt{1+({{\mathcal{H}}}^{(0)})_x^2}\big)({{\mathcal{H}}}^{(0)}_x,-1)$ is the leading-order unit normal to the interface. On the solid surface $\partial\omega_s(t)$ ,

(3.18e) \begin{align} \bar{{\textbf{u}}}^{(0)}={\textbf{0}},\end{align}

along with periodicity of $\bar{p}^{(0)}$ and $\bar{{\textbf{u}}}^{(0)}$ in x and t. We recall that $\bar{{\textbf{u}}}^{(0)}$ and $p^{(0)}$ are allowed to be functions of x, z, t and T, but not of Z. However, we see from (3.18b) that $\bar{p}^{(0)}$ is independent of x and z.

Integrating equation (3.18a) for ${\textbf{u}}^{(0)}$ over the cell $\omega_{{{\mathcal{H}}}^{(0)}}(t)$ and using the divergence theorem, we find that

(3.19) \begin{align} 0=\int_{\partial\omega_{{{\mathcal{H}}}^{(0)}}}{\textbf{u}}^{(0)}\cdot{\textbf{n}}\,\mathrm{d}s.\end{align}

Using the periodicity in x and boundary conditions (3.18c) on $z={{\mathcal{H}}}^{(0)}(x,t)$ and (3.18e) on the solid, we see that

(3.20) \begin{align} \lim_{z_\Gamma\rightarrow\infty}\int_{\Gamma}\bar{{\textbf{u}}}^{(0)}\cdot{\textbf{n}}\,\mathrm{d}s=-\int_{z={{\mathcal{H}}}^{(0)}}\frac{(1-\delta)(H_T+{{\mathcal{H}}}_t^{(0)})}{\sqrt{1+\left({{\mathcal{H}}}_x^{(0)}\right)^2}}\,\mathrm{d}s ,\end{align}

which relates the flux of material through the evaporating interface to the far-field flux leaving the inner layer through the curve $\Gamma$ , as the height, $z_\Gamma$ of this curve is sent to infinity. We note (again by the divergence theorem, by constructing a closed region bounded by two choices $\Gamma_1$ and $\Gamma_2$ for $\Gamma$ , and by the curves $x=\pm1/2$ connecting them) that the left-hand side of (3.20) is independent of the choice of curve $\Gamma$ , with normal ${\textbf{n}}$ , so long as this connects $x=-{1/2}$ to $1/2$ , passing through the gas-filled pore space $\omega_{{{\mathcal{H}}}^{(0)}}$ . We average (3.20) over the period $[0,1/H_T]$ , say, of t to find that

(3.21) \begin{equation} \lim_{z_\Gamma\rightarrow\infty}H_T\int_{t=0}^{1/H_T}\int_{\Gamma}\bar{{\textbf{u}}}^{(0)}\cdot{\textbf{n}}\,\mathrm{d}s\,\mathrm{d}t =-(1-\delta)\mathcal{F}H_T,\end{equation}

where we have defined

(3.22) \begin{align} \mathcal{F} &=\int_{t=0}^{1/H_T} \int_{z={{\mathcal{H}}}^{(0)}}\frac{H_T+{{\mathcal{H}}}_t^{(0)}}{\sqrt{1+\left({{\mathcal{H}}}_x^{(0)}\right)^2}}\,\mathrm{d}s\,\mathrm{d}t =\int_{t=0}^{1/H_T}\int_{z={{\mathcal{H}}}^{(0)}}H_T+{{\mathcal{H}}}_t^{(0)}\,\mathrm{d}x\,\mathrm{d}t\nonumber\\ &=H_T\int_{t=0}^{1/H_T}\int_{z={{\mathcal{H}}}^{(0)}}\,\mathrm{d}x\,\mathrm{d}t,\end{align}

where we have used the fact that $H_T$ is independent of x, and that we may choose H and ${\mathcal{H}}$ such that the average interface velocity perturbation is zero. We interpret this parameter $\mathcal{F}$ as the total flux of material through the interface in a time period.

The value of $\mathcal{F}$ is not clear from considering only the dynamics of the gas. However, we may determine $\mathcal{F}$ by considering the conservation of mass of the liquid. Since we have assumed that the liquid does not flow, the total flux of liquid that passes through the evaporating interface over the time period $1/H_T$ (the time taken for the interface to move down through one repetition of the periodic pore-scale geometry) must equal the volume of liquid contained in that periodic cell. This is precisely the porosity, $\phi=|\omega_f|/|\omega_c|$ , of the porous medium, which is equivalently the cell volume occupied by fluid (and which we assume is constant).

More formally, we consider the liquid occupied region $\omega_l$ , illustrated in Figure 4, between $z=-1$ and the leading-order evaporating interface at $z={{\mathcal{H}}}^{(0)}$ . The rate of change of liquid mass contained in $\omega_l$ is given by:

(3.23) \begin{equation} R=\int_{\{z=-1\}\setminus\omega_s}\,\mathrm{d}x-\int_{z={{\mathcal{H}}}^{(0)}}\frac{H_T+{{\mathcal{H}}}_t^{(0)}}{\sqrt{1+({{\mathcal{H}}}_x^{(0)})^2}} \,\mathrm{d}s,\end{equation}

which is the rate at which liquid is gained as the liquid and solid move up through $z=-1$ (due to our moving coordinate system) minus the rate at which it is lost through the evaporating interface $z={{\mathcal{H}}}^{(0)}$ . The total liquid lost or gained over the time period $1/H_T$ must be zero as the process is periodic, and so

(3.24) \begin{align} 0&=\int_{t=0}^{1/H_T}R\,\mathrm{d}t=\int_{t=0}^{1/H_T}\left[\int_{\{z=-1\}\setminus\omega_s}\,\mathrm{d}x-\int_{z={{\mathcal{H}}}^{(0)}}\frac{H_T+{{\mathcal{H}}}_t^{(0)}}{\sqrt{1+({{\mathcal{H}}}_x^{(0)})^2}} \,\mathrm{d}s\right]\,\mathrm{d}t.\end{align}

The second integral term here is $\mathcal{F}$ by the definition (3.22), while the first is precisely the porosity, $\phi$ , of the porous medium. We therefore conclude that

(3.25) \begin{equation} \mathcal{F}=\phi=\iint_{\omega_f(t)}\,\mathrm{d}\xi\mathrm{d}\eta,\end{equation}

is the porosity of the porous medium.

Figure 4.

Schematic showing the liquid region $\omega_l$ , consisting of the pore space between $z=-1$ and $z={{\mathcal{H}}}^{(0)}$ .

We now return to our expression (3.21) of the average velocity flux leaving the inner region. Since the left-hand side of (3.21) is independent of the choice of $\Gamma$ , we may match directly with the outer problem by letting the height, $z_{\Gamma}$ of $\Gamma$ become large, so that

(3.26) \begin{equation} \lim_{z_\Gamma\rightarrow\infty}\left( \int_{\Gamma}\bar{{\textbf{u}}}^{(0)}\cdot{\textbf{n}}\,\mathrm{d}s \right)=\left( \int_{\Gamma}{\textbf{u}}^{(0)}\cdot{\textbf{n}}\,\mathrm{d}s \right)\bigg|_{Z=0},\end{equation}

where on the right-hand side we consider $\Gamma$ to be a curve within the outer cell $\omega_f$ , joining $\xi=-1/2$ to $1/2$ . We choose $\Gamma$ to be the flat curve at $\eta=\eta_*$ , say, for some $\eta_*\in[-1/2,1/2]$ , wherever this lies within $\omega_f$ , and to follow the solid boundary $\partial\omega_s$ wherever the solid obstructs the curve $\eta=\eta_*$ . Since the integral along $\Gamma$ on the right-hand side of (3.26) is independent of the value of $\eta_*$ , this integral must equal its average over all possible $\eta_*\in[-1/2,1/2]$ . Thus, the left-hand side of (3.21) is

(3.27) \begin{align} \lim_{z_\Gamma\rightarrow\infty}\left(H_T\int_{t=0}^{1/H_T}\int_{\Gamma}\bar{{\textbf{u}}}^{(0)}\cdot{\textbf{n}}\,\mathrm{d}s\,\mathrm{d}t\right)&=\left(H_T\int_{t=0}^{1/H_T}\int_{\Gamma}{\textbf{u}}^{(0)}\cdot{\textbf{n}}\,\mathrm{d}s\,\mathrm{d}t\right)\bigg|_{Z=0}\nonumber\\ &=\left(H_T\int_{t=0}^{1/H_T}\iint_{\omega_f}{\textbf{u}}^{(0)}\cdot{\textbf{e}}_2\,\mathrm{d}\xi\mathrm{d}\eta\,\mathrm{d}t\right)\bigg|_{Z=0}=q|_{Z=0},\end{align}

which is precisely the (vertical) Darcy velocity. Substituting back into (3.21) (and using $\mathcal{F}=\phi$ ), we find the flux condition for the outer problem, namely

(3.28) \begin{equation} q=-(1-\delta)\phi H_T \qquad \text{ at }Z=0,\end{equation}

which provides the boundary condition for our outer, Darcy flow, problem.

We note that we do not need to find a macroscale boundary condition analogous to the no-tangential-slip condition (3.18d) that holds on the microscale, since the macroscale flow equations (3.13) and (3.17) only require one boundary condition for q on $Z=0$ (even in higher spatial dimensions), which is the normal flux condition (3.28). Since we do not need to explicitly compute the microscale velocity field $\bar{{\textbf{u}}}^{(0)}$ , the condition (3.18d) has not been used explicitly in the analysis, although it is formally needed for closure of the leading-order microscale system (3.18).

3.3.1. Outer region

We first analyse the vapour advection–diffusion problem (3.2c) with (3.2e), in the outer region of the domain. At leading order in $\epsilon$ , (3.2c) becomes

(3.31a) \begin{equation} \nabla_\xi^2\rho^{(0)}=0,\end{equation}

and the boundary condition (3.2e) becomes

(3.31b) \begin{equation} \nabla_\xi\rho^{(0)}\cdot{\textbf{n}}_s=0 \qquad \text{ on }\partial\omega_s(t),\end{equation}

which must be solved along with periodicity of $\rho^{(0)}$ in $\xi$ and $\eta$ . We multiply the equation (3.31a) by $\rho^{(0)}$ and integrate over the cell, using periodicity in $\xi$ and $\eta$ and the boundary condition to show that $\rho^{(0)}(t,Z,T)$ is independent of the microscale spatial variables $\xi$ and $\eta$ . At $O(\epsilon^{1/2})$ we obtain an identical problem for $\rho^{(1/2)}$ , and therefore find that $\rho^{(1/2)}(t,Z,T)$ is also independent of $\xi,\eta$ .

At $O(\epsilon)$ the equation (3.2c) reads

(3.32a) \begin{equation} \delta(\rho^{(0)}_t-H_T\rho^{(0)}_\eta)+{\textbf{u}}^{(0)}\cdot\nabla_\xi\rho^{(0)}=\nabla_\xi^2\rho^{(1)}+\left(\nabla_\xi\cdot{\textbf{e}}_2\frac{\partial}{\partial Z}+{\textbf{e}}_2\frac{\partial}{\partial Z}\cdot\nabla_\xi\right)\rho^{(0)},\end{equation}

subject to the periodicity of $\rho^{(1)}$ in $\xi$ , $\eta$ , and t and, from (3.2e), the boundary condition:

(3.32b) \begin{equation} (\nabla_\xi\rho^{(1)}+{\textbf{e}}_2\rho^{(0)}_Z)\cdot{\textbf{n}}_s=0 \qquad \text{ on }\partial\omega_s(t).\end{equation}

We note that the advection term at this order is zero since $\rho^{(0)}$ is independent of the micro-lengthscale and so, as in [Reference Luckins, Breward, Griffiths and Wilmott16], we integrate (3.32a) over the cell to find that $\rho^{(0)}_t=0$ , and thus $\rho^{(0)}$ is a function of Z and T only.

The $O(\epsilon)$ problem therefore reduces to $\nabla_\xi^2\rho^{(1)}=0$ with the boundary condition (3.32b). We recall that $\rho^{(1)}$ is a function of $\xi,\eta,t,Z$ and T. Exploiting the linearity of the system, we write

(3.33) \begin{equation} \rho^{(1)}=W\rho^{(0)}_Z+F^{(0)},\end{equation}

where $F^{(0)}(t,Z,T)$ is independent of $\xi$ and $\eta$ , and $W(\xi,\eta,t)$ satisfies the cell problem:

(3.34a) \begin{align} \nabla_\xi^2W=0 \qquad &\text{ in }\omega_f(t),\end{align}

(3.34b) \begin{align} (\nabla_\xi W+{\textbf{e}}_2)\cdot{\textbf{n}}_s=0 \qquad &\text{ on }\partial\omega_s(t),\end{align}

(3.34c) \begin{align} W \text{ periodic in }\xi,\eta&\text{ and }t.\end{align}

We will not need any information from the order $O(\epsilon^{3/2})$ problem in the outer region, since the $O(\epsilon^2)$ problem (below) has no dependence on $\rho^{(3/2)}$ and neither will we need information from $O(\epsilon^{3/2})$ in order to derive the leading-order effective interface conditions. We therefore do not study the equations at this order. At $O(\epsilon^2)$ , we find that (3.2c) becomes

(3.35a) \begin{align} \delta\left(\rho^{(0)}_T-H_T\rho^{(0)}_Z+\rho^{(1)}_t-H_T\rho^{(1)}_\eta\right)&+\nabla_\xi\cdot\left(\rho^{(1)}{\textbf{u}}^{(0)}+\rho^{(0)}{\textbf{u}}^{(1)}-\left(\nabla_\xi\rho^{(2)}+{\textbf{e}}_2\rho^{(1)}_Z\right)\right)\nonumber\\[5pt] &+{\textbf{e}}_2\frac{\partial}{\partial Z}\cdot\left(\rho^{(0)}{\textbf{u}}^{(0)}-\left(\nabla_\xi\rho^{(1)}+{\textbf{e}}_2\rho^{(0)}_Z\right)\right)=0,\end{align}

and the boundary condition (3.2e) on $\partial\omega_s(t)$ becomes

(3.35b) \begin{equation} \left(\nabla_\xi\rho^{(2)}+{\textbf{e}}_2\rho_Z^{(1)}\right)\cdot{\textbf{n}}_s=0,\end{equation}

which must be solved along with the periodicity of $\rho^{(2)}$ . Integrating (3.35a) over the fluid-occupied region of the cell, $\omega_f(t)$ , using the divergence theorem, and applying the boundary and periodicity conditions for both $\rho^{(i)}$ and for ${\textbf{u}}^{(i)}$ , we find

(3.36) \begin{align} \delta\phi(\rho^{(0)}_T-H_T\rho^{(0)}_Z)&+\delta\frac{\mathrm{d}}{\mathrm{d}t}\left(\iint_{\omega_f(t)}\rho^{(1)}\,\mathrm{d}\xi\mathrm{d}\eta\right) \nonumber\\ & +\iint_{\omega_f(t)}{\textbf{e}}_2\frac{\partial}{\partial Z}\cdot\left(\rho^{(0)}{\textbf{u}}^{(0)}-(\nabla_\xi\rho^{(1)}+{\textbf{e}}_2\rho^{(0)}_Z)\right)\,\mathrm{d}\xi\mathrm{d}\eta=0,\end{align}

where we have used a transport theorem in t, which reads, for a general function F,

(3.37) \begin{align} \frac{\mathrm{d}}{\mathrm{d}t}\left(\iint_{\omega_f(t)}F\,\mathrm{d}\xi\mathrm{d}\eta\right)&=\iint_{\omega_f(t)}F_t\,\mathrm{d}\xi\mathrm{d}\eta-\int_{\partial\omega_s(t)}FH_T{\textbf{e}}_2\cdot{\textbf{n}}_s\,\mathrm{d}s\nonumber\\ &=\iint_{\omega_f(t)}F_t\,\mathrm{d}\xi\mathrm{d}\eta-\iint_{\omega_f(t)}H_TF_\eta\,\mathrm{d}\xi\mathrm{d}\eta.\end{align}

This transport theorem encorporates the contributions to the total time derivative of the integral due to the time derivative of the integrand and the changing shape of the integration domain $\omega_f(t)$ . Since $\omega_f(t)$ changes by the solid inclusions moving up through the domain (in the ${\textbf{e}}_2$ direction), there is no ${\textbf{e}}_1$ term. The second line of (3.37) follows from the divergence theorem.

Using the form of $\rho^{(1)}$ given in (3.33) and the fact that $\rho^{(0)}$ is independent of $\xi$ and $\eta$ , equation (3.36) becomes

(3.38) \begin{align} \delta\phi(\rho^{(0)}_T-H_T\rho^{(0)}_Z)&+\delta\frac{\mathrm{d}}{\mathrm{d}t}\left(\iint_{\omega_f(t)}\rho^{(1)}\,\mathrm{d}\xi\mathrm{d}\eta\right) \nonumber\\ & ={\textbf{e}}_2\frac{\partial}{\partial Z}\cdot \left(\rho^{(0)}\iint_{\omega_f(t)}{\textbf{u}}^{(0)}\,\mathrm{d}\xi\mathrm{d}\eta-\rho^{(0)}_Z\iint_{\omega_f(t)}(\nabla_\xi W+{\textbf{e}}_2)\,\mathrm{d}\xi\mathrm{d}\eta\right).\end{align}

Averaging over t, the total-time-derivative term integrates to zero by the periodicity of $\rho^{(1)}$ in t, and so we arrive at a homogenised advection–diffusion equation describing the evolution of $\rho^{(0)}$ , namely

(3.39) \begin{align} \delta\phi&(\rho^{(0)}_T-H_T\rho^{(0)}_Z)+\frac{\partial}{\partial Z}\left(\rho^{(0)}q-\mathcal{D}\rho^{(0)}_Z\right)=0.\end{align}

We note that, although W and the cell structure $\omega_f$ vary with t (as the solid microstructure moves up through the domain), due to the periodicity of W over the cell and since the volume of the fluid-occupied region of the cell, $\omega_f$ , remains uniform, the effective diffusivity of the medium, taking into account the solid microstructure, is given by:

(3.40) \begin{equation} \mathcal{D}\;:\!=\;{\textbf{e}}_2\cdot\left(\iint_{\omega_f(t)}\left(\nabla_\xi W+{\textbf{e}}_2\right)\,\mathrm{d}\xi\mathrm{d}\eta\right)=\iint_{\omega_f(t)}\left(W_\eta+1\right)\,\mathrm{d}\xi\mathrm{d}\eta,\end{equation}

and is independent of t.

3.3.2. Intermediate region

We next consider the intermediate region, where the model is given by (3.30). The analysis of the intermediate region is closely linked with that of the inner region, which we will present in Section 3.3.3 below. We will require results from the inner region in order to complete the analysis in the intermediate region; these are highlighted in the discussion.

At O(1), equation (3.30a) becomes simply $\nabla_\xi^2\tilde{\rho}^{(0)}=0$ with (3.30b) giving homogeneous Neumann boundary conditions on the solid, along with the periodicity in t and at the cell boundary. Thus, we conclude, as in the outer region, that $\tilde{\rho}^{(0)}$ is independent of $\xi$ and $\eta$ .

At $O(\epsilon^{1/2})$ we find the problem (3.30a) in $\omega_f$ becomes

(3.41a) \begin{equation} \nabla_\xi^2\tilde{\rho}^{(1/2)}=0,\end{equation}

with periodicity and the boundary condition (3.30b) reading

(3.41b) \begin{equation} \left(\nabla_\xi\tilde{\rho}^{(1/2)}+{\textbf{e}}_2\tilde{\rho}^{(0)}_{\tilde{z}}\right)\cdot{\textbf{n}}_s=0 \qquad \text{ on }\partial\omega_s.\end{equation}

As in the outer problem (at $O(\epsilon)$ ), we can exploit the linearity of (3.41) to write $\tilde{\rho}^{(1/2)}$ , which we recall is allowed to be a function of $\xi,\eta,t,\tilde{z}$ and T, as:

(3.42) \begin{equation} \tilde{\rho}^{(1/2)}=W\tilde{\rho}^{(0)}_{\tilde{z}}+\tilde{f}^{(1/2)},\end{equation}

where $W(\xi,\eta,t)$ is the solution of the cell problem (3.34), and $\tilde{f}^{(1/2)}$ is independent of $\xi$ and $\eta$ .

The $O(\epsilon)$ version of (3.30a) is

(3.43a) \begin{equation} \delta \tilde{\rho}^{(0)}_t=\nabla_\xi^2\tilde{\rho}^{(1)}+\left(\nabla_\xi\cdot{\textbf{e}}_2\frac{\partial}{\partial \tilde{z}}+{\textbf{e}}_2\frac{\partial}{\partial \tilde{z}}\cdot\nabla_\xi\right)\tilde{\rho}^{(1/2)},\end{equation}

in $\omega_f$ , along with periodicity and the boundary condition (3.30b), which at $O(\epsilon)$ reads

(3.43b) \begin{equation} \left(\nabla_\xi\tilde{\rho}^{(1)}+{\textbf{e}}_2\tilde{\rho}^{(1/2)}_{\tilde{z}}\right)\cdot{\textbf{n}}_s=0 \qquad \text{ on }\partial\omega_s.\end{equation}

There are no advection terms in (3.43a) since $\nabla_\xi\tilde{\rho}^{(0)}=0$ . Similarly to the derivation of (3.39) in the outer problem (at $O(\epsilon^2)$ ), we now integrate (3.43a) over $\omega_f$ , using (i) the fact that $\tilde{\rho}^{(0)}$ is independent of $\xi$ and $\eta$ , (ii) the divergence theorem, (iii) the boundary condition (3.43b), (iv) the periodicity over the cell, and (v) the form (3.42) of $\tilde{\rho}^{(1/2)}$ . The outcome is that $\tilde{\rho}^{(0)}$ satisfies the diffusion equation:

(3.44) \begin{equation} \delta\phi\tilde{\rho}^{(0)}_t=\mathcal{D}\tilde{\rho}^{(0)}_{\tilde{z}\tilde{z}},\end{equation}

on the fast timescale t, and the intermediate lengthscale $\tilde{z}$ . Here $\mathcal{D}$ , the effective diffusivity, is given by (3.40). We note that the solution $\tilde{\rho}^{(0)}$ must be periodic in t and, by matching with the outer problem at leading order, we must have $\tilde{\rho}^{(0)}\rightarrow\rho^{(0)}|_{Z=0}$ as $\tilde{z}\rightarrow\infty$ . We may also match with the inner region at leading order. In Section 3.3.3, we will show that this matching gives the condition $\tilde{\rho}^{(0)}_{\tilde{z}}=0$ at $\tilde{z}=0$ . The solution of (3.44) is uniquely determined by these conditions (this uniqueness of solution may be shown by an energy method, in a similar way to in the Appendix of [Reference Luckins, Breward, Griffiths and Wilmott16]), and is given by:

(3.45) \begin{equation} \tilde{\rho}^{(0)}=\rho^{(0)}|_{Z=0}.\end{equation}

We therefore conclude that $\tilde{\rho}^{(0)}$ is, in fact, independent both of $\tilde{z}$ and of t (since $\rho^{(0)}$ is independent of t). Furthermore, since $\tilde{\rho}^{(0)}_{\tilde{z}}$ is therefore everywhere 0, from (3.42) we see that $\tilde{\rho}^{(1/2)}$ is in fact independent of $\xi$ and $\eta$ .

With this knowledge we return to the $O(\epsilon)$ problem (3.43), and we see that (3.43a) reduces to simply $\nabla_\xi^2\tilde{\rho}^{(1)}=0$ . Arguing again by linearity, we see (3.43) therefore has solution:

(3.46) \begin{equation} \tilde{\rho}^{(1)}=W\tilde{\rho}^{(1/2)}_{\tilde{z}}+\tilde{f}^{1},\end{equation}

where $W(\xi,\eta,t)$ is the solution of the cell problem (3.34), and $\tilde{f}^{1}$ is independent of $\xi$ and $\eta$ .

The final order of interest is $O(\epsilon^{3/2})$ . Here, we find that (3.30a) becomes, in $\omega_f$ ,

(3.47a) \begin{align} \delta\left(\tilde{\rho}^{(1/2)}_t-H_T\tilde{\rho}^{(1/2)}_\eta\right)=\nabla_\xi\cdot\left(\nabla_\xi\tilde{\rho}^{(3/2)}+{\textbf{e}}_2\tilde{\rho}^{(1)}_{\tilde{z}}\right)+{\textbf{e}}_2\frac{\partial}{\partial\tilde{z}}\cdot\nabla_\xi\tilde{\rho}^{(1)}+\tilde{\rho}^{(1/2)}_{\tilde{z}\tilde{z}}\end{align}

(where there are no additional advection terms since both $\tilde{\rho}^{(0)}$ and $\tilde{\rho}^{(1/2)}$ are independent of the spatial microscale, and $\tilde{\rho}^{(0)}_{\tilde{z}}=0$ ). We also have the usual periodicity, while the boundary condition (3.30b) reads

(3.47b) \begin{equation} \left(\nabla_\xi\tilde{\rho}^{(3/2)}+{\textbf{e}}_2\tilde{\rho}^{(1)}_{\tilde{z}}\right)\cdot{\textbf{n}}_s=0 \qquad \text{ on }\partial\omega_s.\end{equation}

Again we argue as for the $O(\epsilon^2)$ problem in the outer region: we integrate the equation (3.47a) over the cell, using the transport theorem (3.37), the form (3.46) and apply both the boundary condition (3.47b) and periodicity, to find that

(3.48a) \begin{equation} \delta\phi\tilde{\rho}^{(1/2)}_t=\mathcal{D}\tilde{\rho}^{(1/2)}_{\tilde{z}\tilde{z}}.\end{equation}

We have therefore derived an effective equation for $\tilde{\rho}^{(1/2)}(t,\tilde{z},T)$ in the intermediate layer. The solution of (3.48a) must be periodic in t. Matching with the outer region, we see that we must have

(3.48b) \begin{equation} \tilde{\rho}^{(1/2)}_{\tilde{z}}\rightarrow\rho^{(0)}_Z|_{Z=0} \qquad\text{ as }\tilde{z}\rightarrow\infty.\end{equation}

We must also match with the inner region to close this problem for $\tilde{\rho}^{(1/2)}$ . As we will discuss in Section 3.3.3, there will be two matching conditions that relate $\tilde{\rho}^{(1/2)}$ and $\tilde{\rho}^{(1/2)}_{\tilde{z}}$ to the inner variables, namely

(3.48c) \begin{equation} \tilde{\rho}^{(1/2)}=\lim_{z\rightarrow\infty}\bar{\rho}^{(1/2)} \quad \text{ or } \quad \tilde{\rho}^{(1/2)}_{\tilde{z}}=\lim_{z\rightarrow\infty}\bar{\rho}^{(1)}_z \qquad \text{ at }\tilde{z}=0.\end{equation}

(These are the matching conditions (3.50c) and (3.50e) given below.) Regardless of which of these two matching conditions (3.48c), we view as the boundary condition closing (3.48), the time-periodic solution of (3.48) is unique. (The other condition (3.48c) then provides a far-field condition for the inner problem.) Unlike for $\tilde{\rho}^{(0)}$ , where the matching with the inner region required the solution of (3.34) to be independent of t and $\tilde{z}$ , we will see in Section 4 that the $\tilde{\rho}^{(1/2)}$ problem (3.48) has a solution with non-trivial dependence on t and $\tilde{z}$ , at least for certain chemistry interface conditions.

In summary, we have seen that the leading-order solution $\tilde{\rho}^{(0)}$ is constant through the intermediate region, but that the $O(\epsilon^{1/2})$ correction, $\tilde{\rho}^{(1/2)}$ , satisfies the effective diffusion equation (3.48a). By the matching (3.48b), we see that this problem (3.48) describes the evolution of the leading-order vapour flux through the intermediate region. It is precisely on this intermediate lengthscale $\tilde{z}$ that temporal oscillations in the leading-order vapour flux coming out of the inner region (off the evaporating interface) are smoothed out by diffusion.

3.3.3. Inner region

We now study the advection–diffusion problem for $\bar{\rho}$ and ${\mathcal{H}}$ in the inner region, namely (3.5c) with (3.5e) and (3.5g), with periodicity in x and t, and the appropriate matching conditions (given by (3.50), discussed below). The problem is closed using the chemistry condition (3.6) on $z={{\mathcal{H}}}^{(0)}$ , although we will not need to use this in the analysis of this section.

We will also require several results from the intermediate region, derived in Section 3.3.2, which we will incorporate by matching between the inner and intermediate regions. We note that this matching between the inner region, where there is a single variable z in the ${\textbf{e}}_2$ direction, and the intermediate region, where we have both $\eta$ and $\tilde{z}$ , is a subtle process. As z grows and we leave the inner region, we still expect fine scale oscillations in $\bar{\rho}$ due to the solid microstructure. In the limit $z\rightarrow\infty$ , we must therefore introduce multiple lengthscales by introducing the additional variable $\eta$ , and assuming that $\bar{\rho}$ depends on both $\eta$ and z independently in this limit. Thus, our derivatives in the limit $z\rightarrow \infty$ must be replaced with the multiple scales ansatz:

(3.49) \begin{equation} \nabla_x\rightarrow \nabla_\xi+{\textbf{e}}_2\frac{\partial}{\partial z} \qquad \text{ as }z\rightarrow\infty.\end{equation}

We derive the Van Dyke matching conditions between the inner and intermediate regions, following [Reference Chapman4], and the conditions we will need for our analysis are as follows:

(3.50a) \begin{align} \lim_{z\rightarrow\infty}(\bar{\rho}^{(0)})&=\tilde{\rho}^{(0)}|_{\tilde{z}=0},\end{align}

(3.50b) \begin{align} \lim_{z\rightarrow\infty}(\bar{\rho}^{(1/2)}_z)&=\tilde{\rho}^{(0)}_{\tilde{z}}|_{\tilde{z}=0},\end{align}

(3.50c) \begin{align} \lim_{z\rightarrow\infty}(\bar{\rho}^{(1/2)}-z\tilde{\rho}^{(0)}_{\tilde{z}}|_{\tilde{z}=0})&=\tilde{\rho}^{(1/2)}|_{\tilde{z}=0},\end{align}

(3.50d) \begin{align} \lim_{z\rightarrow\infty}(\bar{\rho}^{(1)}_{zz})&=\tilde{\rho}^{(0)}_{\tilde{z}\tilde{z}}|_{\tilde{z}=0},\end{align}

(3.50e) \begin{align} \lim_{z\rightarrow\infty}\left(\bar{\rho}^{(1)}_z-\frac{z}{2}\tilde{\rho}^{(0)}_{\tilde{z}\tilde{z}}|_{\tilde{z}=0}\right)&=\tilde{\rho}^{(1/2)}_{\tilde{z}}|_{\tilde{z}=0},\end{align}

(3.50f) \begin{align} \lim_{z\rightarrow\infty}\left(\bar{\rho}^{(1)}-z\tilde{\rho}^{(1/2)}_{\tilde{z}}|_{\tilde{z}=0}-\frac{z^2}{2}\tilde{\rho}^{(0)}_{\tilde{z}\tilde{z}}|_{\tilde{z}=0}\right)&=\tilde{\rho}^{(1)}|_{\tilde{z}=0}.\end{align}

We note that the factors of z and $z^2$ in these matching conditions (3.50) are the growing lengthscale z, not $\eta$ , and we emphasise that $ \bar{\rho}_z^i=\bar{\rho}_z^i(\eta,z)$ in this large-z limit.

At leading order, the inner advection–diffusion problem (3.5c), in $\omega_{{{\mathcal{H}}}^{(0)}}$ , becomes

(3.51a) \begin{equation} \nabla_x^2\bar{\rho}^{(0)}=0,\end{equation}

which must be solved with periodicity in x and t, while the boundary conditions (3.5e) and (3.5g) become

(3.51b) \begin{align} \nabla_x\bar{\rho}^{(0)}\cdot{\textbf{n}}_s=0 \qquad &\text{ on }\partial\omega_s, \end{align}

(3.51c) \begin{align} \nabla_x\bar{\rho}^{(0)}\cdot{\textbf{m}}^{(0)}=0, \qquad &\text{ on }z={{\mathcal{H}}}^{(0)}. \end{align}

Multiplying (3.51a) by $\bar{\rho}^{(0)}$ , integrating over the cell (using divergence theorem), and using the no-flux boundary conditions (3.51b)–(3.51c) and periodicity, we find that

(3.52) \begin{equation} \iint_{\omega_{{{\mathcal{H}}}^{(0)}}}|\nabla_x\bar{\rho}^{(0)}|^2\,\mathrm{d}x\mathrm{d}z=\lim_{z\rightarrow\infty}\int_{\Gamma}\bar{\rho}^{(0)}\nabla_x\bar{\rho}^{(0)}\cdot{\textbf{n}}\,\mathrm{d}s,\end{equation}

where, as in Section 3.2.2, $\Gamma$ is any curve through $\omega_{{{\mathcal{H}}}^{(0)}}$ from $x=-1/2$ to $x=1/2$ , at height z, with normal ${\textbf{n}}$ . As we take the limit $z\rightarrow\infty$ , we introduce the multiple scales via (3.49) and use the matching (3.50a) so that (3.52) becomes

(3.53) \begin{equation} \iint_{\omega_{{{\mathcal{H}}}^{(0)}}}|\nabla_x\bar{\rho}^{(0)}|^2\,\mathrm{d}x\mathrm{d}z=\int_{\Gamma}\tilde{\rho}^{(0)}(\nabla_\xi\tilde{\rho}^{(0)}+{\textbf{e}}_2\tilde{\rho}^{(0)}_z)\cdot{\textbf{n}}\,\mathrm{d}s.\end{equation}

In the intermediate layer, we found that the leading-order density $\tilde{\rho}^{(0)}$ is a function only of t and T. Thus, the right-hand side of (3.53) is zero, and we conclude from (3.53) that $\bar{\rho}^{(0)}$ is independent of x and z. Furthermore, since in the intermediate layer $\tilde{\rho}^{(0)}$ is independent of $\tilde{z}$ , we may match directly through the intermediate layer at this order, to see that

(3.54) \begin{equation} \bar{\rho}^{(0)}=\tilde{\rho}^{(0)}|_{\tilde{z}=0}=\lim_{\tilde{z}\rightarrow\infty}\tilde{\rho}^{(0)}=\rho^{(0)}|_{Z=0}.\end{equation}

Thus, since $\rho^{(0)}$ in the outer region is independent of t, we also know that $\bar{\rho}^{(0)}$ is independent of t.

At the next order, $O(\epsilon^{1/2})$ , of the inner problem (3.5c) we find the same problem for $\bar{\rho}^{(1/2)}$ as we had for $\bar{\rho}^{(0)}$ at leading order, namely, in $\omega_{{{\mathcal{H}}}^{(0)}}$ ,

(3.55a) \begin{equation} \nabla_x^2\bar{\rho}^{(1/2)}=0,\end{equation}

which we solve with periodicity in x and t, and the boundary conditions (3.5e) and (3.5g) become

(3.55b) \begin{align} \nabla_x\bar{\rho}^{(1/2)}\cdot{\textbf{n}}_s=0, \qquad &\text{ on }\partial\omega_s, \end{align}

(3.55c) \begin{align} \nabla_x\bar{\rho}^{(1/2)}\cdot{\textbf{m}}^{(0)}=0, \qquad &\text{ on }z={{\mathcal{H}}}^{(0)}.\end{align}

We might expect there to be additional terms in the $O(\epsilon^{1/2})$ problem, from expanding ${{\mathcal{H}}}={{\mathcal{H}}}^{(0)}+\epsilon^{1/2}{{\mathcal{H}}}^{(1/2)}+...$ and from evaluating the boundary conditions at ${{\mathcal{H}}}^{(0)}$ rather than ${{\mathcal{H}}}$ . However, these terms are zero, since $\bar{\rho}^{(0)}$ is independent of the microscale. Integrating (3.55a) over the domain $\omega_{{{\mathcal{H}}}^{(0)}}$ , and using the divergence theorem, periodicity and the boundary conditions (3.55b)–(3.55c), we find that

(3.56) \begin{equation} 0=\lim_{z\rightarrow\infty}\int_{\Gamma}\nabla_x\bar{\rho}^{(1/2)}\cdot{\textbf{n}}\,\mathrm{d}s=\lim_{z\rightarrow\infty}\int_{\Gamma}\left(\nabla_\xi+{\textbf{e}}_2\frac{\partial}{\partial z}\right)\bar{\rho}^{(1/2)}\cdot{\textbf{n}}\,\mathrm{d}s,\end{equation}

where we have made use of the multiple scales ansatz (3.49). Using the matching condition (3.50c), we see that

(3.57) \begin{equation} \bar{\rho}^{(1/2)}\sim z\tilde{\rho}^{(0)}_{\tilde{z}}|_{\tilde{z}=0}+\tilde{\rho}^{(1/2)}|_{\tilde{z}=0}=\tilde{\rho}^{(0)}_{\tilde{z}}|_{\tilde{z}=0}\left(z+W(\xi,\eta,t)\right) \qquad \text{ as }z\rightarrow\infty,\end{equation}

where for the equality we have used the form (3.42) of $\tilde{\rho}^{(1/2)}$ . (We note that the form of $\bar{\rho}^{(1/2)}$ given by (3.57) is compatible with the matching condition (3.50b) precisely since the z derivative in (3.50b) is only with respect to the growing z variable, and not $\eta$ .) Substituting (3.57) into (3.56), and since $\tilde{\rho}^{(0)}$ is independent of $\xi$ and $\eta$ , we have

(3.58) \begin{equation} 0=\tilde{\rho}^{(0)}_{\tilde{z}}|_{\tilde{z}=0}\int_{\Gamma}\left(\nabla_\xi W+{\textbf{e}}_2\right)\cdot{\textbf{n}}\,\mathrm{d}s.\end{equation}

The integral on the right-hand side is independent of the choice of curve $\Gamma$ by definition of W, and is non-zero, and thus we conclude that $\tilde{\rho}^{(0)}_{\tilde{z}}|_{\tilde{z}=0}=0$ . (This fact was used in the analysis of the intermediate region, to show that $\tilde{\rho}^{(0)}$ is in fact everywhere independent of $\tilde{z}$ , and that $\tilde{\rho}^{(1/2)}$ is independent of $\xi$ and $\eta$ .)

Returning to the $O(\epsilon^{1/2})$ inner problem (3.55) once more, we multiply (3.55a) through by $\bar{\rho}^{(1/2)}$ and integrate over $\omega_{{{\mathcal{H}}}^{(0)}}$ (much as for the O(1) inner problem), to find

(3.59) \begin{align} \iint_{\omega_{{{\mathcal{H}}}^{(0)}}}|\nabla_x\bar{\rho}^{(1/2)}|^2\,\mathrm{d}x\mathrm{d}z&=\lim_{z\rightarrow\infty}\int_{\Gamma}\bar{\rho}^{(1/2)}\nabla_x\bar{\rho}^{(1/2)}\cdot{\textbf{n}}\,\mathrm{d}s\nonumber\\ &=\lim_{z\rightarrow\infty}\int_{\Gamma}\bar{\rho}^{(1/2)}\left(\nabla_\xi+{\textbf{e}}_2\frac{\partial}{\partial z}\right)\bar{\rho}^{(1/2)}\cdot{\textbf{n}}\,\mathrm{d}s\nonumber\\ &=\lim_{z\rightarrow\infty}\int_{\Gamma}(\tilde{\rho}^{0}_{\tilde{z}}|_{\tilde{z}=0})^2\left(W+z\right)\left(\nabla_\xi W+{\textbf{e}}_2\right)\cdot{\textbf{n}}\,\mathrm{d}s,\end{align}

again using (3.49), and (3.57). Since $\tilde{\rho}^{(0)}_{\tilde{z}}|_{\tilde{z}=0}=0$ , the right-hand side is 0 and thus $\bar{\rho}^{(1/2)}$ is also independent of $\xi$ and $\eta$ .

The final order of interest in the inner region is $O(\epsilon)$ . At this order, (3.5c) becomes, in $\omega_{{{\mathcal{H}}}^{(0)}}$ ,

(3.60a) \begin{equation}\nabla_x^2\bar{\rho}^{(1)}=0,\end{equation}

which must be solved with periodicity in x and t and the boundary conditions (3.5e) and (3.5g), which become

(3.60b) \begin{align} \nabla_x\bar{\rho}^{(1)}\cdot{\textbf{n}}_s=0, \qquad &\text{ on }\partial\omega_s, \end{align}

(3.60c) \begin{align} \nabla_x\bar{\rho}^{(1)}\cdot{\textbf{m}}^{(0)}=(1-\nu\bar{\rho}^{(0)})\frac{H_T+{{\mathcal{H}}}^{(0)}_t}{\sqrt{1+\left({{\mathcal{H}}}^{(0)}_x\right)^2}}, \qquad &\text{ on }z={{\mathcal{H}}}^{(0)}.\end{align}

Again we have used the fact that $\bar{\rho}^{(0)}$ is independent of x, z and t, as well as the fact that $\bar{{\textbf{u}}}^{(0)}$ is divergence free in the microscale variables (as shown in Section 3.2.2). Since $\bar{\rho}^{(1/2)}$ is independent of the spatial microscale, we have no correction terms due to using the domain $\omega_{{{\mathcal{H}}}^{(0)}}$ rather than $\omega_{{\mathcal{H}}}$ . The problem is closed using the matching condition (3.50f) as $z\rightarrow\infty$ , which, since we know $\tilde{\rho}^{(1)}=W\tilde{\rho}^{(1/2)}+\tilde{f}^{(1)}$ in the intermediate region, is given by:

(3.60d) \begin{equation} \lim_{z\rightarrow\infty}(\bar{\rho}^{(1)}-z\tilde{\rho}^{(1/2)}_{\tilde{z}}|_{\tilde{z}=0})=\tilde{\rho}^{(1/2)}_{\tilde{z}}|_{\tilde{z}=0}W(\xi,\eta,t)+\tilde{f}^{(1)}_{\tilde{z}}|_{\tilde{z}=0}.\end{equation}

The problem (3.60) along with the appropriate chemistry boundary condition forms a closed system for $\bar{\rho}^{(1)}$ and ${{\mathcal{H}}}^{(0)}$ , given in terms of $H_T$ and $\tilde{\rho}^{(1/2)}_{\tilde{z}}|_{\tilde{z}=0}$ .

However, without computing $\bar{\rho}^{(1)}$ and ${{\mathcal{H}}}^{(0)}$ we may derive the effective vapour–flux interface condition for the macroscale problem. Integrating (3.60a) over $\omega_{{{\mathcal{H}}}^{(0)}}$ , using the divergence theorem, and the periodicity and boundary conditions (3.60b)–(3.60c), we find that the total diffusive flux through the evaporating interface equals the flux out of the inner cell as $z\rightarrow\infty$ , so that

(3.61) \begin{align} \int_{z={{\mathcal{H}}}^{(0)}}(1-\nu\bar{\rho}^{(0)})\frac{H_T+{{\mathcal{H}}}^{(0)}_t}{\sqrt{1+({{\mathcal{H}}}^{(0)}_x)^2}}\,\mathrm{d}s&=\lim_{z\rightarrow\infty}\int_{\Gamma}\nabla_x\bar{\rho}^{(1)}\cdot{\textbf{n}}\,\mathrm{d}s\nonumber\\ &=\lim_{z\rightarrow\infty}\int_{\Gamma}\left(\nabla_\xi+{\textbf{e}}_2\frac{\partial}{\partial z}\right)\bar{\rho}^{(1)}\cdot{\textbf{n}}\,\mathrm{d}s\nonumber\\ &=\tilde{\rho}^{(1/2)}_{\tilde{z}}|_{\tilde{z}=0}\int_{\Gamma}\left(\nabla_\xi W+{\textbf{e}}_2\right)\cdot{\textbf{n}}\,\mathrm{d}s,\end{align}

where for the final equality we have used the matching condition (3.60d), and the fact that $\tilde{\rho}^{(1/2)}$ is independent of $\xi$ and $\eta$ .

Since W satisfies the cell problem (3.34), the function ${\textbf{F}}\;:\!=\;\nabla_\xi W+{\textbf{e}}_2$ is divergence free on the cell $\omega_f$ , and ${\textbf{F}}\cdot{\textbf{n}}_s=0$ on the solid $\partial\omega_s$ . The integral over the curve $\Gamma$ in (3.61) is therefore independent of the choice of $\Gamma$ (by application of the divergence theorem). Similarly to the analysis of Section 3.2.2, by choosing $\Gamma$ to be the flat curve $\eta=\eta^*$ where this lies within $\omega_f$ , and on the surface $\partial\omega_s$ wherever the solid obstructs the line $\eta=\eta^*$ , we see that the integral in (3.61),

(3.62) \begin{equation} \int_{\Gamma}\left(\nabla_\xi W+{\textbf{e}}_2\right)\cdot{\textbf{n}}\,\mathrm{d}s=\int_{\{\eta=\eta^*\}\cap\omega_f}W_\eta+1\,\mathrm{d}\xi,\end{equation}

is independent of the value of $\eta^*$ . Thus, this integral is equal to its average over $\eta^*$ ,

(3.63) \begin{equation} \int_{\Gamma}\left(\nabla_\xi W+{\textbf{e}}_2\right)\cdot{\textbf{n}}\,\mathrm{d}s=\iint_{\omega_f}W_\eta+1\,\mathrm{d}\xi\mathrm{d}\eta=\mathcal{D},\end{equation}

and so is precisely the effective diffusivity $\mathcal{D}$ . Substituting this into (3.61), we find

(3.64) \begin{equation} (1-\nu\bar{\rho}^{(0)})\int_{z={{\mathcal{H}}}^{(0)}}\frac{H_T+{{\mathcal{H}}}^{(0)}_t}{\sqrt{1+({{\mathcal{H}}}^{(0)}_x)^2}}\,\mathrm{d}s=\mathcal{D}\tilde{\rho}^{(1/2)}_{\tilde{z}}|_{\tilde{z}=0},\end{equation}

where we have used the fact that $\bar{\rho}^{(0)}$ is independent of x and z to rearrange the left-hand side. This equation (3.64) relates the instantaneous leading-order diffusive flux of vapour entering the intermediate layer, to the flux of vapour off the interface. We note that the integral on the left of (3.64) is the instantaneous total interface velocity, and that this depends on t, since the shape and velocity of the microscale interface, $z={{\mathcal{H}}}^{(0)}$ , in general vary with t. This is the reason why we need an intermediate layer: we could not match directly between the inner and outer regions because the leading-order outer density flux $\rho^{(0)}_Z$ is independent of the micro-timescale t. The role of the intermediate region is that it allows these temporal oscillations to be smoothed out.

We now average (3.64) in t, using the fact that $\bar{\rho}^{(0)}$ is independent of t and the definition (3.22) of $\mathcal{F}$ (which from our discussion in Section 3.2.2 we know to be given by $\mathcal{F}=\phi$ ), to simplify the left-hand side, to give

(3.65) \begin{equation} (1-\nu\bar{\rho}^{(0)})\phi H_T=\mathcal{D}H_T\int_{t=0}^{1/H_T}\tilde{\rho}^{(1/2)}_{\tilde{z}}|_{\tilde{z}=0}\,\mathrm{d}t.\end{equation}

We note that by integrating (3.48a) over the period of t, and using the periodicity of $\tilde{\rho}^{(1/2)}$ , we must have

(3.66) \begin{equation} 0=H_T\int_{t=0}^{1/H_T}\mathcal{D}\tilde{\rho}^{(1/2)}_{\tilde{z}\tilde{z}}\,\mathrm{d}t=\left(H_T\int_{t=0}^{1/H_T}\mathcal{D}\tilde{\rho}^{(1/2)}_{\tilde{z}}\,\mathrm{d}t\right)_{\tilde{z}},\end{equation}

so that the time-averaged diffusive flux of vapour through the intermediate region is independent of $\tilde{z}$ . In particular, we may therefore replace $\tilde{z}=0$ with the limit $\tilde{z}\rightarrow\infty$ in (3.65). We then apply the matching condition:

(3.67) \begin{equation} \lim_{\tilde{z}\rightarrow\infty}\tilde{\rho}^{(1/2)}_{\tilde{z}}=\rho^{(0)}_Z|_{Z=0},\end{equation}

and the fact that $\rho^{(0)}_Z$ is independent of t to find that

(3.68) \begin{align} (1-\nu\bar{\rho}^{(0)})\phi H_T =\mathcal{D}\rho^{(0)}_Z|_{Z=0}.\end{align}

Finally, using (3.54), we see that (3.68) is an effective boundary condition for the macroscale leading-order vapour density $\rho^{(0)}$ , namely

(3.69) \begin{equation} (1-\nu\rho^{(0)})\phi H_T=\mathcal{D}\rho^{(0)}_Z \qquad \text{ at }Z=0.\end{equation}

4.1.1. Case $\alpha=O(\epsilon)$ or smaller

If $\alpha\ll\epsilon^{1/2}$ then at $O(\epsilon^{1/2})$ , the chemistry interface condition (4.1) becomes

(4.4) \begin{equation} \bar{\rho}^{(1/2)}=0 \qquad \text{ on }z={{\mathcal{H}}}^{(0)}.\end{equation}

Since we know that $\bar{\rho}^{(1/2)}$ is independent of the spatial variables x and z, we conclude that $\bar{\rho}^{(1/2)}\equiv 0$ is always zero throughout the inner region. This is surprising: we have no $O(\epsilon^{1/2})$ correction term in the inner region.

Furthermore, we can close the problem (3.48a) for $\tilde{\rho}^{(1/2)}$ in the intermediate region with the Dirichlet boundary condition (3.48c, left), which is

(4.5) \begin{equation} \tilde{\rho}^{(1/2)}|_{\tilde{z}=0}=\lim_{z\rightarrow\infty}\bar{\rho}^{(1/2)}=0.\end{equation}

The unique time-periodic solution of (3.48) is then

(4.6) \begin{equation} \tilde{\rho}^{(1/2)}=\tilde{z}\rho^{(0)}_Z|_{Z=0},\end{equation}

which is, in fact, independent of t. The intermediate region in this case is not performing its expected function of smoothing temporal oscillations in the flux off the interface. Indeed, the flux of vapour through the intermediate region, $\rho^{(0)}_Z|_{Z=0}$ , is constant: the intermediate region does nothing at this order. Since (4.6) holds, the far-field condition (3.60d) for the inner region flux problem (3.60) becomes simply

(4.7) \begin{equation} \bar{\rho}^{(1)}\sim\rho^{(0)}_Z|_{Z=0}(z+W) \qquad \text{ as }z\rightarrow\infty.\end{equation}

Thus, in the case when $\alpha\ll\epsilon^{1/2}$ , we could have greatly simplified the analysis of Section 3.3, as we do not need an intermediate layer or an expansion in terms of $O(\epsilon^{1/2})$ at all.

Substituting (4.6) into (3.64) (and using the fact that $\bar{\rho}^{(0)}=1$ ), we find that the instantaneous total flux of vapour coming off the interface on the microscale, namely

(4.8) \begin{equation} (1-\nu)\int_{z={{\mathcal{H}}}^{(0)}}\frac{H_T+{{\mathcal{H}}}^{(0)}_t}{\sqrt{1+\left({{\mathcal{H}}}^{(0)}_x\right)^2}}\,\mathrm{d}s=\mathcal{D}\rho^{(0)}_Z|_{Z=0},\end{equation}

is independent of t (since the right-hand side is independent of t). The shape and/or speed of the microscale interface $z={{\mathcal{H}}}^{(0)}(x,t)$ must therefore vary in time as the solid microstructure moves up through the cell in order to maintain this constant total flux off the interface. For instance, at times when the fluid-occupied fraction of the curve $z=0$ is smaller than average, the interface $z={{\mathcal{H}}}^{(0)}(x,t)$ must move faster than average in order to maintain the same total evaporation flux. The shape and velocity of the microscale interface $z={{\mathcal{H}}}^{(0)}$ may be found, along with $\bar{\rho}^{(1)}$ , by solving the problem (3.60). However, we note that there is no need to compute this solution in order to understand the macroscale evaporation process, since the homogenised model is not affected in any way by the details of the microscale interface motion. Indeed, the macroscale, outer process informs the inner: the microscale interface shape and speed adapt to maintain the t-independent vapour flux drawn off the interface by the macroscale density gradient.

4.1.2. Case $\alpha=O(\epsilon^{1/2})$

We now suppose that $\alpha=O(\epsilon^{1/2})$ and write $\alpha=\beta\epsilon^{1/2}$ with $\beta=O(1)$ . While $\bar{\rho}^{(0)}$ satisfies the Dirichlet condition (4.2) at leading order, at $O(\epsilon^{1/2})$ (4.1) becomes

(4.9) \begin{equation} \beta\frac{H_T+{{\mathcal{H}}}^{(0)}_t}{\sqrt{1+\left({{\mathcal{H}}}^{(0)}_x\right)^2}}=\bar{\rho}^{(1/2)} \qquad \text{ on }z={{\mathcal{H}}}^{(0)}.\end{equation}

We recall that, like $\bar{\rho}^{(0)}$ , $\bar{\rho}^{(1/2)}$ is independent of x and z (although unlike $\bar{\rho}^{(0)}$ it may depend on t).

The value of $\bar{\rho}^{(1/2)}(t)=\tilde{\rho}^{(1/2)}|_{\tilde{z}=0}$ is found by solving the diffusion equation for $\tilde{\rho}^{(1/2)}$ in the intermediate region (3.48) (with the boundary condition (3.48c, right)). The two problems (3.60) and (3.48) are therefore fully coupled by both (4.9) and the matching condition (3.60d) and must be solved simultaneously in order to understand the microscale interface motion. We note that we can simplify this problem by deriving a new interface condition for $\tilde{\rho}^{(1/2)}$ that holds at $\tilde{z}=0$ , rather than matching with $\bar{\rho}^{(1)}$ in the inner region. Specifically, we note from (3.64) that the instantaneous flux of vapour out of the inner region is

(4.10) \begin{equation} \mathcal{D}\tilde{\rho}^{(1/2)}_{\tilde{z}}|_{\tilde{z}=0}=(1-\nu)\int_{z={{\mathcal{H}}}^{(0)}}\frac{H_T+{{\mathcal{H}}}^{(0)}_t}{\sqrt{1+({{\mathcal{H}}}^{(0)}_x)^2}}\,\mathrm{d}s=\frac{(1-\nu)}{\beta}L\bar{\rho}^{(1/2)}=\frac{(1-\nu)}{\beta}L\tilde{\rho}^{(1/2)}_{\tilde{z}=0},\end{equation}

where $L=\int_{z={{\mathcal{H}}}^{(0)}}\,\mathrm{d}s$ is the instantaneous length of the microscale interface. Equation (4.10) provides a Robin boundary condition for the problem (3.48a), replacing the matching conditions (3.48c).

In particular, we can use this expression for the instantaneous vapour flux (4.10) to argue that the intermediate region is needed in this case of $\alpha=O(\epsilon^{1/2})$ , unlike in the case $\alpha=O(\epsilon)$ above. We demonstrate that the intermediate region is needed using a contradiction, and so suppose that the intermediate region is not required, that is, that the flux through the intermediate region is constant in t and equal to the far-field flux $\rho^{(0)}_Z|_{Z=0}$ . The solution $\tilde{\rho}^{1/2}$ of (3.48) then takes form:

(4.11) \begin{equation} \tilde{\rho}^{(1/2)}=\rho^{(0)}_Z|_{Z=0}\tilde{z}+\bar{\rho}^{(1/2)}(t).\end{equation}

From (4.10), we see that $L\bar{\rho}^{(1/2)}$ must be independent of t, and since for a general pore-scale geometry L varies in t, we therefore require that $\bar{\rho}^{(1/2)}$ is non-constant in t (noting from (4.9) that we must have $\bar{\rho}^{(1/2)}$ non-zero in order for there to be an interface motion at all). This leads to the required contradiction, as $\tilde{\rho}^{(1/2)}$ given by (4.11), with $\bar{\rho}^{(1/2)}$ non-constant in t does not satisfy the diffusion equation (3.48a).

Thus, the instantaneous flux of vapour out of the inner region cannot be independent of t, and so the intermediate region is needed to smooth the temporal oscillations in the vapour flux in this case $\alpha=O(\epsilon^{1/2})$ .