3.1 The bulk equations for $\mathsf{L}_{\mathsf{d}}=O(\unicode[STIX]{x1D700})$

We first do asymptotic analysis for the CHNS system far from the boundary. The analysis is the same as that for two-phase flow without contact lines. We will state the key steps of the analysis and illustrate the main results. In the next subsection, the bulk analysis here will be combined with the analysis near the boundary to derive the sharp-interface limits of the GNBC.

Let $\mathsf{L}_{\mathsf{d}}=\unicode[STIX]{x1D700}l_{d}$ , where $l_{d}$ is a proportional constant independent of $\unicode[STIX]{x1D700}$ . Consider the CHNS system far from the boundary of $\unicode[STIX]{x1D6FA}$ . We first do outer expansions far from the interface $\unicode[STIX]{x1D6E4}$ , then we consider inner expansions near $\unicode[STIX]{x1D6E4}$ . Combining them together, we will obtain the sharp-interface limit of the CHNS system in $\unicode[STIX]{x1D6FA}$ .

Outer expansions. Far from the two-phase interface $\unicode[STIX]{x1D6E4}$ , we use the following ansatz:

(3.2) $$\begin{eqnarray}\left.\begin{array}{@{}c@{}}\boldsymbol{v}_{\unicode[STIX]{x1D700}}^{\pm }=\boldsymbol{v}_{0}^{\pm }+\unicode[STIX]{x1D700}\boldsymbol{v}_{1}^{\pm }+\unicode[STIX]{x1D700}^{2}\boldsymbol{v}_{2}^{\pm }+\cdots \,,\\ \unicode[STIX]{x1D719}_{\unicode[STIX]{x1D700}}^{\pm }=\unicode[STIX]{x1D719}_{0}^{\pm }+\unicode[STIX]{x1D700}\unicode[STIX]{x1D719}_{1}^{\pm }+\unicode[STIX]{x1D700}^{2}\unicode[STIX]{x1D719}_{2}^{\pm }+\cdots \,,\\ p_{\unicode[STIX]{x1D700}}^{\pm }=p_{0}^{\pm }+\unicode[STIX]{x1D700}p_{1}^{\pm }+\unicode[STIX]{x1D700}^{2}p_{2}^{\pm }+\cdots \,.\end{array}\right\}\end{eqnarray}$$

Here $f^{\pm }$ denotes the restriction of a function $f$ in $\unicode[STIX]{x1D6FA}^{+}$ and $\unicode[STIX]{x1D6FA}^{-}$ respectively. For $\unicode[STIX]{x1D707}_{\unicode[STIX]{x1D700}}$ , we easily have

(3.3) $$\begin{eqnarray}\unicode[STIX]{x1D707}_{\unicode[STIX]{x1D700}}^{\pm }=\unicode[STIX]{x1D700}^{-1}\unicode[STIX]{x1D707}_{0}^{\pm }+\unicode[STIX]{x1D707}_{1}^{\pm }+\unicode[STIX]{x1D700}\unicode[STIX]{x1D707}_{2}^{\pm }+\cdots \,,\end{eqnarray}$$

where

(3.4) $$\begin{eqnarray}\unicode[STIX]{x1D707}_{0}^{\pm }=-\unicode[STIX]{x1D719}_{0}^{\pm }+(\unicode[STIX]{x1D719}_{0}^{\pm })^{3}.\end{eqnarray}$$

We substitute the above expansions in the CHNS system (2.6). The leading order of the first equation of (2.6) gives

(3.5) $$\begin{eqnarray}{\displaystyle \frac{\unicode[STIX]{x2202}\unicode[STIX]{x1D719}_{0}^{\pm }}{\unicode[STIX]{x2202}t}}+\boldsymbol{v}_{0}\boldsymbol{\cdot }\unicode[STIX]{x1D735}\unicode[STIX]{x1D719}_{0}^{\pm }=l_{d}\unicode[STIX]{x0394}\unicode[STIX]{x1D707}_{0}^{\pm }.\end{eqnarray}$$

The leading order of the second equation of (2.6) gives

(3.6) $$\begin{eqnarray}\unicode[STIX]{x1D707}_{0}^{\pm }\unicode[STIX]{x1D735}\unicode[STIX]{x1D719}_{0}^{\pm }=0.\end{eqnarray}$$

More precisely, we have

(3.7) $$\begin{eqnarray}(-\unicode[STIX]{x1D719}_{0}^{\pm }+(\unicode[STIX]{x1D719}_{0}^{\pm })^{3})\unicode[STIX]{x1D735}\unicode[STIX]{x1D719}_{0}^{\pm }=\unicode[STIX]{x1D735}\left({\displaystyle \frac{(1-(\unicode[STIX]{x1D719}_{0}^{\pm })^{2})^{2}}{4}}\right)=0.\end{eqnarray}$$

This implies that

(3.8) $$\begin{eqnarray}\unicode[STIX]{x1D719}_{0}^{\pm }=c_{\pm }\quad \text{in }\unicode[STIX]{x1D6FA}^{\pm },\end{eqnarray}$$

where $c_{\pm }$ are two constants such that $c_{+}>0$ and $c_{-}<0$ . For the third equation of (2.6), at the leading order, we have

(3.9) $$\begin{eqnarray}\unicode[STIX]{x1D735}\boldsymbol{\cdot }\boldsymbol{v}_{0}^{\pm }=0.\end{eqnarray}$$

By direct calculations, we also have the next order of the second equation of (2.6) as

(3.10) $$\begin{eqnarray}Re\left[{\displaystyle \frac{\unicode[STIX]{x2202}\boldsymbol{v}_{0}^{\pm }}{\unicode[STIX]{x2202}t}}+(\boldsymbol{v}_{0}^{\pm }\boldsymbol{\cdot }\unicode[STIX]{x1D735})\boldsymbol{v}_{0}^{\pm }\right]=\boldsymbol{F}-\unicode[STIX]{x1D735}p_{0}^{\pm }+\unicode[STIX]{x0394}\boldsymbol{v}_{0}^{\pm }+\unicode[STIX]{x1D707}_{0}^{\pm }\unicode[STIX]{x1D735}\unicode[STIX]{x1D719}_{1}^{\pm }.\end{eqnarray}$$

Here we have used the fact that $\unicode[STIX]{x1D719}_{0}^{\pm }$ are constants in $\unicode[STIX]{x1D6FA}^{\pm }$ .

Inner expansions. The outer expansions in $\unicode[STIX]{x1D6FA}^{+}$ and $\unicode[STIX]{x1D6FA}^{-}$ are connected by the transition layer near the interface $\unicode[STIX]{x1D6E4}$ . We will consider the so-called inner expansions near $\unicode[STIX]{x1D6E4}$ . Let $d(x)$ be signed distance to $\unicode[STIX]{x1D6E4}$ , which is well defined near the interface. Then the unit normal of the interface pointing to $\unicode[STIX]{x1D6FA}^{+}$ is given by $\boldsymbol{n}=\unicode[STIX]{x1D735}d$ . We introduce a new rescaled variable

(3.11) $$\begin{eqnarray}\unicode[STIX]{x1D709}={\displaystyle \frac{d(x)}{\unicode[STIX]{x1D700}}}.\end{eqnarray}$$

A signed distance function to define a stretched variable is widely used in asymptotic analysis for phase-field equations, e.g. Pego (Reference Pego1989), Lowengrub & Truskinovsky (Reference Lowengrub and Truskinovsky1998), Wang & Wang (Reference Wang and Wang2007) and Xu & Wang (Reference Xu and Wang2011). The geometric properties such as the unit normal vector and the mean curvature can be computed by $d(x)$ directly. In principle, the analysis is equivalent to that using a curvilinear coordinate system as in Magaletti et al. (Reference Magaletti, Picano, Chinappi, Marino and Casciola2013). For any function $f(x)$ (e.g. $f=\boldsymbol{v}_{\unicode[STIX]{x1D700}},p_{\unicode[STIX]{x1D700}},\unicode[STIX]{x1D719}_{\unicode[STIX]{x1D700}}$ ), we can rewrite it as

(3.12) $$\begin{eqnarray}f(x)=\tilde{f}(x,\unicode[STIX]{x1D709}).\end{eqnarray}$$

Then we have

(3.13) $$\begin{eqnarray}\left.\begin{array}{@{}c@{}}\unicode[STIX]{x1D735}f=\unicode[STIX]{x1D735}_{x}\tilde{f}+\unicode[STIX]{x1D700}^{-1}\unicode[STIX]{x2202}_{\unicode[STIX]{x1D709}}\tilde{f}\boldsymbol{n},\\ \unicode[STIX]{x0394}f=\unicode[STIX]{x1D6E5}_{x}\tilde{f}+\unicode[STIX]{x1D700}^{-1}\unicode[STIX]{x2202}_{\unicode[STIX]{x1D709}}\tilde{f}\unicode[STIX]{x1D705}+2\unicode[STIX]{x1D700}^{-1}(\boldsymbol{n}\boldsymbol{\cdot }\unicode[STIX]{x1D735}_{x})\unicode[STIX]{x2202}_{\unicode[STIX]{x1D709}}\tilde{f}+\unicode[STIX]{x1D700}^{-2}\unicode[STIX]{x2202}_{\unicode[STIX]{x1D709}\unicode[STIX]{x1D709}}\tilde{f},\\ \unicode[STIX]{x2202}_{t}f=\unicode[STIX]{x2202}_{t}\tilde{f}+\unicode[STIX]{x1D700}^{-1}\unicode[STIX]{x2202}_{\unicode[STIX]{x1D709}}\tilde{f}\unicode[STIX]{x2202}_{t}d_{\unicode[STIX]{x1D700}}.\end{array}\right\}\end{eqnarray}$$

Here the $\unicode[STIX]{x1D735}_{x}$ and $\unicode[STIX]{x1D6E5}_{x}$ stand for the gradient and Laplace operator with respect to $x$ , respectively. In (3.13) we use the fact that $\unicode[STIX]{x1D735}_{x}\boldsymbol{\cdot }\boldsymbol{n}=\unicode[STIX]{x1D705}$ . $\unicode[STIX]{x1D705}(x)$ for $x\in \unicode[STIX]{x1D6E4}(t)$ is the mean curvature of the interface and is positive (respectively negative) if the domain $\unicode[STIX]{x1D6FA}_{-}$ is convex (respectively concave) near $x$ .

In the inner region, we assume that

(3.14) $$\begin{eqnarray}\left.\begin{array}{@{}c@{}}\tilde{\boldsymbol{v}}_{\unicode[STIX]{x1D700}}=\tilde{\boldsymbol{v}}_{0}+\unicode[STIX]{x1D700}\tilde{\boldsymbol{v}}_{1}+\unicode[STIX]{x1D700}^{2}\tilde{\boldsymbol{v}}_{2}+\cdots \,,\\ \tilde{\unicode[STIX]{x1D719}}_{\unicode[STIX]{x1D700}}=\tilde{\unicode[STIX]{x1D719}}_{0}+\unicode[STIX]{x1D700}\tilde{\unicode[STIX]{x1D719}}_{1}+\unicode[STIX]{x1D700}^{2}\tilde{\unicode[STIX]{x1D719}}_{2}+\cdots \,,\\ \tilde{p}_{\unicode[STIX]{x1D700}}=\tilde{p}_{0}+\unicode[STIX]{x1D700}\tilde{p}_{1}+\unicode[STIX]{x1D700}^{2}\tilde{p}_{2}+\cdots \,.\end{array}\right\}\end{eqnarray}$$

A direct expansion for the chemical potential $\tilde{\unicode[STIX]{x1D707}}_{\unicode[STIX]{x1D700}}$ gives

(3.15) $$\begin{eqnarray}\tilde{\unicode[STIX]{x1D707}}_{\unicode[STIX]{x1D700}}=\unicode[STIX]{x1D700}^{-1}\tilde{\unicode[STIX]{x1D707}}_{0}+\tilde{\unicode[STIX]{x1D707}}_{1}+\unicode[STIX]{x1D700}\tilde{\unicode[STIX]{x1D707}}_{2}+\cdots \,,\end{eqnarray}$$

with

(3.16) $$\begin{eqnarray}\displaystyle & \displaystyle \tilde{\unicode[STIX]{x1D707}}_{0}=-\unicode[STIX]{x2202}_{\unicode[STIX]{x1D709}\unicode[STIX]{x1D709}}\tilde{\unicode[STIX]{x1D719}}_{0}-\tilde{\unicode[STIX]{x1D719}}_{0}+\tilde{\unicode[STIX]{x1D719}}_{0}^{3}, & \displaystyle\end{eqnarray}$$

(3.17) $$\begin{eqnarray}\displaystyle & \displaystyle \tilde{\unicode[STIX]{x1D707}}_{1}=-\unicode[STIX]{x2202}_{\unicode[STIX]{x1D709}\unicode[STIX]{x1D709}}\tilde{\unicode[STIX]{x1D719}}_{1}-\unicode[STIX]{x2202}_{\unicode[STIX]{x1D709}}\tilde{\unicode[STIX]{x1D719}}_{0}\unicode[STIX]{x1D705}+2(\boldsymbol{n}\boldsymbol{\cdot }\unicode[STIX]{x1D735}_{x})\unicode[STIX]{x2202}_{\unicode[STIX]{x1D709}}\tilde{\unicode[STIX]{x1D719}}_{0}-\tilde{\unicode[STIX]{x1D719}}_{1}+3\tilde{\unicode[STIX]{x1D719}}_{0}^{2}\tilde{\unicode[STIX]{x1D719}}_{1}. & \displaystyle\end{eqnarray}$$

We substitute the above expansions into the system (2.6). Using the fact that $\mathsf{L}_{\mathsf{d}}=\unicode[STIX]{x1D700}l_{d}$ , at the leading order, we have

(3.18) $$\begin{eqnarray}\left.\begin{array}{@{}c@{}}\unicode[STIX]{x2202}_{\unicode[STIX]{x1D709}\unicode[STIX]{x1D709}}\tilde{\unicode[STIX]{x1D707}}_{0}=0,\\ \unicode[STIX]{x2202}_{\unicode[STIX]{x1D709}\unicode[STIX]{x1D709}}\tilde{\boldsymbol{v}}_{0}+\mathsf{B}\tilde{\unicode[STIX]{x1D707}}_{0}\unicode[STIX]{x2202}_{\unicode[STIX]{x1D709}}\tilde{\unicode[STIX]{x1D719}}_{0}\boldsymbol{n}=0,\\ \boldsymbol{n}\boldsymbol{\cdot }\unicode[STIX]{x2202}_{\unicode[STIX]{x1D709}}\tilde{\boldsymbol{v}}_{0}=0.\end{array}\right\}\end{eqnarray}$$

The next order is

(3.19) $$\begin{eqnarray}\left.\begin{array}{@{}l@{}}\unicode[STIX]{x2202}_{t}d\unicode[STIX]{x2202}_{\unicode[STIX]{x1D709}}\tilde{\unicode[STIX]{x1D719}}_{0}+\tilde{\boldsymbol{v}}_{0}\boldsymbol{\cdot }\boldsymbol{n}\unicode[STIX]{x2202}_{\unicode[STIX]{x1D709}}\tilde{\unicode[STIX]{x1D719}}_{0}=l_{d}(\unicode[STIX]{x2202}_{\unicode[STIX]{x1D709}\unicode[STIX]{x1D709}}\tilde{\unicode[STIX]{x1D707}}_{1}+\unicode[STIX]{x1D705}\unicode[STIX]{x2202}_{\unicode[STIX]{x1D709}}\tilde{\unicode[STIX]{x1D707}}_{0}+2(\boldsymbol{n}\boldsymbol{\cdot }\unicode[STIX]{x1D735}_{x})\unicode[STIX]{x2202}_{\unicode[STIX]{x1D709}}\tilde{\unicode[STIX]{x1D707}}_{0}),\\ \tilde{\boldsymbol{v}}_{0}\boldsymbol{\cdot }\boldsymbol{n}\unicode[STIX]{x2202}_{\unicode[STIX]{x1D709}}\tilde{\boldsymbol{v}}_{0}=-\unicode[STIX]{x2202}_{\unicode[STIX]{x1D709}}\tilde{p}_{0}\boldsymbol{n}+\unicode[STIX]{x2202}_{\unicode[STIX]{x1D709}\unicode[STIX]{x1D709}}\tilde{\boldsymbol{v}}_{1}+\unicode[STIX]{x2202}_{\unicode[STIX]{x1D709}}\tilde{\boldsymbol{v}}_{0}\unicode[STIX]{x1D705}+2(\boldsymbol{n}\boldsymbol{\cdot }\unicode[STIX]{x1D735}_{x})\unicode[STIX]{x2202}_{\unicode[STIX]{x1D709}}\tilde{\boldsymbol{v}}_{0}\\ \quad +\,\mathsf{B}(\tilde{\unicode[STIX]{x1D707}}_{1}\unicode[STIX]{x2202}_{\unicode[STIX]{x1D709}}\tilde{\unicode[STIX]{x1D719}}_{0}\boldsymbol{n}+\tilde{\unicode[STIX]{x1D707}}_{0}\unicode[STIX]{x2202}_{\unicode[STIX]{x1D709}}\tilde{\unicode[STIX]{x1D719}}_{1}\boldsymbol{n}+\tilde{\unicode[STIX]{x1D707}}_{0}\unicode[STIX]{x1D735}_{x}\tilde{\unicode[STIX]{x1D719}}_{0}),\\ \boldsymbol{n}\boldsymbol{\cdot }\unicode[STIX]{x2202}_{\unicode[STIX]{x1D709}}\tilde{\boldsymbol{v}}_{1}+\unicode[STIX]{x1D735}_{x}\boldsymbol{\cdot }\tilde{\boldsymbol{v}}_{0}=0.\end{array}\right\}\end{eqnarray}$$

Matching conditions. We need the following matching conditions for inner and outer expansions:

(3.20) $$\begin{eqnarray}\displaystyle & \displaystyle \lim _{\unicode[STIX]{x1D709}\rightarrow \pm \infty }\tilde{f}_{i}(x,\unicode[STIX]{x1D709})=f_{i}^{\pm }(x), & \displaystyle\end{eqnarray}$$

(3.21) $$\begin{eqnarray}\displaystyle & \displaystyle \lim _{\unicode[STIX]{x1D709}\rightarrow \pm \infty }(\unicode[STIX]{x1D735}_{x}\tilde{f}_{i}(x,\unicode[STIX]{x1D709})+\unicode[STIX]{x2202}_{\unicode[STIX]{x1D709}}\tilde{f}_{i+1}(x,\unicode[STIX]{x1D709})\boldsymbol{n})=\unicode[STIX]{x1D735}f_{i}^{\pm }(x). & \displaystyle\end{eqnarray}$$

In the following, we will derive the sharp-interface limit of the CHNS system (2.6) by the above inner and outer expansions. From the first equation of (3.18), we know that $\tilde{\unicode[STIX]{x1D707}}_{0}$ is a linear function of $\unicode[STIX]{x1D709}$ , which can be written as $\tilde{\unicode[STIX]{x1D707}}_{0}(\unicode[STIX]{x1D709})=c_{1}\unicode[STIX]{x1D709}+c_{0},$ where $c_{0}$ and $c_{1}$ are independent of $\unicode[STIX]{x1D709}$ . Since $\lim _{\unicode[STIX]{x1D709}\rightarrow \pm \infty }\tilde{\unicode[STIX]{x1D707}}_{0}=\unicode[STIX]{x1D707}^{\pm }$ is bounded, we have $c_{1}=0$ . Therefore

(3.22) $$\begin{eqnarray}\tilde{\unicode[STIX]{x1D707}}_{0}=c_{0}.\end{eqnarray}$$

Then the second equation of (3.18) is reduced to

(3.23) $$\begin{eqnarray}\unicode[STIX]{x2202}_{\unicode[STIX]{x1D709}\unicode[STIX]{x1D709}}\tilde{\boldsymbol{v}}_{0}+\mathsf{B}c_{0}\unicode[STIX]{x2202}_{\unicode[STIX]{x1D709}}\tilde{\unicode[STIX]{x1D719}}_{0}\boldsymbol{n}=0.\end{eqnarray}$$

We integrate the equation with respect to $\unicode[STIX]{x1D709}$ in $(-\infty ,\infty )$ and obtain

(3.24) $$\begin{eqnarray}\unicode[STIX]{x2202}_{\unicode[STIX]{x1D709}}\tilde{\boldsymbol{v}}_{0}|_{-\infty }^{\infty }+\mathsf{B}c_{0}\tilde{\unicode[STIX]{x1D719}}_{0}|_{-\infty }^{\infty }\boldsymbol{n}=0.\end{eqnarray}$$

The inner product of the equation with $\boldsymbol{n}$ gives

(3.25) $$\begin{eqnarray}\boldsymbol{n}\boldsymbol{\cdot }\unicode[STIX]{x2202}_{\unicode[STIX]{x1D709}}\tilde{\boldsymbol{v}}_{0}|_{-\infty }^{\infty }+\mathsf{B}c_{0}\tilde{\unicode[STIX]{x1D719}}_{0}|_{-\infty }^{\infty }=0.\end{eqnarray}$$

By the third equation of (3.18), we obtain that

(3.26) $$\begin{eqnarray}c_{0}\tilde{\unicode[STIX]{x1D719}}_{0}|_{-\infty }^{\infty }=0.\end{eqnarray}$$

By the matching condition, we have

(3.27) $$\begin{eqnarray}c_{0}(\unicode[STIX]{x1D719}_{0}^{+}-\unicode[STIX]{x1D719}_{0}^{-})=0.\end{eqnarray}$$

Notice that $c_{+}=\unicode[STIX]{x1D719}_{0}^{+}>0>\unicode[STIX]{x1D719}_{0}^{-}=c_{-}$ ; we immediately have $c_{0}=0$ , or equivalently

(3.28) $$\begin{eqnarray}\tilde{\unicode[STIX]{x1D707}}_{0}=0.\end{eqnarray}$$

By the equation (3.16), we have

(3.29) $$\begin{eqnarray}-\unicode[STIX]{x2202}_{\unicode[STIX]{x1D709}\unicode[STIX]{x1D709}}\tilde{\unicode[STIX]{x1D719}}_{0}-\tilde{\unicode[STIX]{x1D719}}_{0}+\tilde{\unicode[STIX]{x1D719}}_{0}^{3}=0.\end{eqnarray}$$

The solvability condition for this equation (Pego Reference Pego1989) leads to

(3.30) $$\begin{eqnarray}\lim _{\unicode[STIX]{x1D709}\rightarrow \pm \infty }\tilde{\unicode[STIX]{x1D719}}_{0}=\pm 1,\end{eqnarray}$$

and the solution of (3.29) is

(3.31) $$\begin{eqnarray}\tilde{\unicode[STIX]{x1D719}}_{0}=\tanh (\unicode[STIX]{x1D709}/\sqrt{2}).\end{eqnarray}$$

This is the profile of the $\tilde{\unicode[STIX]{x1D719}}_{0}$ in the diffuse-interface layer. By the matching condition $\lim _{\unicode[STIX]{x1D709}\rightarrow \pm \infty }\tilde{\unicode[STIX]{x1D719}}_{0}=\unicode[STIX]{x1D719}_{0}^{\pm }$ and (3.8), we have

(3.32) $$\begin{eqnarray}\unicode[STIX]{x1D719}_{0}^{\pm }(x)=c_{\pm }=\pm 1,\quad \text{in}~\unicode[STIX]{x1D6FA}^{\pm }.\end{eqnarray}$$

This will lead to $\unicode[STIX]{x1D707}_{0}^{\pm }=0$ . Therefore the equation (3.10) is reduced to

(3.33) $$\begin{eqnarray}Re\left({\displaystyle \frac{\unicode[STIX]{x2202}\boldsymbol{v}_{0}^{\pm }}{\unicode[STIX]{x2202}t}}+(\boldsymbol{v}_{0}^{\pm }\boldsymbol{\cdot }\unicode[STIX]{x1D735})\boldsymbol{v}_{0}^{\pm }\right)=\boldsymbol{F}-\unicode[STIX]{x1D735}p_{0}^{\pm }+\unicode[STIX]{x0394}\boldsymbol{v}_{0}^{\pm }.\end{eqnarray}$$

Together with (3.9), this is the standard incompressible Navier–Stokes equation in $\unicode[STIX]{x1D6FA}^{\pm }$ .

We then derive the jump conditions on the interface $\unicode[STIX]{x1D6E4}$ . Noticing (3.28), the second equation of (3.18) is reduced to

(3.34) $$\begin{eqnarray}\unicode[STIX]{x2202}_{\unicode[STIX]{x1D709}\unicode[STIX]{x1D709}}\tilde{\boldsymbol{v}}_{0}=0.\end{eqnarray}$$

By a similar argument as for $\tilde{\unicode[STIX]{x1D707}}_{0}$ in (3.22), we know that $\tilde{\boldsymbol{v}}_{0}$ is independent of $\unicode[STIX]{x1D709}$ , or

(3.35) $$\begin{eqnarray}\tilde{\boldsymbol{v}}_{0}(x,\unicode[STIX]{x1D709})=\tilde{\boldsymbol{v}}_{0}(x).\end{eqnarray}$$

By the matching condition for $\tilde{\boldsymbol{v}}_{0}$ , we obtain

(3.36) $$\begin{eqnarray}[\boldsymbol{v}_{0}]=0,\end{eqnarray}$$

where $[f]=f^{+}-f^{-}$ denotes the jump of a function $f$ across the interface $\unicode[STIX]{x1D6E4}$ . The equation (3.36) implies that the velocity is continuous across $\unicode[STIX]{x1D6E4}$ .

Similarly, the first equation of (3.19) is reduced to

(3.37) $$\begin{eqnarray}\unicode[STIX]{x2202}_{t}d\unicode[STIX]{x2202}_{\unicode[STIX]{x1D709}}\tilde{\unicode[STIX]{x1D719}}_{0}+\tilde{\boldsymbol{v}}_{0}\boldsymbol{\cdot }\boldsymbol{n}\unicode[STIX]{x2202}_{\unicode[STIX]{x1D709}}\tilde{\unicode[STIX]{x1D719}}_{0}=l_{d}\unicode[STIX]{x2202}_{\unicode[STIX]{x1D709}\unicode[STIX]{x1D709}}\tilde{\unicode[STIX]{x1D707}}_{1}.\end{eqnarray}$$

Integrating the equation in $(-\infty ,\infty )$ and using the matching condition for $\tilde{\unicode[STIX]{x1D719}}_{0}$ and $\tilde{\unicode[STIX]{x1D707}}_{1}$ , we obtain

(3.38) $$\begin{eqnarray}\unicode[STIX]{x2202}_{t}d+\tilde{\boldsymbol{v}}_{0}\boldsymbol{\cdot }\boldsymbol{n}=0.\end{eqnarray}$$

This implies that the normal velocity of the interface $\unicode[STIX]{x1D6E4}$ is

(3.39) $$\begin{eqnarray}V_{n}=\boldsymbol{v}_{0}\boldsymbol{\cdot }\boldsymbol{n}.\end{eqnarray}$$

We then show the jump condition for the viscous stress. Noticing (3.28) and (3.35), the second equation of (3.19) can be reduced to

(3.40) $$\begin{eqnarray}-\unicode[STIX]{x2202}_{\unicode[STIX]{x1D709}}\tilde{p}_{0}\boldsymbol{n}+\unicode[STIX]{x2202}_{\unicode[STIX]{x1D709}\unicode[STIX]{x1D709}}\tilde{\boldsymbol{v}}_{1}+\mathsf{B}\unicode[STIX]{x1D707}_{1}\unicode[STIX]{x2202}_{\unicode[STIX]{x1D709}}\tilde{\unicode[STIX]{x1D719}}_{0}\boldsymbol{n}=0.\end{eqnarray}$$

By the equation (3.17), we have

(3.41) $$\begin{eqnarray}\int _{-\infty }^{\infty }\tilde{\unicode[STIX]{x1D707}}_{1}\unicode[STIX]{x2202}_{\unicode[STIX]{x1D709}}\tilde{\unicode[STIX]{x1D719}}_{0}\,\text{d}\unicode[STIX]{x1D709}=-\int _{-\infty }^{\infty }(\unicode[STIX]{x2202}_{\unicode[STIX]{x1D709}}\tilde{\unicode[STIX]{x1D719}}_{0})^{2}\,\text{d}\unicode[STIX]{x1D709}\unicode[STIX]{x1D705}+\int _{-\infty }^{\infty }(-\unicode[STIX]{x2202}_{\unicode[STIX]{x1D709}\unicode[STIX]{x1D709}}\tilde{\unicode[STIX]{x1D719}}_{1}-\tilde{\unicode[STIX]{x1D719}}_{1}+3\tilde{\unicode[STIX]{x1D719}}_{0}^{2}\tilde{\unicode[STIX]{x1D719}}_{1})\unicode[STIX]{x2202}_{\unicode[STIX]{x1D709}}\tilde{\unicode[STIX]{x1D719}}_{0}\,\text{d}\unicode[STIX]{x1D709}.\end{eqnarray}$$

Integration by parts for the second term on the right-hand side of the equation leads to

(3.42) $$\begin{eqnarray}\int _{-\infty }^{\infty }(-\unicode[STIX]{x2202}_{\unicode[STIX]{x1D709}\unicode[STIX]{x1D709}}\tilde{\unicode[STIX]{x1D719}}_{1}-\tilde{\unicode[STIX]{x1D719}}_{1}+3\tilde{\unicode[STIX]{x1D719}}_{0}^{2}\tilde{\unicode[STIX]{x1D719}}_{1})\unicode[STIX]{x2202}_{\unicode[STIX]{x1D709}}\tilde{\unicode[STIX]{x1D719}}_{0}\,\text{d}\unicode[STIX]{x1D709}=\int _{-\infty }^{\infty }(\unicode[STIX]{x2202}_{\unicode[STIX]{x1D709}\unicode[STIX]{x1D709}}\tilde{\unicode[STIX]{x1D719}}_{0}+\tilde{\unicode[STIX]{x1D719}}_{0}-\tilde{\unicode[STIX]{x1D719}}_{0}^{3})\unicode[STIX]{x2202}_{\unicode[STIX]{x1D709}}\tilde{\unicode[STIX]{x1D719}}_{1}\,\text{d}\unicode[STIX]{x1D709}=0.\end{eqnarray}$$

Here we use the equation (3.29). Then (3.41) is reduced to

(3.43) $$\begin{eqnarray}\int _{-\infty }^{\infty }\tilde{\unicode[STIX]{x1D707}}_{1}\unicode[STIX]{x2202}_{\unicode[STIX]{x1D709}}\tilde{\unicode[STIX]{x1D719}}_{0}\,\text{d}\unicode[STIX]{x1D709}=-\unicode[STIX]{x1D70E}\unicode[STIX]{x1D705},\end{eqnarray}$$

with $\unicode[STIX]{x1D70E}=\int _{-\infty }^{\infty }(\unicode[STIX]{x2202}_{\unicode[STIX]{x1D709}}\tilde{\unicode[STIX]{x1D719}}_{0})^{2}\,\text{d}\unicode[STIX]{x1D709}=\int _{-\infty }^{\infty }(\unicode[STIX]{x2202}_{\unicode[STIX]{x1D709}}\tanh (\unicode[STIX]{x1D709}/\sqrt{2}))^{2}\,\text{d}\unicode[STIX]{x1D709}=2\sqrt{2}/3$ . Then we integrate the equation (3.40) on $\unicode[STIX]{x1D709}$ in $(-\infty ,\infty )$ , and notice the matching condition

(3.44a,b ) $$\begin{eqnarray}\lim _{\unicode[STIX]{x1D709}\rightarrow \pm \infty }\tilde{p}=p^{\pm },\quad \lim _{\unicode[STIX]{x1D709}\rightarrow \pm \infty }\unicode[STIX]{x2202}_{\unicode[STIX]{x1D709}}\tilde{\boldsymbol{v}}_{1}=\boldsymbol{n}\boldsymbol{\cdot }\unicode[STIX]{x1D735}\boldsymbol{v}_{0}^{\pm }.\end{eqnarray}$$

We are led to

(3.45) $$\begin{eqnarray}[-p_{0}\boldsymbol{n}+(\boldsymbol{n}\boldsymbol{\cdot }\unicode[STIX]{x1D735})\boldsymbol{v}_{0}]=\mathsf{B}\unicode[STIX]{x1D70E}\unicode[STIX]{x1D705}\boldsymbol{n}.\end{eqnarray}$$

This is the jump condition for pressure and stress (Magaletti et al. Reference Magaletti, Picano, Chinappi, Marino and Casciola2013).

Combining the above analysis, at the leading order, we obtain the standard Navier–Stokes equation for two-phase immiscible flow:

(3.46) $$\begin{eqnarray}\left.\begin{array}{@{}c@{}}Re\left({\displaystyle \frac{\unicode[STIX]{x2202}\boldsymbol{v}_{0}^{\pm }}{\unicode[STIX]{x2202}t}}+(\boldsymbol{v}_{0}^{\pm }\boldsymbol{\cdot }\unicode[STIX]{x1D735})\boldsymbol{v}_{0}^{\pm }\right)=\boldsymbol{F}-\unicode[STIX]{x1D735}p_{0}^{\pm }+\unicode[STIX]{x0394}\boldsymbol{v}_{0}^{\pm },\quad \text{in }\unicode[STIX]{x1D6FA}^{\pm },\\ \unicode[STIX]{x1D735}\boldsymbol{\cdot }\boldsymbol{v}_{0}^{\pm }=0,\quad \text{in }\unicode[STIX]{x1D6FA}^{\pm },\\ \text{}[\boldsymbol{v}_{0}]=0,\quad \text{on }\unicode[STIX]{x1D6E4},\\ \text{}[-p_{0}\boldsymbol{n}+(\boldsymbol{n}\boldsymbol{\cdot }\unicode[STIX]{x1D735})\boldsymbol{v}_{0}]=\mathsf{B}\unicode[STIX]{x1D70E}\unicode[STIX]{x1D705}\boldsymbol{n},\quad \text{on }\unicode[STIX]{x1D6E4},\\ V_{n}=\boldsymbol{v}_{0}\boldsymbol{\cdot }\boldsymbol{n},\quad \text{on }\unicode[STIX]{x1D6E4}.\end{array}\right\}\end{eqnarray}$$

3.2 The boundary conditions for $\mathsf{L}_{\mathsf{d}}=O(\unicode[STIX]{x1D700})$

We now consider the sharp-interface limit of the boundary condition (2.7). We consider three different choices for the relaxation parameter $\mathsf{V}_{\mathsf{s}}=O(\unicode[STIX]{x1D700}^{b})$ , $b=0,-1,-2$ . We show that they correspond to different boundary conditions in the sharp-interface limit.

Case I: $\mathsf{V}_{\mathsf{s}}=O(1)$ . Here we assume that $\mathsf{V}_{\mathsf{s}}$ is a constant independent of the Cahn number  $\unicode[STIX]{x1D700}$ .

Outer expansion. Far from the moving contact line, we can use the same outer expansions as in the bulk. By applying the expansions (3.2)–(2.7) and considering the leading order terms, we easily have the boundary condition for chemical potential

(3.47) $$\begin{eqnarray}\unicode[STIX]{x2202}_{n}\unicode[STIX]{x1D707}_{0}^{\pm }=0.\end{eqnarray}$$

Notice that $\unicode[STIX]{x1D707}_{0}^{\pm }=0$ and $\unicode[STIX]{x1D719}_{0}^{\pm }=\pm 1$ in $\unicode[STIX]{x1D6FA}^{\pm }$ ; we obtain

(3.48a,b ) $$\begin{eqnarray}\unicode[STIX]{x1D719}_{0}^{\pm }=\pm 1,\quad \unicode[STIX]{x1D707}_{0}^{\pm }=0,\quad \text{on }\unicode[STIX]{x1D6E4}_{S}^{\pm }.\end{eqnarray}$$

For the Navier slip boundary condition, we have

(3.49a,b ) $$\begin{eqnarray}\mathsf{l}_{\mathsf{s}}^{-1}(v_{0,\unicode[STIX]{x1D70F}}-v_{w})=-\unicode[STIX]{x2202}_{n}v_{0,\unicode[STIX]{x1D70F}},\quad \boldsymbol{v}_{0}\boldsymbol{\cdot }\boldsymbol{n}_{S}=0,\quad \text{on }\unicode[STIX]{x1D6E4}_{S}^{\pm }.\end{eqnarray}$$

Here $v_{0,\unicode[STIX]{x1D70F}}=\boldsymbol{v}_{0}-(\boldsymbol{v}_{0}\boldsymbol{\cdot }\boldsymbol{n}_{S})\boldsymbol{n}_{S}$ is the tangential velocity, and $\unicode[STIX]{x1D6E4}_{S}^{\pm }=\unicode[STIX]{x1D6E4}_{S}\cap \unicode[STIX]{x2202}\unicode[STIX]{x1D6FA}^{\pm }$ is the boundary of $\unicode[STIX]{x1D6FA}^{\pm }$ on the solid surface $\unicode[STIX]{x1D6E4}_{S}$ .

Inner expansion. We consider the boundary condition near the contact line. Here we denote the outer normal of the solid surface $\unicode[STIX]{x1D6E4}_{S}$ by $\boldsymbol{n}_{S}$ and the normal of the contact line in the tangential surface of $\unicode[STIX]{x1D6E4}_{S}$ by $\boldsymbol{m}$ (as shown in figure 1). Since the functions at the contact line need to be matched to the outer expansions on $\unicode[STIX]{x1D6E4}_{S}^{\pm }$ and also to the expansions inside the domain $\unicode[STIX]{x1D6FA}$ , it is convenient to introduce a different inner expansion near the contact line $L:=\{x\in \unicode[STIX]{x1D6E4}_{S}\,|\,\unicode[STIX]{x1D719}_{\unicode[STIX]{x1D700}}(x)=0\}$ as follows. Near the contact line, we introduce two fast changing parameters,

(3.50a,b ) $$\begin{eqnarray}\unicode[STIX]{x1D71A}={\displaystyle \frac{(x-x_{0})\boldsymbol{\cdot }\boldsymbol{m}}{\unicode[STIX]{x1D700}}},\quad \unicode[STIX]{x1D701}={\displaystyle \frac{(x-x_{0})\boldsymbol{\cdot }\boldsymbol{n}_{S}}{\unicode[STIX]{x1D700}}},\end{eqnarray}$$

with $x_{0}\in L$ . For any function $f(x)$ , near $x_{0}$ , it can be written as a function in $(x,\unicode[STIX]{x1D70C},\unicode[STIX]{x1D701})$ as

(3.51) $$\begin{eqnarray}f(x)=\hat{f}(x,\unicode[STIX]{x1D71A},\unicode[STIX]{x1D701}).\end{eqnarray}$$

The derivative of $f$ is then rewritten as

(3.52) $$\begin{eqnarray}\left.\begin{array}{@{}c@{}}\unicode[STIX]{x1D735}f=\unicode[STIX]{x1D735}_{x}\hat{f}+\unicode[STIX]{x1D700}^{-1}\boldsymbol{n}_{S}\unicode[STIX]{x2202}_{\unicode[STIX]{x1D701}}\hat{f}+\unicode[STIX]{x1D700}^{-1}\boldsymbol{m}\unicode[STIX]{x2202}_{\unicode[STIX]{x1D71A}}\hat{f},\\ \unicode[STIX]{x0394}f=\unicode[STIX]{x1D6E5}_{x}\hat{f}+\unicode[STIX]{x1D700}^{-1}(\unicode[STIX]{x1D735}\boldsymbol{\cdot }\boldsymbol{n}_{S}\unicode[STIX]{x2202}_{\unicode[STIX]{x1D701}}\hat{f}+\unicode[STIX]{x1D735}\boldsymbol{\cdot }\boldsymbol{m}\unicode[STIX]{x2202}_{\unicode[STIX]{x1D71A}}\hat{f}+2\unicode[STIX]{x2202}_{n\unicode[STIX]{x1D701}}\hat{f}+2\unicode[STIX]{x2202}_{m\unicode[STIX]{x1D71A}}\hat{f})+\unicode[STIX]{x1D700}^{-2}(\unicode[STIX]{x2202}_{\unicode[STIX]{x1D701}\unicode[STIX]{x1D701}}\hat{f}+\unicode[STIX]{x2202}_{\unicode[STIX]{x1D71A}\unicode[STIX]{x1D71A}}\hat{f}),\end{array}\right\}\end{eqnarray}$$

and the boundary derivative of $f$ is given by

(3.53a,b ) $$\begin{eqnarray}\unicode[STIX]{x2202}_{n}f=\unicode[STIX]{x2202}_{n}\hat{f}+\unicode[STIX]{x1D700}^{-1}\unicode[STIX]{x2202}_{\unicode[STIX]{x1D701}}\hat{f},\quad \unicode[STIX]{x2202}_{\unicode[STIX]{x1D70F}}f=\unicode[STIX]{x2202}_{\unicode[STIX]{x1D70F}}\hat{f}+\unicode[STIX]{x1D700}^{-1}\unicode[STIX]{x2202}_{\unicode[STIX]{x1D71A}}\hat{f}\boldsymbol{m}.\end{eqnarray}$$

We also have

(3.54) $$\begin{eqnarray}\unicode[STIX]{x2202}_{t}f=\unicode[STIX]{x2202}_{t}\hat{f}-\unicode[STIX]{x1D700}^{-1}\unicode[STIX]{x2202}_{\unicode[STIX]{x1D71A}}\hat{f}\unicode[STIX]{x2202}_{t}x_{0}\boldsymbol{\cdot }\boldsymbol{m}.\end{eqnarray}$$

Similarly as before, we assume that

(3.55) $$\begin{eqnarray}\left.\begin{array}{@{}c@{}}\hat{\boldsymbol{v}}_{\unicode[STIX]{x1D700}}=\hat{\boldsymbol{v}}_{0}+\unicode[STIX]{x1D700}\hat{\boldsymbol{v}}_{1}+\unicode[STIX]{x1D700}^{2}\hat{\boldsymbol{v}}_{2}+\cdots \,,\\ \hat{\unicode[STIX]{x1D719}}_{\unicode[STIX]{x1D700}}=\hat{\unicode[STIX]{x1D719}}_{0}+\unicode[STIX]{x1D700}\hat{\unicode[STIX]{x1D719}}_{1}+\unicode[STIX]{x1D700}^{2}\hat{\unicode[STIX]{x1D719}}_{2}+\cdots \,.\end{array}\right\}\end{eqnarray}$$

Direct computations give

(3.56) $$\begin{eqnarray}\hat{\unicode[STIX]{x1D707}}_{\unicode[STIX]{x1D700}}=\unicode[STIX]{x1D700}^{-1}\hat{\unicode[STIX]{x1D707}}_{0}+\hat{\unicode[STIX]{x1D707}}_{1}+\cdots \,,\end{eqnarray}$$

with $\hat{\unicode[STIX]{x1D707}}_{0}=-(\unicode[STIX]{x2202}_{\unicode[STIX]{x1D701}\unicode[STIX]{x1D701}}+\unicode[STIX]{x2202}_{\unicode[STIX]{x1D71A}\unicode[STIX]{x1D71A}})\hat{\unicode[STIX]{x1D719}}_{0}-\hat{\unicode[STIX]{x1D719}}_{0}+\hat{\unicode[STIX]{x1D719}}_{0}^{3}$ , and

(3.57) $$\begin{eqnarray}\hat{{\mathcal{L}}}=\hat{{\mathcal{L}}}_{0}+\unicode[STIX]{x1D700}\hat{{\mathcal{L}}}_{1}+\cdots \,,\end{eqnarray}$$

with $\hat{{\mathcal{L}}}_{0}=\unicode[STIX]{x2202}_{\unicode[STIX]{x1D701}}\hat{\unicode[STIX]{x1D719}}_{0}+\unicode[STIX]{x2202}\unicode[STIX]{x1D6FE}_{wf}/\unicode[STIX]{x2202}\unicode[STIX]{x1D719}(\hat{\unicode[STIX]{x1D719}}_{0})$ .

Substitute the expansions into the CHNS system (2.6) and the boundary condition (2.7). At the leading order we have the equation

(3.58) $$\begin{eqnarray}\left.\begin{array}{@{}c@{}}(\unicode[STIX]{x2202}_{\unicode[STIX]{x1D701}\unicode[STIX]{x1D701}}+\unicode[STIX]{x2202}_{\unicode[STIX]{x1D71A}\unicode[STIX]{x1D71A}})\hat{\unicode[STIX]{x1D707}}_{0}=0,\\ \mathsf{B}\hat{\unicode[STIX]{x1D707}}_{0}(\unicode[STIX]{x2202}_{\unicode[STIX]{x1D701}}\hat{\unicode[STIX]{x1D719}}_{0}\boldsymbol{n}_{S}+\unicode[STIX]{x2202}_{\unicode[STIX]{x1D71A}}\hat{\unicode[STIX]{x1D719}}_{0}\boldsymbol{m})+(\unicode[STIX]{x2202}_{\unicode[STIX]{x1D701}\unicode[STIX]{x1D701}}+\unicode[STIX]{x2202}_{\unicode[STIX]{x1D71A}\unicode[STIX]{x1D71A}})\hat{\boldsymbol{v}}_{0}=0,\\ \boldsymbol{n}_{S}\boldsymbol{\cdot }\unicode[STIX]{x2202}_{\unicode[STIX]{x1D701}}\hat{\boldsymbol{v}}_{0}+\boldsymbol{m}\boldsymbol{\cdot }\unicode[STIX]{x2202}_{\unicode[STIX]{x1D71A}}\hat{\boldsymbol{v}}_{0}=0,\end{array}\right\}\end{eqnarray}$$

and the boundary condition on $\unicode[STIX]{x1D6E4}_{S}$ :

(3.59) $$\begin{eqnarray}\left.\begin{array}{@{}c@{}}-\unicode[STIX]{x2202}_{t}x_{0}\boldsymbol{\cdot }\boldsymbol{m}+\hat{v}_{0,\unicode[STIX]{x1D70F}}\boldsymbol{\cdot }\boldsymbol{m}=0,\\ \unicode[STIX]{x1D700}\mathsf{l}_{\mathsf{s}}^{-1}(\hat{v}_{0,\unicode[STIX]{x1D70F}}-v_{w})+\unicode[STIX]{x1D700}\unicode[STIX]{x2202}_{n}\hat{v}_{0,\unicode[STIX]{x1D70F}}+\unicode[STIX]{x2202}_{\unicode[STIX]{x1D701}}\hat{v}_{0,\unicode[STIX]{x1D70F}}=\mathsf{B}\hat{{\mathcal{L}}}_{0}\unicode[STIX]{x2202}_{\unicode[STIX]{x1D71A}}\hat{\unicode[STIX]{x1D719}}_{0}\boldsymbol{m},\\ \unicode[STIX]{x2202}_{\unicode[STIX]{x1D701}}\hat{\unicode[STIX]{x1D707}}_{0}=0,\quad \hat{\boldsymbol{v}}_{0}\boldsymbol{\cdot }\boldsymbol{n}_{S}=0.\end{array}\right\}\end{eqnarray}$$

Here in the second equation of (3.59), we keep the terms $\unicode[STIX]{x1D700}\mathsf{l}_{\mathsf{s}}^{-1}(\hat{v}_{0,\unicode[STIX]{x1D70F}}-v_{w})+\unicode[STIX]{x1D700}\unicode[STIX]{x2202}_{n}\hat{v}_{0,\unicode[STIX]{x1D70F}}$ , since the slip velocity in the vicinity of the moving contact line might be large compared with the outer region (Qian et al. Reference Qian, Wang and Sheng2004).

Matching condition. We have the matching condition

(3.60) $$\begin{eqnarray}\left.\begin{array}{@{}c@{}}\displaystyle \lim _{\unicode[STIX]{x1D701}\rightarrow -\infty }\hat{f}(x_{0},\unicode[STIX]{x1D71A},\unicode[STIX]{x1D701})=\lim _{x\rightarrow x_{0}}\tilde{f}(x,\unicode[STIX]{x1D709}),\\ \displaystyle \lim _{\unicode[STIX]{x1D71A}\rightarrow \pm \infty }\hat{f}(x_{0},\unicode[STIX]{x1D71A},\unicode[STIX]{x1D701})=\lim _{x\rightarrow x_{0}}f^{\pm }(x).\end{array}\right\}\end{eqnarray}$$

In the first equation, $x$ denotes a point on $\unicode[STIX]{x1D6E4}$ approaching the contact point $x_{0}$ (as shown in figure 2). In the second equation $x$ denotes a point in $\unicode[STIX]{x1D6FA}^{\pm }$ approaching to  $x_{0}$ .

Figure 2.

The inner expansion near the contact point $x_{0}$ (in the rectangle). The matching occurs on the three sides (upper, left and right).

In the following, we will use these equations to derive the condition for moving contact lines. By the matching relation for $\unicode[STIX]{x1D707}$ ,

(3.61a,b ) $$\begin{eqnarray}\lim _{\unicode[STIX]{x1D701}\rightarrow -\infty }\hat{\unicode[STIX]{x1D707}}_{0}=\tilde{\unicode[STIX]{x1D707}}_{0}=0,\quad \lim _{\unicode[STIX]{x1D71A}\rightarrow \pm \infty }\hat{\unicode[STIX]{x1D707}}_{0}=\unicode[STIX]{x1D707}^{\pm }=0.\end{eqnarray}$$

The first equation of (3.58) implies that

(3.62) $$\begin{eqnarray}\hat{\unicode[STIX]{x1D707}}_{0}=0.\end{eqnarray}$$

This means

(3.63) $$\begin{eqnarray}-(\unicode[STIX]{x2202}_{\unicode[STIX]{x1D701}\unicode[STIX]{x1D701}}+\unicode[STIX]{x2202}_{\unicode[STIX]{x1D71A}\unicode[STIX]{x1D71A}})\hat{\unicode[STIX]{x1D719}}_{0}-\hat{\unicode[STIX]{x1D719}}_{0}+\hat{\unicode[STIX]{x1D719}}_{0}^{3}=0.\end{eqnarray}$$

We also have the matching condition for $\hat{\unicode[STIX]{x1D719}}_{0}$ ,

(3.64a,b ) $$\begin{eqnarray}\lim _{\unicode[STIX]{x1D701}\rightarrow -\infty }\hat{\unicode[STIX]{x1D719}}_{0}=\tilde{\unicode[STIX]{x1D719}}_{0}(\unicode[STIX]{x1D709}),\quad \lim _{\unicode[STIX]{x1D71A}\rightarrow \pm \infty }\hat{\unicode[STIX]{x1D719}}_{0}=\unicode[STIX]{x1D719}_{0}^{\pm }=\pm 1.\end{eqnarray}$$

By (3.29), it is easy to see that

(3.65) $$\begin{eqnarray}\hat{\unicode[STIX]{x1D719}}_{0}(\unicode[STIX]{x1D71A},\unicode[STIX]{x1D701})=\tilde{\unicode[STIX]{x1D719}}_{0}(\unicode[STIX]{x1D709})\end{eqnarray}$$

satisfies the equation (3.63) and the matching condition if the relation

(3.66) $$\begin{eqnarray}\unicode[STIX]{x1D709}=\unicode[STIX]{x1D71A}\sin \unicode[STIX]{x1D703}_{d}-\unicode[STIX]{x1D701}\cos \unicode[STIX]{x1D703}_{d}\end{eqnarray}$$

holds. Here $\unicode[STIX]{x1D703}_{d}$ is the dynamic contact angle (see figure 1). This leads to the relation

(3.67a,b ) $$\begin{eqnarray}\unicode[STIX]{x2202}_{\unicode[STIX]{x1D71A}}\hat{\unicode[STIX]{x1D719}}_{0}=\unicode[STIX]{x2202}_{\unicode[STIX]{x1D709}}\tilde{\unicode[STIX]{x1D719}}_{0}\sin \unicode[STIX]{x1D703}_{d},\quad \unicode[STIX]{x2202}_{\unicode[STIX]{x1D701}}\hat{\unicode[STIX]{x1D719}}_{0}=-\unicode[STIX]{x2202}_{\unicode[STIX]{x1D709}}\tilde{\unicode[STIX]{x1D719}}_{0}\cos \unicode[STIX]{x1D703}_{d}.\end{eqnarray}$$

In addition, (3.62) and the second equation of (3.58) implies that

(3.68) $$\begin{eqnarray}(\unicode[STIX]{x2202}_{\unicode[STIX]{x1D701}\unicode[STIX]{x1D701}}+\unicode[STIX]{x2202}_{\unicode[STIX]{x1D71A}\unicode[STIX]{x1D71A}})\hat{\boldsymbol{v}}_{0}=0.\end{eqnarray}$$

We also have the matching condition $\lim _{\unicode[STIX]{x1D701}\rightarrow -\infty }\hat{\boldsymbol{v}}_{0}=\lim _{x\rightarrow x_{0}}\tilde{\boldsymbol{v}}_{0}=\boldsymbol{v}_{0}(x_{0})$ and the boundary condition $\hat{\boldsymbol{v}}_{0}\boldsymbol{\cdot }\boldsymbol{n}_{S}=0$ . It is easy to see that

(3.69) $$\begin{eqnarray}\hat{\boldsymbol{v}}_{0}=\boldsymbol{v}_{0}(x_{0}),\end{eqnarray}$$

where

(3.70) $$\begin{eqnarray}\boldsymbol{v}_{0}(x_{0})\boldsymbol{\cdot }\boldsymbol{n}_{S}=0\end{eqnarray}$$

is a solution of the above equation, i.e. $\hat{\boldsymbol{v}}_{0}$ is independent of $\unicode[STIX]{x1D701}$ and $\unicode[STIX]{x1D71A}$ .

Using the above relations, we will derive the moving contact line condition. The first equation of (3.59) implies that

(3.71) $$\begin{eqnarray}\unicode[STIX]{x2202}_{t}x_{0}\boldsymbol{\cdot }\boldsymbol{m}=v_{\unicode[STIX]{x1D70F}}(x_{0})\boldsymbol{\cdot }\boldsymbol{m},\end{eqnarray}$$

since $\hat{v}_{\unicode[STIX]{x1D70F}}=\hat{\boldsymbol{v}}_{0}-(\hat{\boldsymbol{v}}_{0}\boldsymbol{\cdot }\boldsymbol{n}_{S})\boldsymbol{n}_{S}=v_{\unicode[STIX]{x1D70F}}$ . This implies that the normal velocity of the moving contact line is equal to the fluid velocity in this direction. The second equation (3.59) can be reduced to

(3.72) $$\begin{eqnarray}\unicode[STIX]{x1D700}\mathsf{l}_{\mathsf{s}}^{-1}(\hat{v}_{0,\unicode[STIX]{x1D70F}}-v_{w})+\unicode[STIX]{x1D700}\unicode[STIX]{x2202}_{n}\hat{v}_{0,\unicode[STIX]{x1D70F}}=\mathsf{B}\hat{{\mathcal{L}}}_{0}\unicode[STIX]{x2202}_{\unicode[STIX]{x1D71A}}\hat{\unicode[STIX]{x1D719}}_{0}\boldsymbol{m}.\end{eqnarray}$$

Integrate this equation with respect to $\unicode[STIX]{x1D71A}$ and we get

(3.73) $$\begin{eqnarray}\int _{-\infty }^{+\infty }\unicode[STIX]{x1D700}\mathsf{l}_{\mathsf{s}}^{-1}(\hat{v}_{0,\unicode[STIX]{x1D70F}}-v_{w})+\unicode[STIX]{x1D700}\unicode[STIX]{x2202}_{n}\hat{v}_{0,\unicode[STIX]{x1D70F}}\,\text{d}\unicode[STIX]{x1D71A}=\int _{-\infty }^{+\infty }\mathsf{B}\left(\unicode[STIX]{x2202}_{\unicode[STIX]{x1D701}}\hat{\unicode[STIX]{x1D719}}_{0}+{\displaystyle \frac{\unicode[STIX]{x2202}\unicode[STIX]{x1D6FE}_{wf}}{\unicode[STIX]{x2202}\unicode[STIX]{x1D719}}}(\hat{\unicode[STIX]{x1D719}}_{0})\right)\unicode[STIX]{x2202}_{\unicode[STIX]{x1D71A}}\hat{\unicode[STIX]{x1D719}}_{0}\,\text{d}\unicode[STIX]{x1D71A}\,\boldsymbol{m}.\end{eqnarray}$$

The left-hand side term is

(3.74) $$\begin{eqnarray}\int _{-\infty }^{+\infty }(\unicode[STIX]{x1D700}\mathsf{l}_{\mathsf{s}}^{-1}(\hat{v}_{0,\unicode[STIX]{x1D70F}}-v_{w})+\unicode[STIX]{x1D700}\unicode[STIX]{x2202}_{n}\hat{v}_{0,\unicode[STIX]{x1D70F}})\,\text{d}\unicode[STIX]{x1D71A}=\int _{interface}(\mathsf{l}_{\mathsf{s}}^{-1}(\hat{v}_{0,\unicode[STIX]{x1D70F}}-v_{w})+\unicode[STIX]{x2202}_{n}\hat{v}_{0,\unicode[STIX]{x1D70F}})\,\text{d}m.\end{eqnarray}$$

The right-hand side term is

(3.75) $$\begin{eqnarray}\displaystyle \int _{-\infty }^{+\infty }\mathsf{B}\left(\unicode[STIX]{x2202}_{\unicode[STIX]{x1D701}}\hat{\unicode[STIX]{x1D719}}_{0}+{\displaystyle \frac{\unicode[STIX]{x2202}\unicode[STIX]{x1D6FE}_{wf}}{\unicode[STIX]{x2202}\unicode[STIX]{x1D719}}}(\hat{\unicode[STIX]{x1D719}}_{0})\right)\unicode[STIX]{x2202}_{\unicode[STIX]{x1D71A}}\hat{\unicode[STIX]{x1D719}}_{0}\,\text{d}\unicode[STIX]{x1D71A} & = & \displaystyle \int _{-\infty }^{+\infty }\mathsf{B}\left(-\unicode[STIX]{x2202}_{\unicode[STIX]{x1D709}}\hat{\unicode[STIX]{x1D719}}_{0}\cos \unicode[STIX]{x1D703}_{d}+{\displaystyle \frac{\unicode[STIX]{x2202}\unicode[STIX]{x1D6FE}_{wf}}{\unicode[STIX]{x2202}\unicode[STIX]{x1D719}}}(\hat{\unicode[STIX]{x1D719}}_{0})\right)\unicode[STIX]{x2202}_{\unicode[STIX]{x1D709}}\hat{\unicode[STIX]{x1D719}}_{0}\,\text{d}\unicode[STIX]{x1D709}\nonumber\\ \displaystyle & = & \displaystyle -\mathsf{B}\unicode[STIX]{x1D70E}\cos \unicode[STIX]{x1D703}_{d}+\mathsf{B}(\unicode[STIX]{x1D6FE}_{wf}(1)-\unicode[STIX]{x1D6FE}_{wf}(-1))\nonumber\\ \displaystyle & = & \displaystyle -\mathsf{B}\unicode[STIX]{x1D70E}(\cos \unicode[STIX]{x1D703}_{d}-\cos \unicode[STIX]{x1D703}_{s}).\end{eqnarray}$$

Here we used the Young equation $\unicode[STIX]{x1D70E}\cos \unicode[STIX]{x1D703}_{s}=\unicode[STIX]{x1D6FE}_{wf}(1)-\unicode[STIX]{x1D6FE}_{wf}(-1)$ and (3.67a,b ). Then the equation (3.73) is reduced to

(3.76) $$\begin{eqnarray}\int _{interface}(\mathsf{l}_{\mathsf{s}}^{-1}(v_{0,\unicode[STIX]{x1D70F}}-v_{w})+\unicode[STIX]{x2202}_{n_{S}}v_{0,\unicode[STIX]{x1D70F}})\,\text{d}m=-\mathsf{B}\unicode[STIX]{x1D70E}(\cos \unicode[STIX]{x1D703}_{d}-\cos \unicode[STIX]{x1D703}_{s})\boldsymbol{m}.\end{eqnarray}$$

This implies that

(3.77) $$\begin{eqnarray}\mathsf{l}_{\mathsf{s}}^{-1}(v_{0,\unicode[STIX]{x1D70F}}-v_{w})+\unicode[STIX]{x2202}_{n_{S}}v_{0,\unicode[STIX]{x1D70F}}=-\mathsf{B}\unicode[STIX]{x1D70E}(\cos \unicode[STIX]{x1D703}_{d}-\cos \unicode[STIX]{x1D703}_{s})\unicode[STIX]{x1D6FF}_{CL}\boldsymbol{m}.\end{eqnarray}$$

Combining the above analysis, at the leading order, we have the boundary condition

(3.78) $$\begin{eqnarray}\left.\begin{array}{@{}c@{}}\boldsymbol{v}_{0}\boldsymbol{\cdot }\boldsymbol{n}_{S}=0,\\ V_{CL}=v_{0,\unicode[STIX]{x1D70F}}\boldsymbol{\cdot }\boldsymbol{m},\\ \mathsf{l}_{\mathsf{s}}^{-1}(v_{0,\unicode[STIX]{x1D70F}}-v_{w})+\unicode[STIX]{x2202}_{n}v_{0,\unicode[STIX]{x1D70F}}=-\mathsf{B}\unicode[STIX]{x1D70E}(\cos \unicode[STIX]{x1D703}_{d}-\cos \unicode[STIX]{x1D703}_{s})\unicode[STIX]{x1D6FF}_{CL}\boldsymbol{m}.\end{array}\right\}\end{eqnarray}$$

This is the sharp-interface version of the generalized Navier slip boundary conditions, which has been used by Gerbeau & Lelievre (Reference Gerbeau and Lelievre2009), Buscaglia & Ausas (Reference Buscaglia and Ausas2011) and Reusken, Xu & Zhang (Reference Reusken, Xu and Zhang2017).

Case II: $\mathsf{V}_{\mathsf{s}}=O(\unicode[STIX]{x1D700}^{-1})$ . In this case, we assume a larger relaxation number $\mathsf{V}_{\mathsf{s}}=\unicode[STIX]{x1D700}^{-1}\unicode[STIX]{x1D6FC}$ . The boundary condition (2.7) will be reduced to

(3.79) $$\begin{eqnarray}\left.\begin{array}{@{}c@{}}{\displaystyle \frac{\unicode[STIX]{x2202}\unicode[STIX]{x1D719}}{\unicode[STIX]{x2202}t}}+v_{\unicode[STIX]{x1D70F}}\unicode[STIX]{x2202}_{\unicode[STIX]{x1D70F}}\unicode[STIX]{x1D719}=-\unicode[STIX]{x1D700}^{-1}\unicode[STIX]{x1D6FC}{\mathcal{L}}(\unicode[STIX]{x1D719}),\\ \mathsf{l}_{\mathsf{s}}^{-1}(v_{\unicode[STIX]{x1D70F}}-v_{w})=-\unicode[STIX]{x2202}_{n}v_{\unicode[STIX]{x1D70F}}+\mathsf{B}{\mathcal{L}}(\unicode[STIX]{x1D719})\unicode[STIX]{x2202}_{\unicode[STIX]{x1D70F}}\unicode[STIX]{x1D719},\\ \unicode[STIX]{x1D735}\unicode[STIX]{x1D707}\boldsymbol{\cdot }\boldsymbol{n}_{S}=0,\quad \boldsymbol{v}\boldsymbol{\cdot }\boldsymbol{n}_{S}=0.\end{array}\right\}\end{eqnarray}$$

We repeat the analysis in Case I with this boundary condition. The only difference is the first equation of (3.79). Using the same inner expansions, the leading order of the first equation of (3.79) is given by

(3.80) $$\begin{eqnarray}\unicode[STIX]{x2202}_{\unicode[STIX]{x1D71A}}\hat{\unicode[STIX]{x1D719}}_{0}(\unicode[STIX]{x2202}_{t}x_{0}\boldsymbol{\cdot }\boldsymbol{m}-\hat{v}_{0,\unicode[STIX]{x1D70F}}\boldsymbol{\cdot }\boldsymbol{m})=\unicode[STIX]{x1D6FC}\hat{{\mathcal{L}}}_{0}.\end{eqnarray}$$

We multiply $\unicode[STIX]{x2202}_{\unicode[STIX]{x1D71A}}\hat{\unicode[STIX]{x1D719}}_{0}$ to both sides of the equation, and integrate the results in $(-\infty ,\infty )$ . Direct calculations give

(3.81) $$\begin{eqnarray}(\unicode[STIX]{x2202}_{t}x_{0}\boldsymbol{\cdot }\boldsymbol{m}-v_{0,\unicode[STIX]{x1D70F}}\boldsymbol{\cdot }\boldsymbol{m})\sin \unicode[STIX]{x1D703}_{d}=-\unicode[STIX]{x1D6FC}(\cos \unicode[STIX]{x1D703}_{d}-\cos \unicode[STIX]{x1D703}_{s}).\end{eqnarray}$$

This implies

(3.82) $$\begin{eqnarray}V_{CL}-v_{0,\unicode[STIX]{x1D70F}}\boldsymbol{\cdot }\boldsymbol{m}=-{\displaystyle \frac{\unicode[STIX]{x1D6FC}}{\sin \unicode[STIX]{x1D703}_{d}}}(\cos \unicode[STIX]{x1D703}_{d}-\cos \unicode[STIX]{x1D703}_{s}),\end{eqnarray}$$

where $V_{CL}$ is the velocity of the contact line in $\boldsymbol{m}$ direction.

Therefore, the boundary condition in this case is given by

(3.83) $$\begin{eqnarray}\left.\begin{array}{@{}c@{}}\mathsf{l}_{\mathsf{s}}^{-1}(v_{0,\unicode[STIX]{x1D70F}}-v_{w})+\unicode[STIX]{x2202}_{n}v_{0,\unicode[STIX]{x1D70F}}=-\mathsf{B}\unicode[STIX]{x1D70E}(\cos \unicode[STIX]{x1D703}_{d}-\cos \unicode[STIX]{x1D703}_{s})\unicode[STIX]{x1D6FF}_{CL}\boldsymbol{m},\quad \boldsymbol{v}_{0}\boldsymbol{\cdot }\boldsymbol{n}_{S}=0,\\ V_{CL}=v_{0,\unicode[STIX]{x1D70F}}\boldsymbol{\cdot }\boldsymbol{m}-{\displaystyle \frac{\unicode[STIX]{x1D6FC}}{\sin \unicode[STIX]{x1D703}_{d}}}(\cos \unicode[STIX]{x1D703}_{d}-\cos \unicode[STIX]{x1D703}_{s}).\end{array}\right\}\end{eqnarray}$$

Here the velocity of the contact line is different from the fluid velocity due to some extra chemical diffusion on the contact line (Jacqmin Reference Jacqmin2000; Yue & Feng Reference Yue and Feng2011b ).

Case III: $\mathsf{V}_{\mathsf{s}}=O(\unicode[STIX]{x1D700}^{-2})$ . In this case, we assume $\mathsf{V}_{\mathsf{s}}=\unicode[STIX]{x1D700}^{-2}\unicode[STIX]{x1D6FC}$ . The boundary condition (2.7) will be reduced to

(3.84) $$\begin{eqnarray}\left.\begin{array}{@{}c@{}}{\displaystyle \frac{\unicode[STIX]{x2202}\unicode[STIX]{x1D719}}{\unicode[STIX]{x2202}t}}+v_{\unicode[STIX]{x1D70F}}\unicode[STIX]{x2202}_{\unicode[STIX]{x1D70F}}\unicode[STIX]{x1D719}=-\unicode[STIX]{x1D700}^{-2}\unicode[STIX]{x1D6FC}{\mathcal{L}}(\unicode[STIX]{x1D719}),\\ \mathsf{l}_{\mathsf{s}}^{-1}(v_{\unicode[STIX]{x1D70F}}-v_{w})=-\unicode[STIX]{x2202}_{n}v_{\unicode[STIX]{x1D70F}}+\mathsf{B}{\mathcal{L}}(\unicode[STIX]{x1D719})\unicode[STIX]{x2202}_{\unicode[STIX]{x1D70F}}\unicode[STIX]{x1D719},\\ \unicode[STIX]{x1D735}\unicode[STIX]{x1D707}\boldsymbol{\cdot }\boldsymbol{n}_{S}=0,\quad \boldsymbol{v}\boldsymbol{\cdot }\boldsymbol{n}_{S}=0.\end{array}\right\}\end{eqnarray}$$

Once again, the only difference is the first equation of (3.84). For inner expansions, the leading order of the first equation of (3.79) is given by

(3.85) $$\begin{eqnarray}\unicode[STIX]{x1D6FC}\hat{{\mathcal{L}}}_{0}(\hat{\unicode[STIX]{x1D719}}_{0})=0.\end{eqnarray}$$

This leads to

(3.86) $$\begin{eqnarray}\cos \unicode[STIX]{x1D703}_{d}=\cos \unicode[STIX]{x1D703}_{s}.\end{eqnarray}$$

The equation implies that the dynamic contact angle is equal to the (static) Young’s angle. Thus, the boundary condition in this case is reduced to

(3.87) $$\begin{eqnarray}\left.\begin{array}{@{}c@{}}\mathsf{l}_{\mathsf{s}}^{-1}(v_{0,\unicode[STIX]{x1D70F}}-v_{w})+\unicode[STIX]{x2202}_{n_{S}}v_{0,\unicode[STIX]{x1D70F}}=0,\quad \boldsymbol{v}_{0}\boldsymbol{\cdot }\boldsymbol{n}_{S}=0,\\ \unicode[STIX]{x1D703}_{d}=\unicode[STIX]{x1D703}_{s}.\end{array}\right\}\end{eqnarray}$$

This boundary condition is used by Renardy et al. (Reference Renardy, Renardy and Li2001) and Spelt (Reference Spelt2005). We see that this condition is correct only for very large wall relaxation case.

3.3 Analysis for the case $\mathsf{L}_{\mathsf{d}}=O(\unicode[STIX]{x1D700}^{2})$ .

In previous analysis, we consider the case $\mathsf{L}_{\mathsf{d}}=O(\unicode[STIX]{x1D700})$ . Now we apply a similar analysis to the case $\mathsf{L}_{\mathsf{d}}=O(\unicode[STIX]{x1D700}^{2})$ , which balances perfectly the driving mechanism of the outer and inner layers and is physically more favourable as shown in Magaletti et al. (Reference Magaletti, Picano, Chinappi, Marino and Casciola2013). We assume that $\mathsf{L}_{\mathsf{d}}=\unicode[STIX]{x1D700}^{2}\tilde{l}_{d}$ , with $\tilde{l}_{d}$ a constant independent of $\unicode[STIX]{x1D700}$ .

The bulk equations. We do a similar analysis to that in § 3.1. We consider the outer expansions and inner expansions and match them together. It is easy to see that the scaling of $\mathsf{L}_{\mathsf{d}}$ affects only the first equation of (2.6). Therefore, the only difference is the expansion of this equation. In the case $\mathsf{L}_{\mathsf{d}}=\unicode[STIX]{x1D700}^{2}\tilde{l}_{d}$ , the leading order of the first equation of (2.6) in outer expansion and inner expansions are given respectively by

(3.88) $$\begin{eqnarray}{\displaystyle \frac{\unicode[STIX]{x2202}\unicode[STIX]{x1D719}_{0}^{\pm }}{\unicode[STIX]{x2202}t}}+\boldsymbol{v}_{0}^{\pm }\boldsymbol{\cdot }\unicode[STIX]{x1D735}\unicode[STIX]{x1D719}_{0}^{\pm }=0,\end{eqnarray}$$

and

(3.89) $$\begin{eqnarray}(\unicode[STIX]{x2202}_{t}d+\tilde{\boldsymbol{v}}_{0}\boldsymbol{\cdot }\boldsymbol{n})\unicode[STIX]{x2202}_{\unicode[STIX]{x1D709}}\tilde{\unicode[STIX]{x1D719}}_{0}=\tilde{l}_{d}\unicode[STIX]{x2202}_{\unicode[STIX]{x1D709}\unicode[STIX]{x1D709}}\tilde{\unicode[STIX]{x1D707}}_{0}.\end{eqnarray}$$

They are used in place of (3.5) and the first equation of (3.18), respectively.

The equation (3.88) is actually equivalent to (3.5), since the latter can be further simplified. More precisely, by (3.4) and (3.8), we easily see that $\unicode[STIX]{x1D707}_{0}^{\pm }$ are constants and thus $\unicode[STIX]{x0394}\unicode[STIX]{x1D707}_{0}^{\pm }=0$ . Then (3.5) is reduced to (3.88).

We can also show that (3.89) can be reduced to the first equation of (3.18). From the third equation of (3.18), we have

(3.90) $$\begin{eqnarray}\unicode[STIX]{x2202}_{\unicode[STIX]{x1D709}}(\tilde{\boldsymbol{v}}_{0}\boldsymbol{\cdot }\boldsymbol{n})=0.\end{eqnarray}$$

This implies that $\tilde{\boldsymbol{v}}_{0}\boldsymbol{\cdot }\boldsymbol{n}$ is independent of $\unicode[STIX]{x1D709}$ . We integrate the equation (3.89) with respect to $\unicode[STIX]{x1D709}$ and obtain

(3.91) $$\begin{eqnarray}(\unicode[STIX]{x2202}_{t}d+\tilde{\boldsymbol{v}}_{0}\boldsymbol{\cdot }\boldsymbol{n})\int _{-\infty }^{\infty }\unicode[STIX]{x2202}_{\unicode[STIX]{x1D709}}\tilde{\unicode[STIX]{x1D719}}_{0}\,\text{d}\unicode[STIX]{x1D709}=\tilde{l}_{d}\int _{-\infty }^{\infty }\unicode[STIX]{x2202}_{\unicode[STIX]{x1D709}\unicode[STIX]{x1D709}}\tilde{\unicode[STIX]{x1D707}}_{0}\,\text{d}\unicode[STIX]{x1D709}.\end{eqnarray}$$

Using the matching condition that $\lim _{\unicode[STIX]{x1D709}\rightarrow \pm \infty }\tilde{\unicode[STIX]{x1D719}}_{0}=\unicode[STIX]{x1D719}_{0}^{\pm }$ , and $\lim _{\unicode[STIX]{x1D709}\rightarrow \pm \infty }\unicode[STIX]{x2202}_{\unicode[STIX]{x1D709}}\tilde{\unicode[STIX]{x1D707}}_{0}=0$ , we have

(3.92) $$\begin{eqnarray}(\unicode[STIX]{x2202}_{t}d+\tilde{\boldsymbol{v}}_{0}\boldsymbol{\cdot }\boldsymbol{n})(\unicode[STIX]{x1D719}_{0}^{+}-\unicode[STIX]{x1D719}_{0}^{-})=0.\end{eqnarray}$$

Thus, we have

(3.93) $$\begin{eqnarray}\unicode[STIX]{x2202}_{t}d_{0}+\tilde{\boldsymbol{v}}_{0}\boldsymbol{\cdot }\boldsymbol{n}=0.\end{eqnarray}$$

Substituting this back into (3.89), we derive the relation that $\unicode[STIX]{x2202}_{\unicode[STIX]{x1D709}\unicode[STIX]{x1D709}}\tilde{\unicode[STIX]{x1D707}}_{0}=0$ , which is the same as the first equation of (3.18).

Therefore, the leading order of the CHNS system in the case $\mathsf{L}_{\mathsf{d}}=O(\unicode[STIX]{x1D700}^{2})$ is the same as that for $\mathsf{L}_{\mathsf{d}}=O(\unicode[STIX]{x1D700})$ . By similar arguments to those in § 3.1, we can derive all the equations in (3.46). This implies that the case $\mathsf{L}_{\mathsf{d}}=O(\unicode[STIX]{x1D700}^{2})$ corresponds to the same sharp-interface limit as in the previous case ( $\mathsf{L}_{\mathsf{d}}=O(\unicode[STIX]{x1D700})$ ). This is already known in the literature (Lowengrub & Truskinovsky Reference Lowengrub and Truskinovsky1998). But this does not mean the two cases have the same appearance in dynamics. The scaling of $\mathsf{L}_{\mathsf{d}}$ affects the relaxation of the transition layer to stationary profile, which might be important in some cases, e.g. in capillary waves or droplet coalescence (Magaletti et al. Reference Magaletti, Picano, Chinappi, Marino and Casciola2013). In the following, we show that the scaling of $\mathsf{L}_{\mathsf{d}}$ also affects the boundary condition of the moving contact lines.

The boundary conditions. We then consider the sharp-interface limit of the boundary conditions. A similar analysis to that in § 3.2 can be done. It turns out that the new choice $\mathsf{L}_{\mathsf{d}}=O(\unicode[STIX]{x1D700}^{2})$ affects only the first equation of (3.58) in inner expansions, which is replaced by

(3.94) $$\begin{eqnarray}\displaystyle \tilde{l}_{d}(\unicode[STIX]{x2202}_{\unicode[STIX]{x1D701}\unicode[STIX]{x1D701}}+\unicode[STIX]{x2202}_{\unicode[STIX]{x1D71A}\unicode[STIX]{x1D71A}})\hat{\unicode[STIX]{x1D707}}_{0} & = & \displaystyle -\boldsymbol{m}\boldsymbol{\cdot }\unicode[STIX]{x2202}_{t}x_{0}\unicode[STIX]{x2202}_{\unicode[STIX]{x1D71A}}\hat{\unicode[STIX]{x1D719}}_{0}+\hat{\boldsymbol{v}}_{0}\boldsymbol{\cdot }\boldsymbol{n}_{S}\unicode[STIX]{x2202}_{\unicode[STIX]{x1D701}}\hat{\unicode[STIX]{x1D719}}_{0}+\hat{\boldsymbol{v}}_{0}\boldsymbol{\cdot }\boldsymbol{m}\unicode[STIX]{x2202}_{\unicode[STIX]{x1D71A}}\hat{\unicode[STIX]{x1D719}}_{0}\nonumber\\ \displaystyle & = & \displaystyle (-\boldsymbol{m}\boldsymbol{\cdot }\unicode[STIX]{x2202}_{t}x_{0}+\hat{\boldsymbol{v}}_{0}\boldsymbol{\cdot }\boldsymbol{m})\unicode[STIX]{x2202}_{\unicode[STIX]{x1D71A}}\hat{\unicode[STIX]{x1D719}}_{0}+\hat{\boldsymbol{v}}_{0}\boldsymbol{\cdot }\boldsymbol{n}_{S}\unicode[STIX]{x2202}_{\unicode[STIX]{x1D701}}\hat{\unicode[STIX]{x1D719}}_{0}.\end{eqnarray}$$

Instead of (3.58), we have

(3.95) $$\begin{eqnarray}\left.\begin{array}{@{}c@{}}\tilde{l}_{d}(\unicode[STIX]{x2202}_{\unicode[STIX]{x1D701}\unicode[STIX]{x1D701}}+\unicode[STIX]{x2202}_{\unicode[STIX]{x1D71A}\unicode[STIX]{x1D71A}})\hat{\unicode[STIX]{x1D707}}_{0}=(-\boldsymbol{m}\boldsymbol{\cdot }\unicode[STIX]{x2202}_{t}x_{0}+\hat{\boldsymbol{v}}_{0}\boldsymbol{\cdot }\boldsymbol{m})\unicode[STIX]{x2202}_{\unicode[STIX]{x1D71A}}\hat{\unicode[STIX]{x1D719}}_{0}+\hat{\boldsymbol{v}}_{0}\boldsymbol{\cdot }\boldsymbol{n}_{S}\unicode[STIX]{x2202}_{\unicode[STIX]{x1D701}}\hat{\unicode[STIX]{x1D719}}_{0},\\ \mathsf{B}\hat{\unicode[STIX]{x1D707}}_{0}(\unicode[STIX]{x2202}_{\unicode[STIX]{x1D701}}\hat{\unicode[STIX]{x1D719}}_{0}\boldsymbol{n}_{S}+\unicode[STIX]{x2202}_{\unicode[STIX]{x1D71A}}\hat{\unicode[STIX]{x1D719}}_{0}\boldsymbol{m})+(\unicode[STIX]{x2202}_{\unicode[STIX]{x1D701}\unicode[STIX]{x1D701}}+\unicode[STIX]{x2202}_{\unicode[STIX]{x1D71A}\unicode[STIX]{x1D71A}})\hat{\boldsymbol{v}}_{0}=0,\\ \boldsymbol{n}_{S}\boldsymbol{\cdot }\unicode[STIX]{x2202}_{\unicode[STIX]{x1D701}}\hat{\boldsymbol{v}}_{0}+\boldsymbol{m}\boldsymbol{\cdot }\unicode[STIX]{x2202}_{\unicode[STIX]{x1D71A}}\hat{\boldsymbol{v}}_{0}=0.\end{array}\right\}\end{eqnarray}$$

As in the previous subsection, we study several choices for $\mathsf{V}_{\mathsf{s}}=O(\unicode[STIX]{x1D700}^{b})$ with $b=0,-1,-2$ and derive the corresponding boundary conditions respectively.

We first consider the case $\mathsf{V}_{\mathsf{s}}=O(1)$ . Then (3.95) will be combined with (3.59). From (3.59), we know that $-\boldsymbol{m}\boldsymbol{\cdot }\unicode[STIX]{x2202}_{t}x_{0}+\hat{\boldsymbol{v}}_{0}\boldsymbol{\cdot }\boldsymbol{m}=0$ and $\hat{\boldsymbol{v}}_{0}\boldsymbol{\cdot }\boldsymbol{n}_{S}=0$ on the boundary $\unicode[STIX]{x1D6E4}_{S}$ . If we assume that the velocity does not have a fast change in the inner layer, the relations can be extended to inside the domain. Then the first equation of (3.95) will reduce to $(\unicode[STIX]{x2202}_{\unicode[STIX]{x1D701}\unicode[STIX]{x1D701}}+\unicode[STIX]{x2202}_{\unicode[STIX]{x1D71A}\unicode[STIX]{x1D71A}})\hat{\unicode[STIX]{x1D707}}_{0}=0$ , the same equation as the first equation of (3.58). Actually, it is easy to check that $\hat{\unicode[STIX]{x1D707}}_{0}=0$ and $\hat{\boldsymbol{v}}_{0}=\boldsymbol{v}_{0}(x_{0})$ (satisfying $\boldsymbol{v}_{0}(x_{0})\boldsymbol{\cdot }\boldsymbol{n}_{S}=0$ and $\boldsymbol{m}\boldsymbol{\cdot }\unicode[STIX]{x2202}_{t}x_{0}=\boldsymbol{v}_{0}(x_{0})\boldsymbol{\cdot }\boldsymbol{m}$ ) are also the solution of (3.95) and (3.59). Similar arguments will lead to the boundary condition (3.78). Therefore, when $\mathsf{V}_{\mathsf{s}}=O(1)$ , the boundary condition in the sharp limit is also given by (3.78), the same as in the case $\mathsf{L}_{\mathsf{d}}=O(\unicode[STIX]{x1D700})$ .

We then analyse the case $\mathsf{V}_{\mathsf{s}}=O(\unicode[STIX]{x1D700}^{-1})$ . By the equation (3.80), the equation (3.95) will be combined with the following boundary condition on $\unicode[STIX]{x1D6E4}_{S}$ :

(3.96) $$\begin{eqnarray}\left.\begin{array}{@{}c@{}}\unicode[STIX]{x2202}_{\unicode[STIX]{x1D71A}}\hat{\unicode[STIX]{x1D719}}_{0}(\unicode[STIX]{x2202}_{t}x_{0}\boldsymbol{\cdot }\boldsymbol{m}-\hat{v}_{0,\unicode[STIX]{x1D70F}}\boldsymbol{\cdot }\boldsymbol{m})=\unicode[STIX]{x1D6FC}\hat{{\mathcal{L}}}_{0},\\ \unicode[STIX]{x1D700}\mathsf{l}_{\mathsf{s}}^{-1}(\hat{v}_{0,\unicode[STIX]{x1D70F}}-v_{w})+\unicode[STIX]{x1D700}\unicode[STIX]{x2202}_{n}\hat{v}_{0,\unicode[STIX]{x1D70F}}+\unicode[STIX]{x2202}_{\unicode[STIX]{x1D701}}\hat{v}_{0,\unicode[STIX]{x1D70F}}=\mathsf{B}\hat{{\mathcal{L}}}_{0}\unicode[STIX]{x2202}_{\unicode[STIX]{x1D71A}}\hat{\unicode[STIX]{x1D719}}_{0}\boldsymbol{m},\\ \unicode[STIX]{x2202}_{\unicode[STIX]{x1D701}}\hat{\unicode[STIX]{x1D707}}_{0}=0,\quad \hat{\boldsymbol{v}}_{0}\boldsymbol{\cdot }\boldsymbol{n}_{S}=0.\end{array}\right\}\end{eqnarray}$$

The equations (3.95)–(3.96) compose a coupled quasi-linear system. In general, it is difficult to obtain the explicit solution. If we still assume $\hat{\unicode[STIX]{x1D707}}_{0}=0$ and $\hat{\boldsymbol{v}}_{0}=\boldsymbol{v}_{0}(x_{0})$ (satisfying $\boldsymbol{v}_{0}(x_{0})\boldsymbol{\cdot }\boldsymbol{n}_{S}=0$ and $\boldsymbol{m}\boldsymbol{\cdot }\unicode[STIX]{x2202}_{t}x_{0}=\boldsymbol{v}_{0}(x_{0})\boldsymbol{\cdot }\boldsymbol{m}$ ) are the solution of (3.95) and (3.96), by the first equation of (3.96), we could derive $\unicode[STIX]{x1D6FC}\hat{{\mathcal{L}}}_{0}=0$ . This will lead to

(3.97) $$\begin{eqnarray}\unicode[STIX]{x1D703}_{d}=\unicode[STIX]{x1D703}_{s}.\end{eqnarray}$$

In addition, similar arguments to those in the previous subsection lead to

(3.98) $$\begin{eqnarray}\left.\begin{array}{@{}c@{}}\mathsf{l}_{\mathsf{s}}^{-1}(v_{0,\unicode[STIX]{x1D70F}}-v_{w})+\unicode[STIX]{x2202}_{n}v_{0,\unicode[STIX]{x1D70F}}=-\mathsf{B}\unicode[STIX]{x1D70E}(\cos \unicode[STIX]{x1D703}_{d}-\cos \unicode[STIX]{x1D703}_{s})\unicode[STIX]{x1D6FF}_{CL}\boldsymbol{m},\quad \boldsymbol{v}_{0}\boldsymbol{\cdot }\boldsymbol{n}_{S}=0,\\ V_{CL}=v_{0,\unicode[STIX]{x1D70F}}\boldsymbol{\cdot }\boldsymbol{m}.\end{array}\right\}\end{eqnarray}$$

Combining them together, we have

(3.99) $$\begin{eqnarray}\left.\begin{array}{@{}c@{}}\mathsf{l}_{\mathsf{s}}^{-1}(v_{0,\unicode[STIX]{x1D70F}}-v_{w})+\unicode[STIX]{x2202}_{n}v_{0,\unicode[STIX]{x1D70F}}=0,\quad \boldsymbol{v}_{0}\boldsymbol{\cdot }\boldsymbol{n}_{S}=0,\\ V_{CL}=v_{0,\unicode[STIX]{x1D70F}}\boldsymbol{\cdot }\boldsymbol{m},\quad \unicode[STIX]{x1D703}_{d}=\unicode[STIX]{x1D703}_{s}.\end{array}\right\}\end{eqnarray}$$

Here, in addition to the condition on the dynamic contact angle that $\unicode[STIX]{x1D703}_{d}=\unicode[STIX]{x1D703}_{s}$ , we have an extra restriction that the velocity of the contact line is equal to the fluid velocity. Both conditions seem consistent with physical intuition, however, only one is needed to make the resulting system well defined mathematically.

The existence of two constraints for the contact line is due to the fact that we have assumed $\hat{\unicode[STIX]{x1D707}}_{0}=0$ and $\hat{\boldsymbol{v}}_{0}=\boldsymbol{v}_{0}(x_{0})$ , which may not be the exact solution of (3.95)–(3.96). Without any assumption on the exact solution (3.95)–(3.96), we can still repeat the similar procedure for the second equation of (3.96) as done in the $\mathsf{L}_{\mathsf{d}}=O(\unicode[STIX]{x1D700})$ case, to get

(3.100) $$\begin{eqnarray}\mathsf{l}_{\mathsf{s}}^{-1}(v_{0,\unicode[STIX]{x1D70F}}-v_{w})+\unicode[STIX]{x2202}_{n_{S}}v_{0,\unicode[STIX]{x1D70F}}=-\mathsf{B}(\unicode[STIX]{x1D70E}^{\prime }\cos \unicode[STIX]{x1D703}_{d}-\unicode[STIX]{x1D70E}\cos \unicode[STIX]{x1D703}_{s})\unicode[STIX]{x1D6FF}_{CL}\boldsymbol{m},\end{eqnarray}$$

where $\unicode[STIX]{x1D70E}^{\prime }=-\int _{-\infty }^{\infty }\unicode[STIX]{x2202}_{\unicode[STIX]{x1D701}}\hat{\unicode[STIX]{x1D719}}_{0}(0,\unicode[STIX]{x1D70C})\unicode[STIX]{x2202}_{\unicode[STIX]{x1D70C}}\hat{\unicode[STIX]{x1D719}}_{0}(0,\unicode[STIX]{x1D70C})\,\text{d}\unicode[STIX]{x1D70C}/\cos \unicode[STIX]{x1D703}_{d}$ . Here $\unicode[STIX]{x1D70E}^{\prime }\cos \unicode[STIX]{x1D703}_{d}$ is the effective force (in the $-\boldsymbol{m}$ direction) of the two-phase interface on the contact line. Notice that we do not get the exact value of $\unicode[STIX]{x1D70E}^{\prime }$ . It is written formally in $\hat{\unicode[STIX]{x1D719}}_{0}(\unicode[STIX]{x1D701},\unicode[STIX]{x1D70C})$ which is part of the solution of the quasi-linear system (3.95)–(3.96). Using a similar procedure to the first equation of (3.96), we get

(3.101) $$\begin{eqnarray}V_{CL}-v_{0,\unicode[STIX]{x1D70F}}\boldsymbol{\cdot }\boldsymbol{m}=-{\displaystyle \frac{\unicode[STIX]{x1D6FC}}{\unicode[STIX]{x1D70E}^{\prime \prime }}}(\unicode[STIX]{x1D70E}^{\prime }\cos \unicode[STIX]{x1D703}_{d}-\unicode[STIX]{x1D70E}\cos \unicode[STIX]{x1D703}_{s}),\end{eqnarray}$$

where $\unicode[STIX]{x1D70E}^{\prime \prime }=\int _{-\infty }^{\infty }(\unicode[STIX]{x2202}_{\unicode[STIX]{x1D70C}}\hat{\unicode[STIX]{x1D719}}_{0}(0,\unicode[STIX]{x1D70C}))^{2}\,\text{d}\unicode[STIX]{x1D70C}$ . In the derivation of this condition, we have assumed that $\hat{v}_{0,\unicode[STIX]{x1D70F}}$ is independent of the fast variable.

The boundary conditions (3.100)–(3.101) include two parameters $\unicode[STIX]{x1D70E}^{\prime }$ and $\unicode[STIX]{x1D70E}^{\prime \prime }$ . They are written in the form of $\hat{\unicode[STIX]{x1D719}}_{0}(\unicode[STIX]{x1D701},\unicode[STIX]{x1D70C})$ . In general, it is difficult to calculate them explicitly. One might need to solve the system (3.95)–(3.96) numerically to compute $\hat{\unicode[STIX]{x1D719}}_{0}(\unicode[STIX]{x1D701},\unicode[STIX]{x1D70C})$ . However, by doing an energy estimation for the system (3.95)–(3.96), we find that $\hat{\unicode[STIX]{x1D719}}_{0}$ is close to the equilibrium state. In this sense, both $\unicode[STIX]{x1D703}_{d}=\unicode[STIX]{x1D703}_{s}$ and $V_{CL}=\boldsymbol{v}_{0,\unicode[STIX]{x1D70F}}\boldsymbol{\cdot }\boldsymbol{m}$ are approximately satisfied.

Finally, we consider the case $\mathsf{V}_{\mathsf{s}}=O(\unicode[STIX]{x1D700}^{-2})$ . Noticing the equation (3.85), the equation (3.95) will be combined with the following boundary condition on $\unicode[STIX]{x1D6E4}_{S}$ :

(3.102) $$\begin{eqnarray}\left.\begin{array}{@{}c@{}}\hat{{\mathcal{L}}}_{0}(\hat{\unicode[STIX]{x1D719}}_{0})=0,\\ \unicode[STIX]{x1D700}\mathsf{l}_{\mathsf{s}}^{-1}(\hat{v}_{0,\unicode[STIX]{x1D70F}}-v_{w})+\unicode[STIX]{x1D700}\unicode[STIX]{x2202}_{n}\hat{v}_{0,\unicode[STIX]{x1D70F}}+\unicode[STIX]{x2202}_{\unicode[STIX]{x1D701}}\hat{v}_{0,\unicode[STIX]{x1D70F}}=0,\\ \unicode[STIX]{x2202}_{\unicode[STIX]{x1D701}}\hat{\unicode[STIX]{x1D707}}_{0}=0,\quad \hat{\boldsymbol{v}}_{0}\boldsymbol{\cdot }\boldsymbol{n}_{S}=0.\end{array}\right\}\end{eqnarray}$$

If we assume the interface near the contact point is flat and not dilated, i.e. $\hat{\unicode[STIX]{x1D719}}_{0}(\unicode[STIX]{x1D701},\unicode[STIX]{x1D71A})=\hat{\unicode[STIX]{x1D713}}(\unicode[STIX]{x1D71A}\sin \unicode[STIX]{x1D703}_{d}-\unicode[STIX]{x1D701}\cos \unicode[STIX]{x1D703}_{d})$ , then the first equation in (3.102) becomes a first order ordinary differential equation. By solving it and using the matching conditions, we get

(3.103) $$\begin{eqnarray}\hat{\unicode[STIX]{x1D719}}_{0}(0,\unicode[STIX]{x1D70C})=\tanh (\unicode[STIX]{x1D70C}\sin \unicode[STIX]{x1D703}_{d}/\sqrt{2}),\end{eqnarray}$$

and

(3.104) $$\begin{eqnarray}\unicode[STIX]{x1D703}_{d}=\unicode[STIX]{x1D703}_{s}.\end{eqnarray}$$

Similarly to the $\mathsf{L}_{\mathsf{d}}=O(\unicode[STIX]{x1D700})$ case, we are led to the boundary condition (3.87).

3.4 Summary of the analysis results

We summarize the main results in this section. For both choices that the mobility parameter $\mathsf{L}_{\mathsf{d}}$ is of order $O(\unicode[STIX]{x1D700})$ and $O(\unicode[STIX]{x1D700}^{2}),$ the sharp-interface limit of the CHNS system (2.6) is the standard two-phase flow equation

(3.105) $$\begin{eqnarray}\left.\begin{array}{@{}c@{}}Re\left({\displaystyle \frac{\unicode[STIX]{x2202}\boldsymbol{v}}{\unicode[STIX]{x2202}t}}+(\boldsymbol{v}\boldsymbol{\cdot }\unicode[STIX]{x1D735})\boldsymbol{v}\right)=\boldsymbol{F}-\unicode[STIX]{x1D735}p+\unicode[STIX]{x0394}\boldsymbol{v}\quad \text{in }\unicode[STIX]{x1D6FA}^{\pm },\\ \unicode[STIX]{x1D735}\boldsymbol{\cdot }\boldsymbol{v}=0,\quad \text{in }\unicode[STIX]{x1D6FA}^{\pm },\\ \text{}[\boldsymbol{v}]=0,\quad \text{on }\unicode[STIX]{x1D6E4},\\ \text{}[-p_{0}\boldsymbol{n}+(\boldsymbol{n}\boldsymbol{\cdot }\unicode[STIX]{x1D735})\boldsymbol{v}_{0}]=\mathsf{B}\unicode[STIX]{x1D70E}\unicode[STIX]{x1D705}\boldsymbol{n},\quad \text{on }\unicode[STIX]{x1D6E4},\\ V_{n}=\boldsymbol{v}\boldsymbol{\cdot }\boldsymbol{n},\quad \text{on }\unicode[STIX]{x1D6E4}.\end{array}\right\}\end{eqnarray}$$

where $V_{n}$ is the normal velocity of the interface of the two-phase flow.

Both the parameters $\mathsf{L}_{\mathsf{d}}$ and $\mathsf{V}_{\mathsf{s}}$ affect the sharp-interface limits for the GNBC:

Case I. When $\mathsf{V}_{\mathsf{s}}=O(1)$ , whether $\mathsf{L}_{\mathsf{d}}=O(\unicode[STIX]{x1D700})$ or $\mathsf{L}_{\mathsf{d}}=O(\unicode[STIX]{x1D700}^{2})$ , the sharp-interface limit of the boundary condition is

(3.106) $$\begin{eqnarray}\left.\begin{array}{@{}c@{}}\mathsf{l}_{\mathsf{s}}^{-1}(v_{\unicode[STIX]{x1D70F}}-v_{w})+\unicode[STIX]{x2202}_{n}v_{\unicode[STIX]{x1D70F}}=-\mathsf{B}\unicode[STIX]{x1D70E}(\cos \unicode[STIX]{x1D703}_{d}-\cos \unicode[STIX]{x1D703}_{s})\unicode[STIX]{x1D6FF}_{CL}\boldsymbol{m},\quad \boldsymbol{v}\boldsymbol{\cdot }\boldsymbol{n}_{S}=0,\\ V_{CL}=v_{\unicode[STIX]{x1D70F}}\boldsymbol{\cdot }\boldsymbol{m}.\end{array}\right\}\end{eqnarray}$$

The first equation of (3.106) is the sharp-interface version of the generalized Navier slip boundary condition. It can be understood in the following way (Qian et al. Reference Qian, Wang and Sheng2003):

(3.107) $$\begin{eqnarray}\mathsf{l}_{\mathsf{s}}^{-1}(v_{\unicode[STIX]{x1D70F}}-v_{w})=-\unicode[STIX]{x2202}_{n}v_{\unicode[STIX]{x1D70F}}-{\displaystyle \frac{1}{\unicode[STIX]{x1D702}}}\unicode[STIX]{x1D70E}_{Y},\end{eqnarray}$$

where $\unicode[STIX]{x1D70E}_{Y}$ is the unbalanced Young’s stress, satisfying

(3.108) $$\begin{eqnarray}{\displaystyle \frac{1}{\unicode[STIX]{x1D702}}}\int _{interface}\unicode[STIX]{x1D70E}_{Y}=\unicode[STIX]{x1D70E}(\cos \unicode[STIX]{x1D703}_{d}-\cos \unicode[STIX]{x1D703}_{s})\boldsymbol{m}.\end{eqnarray}$$

As shown in the molecular dynamics simulations by Qian et al. (Reference Qian, Wang and Sheng2003), the unbalanced Young force might lead to near-complete slip of the fluid in the vicinity of the contact line.

The second equation of (3.106) implies that the velocity of the contact line is equal to the fluid velocity.

Case II. When $\mathsf{V}_{\mathsf{s}}=O(\unicode[STIX]{x1D700}^{-1})$ and $\mathsf{L}_{\mathsf{d}}=O(\unicode[STIX]{x1D700})$ , the sharp-interface limit of the GNBC is

(3.109) $$\begin{eqnarray}\left.\begin{array}{@{}c@{}}\mathsf{l}_{\mathsf{s}}^{-1}(v_{\unicode[STIX]{x1D70F}}-v_{w})+\unicode[STIX]{x2202}_{n}v_{\unicode[STIX]{x1D70F}}=-\mathsf{B}\unicode[STIX]{x1D70E}(\cos \unicode[STIX]{x1D703}_{d}-\cos \unicode[STIX]{x1D703}_{s})\unicode[STIX]{x1D6FF}_{CL}\boldsymbol{m},\quad \boldsymbol{v}\boldsymbol{\cdot }\boldsymbol{n}_{S}=0,\\ V_{CL}=v_{\unicode[STIX]{x1D70F}}\boldsymbol{\cdot }\boldsymbol{m}-{\displaystyle \frac{\unicode[STIX]{x1D6FC}}{\sin \unicode[STIX]{x1D703}_{d}}}(\cos \unicode[STIX]{x1D703}_{d}-\cos \unicode[STIX]{x1D703}_{s}),\end{array}\right\}\end{eqnarray}$$

where $\unicode[STIX]{x1D6FC}=\unicode[STIX]{x1D700}\mathsf{V}_{\mathsf{s}}$ is a constant. The first equation in (3.109) is the same as the previous case. The second equation in (3.109) implies that the motion of the contact line is not only determined by the fluid velocity, but also by the chemical diffusion on the boundary.

From the second equation of (3.109), when $\unicode[STIX]{x1D703}_{d}$ does not change much from $\unicode[STIX]{x1D703}_{s}$ , we have

(3.110) $$\begin{eqnarray}\cos \unicode[STIX]{x1D703}_{d}-\cos \unicode[STIX]{x1D703}_{s}\approx -(\sin \unicode[STIX]{x1D703}_{d})(\unicode[STIX]{x1D703}_{d}-\unicode[STIX]{x1D703}_{s})+\text{higher-order terms}.\end{eqnarray}$$

Then the second equation implies that $V_{CL}=v_{\unicode[STIX]{x1D70F}}\boldsymbol{\cdot }\boldsymbol{m}+\unicode[STIX]{x1D6FC}(\unicode[STIX]{x1D703}_{d}-\unicode[STIX]{x1D703}_{s}).$ This implies that the slip velocity $V_{CL}^{slip}\propto (\unicode[STIX]{x1D703}_{d}-\unicode[STIX]{x1D703}_{s})$ , which is similar to the boundary condition derived by Ren & E (Reference Ren and E2007).

Case III. When $\mathsf{V}_{\mathsf{s}}=O(\unicode[STIX]{x1D700}^{-2})$ , whether $\mathsf{L}_{\mathsf{d}}=O(\unicode[STIX]{x1D700})$ or $\mathsf{L}_{\mathsf{d}}=O(\unicode[STIX]{x1D700}^{2})$ , the sharp-interface limit of the boundary condition is

(3.111) $$\begin{eqnarray}\left.\begin{array}{@{}c@{}}\mathsf{l}_{\mathsf{s}}^{-1}(v_{\unicode[STIX]{x1D70F}}-v_{w})+\unicode[STIX]{x2202}_{n}v_{\unicode[STIX]{x1D70F}}=0,\quad \boldsymbol{v}\boldsymbol{\cdot }\boldsymbol{n}_{S}=0,\\ \unicode[STIX]{x1D703}_{d}=\unicode[STIX]{x1D703}_{s}.\end{array}\right\}\end{eqnarray}$$

The boundary condition is different from the previous two cases. Here the standard Navier slip boundary condition is used on the solid boundary and the dynamic contact angle is equal to the Young’s angle. This boundary condition has been used by Spelt (Reference Spelt2005).

Case IV. When $\mathsf{V}_{\mathsf{s}}=O(\unicode[STIX]{x1D700}^{-1})$ , $\mathsf{L}_{\mathsf{d}}=O(\unicode[STIX]{x1D700}^{2})$ , the sharp-interface limit of the boundary condition is

(3.112) $$\begin{eqnarray}\left.\begin{array}{@{}c@{}}\mathsf{l}_{\mathsf{s}}^{-1}(v_{\unicode[STIX]{x1D70F}}-v_{w})+\unicode[STIX]{x2202}_{n}v_{\unicode[STIX]{x1D70F}}=-\mathsf{B}(\unicode[STIX]{x1D70E}^{\prime }\cos \unicode[STIX]{x1D703}_{d}-\unicode[STIX]{x1D70E}\cos \unicode[STIX]{x1D703}_{s})\unicode[STIX]{x1D6FF}_{CL}\boldsymbol{m},\quad \boldsymbol{v}\boldsymbol{\cdot }\boldsymbol{n}_{S}=0,\\ V_{CL}=v_{\unicode[STIX]{x1D70F}}\boldsymbol{\cdot }\boldsymbol{m}-{\displaystyle \frac{\unicode[STIX]{x1D6FC}}{\unicode[STIX]{x1D70E}^{\prime \prime }}}(\unicode[STIX]{x1D70E}^{\prime }\cos \unicode[STIX]{x1D703}_{d}-\unicode[STIX]{x1D70E}\cos \unicode[STIX]{x1D703}_{s}),\end{array}\right\}\end{eqnarray}$$

where $\unicode[STIX]{x1D70E}^{\prime }$ and $\unicode[STIX]{x1D70E}^{\prime \prime }$ are two parameters that may depend on a solution of a quasi-linear system (3.95)–(3.96). In general, it is hard to write down explicit formulas for them.

From the above analysis, we have shown the sharp-interface limits (at leading order) for the CHNS equation with the GNBC. For different choices of the mobility number and the relaxation parameter, we obtain different boundary conditions for moving contact lines. In applications, one could choose the parameters according to one’s own purpose. We would like to remark that we did not consider the effects of different scalings of slip length $l_{s}$ with respect to $\unicode[STIX]{x1D700}$ . We take $l_{s}$ as a constant (not going to zero as $\unicode[STIX]{x1D700}$ goes to zero). The reason is that the slip length is important to avoid the contact line paradox. It is also found that the slip velocity has a power-law profile in a fairly large region near the contact line, as shown in Qian et al. (Reference Qian, Wang and Sheng2004). Nevertheless, the effect of the slip length on the sharp-interface limit is an important issue and deserves further study. Moreover, the analysis does not show the convergence rate of the sharp-interface limits, which might be important in real applications. In the next section, we will do numerical simulations for the various choices of the mobility parameter and the relaxation parameter to verify the analytical results and investigate the convergence rates in these cases.