Numerical Treatment of Stokes Solvent Flow and Solute–Solvent Interfacial Dynamics for Nonpolar Molecules

Abstract

We design and implement numerical methods for the incompressible Stokes solvent flow and solute–solvent interface motion for nonpolar molecules in aqueous solvent. The balance of viscous force, surface tension, and van der Waals type dispersive force leads to a traction boundary condition on the solute–solvent interface. To allow the change of solute volume, we design special numerical boundary conditions on the boundary of a computational domain through a consistency condition. We use a finite difference ghost fluid scheme to discretize the Stokes equation with such boundary conditions. The method is tested to have a second-order accuracy. We combine this ghost fluid method with the level-set method to simulate the motion of the solute–solvent interface that is governed by the solvent fluid velocity. Numerical examples show that our method can predict accurately the blow up time for a test example of curvature flow and reproduce the polymodal (e.g., dry and wet) states of hydration of some simple model molecular systems.

Appendix

In the appendix, we provide details of the ghost fluid discretization on the interface. First of all, with the notations $\mathbf{n}=(n_1,n_2)$ and ${\varvec{\tau }}=(-n_2,n_1)$, the traction boundary conditions (2.4) read

$$\begin{aligned} {\left\{ \begin{array}{ll} 2n_1^2u_x+2n_1n_2(u_y+v_x)+2n_2^2v_y-p=f_{\perp },\\ -2n_1n_2u_x+(n_1^2-n_2^2)(u_y+v_x)+2n_1n_2v_y=f_{\parallel }, \end{array}\right. } \end{aligned}$$

where $f_{\perp }=\mathbf{f}\cdot \mathbf{n}$ and $f_{\parallel }=\mathbf{f}\cdot {\varvec{\tau }}$. Some straightforward algebraic calculations together with the incompressibility condition (2.2) lead to

$$\begin{aligned}&2u_x-(n_1^2-n_2^2)p=(n_1^2-n_2^2)f_{\perp }-2n_1n_2f_{\parallel }, \end{aligned}$$

(6.1)

$$\begin{aligned}&2(n_2^2-n_1^2)v_y+2n_1n_2(u_y+v_x)-p=f_{\perp }. \end{aligned}$$

(6.2)

For any ghost velocity point $\mathbf{x}$, we find a point $\mathbf{x}^*\in \Gamma$, such that $|\mathbf{x}-\mathbf{x}^*|=dist(\mathbf{x},\Gamma )$. We call $\mathbf{x}^*$ a projection point of $\mathbf{x}$ onto $\Gamma$. We then discretize Eq. (6.1) at each projection point $\mathbf{x}^*_{i-1/2,j}$ corresponding to each ghost velocity point $\mathbf{x}_{i-1/2,j}$ of u. Similarly, we discretize Eq. (6.2) at each projection point $\mathbf{x}^*_{i,j-1/2}$ corresponding to each ghost velocity point $\mathbf{x}_{i,j-1/2}$ of v.

To obtain a second-order convergence scheme for u, v, and p up to $\Gamma$, we design a third-order discretization of $u_x$, $u_y$, $v_x$, $v_y$, and p in the fluid region. This requires 10 stencil points for u, v, and 6 stencil points for p. We denote by $S(u,\mathbf{x}^*,r)$, $S(v,\mathbf{x}^*,r)$, and $S(p,\mathbf{x}^*,r)$ respectively, the sets of u, v and p stencil points for discretizing $\nabla u$, $\nabla v$ and p with an order r at the projection point $\mathbf{x}^*_{i,j-1/2}$. Then

$$\begin{aligned} |S(u,\mathbf{x}^*,r)|=\frac{(r+1)(r+2)}{2},\ |S(v,\mathbf{x}^*,r)|=\frac{(r+1)(r+2)}{2},\ \text { and } |S(p,\mathbf{x}^*,r)|=\frac{r(r+1)}{2}. \end{aligned}$$

In choosing these stencil points, we follow three criteria: (1) Each of these stencil points needs to be either a ghost point or a fluid point; (2) The stencil points need to include $\mathbf{x}$, that is $\mathbf{x}\!\in \! S(u,\mathbf{x}^*,r)\!\cup \! S(v,\mathbf{x}^*,r)\!\cup \! S(p,\mathbf{x}^*,r)$; (3) All $S(u,\mathbf{x}^*,r)$, $S(v,\mathbf{x}^*,r)$, and $S(p,\mathbf{x}^*,r)$ satisfy

$$\begin{aligned} {\left\{ \begin{array}{ll} max\{i: \mathbf{x}_{i,j}\in S(u,\mathbf{x}^*,r)\}-min\{i: \mathbf{x}_{i,j}\in S(u,\mathbf{x}^*,r)\}=r,\\ max\{j: \mathbf{x}_{i,j}\in S(u,\mathbf{x}^*,r)\}-min\{j: \mathbf{x}_{i,j}\in S(u,\mathbf{x}^*,r)\}=r,\\ max\{i: \mathbf{x}_{i,j}\in S(v,\mathbf{x}^*,r)\}-min\{i: \mathbf{x}_{i,j}\in S(v,\mathbf{x}^*,r)\}=r,\\ max\{j: \mathbf{x}_{i,j}\in S(v,\mathbf{x}^*,r)\}-min\{j: \mathbf{x}_{i,j}\in S(v,\mathbf{x}^*,r)\}=r,\\ max\{i: \mathbf{x}_{i,j}\in S(p,\mathbf{x}^*,r)\}-min\{i: \mathbf{x}_{i,j}\in S(p,\mathbf{x}^*,r)\}=r-1,\\ max\{j: \mathbf{x}_{i,j}\in S(p,\mathbf{x}^*,r)\}-min\{j: \mathbf{x}_{i,j}\in S(p,\mathbf{x}^*,r)\}=r-1. \end{array}\right. } \end{aligned}$$

(6.3)

The criterion (3) is important for the invertibility of matrix A in Eq. (6.4), as shall be discussed later.

We now describe the process of constructing $S(u,\mathbf{x}^*,r)$, $S(v,\mathbf{x}^*,r)$, and $S(p,\mathbf{x}^*,r)$, and the corresponding schemes. In this process, we use the following notations

$$\begin{aligned} (\lambda _1h_x, \lambda _2h_y)=\mathbf{x}^*-\mathbf{x},\quad (s_1, s_2)=(\mathrm{sign}(\lambda _1),\mathrm{sign}(\lambda _2)). \end{aligned}$$

It is easy to see that $|\lambda _1|<1$ and $|\lambda _2|<1$. Since a ghost point is a neighbor to a fluid point, at least one of its neighbor needs to be in $\Omega _+$. We name a neighbor of $\mathbf{x}$ a check point, if that neighbor point is in $\Omega _+$. There are totally six different cases for the combination of ghost point and check point: (a) $\mathbf{x}=\mathbf{x}_{i-1/2,j}$ and $\phi (\mathbf{x}_{i-1/2+s_1,j})>0$; (b) $\mathbf{x}=\mathbf{x}_{i-1/2,j}$ and $\phi (\mathbf{x}_{i-1/2,j+s_2})>0$; (c) $\mathbf{x}=\mathbf{x}_{i-1/2,j}$ and $\phi (\mathbf{x}_{i-1/2+s_1/2,j+s_2/2})>0$; (d) $\mathbf{x}=\mathbf{x}_{i,j-1/2}$ and $\phi (\mathbf{x}_{i+s_1,j-1/2})>0$; (e) $\mathbf{x}=\mathbf{x}_{i,j-1/2}$ and $\phi (\mathbf{x}_{i,j-1/2+s_2})>0$; (f) $\mathbf{x}=\mathbf{x}_{i,j-1/2}$ and $\phi (\mathbf{x}_{i+s_1/2,j-1/2+s_2/2})>0$. For each of these cases, we obtain the corresponding $S(u,\mathbf{x}^*,r)$, $S(v,\mathbf{x}^*,r)$, and $S(p,\mathbf{x}^*,r)$ with $r\in \{1, 2, 3\}$. In Fig. 12, we schematically plot all six cases and the corresponding $S(u,\mathbf{x}^*,r)$, $S(v,\mathbf{x}^*,r)$, and $S(p,\mathbf{x}^*,r)$ for $r=3$. Notice that for cases (a)–(c), Eq. (6.1) is discretized, and $S(v,\mathbf{x}^*,r)$ is not needed.

Fig. 12

Different cases of combination of a ghost point and a check point

We now describe the steps of constructing a third-order discretization scheme for case (a), whereas all other cases just follow tediously. We consider a u ghost point $\mathbf{x}_{i-1/2,j}$ with its projection point $\mathbf{x}^*$. Let us denote

$$\begin{aligned} \mathbf{X}=\left( \mathbf{x}_0, \mathbf{x}_1, \mathbf{x}_2, \mathbf{x}_3, \mathbf{x}_4, \mathbf{x}_5, \mathbf{x}_6, \mathbf{x}_7, \mathbf{x}_8, \mathbf{x}_9\right) ^T=\left( \mathbf{x}_{i-\frac{1}{2},j}, \mathbf{x}_{i-\frac{1}{2}+s_1,j}, \mathbf{x}_{i-\frac{1}{2}-s_1,j+s_2}, \mathbf{x}_{i-\frac{1}{2},j+s_2}, \right. \\ \left. \mathbf{x}_{i-\frac{1}{2}+s_1,j+s_2}, \mathbf{x}_{i-\frac{1}{2}+2s_1,j+s_2}, \mathbf{x}_{i-\frac{1}{2},j+2s_2}, \mathbf{x}_{i-\frac{1}{2}+s_1,j+2s_2}, \mathbf{x}_{i-\frac{1}{2}+2s_1,j+2s_2}, \mathbf{x}_{i-\frac{1}{2},j+3s_2}\right) ^T, \end{aligned}$$

and $\mathbf{F}=\left( u, u_x, u_y, u_{xx}, u_{xy}, u_{yy}, u_{xxx}, u_{xyy}, u_{xxy}, u_{yyy}\right) ^T$, where $(\cdot )^T$ denotes transpose. Then a third-order Taylor expansion leads to a linear system $\mathbf{X}=A\mathbf{F}(\mathbf{x}^*)$, where

$$\begin{aligned} A= \left[ \begin{array}{cccccccccc}1 &{} h_{x,1} &{} h_{y,1} &{} h^2_{x,1}/2 &{} h_{x,1}h_{y,1} &{} h^{2}_{y,1}/2 &{} h^3_{x,1}/6 &{} h^2_{x,1}h_{y,1}/2 &{} h_{x,1}h^2_{y,1}/2 &{} h^3_{y,1}/6 \\ 1 &{} h_{x,2} &{} h_{y,2} &{} h^2_{x,2}/2 &{} h_{x,2}h_{y,2} &{} h^{2}_{y,2}/2 &{} h^3_{x,2}/6 &{} h^2_{x,2}h_{y,2}/2 &{} h_{x,2}h^2_{y,2}/2 &{} h^3_{y,2}/6 \\ 1 &{} h_{x,3} &{} h_{y,3} &{} h^2_{x,3}/2 &{} h_{x,3}h_{y,3} &{} h^{2}_{y,3}/2 &{} h^3_{x,3}/6 &{} h^2_{x,3}h_{y,3}/2 &{} h_{x,3}h^2_{y,3}/2 &{} h^3_{y,3}/6 \\ 1 &{} h_{x,4} &{} h_{y,4} &{} h^2_{x,4}/2 &{} h_{x,4}h_{y,4} &{} h^{2}_{y,4}/2 &{} h^3_{x,4}/6 &{} h^2_{x,4}h_{y,4}/2 &{} h_{x,4}h^2_{y,4}/2 &{} h^3_{y,4}/6 \\ 1 &{} h_{x,5} &{} h_{y,5} &{} h^2_{x,5}/2 &{} h_{x,5}h_{y,5} &{} h^{2}_{y,5}/2 &{} h^3_{x,5}/6 &{} h^2_{x,5}h_{y,5}/2 &{} h_{x,5}h^2_{y,5}/2 &{} h^3_{y,5}/6 \\ 1 &{} h_{x,6} &{} h_{y,6} &{} h^2_{x,6}/2 &{} h_{x,6}h_{y,6} &{} h^{2}_{y,6}/2 &{} h^3_{x,6}/6 &{} h^2_{x,6}h_{y,6}/2 &{} h_{x,6}h^2_{y,6}/2 &{} h^3_{y,6}/6 \\ 1 &{} h_{x,7} &{} h_{y,7} &{} h^2_{x,7}/2 &{} h_{x,7}h_{y,7} &{} h^{2}_{y,7}/2 &{} h^3_{x,7}/6 &{} h^2_{x,7}h_{y,7}/2 &{} h_{x,7}h^2_{y,7}/2 &{} h^3_{y,7}/6 \\ 1 &{} h_{x,8} &{} h_{y,8} &{} h^2_{x,8}/2 &{} h_{x,8}h_{y,8} &{} h^{2}_{y,8}/2 &{} h^3_{x,8}/6 &{} h^2_{x,8}h_{y,8}/2 &{} h_{x,8}h^2_{y,8}/2 &{} h^3_{y,8}/6 \\ 1 &{} h_{x,9} &{} h_{y,9} &{} h^2_{x,9}/2 &{} h_{x,9}h_{y,9} &{} h^{2}_{y,9}/2 &{} h^3_{x,9}/6 &{} h^2_{x,9}h_{y,9}/2 &{} h_{x,9}h^2_{y,9}/2 &{} h^3_{y,9}/6 \\ \end{array}\right] , \end{aligned}$$

(6.4)

with $(h_{x,k}, h_{y,k}):=\mathbf{x}_k-\mathbf{x}^*$. A sufficient condition to have A invertible is (6.3). Inverting A symbolically, we get $\mathbf{F}(\mathbf{x}^*)=A^{-1}\mathbf{X}$, and the second entry of $\mathbf{F}(\mathbf{x}^*)$ corresponds to a third-order discretization of $u_x(\mathbf{x}^*)$:

$$\begin{aligned}&u_x(\mathbf{x}^*)=-\frac{(\lambda _2-s_2)(\lambda _2-2s_2)}{2s_1h_x}u_0+\frac{(\lambda _2-s_2)(\lambda _2-2s_2)}{2s_1h_x}u_1-\frac{3\lambda _1^2-6\lambda _1s_1+2}{6s_1h_x}u_2\\&\quad -\frac{s_1(\lambda _1+\lambda _1\lambda _2 s_2)+3\lambda _2s_2/2-\lambda _2^2-3\lambda _1^2/2}{s_1h_x}u_3\\&\quad -\frac{s_1(\lambda _1-2\lambda _1\lambda _2 s_2)+\lambda _2^2- \lambda _2s_2-1+3\lambda _1^2/2}{s_1h_x}u_4\\&\quad +\frac{\lambda _2/2-2s_2/3+\lambda _1^2s_2/2-s_1\lambda _1\lambda _2+s_1\lambda _1s_2}{s_1s_2h_x}u_5-\frac{(\lambda _2-s_2)(\lambda _2s_1-2\lambda _1s_2+s_1s_2)}{2h_x}u_6\\&\quad +\frac{(\lambda _2-s_2)(\lambda _2s_1-4\lambda _1s_2+2s_1s_2)}{2h_x}u_7+\frac{(\lambda _2-s_2)(2\lambda _1-s_1)}{2s_2h_x}u_8, \end{aligned}$$

where the notation $u(\mathbf{X})=\left( u_0, u_1, u_2, u_3, u_4, u_5, u_6, u_7, u_8, u_9 \right) ^T$ is used. The third entry of $\mathbf{F}(\mathbf{x}^*)^t$ gives a third-order discretization of $u_y(\mathbf{x}^*)$. However, it is not necessary in discretizing Eq. (6.1).

Similarly, one can derive a third-order discretization scheme for $p(\mathbf{x}^*)$:

$$\begin{aligned}&p(\mathbf{x}^*)=\left( \frac{3}{8}-\lambda _1s_1+\frac{\lambda _1^2}{2}\right) p_0-\frac{(2\lambda _1-3s_1)(2\lambda _1s_2-2\lambda _2s_1+3s_1s_2)}{4s_2}p_1\\&\quad +\frac{((2\lambda _1s_2 - 2\lambda _2s_1 + s_1s_2)(2\lambda _1s_2 - 2\lambda _2s_1 + 3s_1s_2))}{8}p_2-\frac{(\lambda _2 - s_2)(2\lambda _1 - 3s_1)}{2s_1s_2}p_3\\&\quad +\frac{(\lambda _2 - s_2)(2\lambda _1s_2 - 2\lambda _2s_1 + 3s_1s_2)}{2*s_1}p_4+\frac{(\lambda _2 - s_2)(\lambda _2 - 2s_2)}{2}p_5, \end{aligned}$$

where $p_0=p(\mathbf{x}_{i-\frac{s_1+1}{2},j+s_2})$, $p_1=p(\mathbf{x}_{i+\frac{s_1-1}{2},j+s_2})$, $p_2=p(\mathbf{x}_{i+\frac{3s_1-1}{2},j+s_2})$, $p_3=p(\mathbf{x}_{i+\frac{s_1-1}{2},j+2s_2})$, $p_4=p(\mathbf{x}_{i+\frac{3s_1-1}{2},j+2s_2})$, $p_5=p(\mathbf{x}_{i+\frac{3s_1-1}{2},j+3s_2})$.

We use this symbolic inverse matrix method to find a third-order scheme for discretizing Eq. (6.1) around $\mathbf{x}^*$. To find a second-order or first-order scheme, we form a smaller matrix A by applying a second or first-order Taylor expansion to $u(\mathbf{x}_k)$, $v(\mathbf{y}_k)$, and $p(\mathbf{z}_k)$ at $\mathbf{x}^*$, with $\mathbf{x}_k\in S(u,\mathbf{x}^*,r)$, $\mathbf{y}_k\in S(v,\mathbf{x}^*,r)$, and $\mathbf{z}_k\in S(p,\mathbf{x}^*,r)$, respectively. Using similar computations, we are able to construct the first, second, and third-order schemes for all six cases (a)–(f), but we do not need to record the ten-page formulas here.