In this section, the macroscopic governing equations are first briefly introduced. Then we offer a detailed explanation of the Strang-splitting discrete unified gas kinetic scheme. The pseudopotential model for DUGKS will be introduced in the final part.

### 2.1 Macroscopic equations

The macroscopic governing equations recovered by the kinetic equation through the Chapman-Enskog theory read

$$\begin{aligned} \frac{\partial {\rho }}{\partial {t}} + \nabla \cdot (\rho \varvec{u}) = 0, \end{aligned}$$

(1a)

$$\begin{aligned} \frac{\partial (\rho \varvec{u})}{\partial {t}} + \nabla \cdot \left( \rho \varvec{u}\otimes \varvec{u}\right) =-\nabla {p}+\nabla \left[ \mu \left( \nabla \varvec{u}+(\nabla \varvec{u})^{T}\right) \right] +\varvec{F}_{s} +\varvec{G}, \end{aligned}$$

(1b)

where *t* represents time, \(\rho\) indicates the fluid density, \(\varvec{u}\) denotes the flow velocity, *p* is the pressure, and \(\mu\) is the dynamic viscosity. \(\varvec{F}_{s}\) stands for the volumetric force that mimics the interaction effects between/among different phases whereas \(\varvec{G}\) indicates the gravitational or buoyant force.

### 2.2 Discrete unified gas kinetic scheme

In present research, the flow field is directly governed by the Boltzmann-BGK equation, which takes the form of

$$\begin{aligned} \frac{\partial {f}}{\partial {t}} + \varvec{\xi }\cdot \nabla _{\varvec{x}}{f} = \Omega \equiv -\frac{{f}-{f}^{\text {E}}}{\tau }, \end{aligned}$$

(2)

where \(f=f(\varvec{x},\varvec{\xi },t)\) is the distribution function (DF) accounting for the particles residing at position \(\varvec{x}\) with a velocity of \(\varvec{\xi }\) at time *t*, \({\tau }\) is the relaxation time, \({f}^{\text {E}}\) is the equilibrium distribution function approached by *f* within each collision. The moments of the distribution function yield the conservative flow variables via

$$\begin{aligned} \rho = \int {f}d\varvec{\xi } = \int {f^{\text {E}}}d\varvec{\xi },\ \rho \varvec{u}=\int \varvec{\xi }{f}d\varvec{\xi }=\int \varvec{\xi }{f^{\text {E}}}d\varvec{\xi }. \end{aligned}$$

(3)

A necessary prerequisite for the numerical evaluation of the moments is the discretization of the velocity space. In present work, the three-point Gauss-Hermite quadrature is employed to determine the discrete particle velocities along each single dimension. In two dimension the discrete velocities can be derived from the tensor product of the single dimensional velocities, which reads

$$\begin{aligned} \varvec{\xi }_{i}=\sqrt{3{c_{s}^{2}}} \left[ \begin{array}{rrrrrrrrr} 0&{}1&{}0&{}-1&{}0&{}1&{}-1&{}-1&{}1\\ 0&{}0&{}1&{}0&{}-1&{}1&{}1&{}-1&{}-1 \end{array}\right] , \end{aligned}$$

where \(\varvec{\xi }_{i}\) is the *i*th discrete velocity and \(c_{s} = 1/\sqrt{3}\) is the model speed of sound. To fulfill the relation of Eq. (3) at the discrete level, the equilibrium DF \(\varvec{f}^{\text {E}}\) in present research is evaluated by

$$\begin{aligned} \varvec{f}^{\text {E}}=\varvec{M}^{-1}\varvec{m}^{\text {E}}, \end{aligned}$$

(4)

where \(\varvec{f}^{\text {E}}=\left\{ {f}_{0}^{\text {E}},{f}_{1}^{\text {E}},\cdots ,{f}_{8}^{\text {E}}\right\} ^{\top }\) represents the column vector constituted by the discrete equilibrium DFs, \(\varvec{M}\) is the transformation matrix defined as

$$\begin{aligned} \varvec{M}=\left[ \begin{array}{rrrrrrrrr} 1 &{} 1 &{} 1 &{} 1 &{} 1 &{} 1 &{} 1 &{} 1 &{} 1 \\ -4 &{} -1 &{} -1 &{} -1 &{} -1 &{} 2 &{} 2 &{} 2 &{} 2 \\ 4 &{} -2 &{} -2 &{} -2 &{} -2 &{} 1 &{} 1 &{} 1 &{} 1 \\ 0 &{} 1 &{} 0 &{} -1 &{} 0 &{} 1 &{} -1 &{} -1 &{} 1 \\ 0 &{} -2 &{} 0 &{} 2 &{} 0 &{} 1 &{} -1 &{} -1 &{} -1 \\ 0 &{} 0 &{} 1 &{} 0 &{} -1 &{} 1 &{} 1 &{} -1 &{} -1 \\ 0 &{} 0 &{} -2 &{} 0 &{} 2 &{} 1 &{} 1 &{} -1 &{} -1 \\ 0 &{} 1 &{} -1 &{} 1 &{} -1 &{} 0 &{} 0 &{} 0 &{} 0 \\ 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 1 &{} -1 &{} 1 &{} -1 \end{array}\right] , \end{aligned}$$

and \(\varvec{m}^{\text {E}}\) signifies the macroscopic equilibria:

$$\begin{aligned} \varvec{m}^{\text {E}}=\rho \left\{ 1,3{\vert \varvec{u}\vert }^{2}-2,{\alpha }-3{\vert \varvec{u}\vert }^{2},u_{x},-u_{x},u_{y},-u_{y},u_{x}^{2}-u_{y}^{2},u_{x}u_{y}\right\} ^{\top }. \end{aligned}$$

Here \(\alpha\) is a free parameter used to eliminate the nonisotropic effects of the scheme [26]. The relations between the conservative variables and the discrete DFs become

$$\begin{aligned} \rho = \sum \limits _{i}f_{i} = \sum \limits _{i}f_{i}^{\text {E}},\rho \varvec{u} = \sum \limits _{i}\varvec{\xi }_{i}f_{i} = \sum \limits _{i}\varvec{\xi }_{i}f_{i}^{\text {E}}. \end{aligned}$$

(5)

With the discretization of velocity space, the discrete velocity Boltzmann-BGK equation takes the following form:

$$\begin{aligned} \frac{\partial {f}_{i}}{\partial {t}} + \varvec{\xi }_{i}\cdot \nabla _{\varvec{x}}{f}_{i} = \Omega _{i} \equiv -\frac{{f}_{i}-{f}^{\text {E}}_{i}}{\tau }. \end{aligned}$$

(6)

To numerically solve Eq. (6), we first subdivide the spatial domain into a set of grid cells and integrate this equation over a certain cell, which yields

$$\begin{aligned} \frac{d}{dt}\int _{V_{c}}f_{i}(\varvec{x},t)d\varvec{x}+\int _{\partial {V}_{c}}\left( \varvec{\xi }\cdot \varvec{n}\right) f_{i}(\varvec{x},t)dS= \int _{V_{c}}\Omega _{i}(\varvec{x},t)d\varvec{x}, \end{aligned}$$

(7)

where \(V_{c}\) represents the integral cell centered at position \(\varvec{x}_{c}\), \(\partial {V_{c}}\) indicates the surface bounding the cell, and \(\varvec{n}\) represents the unit vector normal to the surface. Integrating Eq. (7) over a time step of length \({\Delta }t = t_{n+1}-t_{n}\) yields

$$\begin{aligned} f_{i}^{n+1}-f_{i}^{n}+\frac{{\Delta }t}{\vert {V_{c}}\vert }{F}_{i}^{n+1/2}=\frac{{\Delta }t}{2}\left[ \Omega _{i}^{n+1} + \Omega _{i}^{n}\right] , \end{aligned}$$

(8)

where \(\vert {V_{c}}\vert\) measures the volume of cell \(V_{c}\), \(f_{i}^{n}\) and \(\Omega _{i}^{n}\) approximate the cell averages of \(V_{c}\) in such a way that

$$\begin{aligned} f_{i}^{n} = \frac{1}{\vert {V_{c}}\vert }\int _{V_{c}}f_{i}(\varvec{x},t_{n})d\varvec{x}, \end{aligned}$$

(9a)

$$\begin{aligned} \Omega _{i}^{n} = \frac{1}{\vert {V_{c}}\vert }\int _{V_{c}}{\Omega }_{i}(\varvec{x},t_{n})d\varvec{x}, \end{aligned}$$

(9b)

\(F_{i}^{n+1/2}\) measures the kinetic flux at the mid time \(t_{n}+\Delta {t}/2\) by

$$\begin{aligned} F_{i}^{n+1/2}=\int _{\partial {V_{c}}}\left( \varvec{\xi }_{i}\cdot \varvec{n}\right) f_{i}(\varvec{x},t_{n}+\Delta {t}/2)dS. \end{aligned}$$

(10)

Note that the midpoint rule is utilized to compute the time integral of the kinetic flux and trapezoidal rule is employed to evaluate the time integral of the collision term in Eq. (8). To obtain a fully explicit evolution equation, two auxiliary distribution functions are introduced:

$$\begin{aligned} \tilde{f}_{i} = f_{i}-\frac{{\Delta }t}{2}\Omega _{i},\tilde{f}_{i}^{+}=f_{i}+\frac{{\Delta }t}{2}\Omega _{i}. \end{aligned}$$

(11)

Substituting Eq. (11) into Eq. (8), we have

$$\begin{aligned} \tilde{f}_{i}^{n+1} = \tilde{f}_{i}^{+,n} {-} \frac{\Delta {t}}{{\vert {V_{c}}\vert }}F_{i}^{n+1/2}, \end{aligned}$$

(12)

which turns out to be fully explicit.

To evaluate the kinetic flux \(F_{i}^{n+1/2}\), the primitive distribution function \(f_{i}(\varvec{x}_{f},t_{n+1/2})\) on cell interface is needed. To this end, we integrate Eq. (6) along the characteristic line over a time step length of \(\delta {t}={\Delta }t/2\):

$$\begin{aligned} f_{i}(\varvec{x}_{f},t_{n}+{\delta }t)-f_{i}(\varvec{x}_{f}-\varvec{\xi }_{i}{\delta {t}},t_{n}) = \frac{{\delta }t}{2}\left[ \Omega _{i}(\varvec{x}_{f},t_{n}+{\delta }t) + \Omega _{i}(\varvec{x}_{f}-\varvec{\xi }_{i}{\delta {t}},t_{n})\right] . \end{aligned}$$

(13)

Here the trapezoidal rule is once again employed to evaluate the time integral of collision term. To realize the explicit treatment of Eq. (13), another two auxiliary distribution functions are introduced as follows:

$$\begin{aligned} \bar{f} = f-\frac{{\delta }t}{2}\Omega ,\bar{f}^{+} = f+\frac{{\delta }t}{2}\Omega . \end{aligned}$$

(14)

Eq. (13) then can be rearranged as

$$\begin{aligned} \bar{f}_{i}(\varvec{x}_{f},t_{n}+{\delta }t) = \bar{f}_{i}^{+}(\varvec{x}_{f}-\varvec{\xi }_{i}{\delta {t}},t_{n}). \end{aligned}$$

(15)

The cell-centered auxiliary distribution function \(f^{+}\) can be constructed according to its definition:

$$\begin{aligned} \bar{f}^{+} = f+\frac{{\delta }t}{2}\Omega = \frac{2\tau -{\delta }t}{2\tau }f + \frac{{\delta }t}{2\tau }f^{\text {E}}. \end{aligned}$$

(16)

The value of \(f^{+}(\varvec{x}_{f}-\varvec{\xi }_{i}{\delta {t}},t_{n})\) can be interpolated from the corresponding cell-centered distribution function [53]. For the face-based reconstruction scheme (FRS), \(f^{+}(\varvec{x}_{f}-\varvec{\xi }_{i}{\delta {t}},t_{n})\) can be evaluated by

$$\begin{aligned} f^{+}_{i}(\varvec{x}_{f}-\varvec{\xi }_{i}{\delta {t}},t_{n}) = f^{+}_{i}(\varvec{x}_{f},t_{n})-\varvec{\xi }_{i}{\delta {t}}\cdot {\nabla }f^{+}_{i}(\varvec{x}_{f},t_{n}). \end{aligned}$$

(17)

For the cell-based reconstruction scheme (CRS), \(f^{+}(\varvec{x}_{f}-\varvec{\xi }_{i}{\delta {t}},t_{n})\) can be computed by

$$\begin{aligned} \bar{f}_{i}^{+}(\varvec{x}_{f}-\varvec{\xi }_{i}{\delta }t)&= \left\{ \begin{array}{ll} \bar{f}_{i}^{+}(\varvec{x}_{\text {L}}) + (\varvec{x}_{f}-\varvec{x}_{\text {L}}-\varvec{\xi }_{i}{\delta }{t})\cdot {\nabla }\bar{f}_{i}^{+}(\varvec{x}_{\text {L}}) + \frac{1}{2}(\varvec{x}_{f}-\varvec{x}_{\text {L}}-\varvec{\xi }_{i}{\delta }{t})^2{:}{\nabla }^2\bar{f}_{i}^{+}(\varvec{x}_{\text {L}}), &{}\varvec{\xi }_{i}\cdot \varvec{n} \le 0,\\ \bar{f}_{i}^{+}(\varvec{x}_{\text {R}}) + (\varvec{x}_{f}-\varvec{x}_{\text {R}}-\varvec{\xi }_{i}{\delta }{t})\cdot {\nabla }\bar{f}_{i}^{+}(\varvec{x}_{\text {R}}) + \frac{1}{2}(\varvec{x}_{f}-\varvec{x}_{\text {R}}-\varvec{\xi }_{i}{\delta }{t})^2{:}{\nabla }^2\bar{f}_{i}^{+}(\varvec{x}_{\text {R}}), &{}\varvec{\xi }_{i}\cdot \varvec{n} \ge 0,\\ \end{array}\right. \end{aligned}$$

(18)

where \(\varvec{x}_{\text {L}}\) and \(\varvec{x}_{\text {R}}\) correspond respectively to the center positions of the two cells adjacent to the interface located at \(\varvec{x}_{f}\). Once the value of \(f^{+}\) is known, the primitive DF at cell interface can be updated by

$$\begin{aligned} f = \frac{2\tau }{2\tau +\delta {t}}\bar{f} + \frac{\delta {t}}{2\tau +\delta {t}}f^{\text {E}}. \end{aligned}$$

(19)

Thereafter the kinetic flux \(F^{n+1/2}\) can be evaluated according to Eq. (10) and the auxiliary distribution function at time \(t_{n+1}\) can be updated via Eq. (12). The cell-averaged primitive DF then can be obtained according to

$$\begin{aligned} f = \frac{2\tau }{2\tau +{\Delta }t}\tilde{f} + \frac{{\Delta }t}{2\tau +{\Delta }t}{f}^{\text {E}}. \end{aligned}$$

(20)

To obtain the primitive DFs in Eq. (19) and (20), the value of equilibrium DFs that depend on the conservative variables should be first determined via Eq. (4). With the help of Eq. (5), the corresponding conservative variables can be evaluated by

$$\begin{aligned} \rho =\sum \limits _{i}f_{i} = \sum \limits _{i}\bar{f}_{i},\ \rho \varvec{u}=\sum \limits _{i}\varvec{\xi }_{i}f_{i} = \sum \limits _{i}\varvec{\xi }_{i}\bar{f}_{i} \end{aligned}$$

(21)

on cell interfaces and by

$$\begin{aligned} \rho =\sum \limits _{i}f_{i} = \sum \limits _{i}\tilde{f}_{i},\ \rho \varvec{u}=\sum \limits _{i}\varvec{\xi }_{i}f_{i} = \sum \limits _{i}\varvec{\xi }_{i}\tilde{f}_{i} \end{aligned}$$

(22)

at cell centers.

To date, the evolution process of DUGKS without considering force effects has been basically explained. To incorporate the influence of external forces, another discrete distribution function \(f_{i}^{\text {S}}\) accounting for the force effects should be introduced:

$$\begin{aligned} \frac{\partial {f}_{i}}{\partial {t}} + \varvec{\xi }_{i}\cdot \nabla _{\varvec{x}}{f}_{i} = \Omega _{i} \equiv -\frac{{f}_{i}-{f}^{\text {E}}_{i}}{\tau } + f_{i}^{\text {S}}. \end{aligned}$$

(23)

To correctly recover the macroscopic equations, the moments of discrete force DF should obey

$$\begin{aligned} \sum \limits _{i}f_{i}^{\text {S}} = 0, \ \sum \limits _{i}\varvec{\xi }_{i}f_{i}^{\text {S}} = \varvec{F}, \ \sum \limits _{i}\varvec{\xi }_{i}\varvec{\xi }_{i}f_{i}^{\text {S}} = \varvec{uF}+\varvec{Fu}, \end{aligned}$$

(24)

where \(\varvec{F}=\varvec{F}_{s}+ \varvec{G}\) is the external force in total. In present research the force DF \(\varvec{f}^{\text {S}}\) is evaluated by

$$\begin{aligned} \varvec{f}^{\text {S}}=\varvec{M}^{-1}\varvec{m}^{\text {S}}, \end{aligned}$$

(25)

where \(\varvec{f}^{\text {S}} = \left\{ {f}_{0}^{\text {S}},{f}_{1}^{\text {S}},\cdots ,{f}_{8}^{\text {S}}\right\}\) represents the column vector constituted by the discrete force DFs, \(\varvec{M}\) is the identical transformation matrix appeared in Eq. (4), and \(\varvec{m}^{\text {S}}\) signifies the macroscopic force terms expressed as

$$\begin{aligned} \varvec{m}^{\text {S}} = \left\{ 0,6\varvec{u}\cdot \varvec{F},-6\varvec{u}\cdot \varvec{F},F_{x},-F_{x},F_{y},-F_{y},2(F_{x}u_{x}-F_{y}u_{y}),F_{x}u_{y}+F_{y}u_{x}\right\} . \end{aligned}$$

To circumvent the force effects on the interface flux, the Strang-splitting scheme is employed to evaluate the force influences. With such a treatment, the force effects are incorporated before and after the DUGKS procedure in a way that

$$\begin{aligned} \frac{\partial {f}_{i}}{{\partial }t} = \frac{1}{2}f_{i}^{\text {S}}, \end{aligned}$$

(26a)

$$\begin{aligned} \frac{\partial {f}_{i}}{\partial {t}} + \varvec{\xi }_{i}\cdot \nabla _{\varvec{x}}{f}_{i} = \Omega _{i} \equiv -\frac{{f}_{i}-{f}^{\text {E}}_{i}}{\tau }, \end{aligned}$$

(26b)

$$\begin{aligned} \frac{\partial {f}_{i}}{{\partial }t} = \frac{1}{2}f_{i}^{\text {S}}. \end{aligned}$$

(26c)

As Eq. (26b) remains identical to Eq. (6), it can be solved by the DUGKS procedure addressed previously. Eq. (26a) and (26c) can be numerically solved by the Euler forward method:

$$\begin{aligned} f_{i}^{*} = f_{i}^{n} + \frac{{\Delta }t}{2}f_{i}^{\text {S},n}. \end{aligned}$$

(27)

The conservative variables should be accordingly updated via

$$\begin{aligned} \rho ^{*}=\rho ^{n}, \varvec{u}^{*} = \varvec{u}^{n} + \frac{{\Delta }t}{2}\frac{\varvec{F}^{n}}{\rho ^{n}}. \end{aligned}$$

(28)

### 2.3 Pseudopotential multiphase model

In the pseudopotential multiphase model, the interaction effects between/among different phases are mimicked by a volumetric force defined as

$$\begin{aligned} \varvec{F}_{s}=-G\psi (\varvec{x})\sum \limits _{i=1}^{N}\omega (\vert \varvec{x}_{i}^{\prime }-\varvec{x}\vert ^{2})\psi (\varvec{x}_{i}^{\prime })\left( \varvec{x}_{i}^{\prime }-\varvec{x}\right) , \end{aligned}$$

(29)

where \(\psi\) represents the interaction potential, *G* indicates the interaction strength, \(\omega\) stands for the weights, \(\varvec{x}_{i}^{\prime }\) denotes the nearby position that is related to \(\varvec{x}\) by \(\varvec{x}_{i}^{\prime } = \varvec{x}+\varvec{\xi }_{i}{\delta }_{t}^{\prime }\), among which \(\varvec{\xi }_{i}\) is the *i*th discrete velocity and \({\delta }_{t}^{\prime }\) is the transporting time. A utilization of nine discrete velocity points leads to the following relation:

$$\begin{aligned} \omega (1) = 1/3,\omega (2) = 1/12, N = 8, {\delta }_{t}^{\prime } = 1. \end{aligned}$$

(30)

In fact, the role of expression \(\sum \limits _{i=1}^{N}\omega (\vert \varvec{x}_{i}^{\prime }-\varvec{x}\vert ^{2})\psi (\varvec{x}_{i}^{\prime })\left( \varvec{x}_{i}^{\prime }-\varvec{x}\right)\) in Eq. (29) is equivalent to evaluating the gradient of \(\psi\) through an isotropic finite-difference scheme [54]. A Taylor expansion of Eq. (29) gives

$$\begin{aligned} \varvec{F}_{s}=-G{\delta }_{x}^{2}\left[ \psi \nabla \psi +\frac{{\delta }_{x}^{2}}{6}\nabla \left( \nabla ^{2}\psi \right) \right] + O(\nabla ^{5}), \end{aligned}$$

(31)

where \({\delta }_{x}={\xi }_{x}{\delta }_{t}^{\prime } = {\xi }_{y}{\delta }_{t}^{\prime } = 1\) measures the grid spacing. To analytically derive the pressure tensor, Eq. (31) could be reformulated as [20]

$$\begin{aligned} \varvec{F}_{s}=-\frac{G{\delta }_{x}^{2}}{2}\nabla \cdot \left( \psi ^{2}\varvec{I}\right) -\frac{G{\delta }_{x}^{4}}{6}\nabla \cdot \left[ \left( \psi \nabla \cdot \nabla \psi +\frac{1}{2}\nabla \psi \cdot \nabla \psi \right) \varvec{I}- \nabla \psi \nabla \psi \right] + O(\nabla ^{5}), \end{aligned}$$

(32)

where \(\varvec{I}\) represents the identity matrix. However, Sbragaglia *et al*. [21] demonstrate that the transformation of Eq. (31) into Eq. (32) does not necessarily guarantee uniqueness. As a matter of fact, Eq. (31) can be reformulated as

$$\begin{aligned} \begin{aligned} \varvec{F}_{s} =&-\frac{G{\delta }_{x}^{2}}{2}\nabla \cdot \left( \psi ^{2}\varvec{I}\right) -\frac{G{\delta }_{x}^{4}}{6}\nabla \cdot \left( a_{1}\nabla \psi \nabla \psi +a_{2}\psi \nabla \nabla \psi \right) \\&-\frac{G{\delta }_{x}^{4}}{6}\nabla \cdot \left( a_{3}\nabla \psi \cdot \nabla \psi +a_{4}\psi \nabla \cdot \nabla \psi \right) \varvec{I} + O(\nabla ^{5}), \end{aligned} \end{aligned}$$

(33)

providing the prefactors satisfy

$$\begin{aligned} \left\{ \begin{array}{l} a_{1} + a_{2} + 2a_{3} =0, \\ a_{1} + a_{4}=0, \\ a_{2} + a_{4}=1. \end{array}\right. \end{aligned}$$

(34)

It can be identified that Eq. (32) represents a special case of Eq. (33) at \(a_{1}=-1,a_{2}=0,a_{3}=1/2,a_{4}=1.\)

The continuum pressure tensor is defined as [26]

$$\begin{aligned} \nabla \cdot \varvec{P}=\nabla \left( \rho {c_{s}^{2}}\right) -\varvec{F}_{s}-\varvec{S}, \end{aligned}$$

(35)

where \(c_{s}\) stands for the model speed of sound and \(\varvec{S}\) represents the additional term introduced from the discretization of DUGKS. Due to the reconstruction approaches utilized, the additional term \(\varvec{S}\) contributed from DUGKS lacks of isotropy. To balance the anisotropic influences, a free parameter \(\alpha\) has been introduced in Eq. (4). As the discretization approaches utilized in DUGKS appear to be complex, it is quite difficult to obtain the general expression of \(\varvec{P}\). Nevertheless, the normal pressure \(P_{n}\) in such a condition could be similarly postulated as [55]

$$\begin{aligned} P_{n} = \rho {RT}+\frac{G{\delta }_{x}^{2}}{2}\psi ^{2}+\frac{G{\delta }_{x}^{4}}{12}\left[ -{k_{1}}\left( \frac{d\psi }{dx}\right) ^{2}+2{k_{2}}\psi \frac{d^{2}\psi }{dx^{2}}\right] , \end{aligned}$$

(36)

where *n* denotes the direction normal to the interface. The normal component of the pressure tensor \(P_{n}\) at the equilibrium state should be equal to the bulk pressure \(p_{0}\) [22]. The mechanical stability condition can then be obtained as [8]

$$\begin{aligned} \int _{{\rho }_{g}}^{{\rho }_{l}}\left( p_{0}-p_{\text {EOS}}\right) \frac{\psi ^{\prime }}{{\psi }^{1+\epsilon }}d{\rho } = 0, \end{aligned}$$

(37)

where \(\psi ^{\prime } = d\psi /d\rho\), \(\epsilon = k_{1}/k_{2}\), \(p_{0}=p(\rho _{l})=p(\rho _{g})\) denotes the bulk pressure and \(p_{\text {EOS}}\) represents the non-ideal equation of state (EOS) in terms of the pseudopotential model:

$$\begin{aligned} p_{\text {EOS}} = \rho {c_{s}^{2}}+\frac{G{\delta }_{x}^{2}}{2}\psi ^{2}. \end{aligned}$$

(38)

Providing the coexistence densities (\(\rho _{l}\) and \(\rho _{g}\)) have been estimated by DUGKS, the value of the produced parameter \(\epsilon\) can then be determined numerically [25]. To consider the effects of various non-ideal equations of state, He and Doolen [19] pointed out that the pseudopotential \(\psi\) should be evaluated as

$$\begin{aligned} \psi =\sqrt{\frac{2(p_{\text {EOS}}-\rho {c_{s}^{2}})}{G{\delta }_{x}^{2}}}, \end{aligned}$$

(39)

where \(p_{\text {EOS}}\) should be one of the non-ideal equations of state in the thermodynamic theory. In present research, the dimensionless van der Waals equation of state (vdW-EOS) expressed as

$$\begin{aligned} p_{\text {EOS}} = \frac{{\rho }T}{1-b\rho }-a\rho ^{2} \end{aligned}$$

(40)

is employed, with \(a=9/392\) and \(b=2/21\). The critical density \(\rho _{c}\) and temperature \(T_{c}\) hold the value of 7/2 and 1/14 in such a condition [56]. With the incorporation of the dimensionless vdW-EOS, the pseudopotential \(\psi\) could be directly calculated through Eq. (39).