1 Introduction

In fluid flow calculation, there are basically two ways to treat the flow field; the first one is to consider the fluid as a number of molecules and in the second case it is considered as a continuum. The first approach is a deterministic approach while in the second approach which is a continuum approach, the velocity, density, pressure, etc. are all defined at every point in the space and time, and the conservation laws lead to Navier–Stokes equations. When the characteristic length of a fluid domain is small, the macroscopic theory is not efficient and may break down. Researchers have dealt with the question whether or not the classical Navier–Stokes equations are valid in channels of small sizes. Observations show that in nanoscale, as the length scale over which the fluid velocity changes approaches the slip length, the fluid can undergo slip at the wall-fluid interface and the usual assumptions of the classical continuum theory with a no-slip boundary condition can break down. From a theoretical point of view, the parameters controlling the degree of slip are still greatly unknown. Previous experimental studies have shown that the boundary slip is highly affected by the solid–fluid interaction energy, the wetting of surface and the surface roughness [1–8]. As experimental observation of flow behavior near the boundary and interface in a nano scale level is extremely difficult, the numerical simulation based on molecular dynamics offer ideal technique for studying the effects of solid–fluid surface properties on the wall slip phenomena. In practice, few surfaces are smooth in actual nano fluidic systems. As the characteristic length of the surface roughness is in the order of molecular structure, the effects of surface roughness on the rheology and fluid slip become important [9]. Mo and Rosenberger [10] studied a 2-D channel with atomically rough walls. Sinusoidal and randomly roughened walls were used in their simulation. Their results indicated that the Knudsen number alone is not a good criterion for the flow behavior near the wall. The ratio of l/A, should also be used to validate the exact behavior of fluid at solid interface. Where l is the molecular mean free path and A is the amplitude of roughness of surface. Jabbarzadeh and Atkinson [11] indicated that the slip length is strongly influenced by the strength of the liquid/solid interaction potential and the degree of commensurability between the liquid particles and wall atoms. Chang and Maginn [12] studied the phenomena of wall slip occurring in the rarefied gases flowing through micro- and nano-channels. They showed that how small differences in slip coefficients can be exploited for the kinetic separation of gases. Gala and Attard [13] investigated the effect of atomic roughness on the slip length at fluid–solid interface for a sheared flow. In their model, atomic roughness was simulated by varying the size and spacing between solid atoms at constant packing fraction while the interaction parameters and the thermodynamic state of the fluid were kept constant. Their results indicated that the fluid–solid slip length exhibits nonmonotonic behavior as the solid structure is varied from smooth to rough. Slip occurred for both smooth and rough surfaces while the fluid stick happened only at some surfaces with intermediate values of roughness. Bing Yang and Zeng-Yang [14] studied the rarefied gas flow in a submicron channel in which surface roughness was represented by arrays of triangular modules. They investigated the effects of rarefaction and surface roughness on the boundary conditions and the flow characteristics. Their study showed the decrease of slip as the roughness of the surface increased. They noticed that the friction constant increased while the surface roughness is increased. Ogata et al. [15] simulated formation of molecularly thin liquid bridge between solid surfaces having molecular-scale roughness by using the molecular dynamics. They found that the layered molecular arrangement inside the liquid bridge was disturbed by the molecular-scale roughness on the surfaces and the projected profiles were able to be approximated by arcs. Priezjev [16] used Molecular dynamics simulations to investigate the influence of molecular-scale surface roughness on the slip behavior in thin liquid films. He found that both periodically and randomly corrugated rigid surfaces reduce the slip length and its shear rate dependence. Kunert and Harting [17] performed simulations of pressure driven flow between two rough plates with and without hydrophobic fluid–wall interaction. They showed that for rough surfaces with a hydrophobic fluid–wall interaction, there is a strong nonlinear effect that leads to an increase of the slip length by a factor of three for small interactions. Ziarani and Mohamad [18] studied the rarified gas flow through a 2-D atomically rough micro channel with molecular dynamics simulation. They indicated that even the roughness topology will affect the flow boundaries. Sofos and Karakasidis [19] further revealed the significant roughness effect on the potential energy distribution, fluid atom localization, streaming velocity profiles, and slip length, which indicate that these effects should always be considered in nanoflow designs. Nikolaos and Dimitris [20] simulated the effects of nano scale surface roughness in a channel undergoing Poiseuille flow with MD simulation. Simulation results indicated a non-linear variation of slip as a factor of roughness amplitude. Kamali and Kharazmi [21] developed a molecular based scheme for simulating of surface roughness effects on nano- and micro-scale flows. Their simulation results suggested that both the wall fluid interaction and surface roughness are important and should be considered simultaneously in determining the nano-structures and profiles of monatomic fluid flow in a nanochannel. Also, their simulation results showed that the roughness and cavitations of the same dimensions induce different local density patterns while the overall average pattern might be the same. Li et al. [22] reviewed the methods and the applications of MD simulation in liquid flows in nanochannels. They focused on the fundamental research and the rationality of the model hypotheses. They summarized the important discoveries in this research field and proposed some worthwhile future research directions. Kamali and Kharazmi [21] examined the surface roughness effects on the fluid flow in a nano channel of simple fluids in hydrophobic and hydrophilic walls by molecular dynamics simulation. Chen et al. [23] conducted a molecular dynamics simulation of slip boundary for fluid flow past a solid surface incorporating roughness effect. Their results indicated that interfacial slip develops if the wall is effectively uncorrugated.

As mentioned earlier, in these articles [1–23], various parameters such as slip length, channel width, surface attraction energy, slip velocity, interface wettability, surface roughness, external force, and heat transfer were studied. From each work different aspects of Poiseuille flow were studied and therefore, different results were obtained that each work could be useful in its position. However, a lack of a generality in terms of surface roughness geometry, the gap between roughness elements (or roughness periodicity), the surface roughness height and the surface attraction energy were observed. Therefore, it seems that this work can make a good contribution to the removal of shortcomings in other works. Hence, the objective of this paper is to examine the effect of these parameters in order to understand the fluid flow behavior in the rough nano channels. The NEMD simulations of simple Lennard–Jones fluids subject to an inlet driving force in a nanochannel were carried out. Also a planer poiseuille flow of liquid argon is studied using molecular dynamics simulation. In the present work, velocity rescaling is used, together with the velocity integrator, for integration of the trajectories of the particles. For this purpose, near the wall region, along the flow-direction, thermostats act on both sides. The interfacial density profiles at different magnitudes of the height roughness are obtained. Also, the effect of gap between roughness elements on the velocity profiles is examined.

2 Simulation

In this paper, the large-scale atomic/molecular massively parallel simulation (LAMMPS) were used, which is an available open-source software for molecular dynamics simulation. This program is written in C++ and was developed at Sandia National Labs. LAMMPS is selected for this study due to its simplicity and ease of operation. Dimensionless physical quantities (Lennard–Jones reduced units) are used for convenience as well as to facilitate scale-up of the system. A simple fluid of argon using Lennard–Jones (12–6) potential as expressed in single atom situation is considered:

$$ \phi (r_{ij} ) = \left\{ {\left. {4\varepsilon \left[ {\left( {\frac{\sigma }{{r_{ij} }}} \right)^{12} - \left( {\frac{\sigma }{{r_{ij} }}} \right)^{6} } \right]} \right\}} \right. $$
(1)

where \( \phi (r_{ij} ) \) is the interaction potential between two particles, ε is the depth of the potential energy that governs the strength of interaction and σ is the finite distance where the interparticle potential is zero and r ij is the distance between the two particles [24, 25]. Table 1 shows the units used in our argon MD simulation [26] where k is the Boltzmann constant. Here after, all quantities will be given in terms of the Lennard–Jones reduced units. Table 1 shows the units used in MD simulation of particles interacting by the Lennard–Jones potential [26]. The simulation region is cuboids with unit lengths of 40, 25.75 and 5 (molecular units) along x-, y- and z-directions respectively as required in LAMMPS input file. Periodic boundary conditions are applied along x-and z-directions while in y-direction a non-periodic and shrink-wrapped boundary is applied. The shrink-wrapped boundary means that the position of the face is set so as to encompass the atoms in that direction; no matter how far they move [27].

Table 1 Reduced physical quantities
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In other words, when a particle enters or leaves the simulation region, an image particle leaves or enters this region, such that the number of particles in the simulation region is always constant [26]. In our simulation, we apply a constant force to each particle along the nanochannel (x-direction). For solid–liquid interactions, the following modified Lennard–Jones potential function is used [28]:

$$ \phi_{w} (r_{ij} ) = \left\{ {\left. {4\varepsilon_{sf} \left[ {\left( {\frac{{\sigma_{sf} }}{z}} \right)^{12} - \left( {\frac{{\sigma_{sf} }}{z}} \right)^{6} } \right]} \right\}} \right. $$
(2)

where the Lorenthz–Berthelot mixing rule [29] is applied to the system for calculating σ sf and ε sf

$$ \varepsilon_{Sf} = \sqrt {\varepsilon_{S} \varepsilon_{f} } ,\quad \sigma_{Sf} = \frac{{\sigma_{S} + \sigma_{f} }}{2} $$
(3)

When a two-body potential model is chosen, the interaction force between a pair of molecules can be derived from the potential model in such a way that the following relation could be used:

$$ \begin{aligned} \mathop F\limits^{ \to }_{i} & = \sum\limits_{i \ne j} {F_{ij} } + {\mathop {F_{ext}}\limits^{ \to }} = m_{i} \frac{{d^{2} r_{i} }}{{dt^{2} }} + {\mathop {F_{ext}}\limits^{ \to }} = m_{i} \frac{{dv_{i} }}{dt} +{\mathop {F_{ext}}\limits^{ \to }} \\ \mathop F\limits^{ \to }_{ij} & = - \overrightarrow {\nabla } (\phi + \phi_{w} ) \\ \end{aligned} $$
(4)

where \( {\mathop {F_{ext}}\limits^{ \to }} \) is the inlet external driving force per molecule in the x direction to characterize the forced flow in nanochannels. By integrating Eq. 4, the position vector r i and velocity vector v i could be found and then the thermodynamic properties of simulation domain can be calculated using r i and v i . Velocity rescaling is done through a velocity rescaling thermostat by LAMMPS which this thermostat is being used in all simulations in this paper to control the temperature. For velocity rescaling the velocities are first updated from the forces acting on the particles and then rescaled at each time step.

Since the total kinetic energy of the system fluctuates, the instantaneous temperature is defined as a fluctuating kinetic energy per particle at each slab, per degree of freedom.

It is obvious that a good intermolecular dynamics program needs a good algorithm to integrate the Newton’s equation of motion. The integration of the equation of motion is straightforward. Velocity Verlet method [30] is used for integration of the Newton’s equation.

The advantage of this algorithm is that the calculations of velocities are in phase with the positions. The initial condition for each molecule is usually assigned by using the Gaussian distribution based on the specified temperature.

The system reaches to equilibrium state after 500,000 time steps and then NEMD simulations are applied. In all simulation cases, more than 2,000,000 time steps were used for data sampling. The fluid is confined to flow between two solid planar walls parallel to xz plane as shown in Fig. 1a, b.

Fig. 1
figure 1

a 2D snapshot of a channel under the presence of nano scale surface roughness. b 2D snapshot of a channel under the presence of nano scale surface roughness

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Each wall consists of atoms forming two [100] planes of an FCC crystal and each atom have a mass of 1. The upper solid surface is smooth while the bottom wall is decorated with roughness with variable height as shown in Fig. 1. It should be noted that any argon atom within the simulation region have interactions with neighboring fluid and wall atoms. The applied force on the fluid atoms is specified to be 0.05 εσ −1 in the x-direction with constant temperature of 1.1 \( \varepsilon k_{B}^{ - 1} \). In this study the numerical experiments were carried out for the gap between roughness elements, roughness aspect ratio and attraction energy varying from d = 3L to d = 0.0, AR = 0.4–0.8σ and ε sf  = 0.2ε to ε sf  = 0.2ε.

3 Results and discussion

In any velocity profile, the colored panes represent the solid surface of the nanochannels and the vertical dashed lines show the height of surface roughness. Figure 2 shows the effects of the surface attraction energy on the velocity profile under the presence of roughness with both triangular and cylindrical roughness elements. As shown the fluid particles velocity near the rough wall in both cases tends to a zero value. This is due to the trapping of fluid particles between the roughness elements. Therefore, slip velocity and the maximum flow velocity decreases, which is due to the surface attraction energies. Figure 3 shows the average density profiles under various wall-fluid interaction parameters. As shown in both rough cases, the density fluctuation amplitude increases as surface attraction energy increases. Comparison of the surface roughness in y/σ = 6.25 with that of y/σ = 2.25, shows that the density value in y/σ = 6.25 is lower that of y/σ = 2.25. This is due to the reduction of surface effects. Figure 4 shows the effect of roughness height on the velocity profiles in both rough cases. As shown the slip velocity and the maximum flow velocity decrease while the roughness height increases. Furthermore, by trapping more particles inside the roughness elements the thickness of the fluid layers will increase. The fluid velocity in this layer is zero. Moreover, the distance that the velocity profile is deviated is equal to the roughness height. Figure 5 shows the fluid velocity profiles under various surface roughness heights in both rough cases. As shown by increasing the surface roughness height, the wall effects penetrate to a greater depth in the fluid inside the channel.

Fig. 2
figure 2

a Effects of surface attraction energies on the velocity profile in triangle geometry. b Effects of surface attraction energies on the velocity profile in cylinder geometry

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Fig. 3
figure 3

a Effects of surface attraction energies on the density profile in triangle geometry. b Effects of surface attraction energies on the density profile in cylinder geometry

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Fig. 4
figure 4

a Effects of height roughness on the velocity profile in triangle geometry. b Effects of height roughness on the velocity profile in cylinder geometry

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Fig. 5
figure 5

a Effects of height roughness on the density profile in triangle geometry. b Effects of height roughness on the density profile in cylinder geometry

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Figure 6 shows a 2D snapshot of the gap between the triangular and the cylindrical roughness elements, where d is gap between roughness elements and L is the base length of the surface roughness. Figure 7 shows that by increasing the number of roughness elements in a unit length in cases (decrease of gap between roughness elements, which have been shown in Fig. 6), the maximum velocity, and the slip velocity will decrees. This is due to the fact that by reducing the gap between the roughness elements; the numbers of particles which are trapped between the roughness elements have lower freedom. Therefore, their velocity will be equal to wall velocity. Figure 8 shows that, in both roughness geometries, when the gap between roughness elements is equal to zero, the first layer of fluid has the lowest density. It also shows that by moving away further from the wall surface the fluid layers become denser. Eventually they converge to the fluid density in y/σ = 8.25. This is due to the fact that, the fluid particles near the rough wall surface, are influenced by stronger repulsion force, and therefore, the particles density near the surface becomes less than that of the other layers of the fluid. Going away further from the channel surface, the gap between the roughness elements increases, therefore, the repulsion force between particles is reduced, and the fluid density increases. When the gap between the roughness elements is 3L, the layer of the fluid has the highest density. This is due to the fact that the reduced repulsive forces between particles which are due to the increase of the gap between roughness elements, causes the density of the first layer of fluid near the wall to decrease. Moreover, the increased gap between the roughness elements decreases the amplitude of the density fluctuations. This is due to the fact that in this case the particles can move more freely.

Fig. 6
figure 6

2D snapshot of gap between triangle and cylinder roughness elements

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Fig. 7
figure 7

a Gap between roughness elements d on the velocity profile in triangle geometry. b Gap between roughness elements d on the velocity profile in cylinder geometry

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Fig. 8
figure 8

a Effects gap between roughness elements on the density profile in triangle geometry. b Effects gap between roughness elements on the density profile in cylinder geometry

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According to Fig. 7 although by increasing the depth of energy and the velocity decreases, the changes in the ratio of energy depth on the velocity are insignificant. This is due to the fact that the increase in depth of energy causes more particles of the fluid to trap between the roughness elements. This, furthermore, leads to an increase in repulsive force between particles. Hence, the fluid particles are affected by both the interaction forces between fluid particles and wall, and the repulsion force between particles. This is the reason why, for high roughness ratios, the effect of interaction force decreases.

3.1 Comparison of a rough surface with cylindrical and triangular roughness elements with a flat channel

Here two roughness geometries are compared with a case that the distance d within roughness elements is zero. According to Fig. 9 the amplitude of the fluctuations near the wall for cylindrical roughness elements is more than that of triangular elements which is a sign of fluid particles trapping between the cylindrical elements. As Fig. 10 shows the maximum velocity in a surface with triangular roughness elements is more than that of a surface with cylindrical roughness elements. This shows that the cylindrical surface roughness affects the flow velocity more than the triangular geometry. This is to the fact that the number of particles that are trapped between cylindrical roughness elements is more than that of triangular elements. Therefore, more fluid particles will have near zero velocities.

Fig. 9
figure 9

Comparison density profile in cylindrical and triangular surface roughness

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Fig. 10
figure 10

Velocity profile for cylindrical and triangular roughness surface in comparison with smooth channels

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4 Conclusions

In this study, the non-equilibrium molecular dynamics simulation of Poiseuille flow of liquid argon in a rough nanochannel is presented. The effects of triangular and cylindrical surface roughness elements under various conditions were studied. These conditions include the surface roughness height, the gap between roughness elements and the surface attraction energy. A summary of some important results obtained as follows:

  • By reducing the gap between the roughness elements the maximum velocity and slip velocity decreases.

  • The effect of reducing the gap between the roughness elements in a surface with triangular roughness elements is more than that of cylindrical one.

  • In comparison with smooth channel, the cylindrical roughness surface has more effect on the flow velocity. This causes more particles to trap between the roughnesses elements near the wall.

  • Increasing the roughness height causes the density fluctuation to penetrate more fluid in depth.

  • Penetration depth of the density fluctuation into the fluid domain is independent of the type of the roughness elements.

The extension of this paper and our previous works [31–34] affords engineers a good option for nanochannel simulation in one and two phase flows. Also, future studies aimed a elucidating the precise nature of boiling flow and two phase heat transfer would be of considerable interest.