- Email: [email protected]
Molecular dynamics simulation of Couette and Poiseuille Water-Copper nanofluid flows in rough and smooth nanochannels with different roughness configurations
Chemical Physics 527 (2019) 110505
Contents lists available at ScienceDirect
Chemical Physics journal homepage: www.elsevier.com/locate/chemphys
Molecular dynamics simulation of Couette and Poiseuille Water-Copper nanofluid flows in rough and smooth nanochannels with different roughness configurations
T
Davood Toghraiea, Maboud Hekmatifarb, Yasaman Salehipoura, Masoud Afrandc,d,
⁎
a
Department of Mechanical Engineering, Khomeinishahr Branch, Islamic Azad University, Khomeinishahr, Iran Young Researchers and Elite Club, Khomeinishahr Branch, Islamic Azad University, Khomeinishahr, Iran c Laboratory of Magnetism and Magnetic Materials, Advanced Institute of Materials Science, Ton Duc Thang University, Ho Chi Minh City, Vietnam d Faculty of Applied Sciences, Ton Duc Thang University, Ho Chi Minh City, Vietnam b
ARTICLE INFO
ABSTRACT
Keywords: Molecular dynamics simulation Couette flow Poiseuille flow Nanofluid Nanochannel
In this study, molecular dynamics simulation of Couette and Poiseuille Water-Copper nanofluid flows in rough and smooth nanochannels was performed. The Lennard-Jones equation is considered as Water-Water intermolecular interaction, while Hamaker’s equation is considered to be the interaction between Water-Copper and Copper-Copper particles. PPPM algorithm is used to calculate the electric potential. It is concluded that increasing the channel height reduces the effect of the surface on the fluid and reduces the flow rate of the nanofluid. Also, the slip velocities on the bottom and top walls remain almost the same. Furthermore, nanoparticles have caused fluctuations in the middle area, which are due to the effects of the surface of the nanoparticles relative to the base fluid of the Water. As expected, the presence of nanoparticles in the middle area and the interaction between the surface and the fluid in this area has caused abnormal fluctuations.
1. Introduction Many studies have been conducted in the field of molecular simulations of macroscopic nanofluids. Researchers have also begun nanoscale researches due to recent applications of nanofluids in some new engineering fields. As a result of extensive development in the field of powerful computing, software systems have been enhanced using numerical solutions and simulation tools. Undoubtedly, the molecular dynamics simulation is one of these tools, which has had a great impact on many fields, such as nanotechnology, heat transfer, fluid mechanics, and physics. Molecular dynamics simulation (MDS) is a bridge between theory and experiment, therefore molecular simulations are employed using theoretical models. In the past decade, the number of published articles on nanofluids has almost doubled every two years, indicating an increase in the interest of researchers in this topic [1–8]. Harmon et al. [9] performed a molecular dynamics simulation of flow past a plate. At fluid velocities large enough to obtain an adequate signal to noise resolution, two counter-circulating vortices were observed behind the obstruction. Xu and Zhou [10] studied liquid argon flow at Platinum surfaces by using MDS. They found out that with an increase in the shear rate, the viscosity increased and the non-
⁎
Newtonian flow appeared. Cao et al. [11] studied the effect of surface roughness on gas flow in microchannels by molecular dynamics simulation. They concluded that the geometry roughness also shows significant effects on the boundary conditions and the friction characteristics. Kim and Darve [12] simulated the electro-osmotic flows in rough wall nanochannels by molecular dynamics simulation. They concluded that along the flow direction, the diffusion of water and ions inside the groove is significantly lowered while it is similar to the bulk value elsewhere. Thomas et al. [13] studied pressure-driven water flow through carbon nanotubes: Insights from molecular dynamics simulation. They predicted the variation of water viscosity and slip length with CNT diameter. Kamali and Kharazmi [14] investigated the surface roughness effects on nanoscale flows by using MDS. The effects of surface roughness and cavitation on the velocity distribution of hydrophobic and hydrophilic wall undergoing Poiseuille flow were presented. Sun et al. [15] simulated the nanofluid’s effective thermal conductivity in high-shear-rate Couette flow. They found out that the conventional correlation is not suitable when the sizes of the suspended particles are reduced to nanometers. Pourali and Maghari [16] investigated non-equilibrium molecular dynamics simulation of thermal conductivity and thermal diffusion of binary mixtures confined in a
Corresponding author at: Ton Duc Thang University, Ho Chi Minh City, Vietnam. E-mail address: [email protected] (M. Afrand).
https://doi.org/10.1016/j.chemphys.2019.110505 Received 2 February 2019; Received in revised form 12 August 2019; Accepted 19 August 2019 Available online 21 August 2019 0301-0104/ © 2019 Elsevier B.V. All rights reserved.
Chemical Physics 527 (2019) 110505
D. Toghraie, et al.
nanochannel. They found out that in very narrow channels, thermal diffusion is small, and it reaches a steady-state value with increasing the channel width. Noorian et al. [8] investigated the effects of checker surface roughness geometry on the flow of liquid argon through nanochannel when the roughness is implemented on the lower channel wall. They found out that as the surface attraction energy or the roughness height increases, the density layering in the near wall is enhanced by higher values or secondary layering phenomena. Dissolutive flow in nanochannels was studied by Miao et al. [17]. Their results showed that in pressure-driven flow, when the dissolubility is low, the dominant dissipation is the viscous dissipation and the theoretical model of insolubility is acceptable. Multiscale dynamic wetting of a droplet on a lyophilic pillar-arrayed surface was carried out by Yuan and Zhao [18]. They used a multiscale combination method of experiments and molecular dynamics simulations.
total number of degrees of freedom to 6. For a particle or quasi-particle like Copper nanoparticles, the number of degrees of freedom is 3. The instantaneous temperature fluctuates like the total kinetic energy and is obtained from the following equation [2], N
T (t ) = i
mi vi2 (t ) kB Nsf
(8)
where Nsf is the total degrees of freedom of the system. The temperature fluctuation is at the order1/ Nfs , i.e., at 102–103. The pressure is calculated using the Virial equation of state [2],
P = kT
rij
3N
i< j
dU (rij ) drij
(9)
The potential force between the particles U is considered for each pair of particles. Therefore, for N particles, the interaction energy is the sum of each pair of particles [2,19–23],
(r N )
2. Theoretical and methodological framework 2.1. Intermolecular interactions
U (r N ) =
The Lennard-Jones equation is considered as Water-Water intermolecular interaction, while Hamaker’s equation is considered to be the interaction between Water-Copper and Copper-Copper particles. The location of ri (t) (t) and momentum pi (t) are obtained by solving the Newton’s equation of motion (2–1) with the initial position and momentum [1],
The most important part of the simulation of a nano-sized system is the correct choice of potential function to minimize the conflicts between nano-system compounds. In this project, the modeling system is a Water-Copper nanofluid and is modeled using molecular dynamics simulation. To model the potential between Water-Water particles, the Lennard -Jones potential function with the SPC/E model is used. Water (H2O) has two Hydrogen atoms, which are associated with an Oxygen atom through a covalent bond.
Fi = mi ai = mi
d2ri dt 2
(1)
where ri is the position of the particle i and mi is the mass of the particle i and ai is the acceleration of particle i. Momentum pi can be defined as follows [2],
The Hydrogen particles have a positive charge and the Oxygen particle has a negative charge. The Coulomb’s term was added to the Lennard-Jones potential in order to have long-range interaction forces that are due to the positive and negative electric charge interactions. The Lennard-Jones potential is used for interaction between particles of Oxygen with another Oxygen particle in the following equations [15],
where vi is the velocity of the particle i. For an isolated system, the total energy (E), which contains kinetic energy and potential energy of the particles or molecules, is constant. The total energy (E) can be expressed in Hamilton form. For N spherical particles, the Hamilton equation follows the following equation,
Pi2 + U (r1 + r2+
+ rn ) = E
i
U (r ) = 4
(3)
pi =
m i ai =
Fi
v ( t + t ) = v (t ) +
a ( t ) + a (t + t ) t + O ( t 2) 2
qo = oo
OO
rOO
rOO
3
6
3
+ ke a=1 b =1 2
qa qb rab
(11)
0.8475 |e|,
= 3.166Å,
qH = 0.4238 |e| oo
= 0.15535kcal/mol
In this paper, PPPM algorithm, which is more appropriate in terms of computational cost than the Ewald method, is used to calculate the electric potential. This method reduces the computing time in comparison to the Ewald sum method. The PPPM method is in the order of O (N log N) and calculates the long-range interaction forces in three stages. The tolerance value of 10-4 is considered to be used in the PPPM method.
(5) (6)
Simulation begins with the initial position and velocity of particles and considering the average temperature. The average kinetic energy for each degree of freedom is as follows [2],
1 2 1 mv = kb T 2 2
12
ke = 8.99 × 109N. m2/C2
(4)
1 a (t ) t 2 + O ( t 4 ) 2
OO
qa and qb are the charges of particles a and b, ke is the Coulomb constant and rab is the distance between the two particles. Coulomb constant, Coulomb and Lennard-Jones parameters are as follows,
In molecular dynamics simulation, the Newton's motion equation (4) can be solved by the algorithm and relay [2],
r ( t + t ) = r ( t ) + v (t ) t +
OO
1
Also, the forces of each particle with potential function are as follows [2],
dU = dri
(10)
2.2. Properties of the SPC/E molecule model
(2)
Pi = mi vi
1 H(r , p) = 2m
u (rij ) i
2.3. Water-Copper interatomic forces
(7)
Based on the equilibrium theorem, the kinetic energy of motion of particles is distributed among all available active degrees of freedom for particles. In this project, the flexible Water molecule has 9 degrees of freedom, i.e., 3 transfer degrees of freedom, 3 rotation degrees of freedom and 3 vibration degrees of freedom. The hardness of the Water molecule in the bonds and angles eliminates vibrations and reduces the
Water-Copper is the second type of interatomic force existing in the nanofluid. The force between the colloid particles (large particles) and the solvent is as follows [15]
U=
2
2Acs a3 3 1 9(a2 r 2)3
(5a6 + 45a4r 2 + 63a2r 4 + 15r 6) 15(a r )6 (a + r )6
6
(12)
Chemical Physics 527 (2019) 110505
D. Toghraie, et al.
where Acs is the Hamaker’s constant, which is equal to 43.2 kcal/mol for Water and a is the radius of Copper colloid. Sigma (σ) is obtained from arithmetic mean of sigma of Copper and Water nanoparticles.
temperature by the relaxation rate square [15] N
HNose
2a1 a2 r2 + ln 2 (a1 a2 )2 r
2a1 a2 + 2 (a1 + a2 ) 2 r
nf
UR = +
A cc 37800
r2
6(a12
7r (a1 + a2) + (r a1
+ 7a1 a2 + a2 )7
a22 )
r 2 + 7r (a1 + a2) + 6(a12 + 7a1 a2 + a22 ) (r + a1 + a2 )7 r 2 + 7r (a1
a2) + 6(a12 7a1 a2 + a22 ) (r + a1 a2 )7
r2
a2) + 6(a12 7a1 a2 + a22 ) (r a1 + a2 )7
7r (a1
U = UA + UR
(14)
=
p
+ (1
)
(15)
bf
In this regard, p and bf are the densities of the nanoparticles and base fluid, respectively. The density of Copper nanoparticles is ρp = 8.94 g/cm3 the Water density is ρbf = 1 g/cm3 [25–30]. Considering SPC/E rigid Water model, the SHAKE algorithm has been used with a time step of 0.05 fs and 5 × 105 steps. The timed step is carefully selected to remove any round-off or truncation errors. According to the papers presented in this field, different values have been selected for the time step, and the most accurate and stable step is obtained at 0.05 fs. All simulations carried out in this research are based on dimensionless values and for making various dimensionless parameters, Table 1 has been used.
(a1 + a2 ) 2 (a1 a2 ) 2 (13a)
6
pi p2 s + U (r N ) + + gkblns 2 2 mi s 2Q
To calculate the density of nanofluids with different percentages of volume fraction of nanoparticles, we use the following equation [15]
The Hamaker’s formula for colloid-colloid was used to model the interaction potential between the Copper nanoparticles [15]
Ac 6 r2
= i=1
2.4. Copper-Copper interatomic forces
UA =
Hoover
3. Results and discussion
(13b) (14)
Molecular dynamics simulation on a three-dimensional system of Water-Copper nanofluid in a Copper nanochannel with dimensions of 4 × 4 × 4 nm3 with different types of surface roughness geometry has been used, which is shown in Fig. 1.
where Acc is Hamaker’s constant, Acs = 107.83 kcal/mol and a1and a2 are the radius of each of the nanoparticles, respectively. 2.5. Molecular dynamics simulation tools
3.1. Couette flow
Large-scale atomic/molecular massively parallel simulator (LAMMPS) is intended to simulate Water-Copper nanofluid system. To solve the motion equations, the initial molecular structure and initial velocity of each particle must be specified. Therefore, the particle spatial path can be observed during simulation. The open-source Packmol software has been used to model limited Water molecules in the nanochannel and the arrangement of Copper nanoparticles in it. The simulation force coefficients have been derived from the Charmm code to calculate the interaction of Hydrogen and Oxygen particles in the Water molecule. After deriving these coefficients and producing the input code of a single Water molecule for the Packmol software, we are going to produce Copper nanoparticles. Modeling Copper nanoparticles is also done using LAMMPS software. The Packmol software uses inputs created for a Water molecule and a nanoparticle and two bottom-up walls fixed at the beginning and end of y-axis of simulation box and, therefore, they put a number of molecules adjusted for Water and a number of nanoparticles designated for Copper nanoparticles heterogeneously between the two walls. The coordinates of initial positions of all particles are created as input data for the LAMMPS program. In addition, the number of molecules selected at each stage of the simulation depends on the different conditions of roughness of the surface and the result of a change in the volume between the two walls and nanoparticles. The initial particle velocity must be determined by selecting the initial temperature set at room temperature (300 K). The initial velocity of particles is randomly selected from the Gaussian distribution. The canonical ensemble (NVT) is used to set the simulated system and have constant number of molecules, constant volume and constant temperature. The total temperature of the system can be set using one of the thermostat methods. Energy and temperature are related, resultantly, the temperature should be kept constant during the simulation. Using a thermostat prevents temperature drifts due to the accumulation of numerical errors during molecular dynamics simulations. The Nosé-Hoover thermostat algorithm has been used to control the temperature at 300 K while keeping the volume constant. This algorithm adds the friction term to the Newton’s equation. In a period of time, changes in the friction coefficient are extracted as relative deviation of the set value of the
3.1.1. Validation of Couette flow According to previous investigations in the field of simulation of flow in nanochannels, simulation of Water-Copper nanofluid flow in nanochannel has not been done so far. To validate the velocity diagram, we have used the results of various simulations that are available in other studies of the researchers. One closely related work in this regard is Jabarzadeh et al. [24] on molecular dynamics simulation of the slip Couette flow of alkanes on a sinusoidal wall. Some of the results are shown in Fig. 2. The result obtained for the Couette flow of nanofluid in this project is almost the same with some differences compared to the probable chart regarding the past investigations. We will continue to discuss these differences and justify them. 3.1.2. Couette flow in a smooth channel In the smooth channel simulation, 529 Copper atoms are considered for each of the bottom and top walls regarding the type of lattice Table 1 Necessary relationships for making dimensionless parameters. Quantity name
Formula
Distance
r =
r
Temperature
T =
kB T
Density
m
Pressure
P =
Time
t =
Volume
V =
V 3
Power
F =
F
Velocity
v =v
Acceleration
3
3
=
a =
P 3 t m
m
a m
Chemical Physics 527 (2019) 110505
D. Toghraie, et al.
Fig. 1. Simulation of the initial geometry of the Water-Copper nanofluid.
structure, lattice constant, density, and molar mass of Copper. Bottom and top wall thickness is 2 Å. One percent of the finite volume between the walls is considered for Copper nanoparticles, and 55 atoms of Copper are placed in this area. Using the Packmol tools, the number of Water molecules for embedding in the finite distance between the nanoparticles and walls is 1907. Moreover, the top wall velocity for the reference mode of Couette flow is considered to be three-dimensional. 3.1.2.1. Effect of channel height. In Fig. 3 the effect of channel height changes on (a) density distribution and (b) velocity distribution in the width of smooth channel is presented. The sharp fluctuations created at side of the lower wall are due to the presence of nanoparticles and the interaction between the two surfaces of the nanoparticle and the lower wall. An increase in the channel height does not affect the amplitude of the density distribution fluctuations of the near-wall areas. Moreover, the level of penetration of density layering remains constant. Depending on the fact that the layering phenomenon depends on the interaction between the wall particles and the fluid near the surface and also on the fluid density, maintaining the degree of the layering penetration is justified. By increasing the channel’s height, the effects of the surface is reduced and depending on the type of flow, the momentum transfer is done through a high surface. Therefore, increasing the channel height reduces the effect of the surface on the fluid and reduces the flow rate of the nanofluid. The slip velocity on the bottom and top wall remains almost the same. A small fracture near the bottom wall is due to the presence of nanoparticles which is located at a short distance from the surface and somehow increases the viscosity and reduces the velocity. Fig. 3. Effect of channel height changes on (a) density distribution and (b) velocity distribution in the width of smooth channel.
3.1.2.2. Effect of initial wall velocity. Fig. 4 shows the effect of the initial velocity change on the (a) density distribution and (b) velocity
Fig. 2. The geometry of the simulation system and the fluid velocity graph in the Jabbarzadeh et al.’s study [] 4
Chemical Physics 527 (2019) 110505
D. Toghraie, et al.
Fig. 4. The effect of the initial velocity change on the (a) density distribution and (b) velocity distribution in the smooth channel width.
Fig. 5. The effect of the change in the volume fraction of nanoparticles on (a) density distribution and (b) velocity distribution in the smooth channel width.
distribution in the smooth channel width. The presence of nanoparticles in the area between two walls has been shown in the density distribution diagram as a sudden increase in the density and also the increase in the amplitude of fluctuations. An increase in the magnitude of the fluctuations near the high surface indicates that as the effect of the surface on the fluid increases, the top wall velocity decreases. Increasing the amplitude of the fluctuations near the bottom wall is not regular and does not follow the high wall velocity reduction. Moreover, the presence of nanoparticles near this surface has caused a sudden increase in fluctuations in this area and is the cause of non-uniformity. Increasing the surface effect can also be found in the velocity distribution. One of the possible reasons for this phenomenon is the freedom to operate more fluid molecules near the walls with the lowest wall velocity as a current stimulus. Fluid molecules near the walls with the lowest top wall velocity act more freely as flow stimulus. Increasing the slip velocity is another clear reason for increasing the surface effect with the top wall velocity reduction. This can be obtained from the velocity distribution chart in Fig. 4.
3.1.2.3. Effect of volume fraction of nanoparticles. Fig. 5 shows the effect of the change in the volume fraction of nanoparticles on (a) density distribution and (b) velocity distribution in the smooth channel width. As shown in Fig. 5, as the volume fraction of nanoparticles increases, the density fluctuations in near-wall areas decrease. In addition, as the volume fraction of the nanoparticles increases, the density fluctuations penetrate into more layers of the fluid. The presence of sharp fluctuations in the middle area of the channel is also due to an increase in the surface relative to the volume in the area. However, the irregular increase in the density fluctuations in areas further away from the walls in different modes of simulation is due to the heterogeneous distribution of nanoparticles in the channel. The reduction in the average velocity along the channel width is clearly evident, which is due to the presence of nanoparticles and to some extent is because of the reduction in nanofluid viscosity in the area. 5
Chemical Physics 527 (2019) 110505
D. Toghraie, et al.
Fig. 6. Modeling of Water-Copper nanofluid in a channel with rectangular roughness.
3.1.3. Couette flow in a rough channel with rectangular roughness The channel roughness, as shown in Fig. 6, is drawn with a rectangular cross-section perpendicular to the flow direction. By adding two roughness elements into each nanochannel wall, the number of atoms in the wall becomes 667. Furthermore, the diameter of the nanoparticle is reduced, and as a result, the number of atoms of each nanoparticle becomes 43. Furthermore, 1813 Water molecules are selected for the input code of the Packmol software. It should be noted that the number of molecules intended for Water is only for rectangular roughness reference mode. Moreover, for different modes of simulation, it is chosen according to the finite volume between the walls and nanoparticles. 3.1.3.1. Effect of initial wall velocity. Fig. 7 shows the effect of initial wall velocity change as a flow stimulus on the (a) density profile and (b) velocity profile in the channel width. In Fig. 7, the reduction in velocity results in an increase in the amplitude of the fluctuations near the wall. However, the amplitude of these fluctuations near the top wall is lesser than the bottom wall. The reason for this increase is the trapping of more nanofluid molecules among roughnesses. As the number of molecules increases between roughnesses, the repulsive intermolecular forces increase and cause the molecules to suddenly set a distance away from each other. Therefore, the density number in these areas is oscillating between low and high. Another variation of the current nanofluid density diagram in a nano-channel with a similar ordinary fluid flow (eg, liquid Argon) is the fluctuations in the mid-channel. Although the amplitude of these fluctuations is low, it is different from the uniformity of the density profile that exists in the middle of the channel with ordinary fluid flow. The presence of nanoparticles and the forces of interaction between the surface of the fluid in this area can be considered as the best reason for these fluctuations. However, due to the low ratio of the outer surface of the nanoparticles to the surface of the top and bottom walls, the amplitude of the oscillations in this area is less. According to Fig. 7(b), we can clearly see the effect of roughness on the velocity graph, especially near the bottom wall. The lack of presence of nanoparticles in the roughness has resulted in the greater effectiveness of the Waterbased fluid particles. Moreover, by increasing the initial velocity of the wall and more momentum transfer of top layers, the slip velocity increases. Slightly higher than the top wall roughness, with the presence of nanoparticles, the average velocity decreases, however, this reduction has emerged near the bottom wall as a fracture in the velocity chart. In the vicinity of the top wall, this change is gradual and this is due to the type of flow and momentum transfer from the top wall, which results in the presence of stronger interaction forces on the base molecules and thus results in lower degree of effectiveness from nanoparticles.
Fig. 7. The effect of initial wall velocity change as a flow stimulus on the (a) density profile and (b) velocity profile in the channel width.
3.1.3.2. Effect of nanoparticle volume fraction. Fig. 8 shows the effect of changing the nanoparticle volume fraction on the (a) density profile and (b) velocity profile in the channel width, According to the density distribution diagram in Fig. 8; the change in the nanoparticle volume fraction does not make much change at the fluctuation range of the near-wall areas. By increasing the volume fraction of nanoparticles,
uniform convergence of the density has decreased in the middle of the graph. While the mean density increases in the middle area of the channel, fluctuations in this area also increase due to increased surface reaction force relative to the base fluid of the fluctuations. The mean density of the nanofluid in the areas near the surface is affected by the 6
Chemical Physics 527 (2019) 110505
D. Toghraie, et al.
Fig. 10. The effect of changing the distance between roughnesses on the (a) density profile and (b) the velocity profile in the channel width. Fig. 8. The effect of changing the nanoparticle volume fraction on the (a) density profile and (b) velocity profile in the channel width.
nanoparticles, momentum transfer occurs from top to bottom layers. This momentum transfer in the areas between surface roughnesses where there are no nanoparticles is more evident and appears as a fracture in the velocity graph. This fracture has been less obvious next to the near-wall areas due to the transfer of the velocity from the wall to
repulsive force between the nanoparticles and the surface. In the velocity chart, the mean velocity decrease is clearly evident across the entire channel width. By increasing the volume fraction of
Fig. 9. Changes in the distance between rectangular roughness elements. 7
Chemical Physics 527 (2019) 110505
D. Toghraie, et al.
the fluid and the presence of higher pressure on the fluid molecules. However, by increasing the fraction of the nanoparticles, this fracture near the top wall becomes gradually more obvious and indicates a greater degree of effectiveness of the flow from surface roughness. 3.1.3.3. The distance between roughness elements. In this section, by preserving the primary dimensions of the rectangular roughness, we increase the number of roughness elements on each surface by dividing the distance between the elements. Resultantly, we can see the effect of reducing the distance between roughness elements. The simulated geometry is shown in Fig. 9. Fig. 10 shows the effect of changing the distance between roughnesses on the (a) density profile and (b) the velocity profile in the channel width. Considering this Figure, we find that the increase in the number of roughness elements on the surface reduces the amplitude of the fluctuations in the near-wall areas. Moreover, as the repulsive force between the particles increases, the mean density in this area decreases. The density profile is not uniform in the middle areas of the channel which is due to the presence of nanoparticles. However, as a result of the averaging of the values obtained for the nanofluid density in the distance between roughness surface of the top and bottom walls shows the value of 0.982 g/cm3 for the mean density. And this value, regardless of the surface effects in the nanoscale, is about 6% different from the theoretical values. Reduction in the slip velocity is clearly evident in the velocity graph of Figs. 4–14 in the near-wall areas. Reducing the number of fluid molecules in this area not only reduces the slip velocity, but also the fracture in the area near the bottom fixed wall also becomes less sloping. This indicates an increase in the surface effect on the distribution of the nanofluid flow rate in the channel. From increasing the impact of the surface by reducing the distance between roughness elements, we conclude that in the top area of the channel (mid-to-top), the nanoscale velocity distribution in the nanochannel with the highest number of roughnesses has the higher mean velocity and in the bottom area of channel area (mid-to-bottom), the mean velocity is lower. Therefore, in the middle area where this velocity variation occurs, there is a share point of velocity among the velocity distribution graphs. This share point is inclined to the top wall. This is due to the type of flow because the momentum transfer from the top surface will increase the force on the trapped particles among roughnesses. However, the momentum transfer from the top surface to the bottom layers becomes less, Fig. 12. The effect of channel height changes on (a) density distribution and (b) velocity distribution in channel width.
and as a result, the velocity zero of bottom wall penetrates into more fluid layers. As a result, surface effects from bottom to top penetrate into more fluid layers. 3.2. Poiseuille flow 3.2.1. Validation To validate the velocity diagram, we have used the results of various simulations that are available in other studies of the researchers (Fig. 11). One closely related work in this field is Jeremy Freid’s study [] in the field of molecular dynamics simulation of Poiseuille flow for fluid Argon on a wall with a rectangular, triangular roughness. 3.2.2. Poiseuille flow in a smooth channel Fig. 12 shows the effect of channel height changes on (a) density distribution and (b) velocity distribution in channel width. In Fig. 12(a), constant fluctuation amplitude in the near-wall areas indicates the ineffectiveness of increasing the channel height on the effects of surface and fluid relative to each other. The phenomenon of density layering and increasing or decreasing fluctuations depends on
Fig. 11. Diagram of fluid flow velocity distribution in Jeremy Fried [] and present work. 8
Chemical Physics 527 (2019) 110505
D. Toghraie, et al.
Fig. 13. The effect of nanoparticle volume fraction change on (a) density distribution and (b) velocity distribution in the channel width.
Fig. 14. The effect of external force changes on (a) density distribution and (b) velocity distribution in the channel width.
changing the various surface and fluid parameters at their contact surface. In this case, surface and fluid components at the contact surface will not change. According to the velocity distribution diagram of Fig. 12, increasing the maximum velocity by increasing the channel’s height indicates greater freedom of action in central layers where maximum velocity occurs. In fact, the higher the channel’s height, the lower the effect of the surface on the fluid. The drop in mean velocity in the channel width during simulation is due to the presence of nanoparticles. The fracture at the beginning of the diagram is also due to the close proximity of the nanoparticle to the surface of the bottom wall during the solution.
experienced sharp fluctuations in this area. The reduction in fluctuations near the top wall also indicates less momentum transfer in the 3 percent volume fraction of nanoparticles. This reduction in momentum transfer in the velocity distribution diagram is also shown as the reduction in mean velocity. The fitting the results for different situations is something that has not been done in this work. Therefore, the fluctuations or abnormal fractures that emerge in the diagram due to the presence of nanoparticles can be clearly seen in the results of smooth nanochannel as well as the results of rough nanochannels. 3.2.3. Poiseuille flow in a channel with rectangular roughness Fig. 14 shows the effect of external force changes on (a) density distribution and (b) velocity distribution in the channel width. As shown in the velocity distribution diagram of Fig. 14, the slip velocity decreases as the external force increases. The fracture in the velocity distribution diagram occurs slightly after the surface roughness, which is due to the presence of nanoparticles. In fact, with an increase in external force, the amount of this fracture in the diagram reduces
3.2.2.1. Effect of nanoparticle volume fraction. Fig. 13 shows the effect of nanoparticle volume fraction change on (a) density distribution and (b) velocity distribution in the channel width. As evident from the sharp increase in fluctuations near the bottom wall of Fig. 13(a), particle agglomeration near the bottom wall has caused a sharp increase in surface-to-volume ratio in these areas; therefore, the diagram has 9
Chemical Physics 527 (2019) 110505
D. Toghraie, et al.
Fig. 15. The effect of nanoparticle volume fraction changes on (a) density distribution and (b) velocity distribution in the channel width.
Fig. 16. The effect of change in the distance between roughness elements on (a) the density distribution and (b) velocity distribution in the channel width.
because the increase in external force reduces the freedom of movement of fluid particles between roughnesses and because, in fact, the cause of this fracture is the reduction in the freedom of movement of fluid particles after the roughness surface, with the presence of nanoparticles. Therefore, in the case of the highest external force, this decrease in freedom becomes less pronounced and the fracture at the beginning and end of the velocity distribution diagram will occur with fewer slopes. Due to the presence of nanoparticles near the bottom wall and the reduction of the freedom of movement of fluid particles in areas near the bottom of the channel, fracture in these areas has a relatively higher slope.
reason why we see a change in the fluctuation amplitude near the wall, and we do not see any changes near the top wall. In fact, in this case, because the particles of the wall surface and the fluid do not undergo any particular changes; therefore, there is no particular change in the amplitude of near-wall fluctuations. However, in the middle of the channel, by increasing volume fraction of nanoparticles, the surface-tovolume ratio increases, and thus the amplitude of the fluctuations increase sharply. In general, the fluctuations in the near-surface areas remain constant unless the nanoparticles are located in these areas. By increasing the nanoparticle volume fraction, the maximum velocity decreases. This shows that as viscosity increases, the number of nanoparticles increases, too. This increased viscosity reduces slip velocity. Thus, by increasing volume fraction of nanoparticles, the effects of surface on fluid increase.
3.2.3.1. Effect of nanoparticle volume fraction. Fig. 15 shows the effect of nanoparticle volume fraction changes on (a) density distribution and (b) velocity distribution in the channel width. It is evident from the density distribution diagram of Fig. 15 that the change in the percentage of nanoparticle volume fraction does not make any changes in the amplitude of near-walls fluctuations. The density depends on the degree of surface and fluid interaction. This is the
3.2.3.2. The distance between roughness elements. Fig. 16 shows the effect of change in the distance between roughness elements on (a) the density distribution and (b) velocity distribution in the channel 10
Chemical Physics 527 (2019) 110505
D. Toghraie, et al.
width. Reducing the density fluctuations amplitude by increasing the number of roughness elements is due to the increased repulsive force between molecules and the possible presence of fewer molecules among roughnesses. In the area after roughness, due to the sharp drop of the repulsive force between particles, the density of the particles suddenly rises. As expected, in Fig. 16(a) the amplitude of fluctuations in the middle layers is oscillating due to the presence of nanoparticles in these areas. Having higher maximum velocity in the reference mode means considering the four roughness elements on the surface due to the lower surface-to-volume ratio compared to the other modes. This means that the greater distance between the roughness elements results in more freedom of movements of the fluid particles in this area and increases the slip velocity in this area. This increase in velocity in the area between the roughness elements causes a fracture with a greater slope on the surface of the roughness. Therefore, the increase of the roughness elements on the surface is accompanied by reducing the effect of the surface on nanofluid; whereas, the effect of the presence of roughness element on the velocity distribution, especially on the maximum velocity, as well as on the fracture slope of the velocity distribution diagram on the roughness surface. 3.2.3.3. Height of roughness elements. Fig. 17 shows the effect of roughness height change on (a) the density distribution and (b) velocity in the channel width. One of the effects of the change in the height of surface roughness on the flow properties is the greater penetration of the density layering in the layers farther from the base wall surface, as can be seen in the density diagram of Fig. 17. By increasing the roughness height, the likelihood of the presence of nanoparticles in roughness increases. Embedding nanoparticles between roughness elements is due to the relatively high surface-tofluid ratio, which results in very high frequency fluctuations in these areas. It is easy to see these fluctuations near the bottom surface. It can be concluded that in nanofluid flow in rough nanochannel, the presence of nanoparticles in each channel area results in non-uniform fluctuations, which violates the prior assumptions of researchers in predicting fluid flow properties in nanochannels. By using nanofluid instead of ordinary fluid, the violation of the prior research in predicting the velocity distribution in the nanochannel is clearly evident at the beginning and end of the velocity distribution diagram. By increasing roughness height, more fluid particles are trapped between roughnesses. Therefore, the slip velocity decreases and with the penetration of the velocity zero of top and bottom wall into the layers farther away from the maximum velocity surface also reduces. Fig. 17. The effect of roughness height change on (a) the density distribution and (b) velocity in the channel width.
3.2.4. Poiseuille flow in a channel with triangular roughness The simulation of the particle’s initial location and the geometry is exactly equal to the same model in the Couette flow.
heterogeneous distribution of nanoparticles in the base fluid causes the placement of two nanoparticles near the bottom wall and these results in high increase in surface-to-fluid ratio in this area. The intense fluctuations created are due to intense interaction between Water molecules and wall surfaces and nanoparticles. Almost without considering the effects of nanoparticles, the amplitude of fluctuations near the walls has not changed much, which to some extent can be seen near the wall surface. The presence of fluctuations in the middle area of the channel is also due to the effects of increasing the surface-to-fluid ratio in these areas, which increases with an increase in the volume fraction of the nanoparticles. The maximum velocity drop is clearly evident in the velocity distribution diagram by increasing the nanoparticle volume fraction. This is somewhat indicative of increased viscosity and, consequently, reduced momentum transfer of fluid by fluid layers. The decrease in the slope of the fracture at the beginning and the end of the flow is also due to the same reduction in momentum transfer.
3.2.4.1. The effect of external force. Fig. 18 shows the effect of external force changes on (a) density distribution and (b) velocity distribution in the channel width. As the external force increases, the force exerting on the fluid particles increases between roughnesses. Moreover, the freedom of movement of these particles greatly reduces, and as a result, the amplitude of the fluctuations in this area decreases. The increase in the amplitude of the fluctuations is also evident due to the presence of Copper nanoparticles in the middle of the channel, as shown in Fig. 18. As the external force increases, the maximum flow velocity increases, which is clearly visible in the velocity diagram. In addition, as the external force increases, the slip velocity also increases. However, due to the momentum transfer, the effect of the nanoparticles on the velocity distribution diagram has decreased and, as a result, the fracture at the beginning and the end of the diagram occurs with less slope. 3.2.4.2. Effect of nanoparticle volume fraction. As shown in Fig. 19, in a simulation with a 2% volume fraction of nanoparticles, the
3.2.4.3. The distance between roughness elements. Simulation of the 11
Chemical Physics 527 (2019) 110505
D. Toghraie, et al.
Fig. 18. The effect of external force changes on (a) density distribution and (b) velocity distribution in the channel width.
Fig. 19. The effect of nanoparticle volume fraction change on (a) density distribution and (b) velocity distribution in the channel width.
initial location of particles and geometry is shown in Fig. 20. Fig. 21 shows the effect of the changes in the distance between the roughness elements on (a) the density distribution and (b) velocity distribution in the channel width. According to Fig. 21, as we expected, a decrease in the slope of the density fluctuations occurs when the distance between the roughness elements decreases. By decreasing the distance between the roughness elements, a reduction in the density fluctuation amplitude occurs. However, interestingly, the rate of variation in the amplitude of fluctuations in the simulation with the same conditions is less than the use of rectangular geometry for roughness, of course, by using rectangular geometry for roughness. This indicates the effect of determining the type of roughness geometry on the rate of variation in flow properties. In other words, it can be stated that during the analysis of the flow at nanoscale, in addition to the Knudsen number and the surface roughness, we should also pay attention to the surface roughness geometry. It should be noted that, the same factor alone can influence the effect of increasing or decreasing the Knudsen number on the flow properties in the channel and justify the difference in results in
comparison with previous assumptions. Clearly, the possibility of the presence of nanoparticles in these areas can be detected from the abnormal fluctuations in the middle of the channel.
• As it is evident, the change in roughness geometry is effective in
reducing the maximum velocity and reducing the slope of fracture at the beginning of the diagram. Regarding the velocity distribution, the slope of fracture of the graph at the beginning and the end of the diagram is mild due to the type of selected geometry. Therefore, by using this, roughness geometry reduces the effect of nanoparticles on the flow characteristics.
4. Conclusion In this study, molecular dynamics simulation of Couette and Poiseuille Water-Copper nanofluid flows in rough and smooth nanochannels with different roughness configurations was performed. Finally, results were as following: 12
Chemical Physics 527 (2019) 110505
D. Toghraie, et al.
Fig. 20. Changes in the distance between roughness elements in triangular roughness.
• Nanoparticles have caused fluctuations in the middle area which are • •
due to the effects of the surface of the nanoparticles relative to the base fluid of the Water. The fluctuations or abnormal fractures that emerge in the diagram due to the presence of nanoparticles can be clearly seen in the results of smooth nanochannel as well as the results of rough nanochannels. The presence of nanoparticles in the middle area and the interaction between the surface and the fluid in this area has caused abnormal fluctuations.
References [1] H. Noorian, D. Toghraie, A.R. Azimian, Molecular dynamics simulation of Poiseuille flow in a rough nano channel with checker surface roughnesses geometry, Heat Mass Transfer 50 (1) (2014) 105–113. [2] D. Toghraie, A.R. Azimian, Nanoscale Poiseuille flow and effects of modified Lennard-Jones potential function, Heat Mass Transfer 46 (7) (2010) 791–801. [3] D. Toghraie, A.R. Azimian, Molecular dynamics simulation of annular flow boiling with the modified Lennard-Jones potential function, Heat Mass Transfer 48 (1) (2012) 141–152. [4] M. Rezaei, A.R. Azimian, D. Toghraie, The surface charge density effect on the electro-osmotic flow in a nanochannel: a molecular dynamics study, Heat Mass Transfer 51 (5) (2015) 661–670. [5] M. Rezaei, A.R. Azimian, D. Toghraie, Molecular dynamics study of an electro-kinetic fluid transport in a charged nanochannel based on the role of the stern layer, Phys. A 426 (2015) 25–34. [6] D. Toghraie, A.R. Azimian, Molecular dynamics simulation of liquid–vapor phase equilibrium by using the modified Lennard-Jones potential function, Heat Mass Transfer 46 (3) (2010) 287–294. [7] D. Toghraie, A.R. Azimian, Molecular dynamics simulation of nonodroplets with the modified Lennard-Jones potential function, Heat Mass Transfer 47 (5) (2011) 579–588. [8] H. Noorian, D. Toghraie, A.R. Azimian, The effects of surface roughness geometry of flow undergoing Poiseuille flow by molecular dynamics simulation, Heat Mass Transfer 50 (1) (2014) 95–104. [9] L. Harmon, G.C. Lie, E. Clementi, Molecular dynamics simulation of flow past a plate, J. Sci. Comput. 1 (2) (1986). [10] J.L. Xu, Z.Q. Zhou, Molecular dynamics simulation of liquid argon flow at platinum surfaces, Heat Mass Transfer 40 (2004) 859–869. [11] Bing-Yang Cao, Min Chen, Zeng-Yuan Guo, Effect of surface roughness on gas flow in microchannels by molecular dynamics simulation, Int. J. Eng. Sci. 44 (2006) 927–937. [12] Daejoong Kim, Eric Darve, Molecular dynamics simulation of electro-osmotic flows in rough wall nanochannels, Phys. Rev. E 73 (2006) 051203. [13] John A. Thomas, Alan J.H. McGaughey, Ottoleo Kuter-Arnebeck, Pressure-driven water flow through carbon nanotubes: insights from molecular dynamics simulation, Int. J. Therm. Sci. 49 (2010) 281–289. [14] Reza Kamali, Ali Kharazmi, Molecular dynamics simulation of surface roughness effects on nanoscale flows, Int. J. Therm. Sci. 50 (2011) 226–232. [15] Chengzhen Sun, Lu. Wen-Qiang, Jie Liu, Bofeng Bai, Molecular dynamics simulation of nanofluid’s effective thermal conductivity in high-shear-rate Couette flow, Int. J. Heat Mass Transfer 54 (2011) 2560–2567. [16] Meisam Pourali, Ali Maghari, Non-equilibrium molecular dynamics simulation of thermal conductivity and thermal diffusion of binary mixtures confined in a nanochannel, Chem. Phys. 444 (2014) 30–38. [17] Q. Miao, Q. Yuan, Y.P. Zhao, Dissolutive flow in nanochannels: transition between plug-like and Poiseuille-like, Microfluid. Nanofluid. 22 (2018) 141. [18] Q. Yuan, Y.P. Zhao, Multiscale dynamic wetting of a droplet on a lyophilic pillararrayed surface, J. Fluid Mech. (2013) 171–188. [19] M. Zarringhalam, H. Ahmadi-Danesh-Ashtiani, D. Toghraie, R. Fazaeli, Molecular dynamic simulation to study the effects of roughness elements with cone geometry on the boiling flow inside a microchannel, Int. J. Heat Mass Transfer 141
Fig. 21. The effect of the changes in the distance between the roughness elements on (a) the density distribution and (b) velocity distribution in the channel width.
• Increasing the channel height reduces the effect of the surface on the fluid and reduces the flow rate of the nanofluid. • The slip velocity on the bottom and top wall remains almost the same.
13
Chemical Physics 527 (2019) 110505
D. Toghraie, et al. (2019) 1–8. [20] P. Alipour, D. Toghraie, A. Karimipour, M. Hajian, Molecular dynamics simulation of fluid flow passing through a nanochannel: effects of geometric shape of roughnesses, J. Mol. Liquids 275 (2019) 192–203. [21] P. Alipour, D. Toghraie, A. Karimipour, M. Hajian, Modeling different structures in perturbed Poiseuille flow in a nanochannel by using of molecular dynamics simulation: Study the equilibrium, Phys. A: Statist. Mech. Appl. 515 (2019) 13–30. [22] M. Tohidi, D. Toghraie, The effect of geometrical parameters, roughness and the number of nanoparticles on the self-diffusion coefficient in Couette flow in a nanochannel by using of molecular dynamics simulation, Phys. B: Condensed Matter 518 (2017) 20–32. [23] D. Toghraie, M. Mokhtari, M. Afrand, Molecular dynamic simulation of copper and platinum nanoparticles Poiseuille flow in a nanochannels, Phys. E: LowDimensional Syst. Nanostruct. 84 (2016) 152–161. [24] Jabbarzadeh, et al., Rheological properties of thin liquid films by molecular dynamics simulations, J. Non-Newtonian Fluid Mech. 69 (1997) 169–193. [25] D. Toghraie, M.M.D. Abdollah, F. Pourfattah, O.A. Akbari, B. Ruhani, Numerical investigation of flow and heat transfer characteristics in smooth, sinusoidal and
[26] [27] [28] [29] [30]
14
zigzag-shaped microchannel with and without nanofluid, J. Therm. Anal. Calorim. 131 (2) (2018) 1757–1766. M. Parsaiemehr, F. Pourfattah, O.A. Akbari, D. Toghraie, G. Sheikhzadeh, Turbulent flow and heat transfer of Water/Al2O3 nanofluid inside a rectangular ribbed channel, Phys. E: Low-Dimensional Syst. Nanostruct. 96 (2018) 73–84. R. Sarlak, S. Yousefzadeh, O.A. Akbari, D. Toghraie, S. Sarlak, The investigation of simultaneous heat transfer of water/Al2O3 nanofluid in a close enclosure by applying homogeneous magnetic field, Int. J. Mech. Sci. 133 (2017) 674–688. A. Akhgar, D. Toghraie, An experimental study on the stability and thermal conductivity of water-ethylene glycol/TiO2-MWCNTs hybrid nanofluid: developing a new correlation, Powder Technol. 338 (2018) 806–818. M. Keyvani, M. Afrand, D. Toghraie, M. Reiszadeh, An experimental study on the thermal conductivity of cerium oxide/ethylene glycol nanofluid: developing a new correlation, J. Mol. Liq. 266 (2018) 211–217. H. Khodadadi, D. Toghraie, A. Karimipour, Effects of nanoparticles to present a statistical model for the viscosity of MgO-Water nanofluid, Powder Technol. 342 (2019) 166–180.
Contents lists available at ScienceDirect
Chemical Physics journal homepage: www.elsevier.com/locate/chemphys
Molecular dynamics simulation of Couette and Poiseuille Water-Copper nanofluid flows in rough and smooth nanochannels with different roughness configurations
T
Davood Toghraiea, Maboud Hekmatifarb, Yasaman Salehipoura, Masoud Afrandc,d,
⁎
a
Department of Mechanical Engineering, Khomeinishahr Branch, Islamic Azad University, Khomeinishahr, Iran Young Researchers and Elite Club, Khomeinishahr Branch, Islamic Azad University, Khomeinishahr, Iran c Laboratory of Magnetism and Magnetic Materials, Advanced Institute of Materials Science, Ton Duc Thang University, Ho Chi Minh City, Vietnam d Faculty of Applied Sciences, Ton Duc Thang University, Ho Chi Minh City, Vietnam b
ARTICLE INFO
ABSTRACT
Keywords: Molecular dynamics simulation Couette flow Poiseuille flow Nanofluid Nanochannel
In this study, molecular dynamics simulation of Couette and Poiseuille Water-Copper nanofluid flows in rough and smooth nanochannels was performed. The Lennard-Jones equation is considered as Water-Water intermolecular interaction, while Hamaker’s equation is considered to be the interaction between Water-Copper and Copper-Copper particles. PPPM algorithm is used to calculate the electric potential. It is concluded that increasing the channel height reduces the effect of the surface on the fluid and reduces the flow rate of the nanofluid. Also, the slip velocities on the bottom and top walls remain almost the same. Furthermore, nanoparticles have caused fluctuations in the middle area, which are due to the effects of the surface of the nanoparticles relative to the base fluid of the Water. As expected, the presence of nanoparticles in the middle area and the interaction between the surface and the fluid in this area has caused abnormal fluctuations.
1. Introduction Many studies have been conducted in the field of molecular simulations of macroscopic nanofluids. Researchers have also begun nanoscale researches due to recent applications of nanofluids in some new engineering fields. As a result of extensive development in the field of powerful computing, software systems have been enhanced using numerical solutions and simulation tools. Undoubtedly, the molecular dynamics simulation is one of these tools, which has had a great impact on many fields, such as nanotechnology, heat transfer, fluid mechanics, and physics. Molecular dynamics simulation (MDS) is a bridge between theory and experiment, therefore molecular simulations are employed using theoretical models. In the past decade, the number of published articles on nanofluids has almost doubled every two years, indicating an increase in the interest of researchers in this topic [1–8]. Harmon et al. [9] performed a molecular dynamics simulation of flow past a plate. At fluid velocities large enough to obtain an adequate signal to noise resolution, two counter-circulating vortices were observed behind the obstruction. Xu and Zhou [10] studied liquid argon flow at Platinum surfaces by using MDS. They found out that with an increase in the shear rate, the viscosity increased and the non-
⁎
Newtonian flow appeared. Cao et al. [11] studied the effect of surface roughness on gas flow in microchannels by molecular dynamics simulation. They concluded that the geometry roughness also shows significant effects on the boundary conditions and the friction characteristics. Kim and Darve [12] simulated the electro-osmotic flows in rough wall nanochannels by molecular dynamics simulation. They concluded that along the flow direction, the diffusion of water and ions inside the groove is significantly lowered while it is similar to the bulk value elsewhere. Thomas et al. [13] studied pressure-driven water flow through carbon nanotubes: Insights from molecular dynamics simulation. They predicted the variation of water viscosity and slip length with CNT diameter. Kamali and Kharazmi [14] investigated the surface roughness effects on nanoscale flows by using MDS. The effects of surface roughness and cavitation on the velocity distribution of hydrophobic and hydrophilic wall undergoing Poiseuille flow were presented. Sun et al. [15] simulated the nanofluid’s effective thermal conductivity in high-shear-rate Couette flow. They found out that the conventional correlation is not suitable when the sizes of the suspended particles are reduced to nanometers. Pourali and Maghari [16] investigated non-equilibrium molecular dynamics simulation of thermal conductivity and thermal diffusion of binary mixtures confined in a
Corresponding author at: Ton Duc Thang University, Ho Chi Minh City, Vietnam. E-mail address: [email protected] (M. Afrand).
https://doi.org/10.1016/j.chemphys.2019.110505 Received 2 February 2019; Received in revised form 12 August 2019; Accepted 19 August 2019 Available online 21 August 2019 0301-0104/ © 2019 Elsevier B.V. All rights reserved.
Chemical Physics 527 (2019) 110505
D. Toghraie, et al.
nanochannel. They found out that in very narrow channels, thermal diffusion is small, and it reaches a steady-state value with increasing the channel width. Noorian et al. [8] investigated the effects of checker surface roughness geometry on the flow of liquid argon through nanochannel when the roughness is implemented on the lower channel wall. They found out that as the surface attraction energy or the roughness height increases, the density layering in the near wall is enhanced by higher values or secondary layering phenomena. Dissolutive flow in nanochannels was studied by Miao et al. [17]. Their results showed that in pressure-driven flow, when the dissolubility is low, the dominant dissipation is the viscous dissipation and the theoretical model of insolubility is acceptable. Multiscale dynamic wetting of a droplet on a lyophilic pillar-arrayed surface was carried out by Yuan and Zhao [18]. They used a multiscale combination method of experiments and molecular dynamics simulations.
total number of degrees of freedom to 6. For a particle or quasi-particle like Copper nanoparticles, the number of degrees of freedom is 3. The instantaneous temperature fluctuates like the total kinetic energy and is obtained from the following equation [2], N
T (t ) = i
mi vi2 (t ) kB Nsf
(8)
where Nsf is the total degrees of freedom of the system. The temperature fluctuation is at the order1/ Nfs , i.e., at 102–103. The pressure is calculated using the Virial equation of state [2],
P = kT
rij
3N
i< j
dU (rij ) drij
(9)
The potential force between the particles U is considered for each pair of particles. Therefore, for N particles, the interaction energy is the sum of each pair of particles [2,19–23],
(r N )
2. Theoretical and methodological framework 2.1. Intermolecular interactions
U (r N ) =
The Lennard-Jones equation is considered as Water-Water intermolecular interaction, while Hamaker’s equation is considered to be the interaction between Water-Copper and Copper-Copper particles. The location of ri (t) (t) and momentum pi (t) are obtained by solving the Newton’s equation of motion (2–1) with the initial position and momentum [1],
The most important part of the simulation of a nano-sized system is the correct choice of potential function to minimize the conflicts between nano-system compounds. In this project, the modeling system is a Water-Copper nanofluid and is modeled using molecular dynamics simulation. To model the potential between Water-Water particles, the Lennard -Jones potential function with the SPC/E model is used. Water (H2O) has two Hydrogen atoms, which are associated with an Oxygen atom through a covalent bond.
Fi = mi ai = mi
d2ri dt 2
(1)
where ri is the position of the particle i and mi is the mass of the particle i and ai is the acceleration of particle i. Momentum pi can be defined as follows [2],
The Hydrogen particles have a positive charge and the Oxygen particle has a negative charge. The Coulomb’s term was added to the Lennard-Jones potential in order to have long-range interaction forces that are due to the positive and negative electric charge interactions. The Lennard-Jones potential is used for interaction between particles of Oxygen with another Oxygen particle in the following equations [15],
where vi is the velocity of the particle i. For an isolated system, the total energy (E), which contains kinetic energy and potential energy of the particles or molecules, is constant. The total energy (E) can be expressed in Hamilton form. For N spherical particles, the Hamilton equation follows the following equation,
Pi2 + U (r1 + r2+
+ rn ) = E
i
U (r ) = 4
(3)
pi =
m i ai =
Fi
v ( t + t ) = v (t ) +
a ( t ) + a (t + t ) t + O ( t 2) 2
qo = oo
OO
rOO
rOO
3
6
3
+ ke a=1 b =1 2
qa qb rab
(11)
0.8475 |e|,
= 3.166Å,
qH = 0.4238 |e| oo
= 0.15535kcal/mol
In this paper, PPPM algorithm, which is more appropriate in terms of computational cost than the Ewald method, is used to calculate the electric potential. This method reduces the computing time in comparison to the Ewald sum method. The PPPM method is in the order of O (N log N) and calculates the long-range interaction forces in three stages. The tolerance value of 10-4 is considered to be used in the PPPM method.
(5) (6)
Simulation begins with the initial position and velocity of particles and considering the average temperature. The average kinetic energy for each degree of freedom is as follows [2],
1 2 1 mv = kb T 2 2
12
ke = 8.99 × 109N. m2/C2
(4)
1 a (t ) t 2 + O ( t 4 ) 2
OO
qa and qb are the charges of particles a and b, ke is the Coulomb constant and rab is the distance between the two particles. Coulomb constant, Coulomb and Lennard-Jones parameters are as follows,
In molecular dynamics simulation, the Newton's motion equation (4) can be solved by the algorithm and relay [2],
r ( t + t ) = r ( t ) + v (t ) t +
OO
1
Also, the forces of each particle with potential function are as follows [2],
dU = dri
(10)
2.2. Properties of the SPC/E molecule model
(2)
Pi = mi vi
1 H(r , p) = 2m
u (rij ) i
2.3. Water-Copper interatomic forces
(7)
Based on the equilibrium theorem, the kinetic energy of motion of particles is distributed among all available active degrees of freedom for particles. In this project, the flexible Water molecule has 9 degrees of freedom, i.e., 3 transfer degrees of freedom, 3 rotation degrees of freedom and 3 vibration degrees of freedom. The hardness of the Water molecule in the bonds and angles eliminates vibrations and reduces the
Water-Copper is the second type of interatomic force existing in the nanofluid. The force between the colloid particles (large particles) and the solvent is as follows [15]
U=
2
2Acs a3 3 1 9(a2 r 2)3
(5a6 + 45a4r 2 + 63a2r 4 + 15r 6) 15(a r )6 (a + r )6
6
(12)
Chemical Physics 527 (2019) 110505
D. Toghraie, et al.
where Acs is the Hamaker’s constant, which is equal to 43.2 kcal/mol for Water and a is the radius of Copper colloid. Sigma (σ) is obtained from arithmetic mean of sigma of Copper and Water nanoparticles.
temperature by the relaxation rate square [15] N
HNose
2a1 a2 r2 + ln 2 (a1 a2 )2 r
2a1 a2 + 2 (a1 + a2 ) 2 r
nf
UR = +
A cc 37800
r2
6(a12
7r (a1 + a2) + (r a1
+ 7a1 a2 + a2 )7
a22 )
r 2 + 7r (a1 + a2) + 6(a12 + 7a1 a2 + a22 ) (r + a1 + a2 )7 r 2 + 7r (a1
a2) + 6(a12 7a1 a2 + a22 ) (r + a1 a2 )7
r2
a2) + 6(a12 7a1 a2 + a22 ) (r a1 + a2 )7
7r (a1
U = UA + UR
(14)
=
p
+ (1
)
(15)
bf
In this regard, p and bf are the densities of the nanoparticles and base fluid, respectively. The density of Copper nanoparticles is ρp = 8.94 g/cm3 the Water density is ρbf = 1 g/cm3 [25–30]. Considering SPC/E rigid Water model, the SHAKE algorithm has been used with a time step of 0.05 fs and 5 × 105 steps. The timed step is carefully selected to remove any round-off or truncation errors. According to the papers presented in this field, different values have been selected for the time step, and the most accurate and stable step is obtained at 0.05 fs. All simulations carried out in this research are based on dimensionless values and for making various dimensionless parameters, Table 1 has been used.
(a1 + a2 ) 2 (a1 a2 ) 2 (13a)
6
pi p2 s + U (r N ) + + gkblns 2 2 mi s 2Q
To calculate the density of nanofluids with different percentages of volume fraction of nanoparticles, we use the following equation [15]
The Hamaker’s formula for colloid-colloid was used to model the interaction potential between the Copper nanoparticles [15]
Ac 6 r2
= i=1
2.4. Copper-Copper interatomic forces
UA =
Hoover
3. Results and discussion
(13b) (14)
Molecular dynamics simulation on a three-dimensional system of Water-Copper nanofluid in a Copper nanochannel with dimensions of 4 × 4 × 4 nm3 with different types of surface roughness geometry has been used, which is shown in Fig. 1.
where Acc is Hamaker’s constant, Acs = 107.83 kcal/mol and a1and a2 are the radius of each of the nanoparticles, respectively. 2.5. Molecular dynamics simulation tools
3.1. Couette flow
Large-scale atomic/molecular massively parallel simulator (LAMMPS) is intended to simulate Water-Copper nanofluid system. To solve the motion equations, the initial molecular structure and initial velocity of each particle must be specified. Therefore, the particle spatial path can be observed during simulation. The open-source Packmol software has been used to model limited Water molecules in the nanochannel and the arrangement of Copper nanoparticles in it. The simulation force coefficients have been derived from the Charmm code to calculate the interaction of Hydrogen and Oxygen particles in the Water molecule. After deriving these coefficients and producing the input code of a single Water molecule for the Packmol software, we are going to produce Copper nanoparticles. Modeling Copper nanoparticles is also done using LAMMPS software. The Packmol software uses inputs created for a Water molecule and a nanoparticle and two bottom-up walls fixed at the beginning and end of y-axis of simulation box and, therefore, they put a number of molecules adjusted for Water and a number of nanoparticles designated for Copper nanoparticles heterogeneously between the two walls. The coordinates of initial positions of all particles are created as input data for the LAMMPS program. In addition, the number of molecules selected at each stage of the simulation depends on the different conditions of roughness of the surface and the result of a change in the volume between the two walls and nanoparticles. The initial particle velocity must be determined by selecting the initial temperature set at room temperature (300 K). The initial velocity of particles is randomly selected from the Gaussian distribution. The canonical ensemble (NVT) is used to set the simulated system and have constant number of molecules, constant volume and constant temperature. The total temperature of the system can be set using one of the thermostat methods. Energy and temperature are related, resultantly, the temperature should be kept constant during the simulation. Using a thermostat prevents temperature drifts due to the accumulation of numerical errors during molecular dynamics simulations. The Nosé-Hoover thermostat algorithm has been used to control the temperature at 300 K while keeping the volume constant. This algorithm adds the friction term to the Newton’s equation. In a period of time, changes in the friction coefficient are extracted as relative deviation of the set value of the
3.1.1. Validation of Couette flow According to previous investigations in the field of simulation of flow in nanochannels, simulation of Water-Copper nanofluid flow in nanochannel has not been done so far. To validate the velocity diagram, we have used the results of various simulations that are available in other studies of the researchers. One closely related work in this regard is Jabarzadeh et al. [24] on molecular dynamics simulation of the slip Couette flow of alkanes on a sinusoidal wall. Some of the results are shown in Fig. 2. The result obtained for the Couette flow of nanofluid in this project is almost the same with some differences compared to the probable chart regarding the past investigations. We will continue to discuss these differences and justify them. 3.1.2. Couette flow in a smooth channel In the smooth channel simulation, 529 Copper atoms are considered for each of the bottom and top walls regarding the type of lattice Table 1 Necessary relationships for making dimensionless parameters. Quantity name
Formula
Distance
r =
r
Temperature
T =
kB T
Density
m
Pressure
P =
Time
t =
Volume
V =
V 3
Power
F =
F
Velocity
v =v
Acceleration
3
3
=
a =
P 3 t m
m
a m
Chemical Physics 527 (2019) 110505
D. Toghraie, et al.
Fig. 1. Simulation of the initial geometry of the Water-Copper nanofluid.
structure, lattice constant, density, and molar mass of Copper. Bottom and top wall thickness is 2 Å. One percent of the finite volume between the walls is considered for Copper nanoparticles, and 55 atoms of Copper are placed in this area. Using the Packmol tools, the number of Water molecules for embedding in the finite distance between the nanoparticles and walls is 1907. Moreover, the top wall velocity for the reference mode of Couette flow is considered to be three-dimensional. 3.1.2.1. Effect of channel height. In Fig. 3 the effect of channel height changes on (a) density distribution and (b) velocity distribution in the width of smooth channel is presented. The sharp fluctuations created at side of the lower wall are due to the presence of nanoparticles and the interaction between the two surfaces of the nanoparticle and the lower wall. An increase in the channel height does not affect the amplitude of the density distribution fluctuations of the near-wall areas. Moreover, the level of penetration of density layering remains constant. Depending on the fact that the layering phenomenon depends on the interaction between the wall particles and the fluid near the surface and also on the fluid density, maintaining the degree of the layering penetration is justified. By increasing the channel’s height, the effects of the surface is reduced and depending on the type of flow, the momentum transfer is done through a high surface. Therefore, increasing the channel height reduces the effect of the surface on the fluid and reduces the flow rate of the nanofluid. The slip velocity on the bottom and top wall remains almost the same. A small fracture near the bottom wall is due to the presence of nanoparticles which is located at a short distance from the surface and somehow increases the viscosity and reduces the velocity. Fig. 3. Effect of channel height changes on (a) density distribution and (b) velocity distribution in the width of smooth channel.
3.1.2.2. Effect of initial wall velocity. Fig. 4 shows the effect of the initial velocity change on the (a) density distribution and (b) velocity
Fig. 2. The geometry of the simulation system and the fluid velocity graph in the Jabbarzadeh et al.’s study [] 4
Chemical Physics 527 (2019) 110505
D. Toghraie, et al.
Fig. 4. The effect of the initial velocity change on the (a) density distribution and (b) velocity distribution in the smooth channel width.
Fig. 5. The effect of the change in the volume fraction of nanoparticles on (a) density distribution and (b) velocity distribution in the smooth channel width.
distribution in the smooth channel width. The presence of nanoparticles in the area between two walls has been shown in the density distribution diagram as a sudden increase in the density and also the increase in the amplitude of fluctuations. An increase in the magnitude of the fluctuations near the high surface indicates that as the effect of the surface on the fluid increases, the top wall velocity decreases. Increasing the amplitude of the fluctuations near the bottom wall is not regular and does not follow the high wall velocity reduction. Moreover, the presence of nanoparticles near this surface has caused a sudden increase in fluctuations in this area and is the cause of non-uniformity. Increasing the surface effect can also be found in the velocity distribution. One of the possible reasons for this phenomenon is the freedom to operate more fluid molecules near the walls with the lowest wall velocity as a current stimulus. Fluid molecules near the walls with the lowest top wall velocity act more freely as flow stimulus. Increasing the slip velocity is another clear reason for increasing the surface effect with the top wall velocity reduction. This can be obtained from the velocity distribution chart in Fig. 4.
3.1.2.3. Effect of volume fraction of nanoparticles. Fig. 5 shows the effect of the change in the volume fraction of nanoparticles on (a) density distribution and (b) velocity distribution in the smooth channel width. As shown in Fig. 5, as the volume fraction of nanoparticles increases, the density fluctuations in near-wall areas decrease. In addition, as the volume fraction of the nanoparticles increases, the density fluctuations penetrate into more layers of the fluid. The presence of sharp fluctuations in the middle area of the channel is also due to an increase in the surface relative to the volume in the area. However, the irregular increase in the density fluctuations in areas further away from the walls in different modes of simulation is due to the heterogeneous distribution of nanoparticles in the channel. The reduction in the average velocity along the channel width is clearly evident, which is due to the presence of nanoparticles and to some extent is because of the reduction in nanofluid viscosity in the area. 5
Chemical Physics 527 (2019) 110505
D. Toghraie, et al.
Fig. 6. Modeling of Water-Copper nanofluid in a channel with rectangular roughness.
3.1.3. Couette flow in a rough channel with rectangular roughness The channel roughness, as shown in Fig. 6, is drawn with a rectangular cross-section perpendicular to the flow direction. By adding two roughness elements into each nanochannel wall, the number of atoms in the wall becomes 667. Furthermore, the diameter of the nanoparticle is reduced, and as a result, the number of atoms of each nanoparticle becomes 43. Furthermore, 1813 Water molecules are selected for the input code of the Packmol software. It should be noted that the number of molecules intended for Water is only for rectangular roughness reference mode. Moreover, for different modes of simulation, it is chosen according to the finite volume between the walls and nanoparticles. 3.1.3.1. Effect of initial wall velocity. Fig. 7 shows the effect of initial wall velocity change as a flow stimulus on the (a) density profile and (b) velocity profile in the channel width. In Fig. 7, the reduction in velocity results in an increase in the amplitude of the fluctuations near the wall. However, the amplitude of these fluctuations near the top wall is lesser than the bottom wall. The reason for this increase is the trapping of more nanofluid molecules among roughnesses. As the number of molecules increases between roughnesses, the repulsive intermolecular forces increase and cause the molecules to suddenly set a distance away from each other. Therefore, the density number in these areas is oscillating between low and high. Another variation of the current nanofluid density diagram in a nano-channel with a similar ordinary fluid flow (eg, liquid Argon) is the fluctuations in the mid-channel. Although the amplitude of these fluctuations is low, it is different from the uniformity of the density profile that exists in the middle of the channel with ordinary fluid flow. The presence of nanoparticles and the forces of interaction between the surface of the fluid in this area can be considered as the best reason for these fluctuations. However, due to the low ratio of the outer surface of the nanoparticles to the surface of the top and bottom walls, the amplitude of the oscillations in this area is less. According to Fig. 7(b), we can clearly see the effect of roughness on the velocity graph, especially near the bottom wall. The lack of presence of nanoparticles in the roughness has resulted in the greater effectiveness of the Waterbased fluid particles. Moreover, by increasing the initial velocity of the wall and more momentum transfer of top layers, the slip velocity increases. Slightly higher than the top wall roughness, with the presence of nanoparticles, the average velocity decreases, however, this reduction has emerged near the bottom wall as a fracture in the velocity chart. In the vicinity of the top wall, this change is gradual and this is due to the type of flow and momentum transfer from the top wall, which results in the presence of stronger interaction forces on the base molecules and thus results in lower degree of effectiveness from nanoparticles.
Fig. 7. The effect of initial wall velocity change as a flow stimulus on the (a) density profile and (b) velocity profile in the channel width.
3.1.3.2. Effect of nanoparticle volume fraction. Fig. 8 shows the effect of changing the nanoparticle volume fraction on the (a) density profile and (b) velocity profile in the channel width, According to the density distribution diagram in Fig. 8; the change in the nanoparticle volume fraction does not make much change at the fluctuation range of the near-wall areas. By increasing the volume fraction of nanoparticles,
uniform convergence of the density has decreased in the middle of the graph. While the mean density increases in the middle area of the channel, fluctuations in this area also increase due to increased surface reaction force relative to the base fluid of the fluctuations. The mean density of the nanofluid in the areas near the surface is affected by the 6
Chemical Physics 527 (2019) 110505
D. Toghraie, et al.
Fig. 10. The effect of changing the distance between roughnesses on the (a) density profile and (b) the velocity profile in the channel width. Fig. 8. The effect of changing the nanoparticle volume fraction on the (a) density profile and (b) velocity profile in the channel width.
nanoparticles, momentum transfer occurs from top to bottom layers. This momentum transfer in the areas between surface roughnesses where there are no nanoparticles is more evident and appears as a fracture in the velocity graph. This fracture has been less obvious next to the near-wall areas due to the transfer of the velocity from the wall to
repulsive force between the nanoparticles and the surface. In the velocity chart, the mean velocity decrease is clearly evident across the entire channel width. By increasing the volume fraction of
Fig. 9. Changes in the distance between rectangular roughness elements. 7
Chemical Physics 527 (2019) 110505
D. Toghraie, et al.
the fluid and the presence of higher pressure on the fluid molecules. However, by increasing the fraction of the nanoparticles, this fracture near the top wall becomes gradually more obvious and indicates a greater degree of effectiveness of the flow from surface roughness. 3.1.3.3. The distance between roughness elements. In this section, by preserving the primary dimensions of the rectangular roughness, we increase the number of roughness elements on each surface by dividing the distance between the elements. Resultantly, we can see the effect of reducing the distance between roughness elements. The simulated geometry is shown in Fig. 9. Fig. 10 shows the effect of changing the distance between roughnesses on the (a) density profile and (b) the velocity profile in the channel width. Considering this Figure, we find that the increase in the number of roughness elements on the surface reduces the amplitude of the fluctuations in the near-wall areas. Moreover, as the repulsive force between the particles increases, the mean density in this area decreases. The density profile is not uniform in the middle areas of the channel which is due to the presence of nanoparticles. However, as a result of the averaging of the values obtained for the nanofluid density in the distance between roughness surface of the top and bottom walls shows the value of 0.982 g/cm3 for the mean density. And this value, regardless of the surface effects in the nanoscale, is about 6% different from the theoretical values. Reduction in the slip velocity is clearly evident in the velocity graph of Figs. 4–14 in the near-wall areas. Reducing the number of fluid molecules in this area not only reduces the slip velocity, but also the fracture in the area near the bottom fixed wall also becomes less sloping. This indicates an increase in the surface effect on the distribution of the nanofluid flow rate in the channel. From increasing the impact of the surface by reducing the distance between roughness elements, we conclude that in the top area of the channel (mid-to-top), the nanoscale velocity distribution in the nanochannel with the highest number of roughnesses has the higher mean velocity and in the bottom area of channel area (mid-to-bottom), the mean velocity is lower. Therefore, in the middle area where this velocity variation occurs, there is a share point of velocity among the velocity distribution graphs. This share point is inclined to the top wall. This is due to the type of flow because the momentum transfer from the top surface will increase the force on the trapped particles among roughnesses. However, the momentum transfer from the top surface to the bottom layers becomes less, Fig. 12. The effect of channel height changes on (a) density distribution and (b) velocity distribution in channel width.
and as a result, the velocity zero of bottom wall penetrates into more fluid layers. As a result, surface effects from bottom to top penetrate into more fluid layers. 3.2. Poiseuille flow 3.2.1. Validation To validate the velocity diagram, we have used the results of various simulations that are available in other studies of the researchers (Fig. 11). One closely related work in this field is Jeremy Freid’s study [] in the field of molecular dynamics simulation of Poiseuille flow for fluid Argon on a wall with a rectangular, triangular roughness. 3.2.2. Poiseuille flow in a smooth channel Fig. 12 shows the effect of channel height changes on (a) density distribution and (b) velocity distribution in channel width. In Fig. 12(a), constant fluctuation amplitude in the near-wall areas indicates the ineffectiveness of increasing the channel height on the effects of surface and fluid relative to each other. The phenomenon of density layering and increasing or decreasing fluctuations depends on
Fig. 11. Diagram of fluid flow velocity distribution in Jeremy Fried [] and present work. 8
Chemical Physics 527 (2019) 110505
D. Toghraie, et al.
Fig. 13. The effect of nanoparticle volume fraction change on (a) density distribution and (b) velocity distribution in the channel width.
Fig. 14. The effect of external force changes on (a) density distribution and (b) velocity distribution in the channel width.
changing the various surface and fluid parameters at their contact surface. In this case, surface and fluid components at the contact surface will not change. According to the velocity distribution diagram of Fig. 12, increasing the maximum velocity by increasing the channel’s height indicates greater freedom of action in central layers where maximum velocity occurs. In fact, the higher the channel’s height, the lower the effect of the surface on the fluid. The drop in mean velocity in the channel width during simulation is due to the presence of nanoparticles. The fracture at the beginning of the diagram is also due to the close proximity of the nanoparticle to the surface of the bottom wall during the solution.
experienced sharp fluctuations in this area. The reduction in fluctuations near the top wall also indicates less momentum transfer in the 3 percent volume fraction of nanoparticles. This reduction in momentum transfer in the velocity distribution diagram is also shown as the reduction in mean velocity. The fitting the results for different situations is something that has not been done in this work. Therefore, the fluctuations or abnormal fractures that emerge in the diagram due to the presence of nanoparticles can be clearly seen in the results of smooth nanochannel as well as the results of rough nanochannels. 3.2.3. Poiseuille flow in a channel with rectangular roughness Fig. 14 shows the effect of external force changes on (a) density distribution and (b) velocity distribution in the channel width. As shown in the velocity distribution diagram of Fig. 14, the slip velocity decreases as the external force increases. The fracture in the velocity distribution diagram occurs slightly after the surface roughness, which is due to the presence of nanoparticles. In fact, with an increase in external force, the amount of this fracture in the diagram reduces
3.2.2.1. Effect of nanoparticle volume fraction. Fig. 13 shows the effect of nanoparticle volume fraction change on (a) density distribution and (b) velocity distribution in the channel width. As evident from the sharp increase in fluctuations near the bottom wall of Fig. 13(a), particle agglomeration near the bottom wall has caused a sharp increase in surface-to-volume ratio in these areas; therefore, the diagram has 9
Chemical Physics 527 (2019) 110505
D. Toghraie, et al.
Fig. 15. The effect of nanoparticle volume fraction changes on (a) density distribution and (b) velocity distribution in the channel width.
Fig. 16. The effect of change in the distance between roughness elements on (a) the density distribution and (b) velocity distribution in the channel width.
because the increase in external force reduces the freedom of movement of fluid particles between roughnesses and because, in fact, the cause of this fracture is the reduction in the freedom of movement of fluid particles after the roughness surface, with the presence of nanoparticles. Therefore, in the case of the highest external force, this decrease in freedom becomes less pronounced and the fracture at the beginning and end of the velocity distribution diagram will occur with fewer slopes. Due to the presence of nanoparticles near the bottom wall and the reduction of the freedom of movement of fluid particles in areas near the bottom of the channel, fracture in these areas has a relatively higher slope.
reason why we see a change in the fluctuation amplitude near the wall, and we do not see any changes near the top wall. In fact, in this case, because the particles of the wall surface and the fluid do not undergo any particular changes; therefore, there is no particular change in the amplitude of near-wall fluctuations. However, in the middle of the channel, by increasing volume fraction of nanoparticles, the surface-tovolume ratio increases, and thus the amplitude of the fluctuations increase sharply. In general, the fluctuations in the near-surface areas remain constant unless the nanoparticles are located in these areas. By increasing the nanoparticle volume fraction, the maximum velocity decreases. This shows that as viscosity increases, the number of nanoparticles increases, too. This increased viscosity reduces slip velocity. Thus, by increasing volume fraction of nanoparticles, the effects of surface on fluid increase.
3.2.3.1. Effect of nanoparticle volume fraction. Fig. 15 shows the effect of nanoparticle volume fraction changes on (a) density distribution and (b) velocity distribution in the channel width. It is evident from the density distribution diagram of Fig. 15 that the change in the percentage of nanoparticle volume fraction does not make any changes in the amplitude of near-walls fluctuations. The density depends on the degree of surface and fluid interaction. This is the
3.2.3.2. The distance between roughness elements. Fig. 16 shows the effect of change in the distance between roughness elements on (a) the density distribution and (b) velocity distribution in the channel 10
Chemical Physics 527 (2019) 110505
D. Toghraie, et al.
width. Reducing the density fluctuations amplitude by increasing the number of roughness elements is due to the increased repulsive force between molecules and the possible presence of fewer molecules among roughnesses. In the area after roughness, due to the sharp drop of the repulsive force between particles, the density of the particles suddenly rises. As expected, in Fig. 16(a) the amplitude of fluctuations in the middle layers is oscillating due to the presence of nanoparticles in these areas. Having higher maximum velocity in the reference mode means considering the four roughness elements on the surface due to the lower surface-to-volume ratio compared to the other modes. This means that the greater distance between the roughness elements results in more freedom of movements of the fluid particles in this area and increases the slip velocity in this area. This increase in velocity in the area between the roughness elements causes a fracture with a greater slope on the surface of the roughness. Therefore, the increase of the roughness elements on the surface is accompanied by reducing the effect of the surface on nanofluid; whereas, the effect of the presence of roughness element on the velocity distribution, especially on the maximum velocity, as well as on the fracture slope of the velocity distribution diagram on the roughness surface. 3.2.3.3. Height of roughness elements. Fig. 17 shows the effect of roughness height change on (a) the density distribution and (b) velocity in the channel width. One of the effects of the change in the height of surface roughness on the flow properties is the greater penetration of the density layering in the layers farther from the base wall surface, as can be seen in the density diagram of Fig. 17. By increasing the roughness height, the likelihood of the presence of nanoparticles in roughness increases. Embedding nanoparticles between roughness elements is due to the relatively high surface-tofluid ratio, which results in very high frequency fluctuations in these areas. It is easy to see these fluctuations near the bottom surface. It can be concluded that in nanofluid flow in rough nanochannel, the presence of nanoparticles in each channel area results in non-uniform fluctuations, which violates the prior assumptions of researchers in predicting fluid flow properties in nanochannels. By using nanofluid instead of ordinary fluid, the violation of the prior research in predicting the velocity distribution in the nanochannel is clearly evident at the beginning and end of the velocity distribution diagram. By increasing roughness height, more fluid particles are trapped between roughnesses. Therefore, the slip velocity decreases and with the penetration of the velocity zero of top and bottom wall into the layers farther away from the maximum velocity surface also reduces. Fig. 17. The effect of roughness height change on (a) the density distribution and (b) velocity in the channel width.
3.2.4. Poiseuille flow in a channel with triangular roughness The simulation of the particle’s initial location and the geometry is exactly equal to the same model in the Couette flow.
heterogeneous distribution of nanoparticles in the base fluid causes the placement of two nanoparticles near the bottom wall and these results in high increase in surface-to-fluid ratio in this area. The intense fluctuations created are due to intense interaction between Water molecules and wall surfaces and nanoparticles. Almost without considering the effects of nanoparticles, the amplitude of fluctuations near the walls has not changed much, which to some extent can be seen near the wall surface. The presence of fluctuations in the middle area of the channel is also due to the effects of increasing the surface-to-fluid ratio in these areas, which increases with an increase in the volume fraction of the nanoparticles. The maximum velocity drop is clearly evident in the velocity distribution diagram by increasing the nanoparticle volume fraction. This is somewhat indicative of increased viscosity and, consequently, reduced momentum transfer of fluid by fluid layers. The decrease in the slope of the fracture at the beginning and the end of the flow is also due to the same reduction in momentum transfer.
3.2.4.1. The effect of external force. Fig. 18 shows the effect of external force changes on (a) density distribution and (b) velocity distribution in the channel width. As the external force increases, the force exerting on the fluid particles increases between roughnesses. Moreover, the freedom of movement of these particles greatly reduces, and as a result, the amplitude of the fluctuations in this area decreases. The increase in the amplitude of the fluctuations is also evident due to the presence of Copper nanoparticles in the middle of the channel, as shown in Fig. 18. As the external force increases, the maximum flow velocity increases, which is clearly visible in the velocity diagram. In addition, as the external force increases, the slip velocity also increases. However, due to the momentum transfer, the effect of the nanoparticles on the velocity distribution diagram has decreased and, as a result, the fracture at the beginning and the end of the diagram occurs with less slope. 3.2.4.2. Effect of nanoparticle volume fraction. As shown in Fig. 19, in a simulation with a 2% volume fraction of nanoparticles, the
3.2.4.3. The distance between roughness elements. Simulation of the 11
Chemical Physics 527 (2019) 110505
D. Toghraie, et al.
Fig. 18. The effect of external force changes on (a) density distribution and (b) velocity distribution in the channel width.
Fig. 19. The effect of nanoparticle volume fraction change on (a) density distribution and (b) velocity distribution in the channel width.
initial location of particles and geometry is shown in Fig. 20. Fig. 21 shows the effect of the changes in the distance between the roughness elements on (a) the density distribution and (b) velocity distribution in the channel width. According to Fig. 21, as we expected, a decrease in the slope of the density fluctuations occurs when the distance between the roughness elements decreases. By decreasing the distance between the roughness elements, a reduction in the density fluctuation amplitude occurs. However, interestingly, the rate of variation in the amplitude of fluctuations in the simulation with the same conditions is less than the use of rectangular geometry for roughness, of course, by using rectangular geometry for roughness. This indicates the effect of determining the type of roughness geometry on the rate of variation in flow properties. In other words, it can be stated that during the analysis of the flow at nanoscale, in addition to the Knudsen number and the surface roughness, we should also pay attention to the surface roughness geometry. It should be noted that, the same factor alone can influence the effect of increasing or decreasing the Knudsen number on the flow properties in the channel and justify the difference in results in
comparison with previous assumptions. Clearly, the possibility of the presence of nanoparticles in these areas can be detected from the abnormal fluctuations in the middle of the channel.
• As it is evident, the change in roughness geometry is effective in
reducing the maximum velocity and reducing the slope of fracture at the beginning of the diagram. Regarding the velocity distribution, the slope of fracture of the graph at the beginning and the end of the diagram is mild due to the type of selected geometry. Therefore, by using this, roughness geometry reduces the effect of nanoparticles on the flow characteristics.
4. Conclusion In this study, molecular dynamics simulation of Couette and Poiseuille Water-Copper nanofluid flows in rough and smooth nanochannels with different roughness configurations was performed. Finally, results were as following: 12
Chemical Physics 527 (2019) 110505
D. Toghraie, et al.
Fig. 20. Changes in the distance between roughness elements in triangular roughness.
• Nanoparticles have caused fluctuations in the middle area which are • •
due to the effects of the surface of the nanoparticles relative to the base fluid of the Water. The fluctuations or abnormal fractures that emerge in the diagram due to the presence of nanoparticles can be clearly seen in the results of smooth nanochannel as well as the results of rough nanochannels. The presence of nanoparticles in the middle area and the interaction between the surface and the fluid in this area has caused abnormal fluctuations.
References [1] H. Noorian, D. Toghraie, A.R. Azimian, Molecular dynamics simulation of Poiseuille flow in a rough nano channel with checker surface roughnesses geometry, Heat Mass Transfer 50 (1) (2014) 105–113. [2] D. Toghraie, A.R. Azimian, Nanoscale Poiseuille flow and effects of modified Lennard-Jones potential function, Heat Mass Transfer 46 (7) (2010) 791–801. [3] D. Toghraie, A.R. Azimian, Molecular dynamics simulation of annular flow boiling with the modified Lennard-Jones potential function, Heat Mass Transfer 48 (1) (2012) 141–152. [4] M. Rezaei, A.R. Azimian, D. Toghraie, The surface charge density effect on the electro-osmotic flow in a nanochannel: a molecular dynamics study, Heat Mass Transfer 51 (5) (2015) 661–670. [5] M. Rezaei, A.R. Azimian, D. Toghraie, Molecular dynamics study of an electro-kinetic fluid transport in a charged nanochannel based on the role of the stern layer, Phys. A 426 (2015) 25–34. [6] D. Toghraie, A.R. Azimian, Molecular dynamics simulation of liquid–vapor phase equilibrium by using the modified Lennard-Jones potential function, Heat Mass Transfer 46 (3) (2010) 287–294. [7] D. Toghraie, A.R. Azimian, Molecular dynamics simulation of nonodroplets with the modified Lennard-Jones potential function, Heat Mass Transfer 47 (5) (2011) 579–588. [8] H. Noorian, D. Toghraie, A.R. Azimian, The effects of surface roughness geometry of flow undergoing Poiseuille flow by molecular dynamics simulation, Heat Mass Transfer 50 (1) (2014) 95–104. [9] L. Harmon, G.C. Lie, E. Clementi, Molecular dynamics simulation of flow past a plate, J. Sci. Comput. 1 (2) (1986). [10] J.L. Xu, Z.Q. Zhou, Molecular dynamics simulation of liquid argon flow at platinum surfaces, Heat Mass Transfer 40 (2004) 859–869. [11] Bing-Yang Cao, Min Chen, Zeng-Yuan Guo, Effect of surface roughness on gas flow in microchannels by molecular dynamics simulation, Int. J. Eng. Sci. 44 (2006) 927–937. [12] Daejoong Kim, Eric Darve, Molecular dynamics simulation of electro-osmotic flows in rough wall nanochannels, Phys. Rev. E 73 (2006) 051203. [13] John A. Thomas, Alan J.H. McGaughey, Ottoleo Kuter-Arnebeck, Pressure-driven water flow through carbon nanotubes: insights from molecular dynamics simulation, Int. J. Therm. Sci. 49 (2010) 281–289. [14] Reza Kamali, Ali Kharazmi, Molecular dynamics simulation of surface roughness effects on nanoscale flows, Int. J. Therm. Sci. 50 (2011) 226–232. [15] Chengzhen Sun, Lu. Wen-Qiang, Jie Liu, Bofeng Bai, Molecular dynamics simulation of nanofluid’s effective thermal conductivity in high-shear-rate Couette flow, Int. J. Heat Mass Transfer 54 (2011) 2560–2567. [16] Meisam Pourali, Ali Maghari, Non-equilibrium molecular dynamics simulation of thermal conductivity and thermal diffusion of binary mixtures confined in a nanochannel, Chem. Phys. 444 (2014) 30–38. [17] Q. Miao, Q. Yuan, Y.P. Zhao, Dissolutive flow in nanochannels: transition between plug-like and Poiseuille-like, Microfluid. Nanofluid. 22 (2018) 141. [18] Q. Yuan, Y.P. Zhao, Multiscale dynamic wetting of a droplet on a lyophilic pillararrayed surface, J. Fluid Mech. (2013) 171–188. [19] M. Zarringhalam, H. Ahmadi-Danesh-Ashtiani, D. Toghraie, R. Fazaeli, Molecular dynamic simulation to study the effects of roughness elements with cone geometry on the boiling flow inside a microchannel, Int. J. Heat Mass Transfer 141
Fig. 21. The effect of the changes in the distance between the roughness elements on (a) the density distribution and (b) velocity distribution in the channel width.
• Increasing the channel height reduces the effect of the surface on the fluid and reduces the flow rate of the nanofluid. • The slip velocity on the bottom and top wall remains almost the same.
13
Chemical Physics 527 (2019) 110505
D. Toghraie, et al. (2019) 1–8. [20] P. Alipour, D. Toghraie, A. Karimipour, M. Hajian, Molecular dynamics simulation of fluid flow passing through a nanochannel: effects of geometric shape of roughnesses, J. Mol. Liquids 275 (2019) 192–203. [21] P. Alipour, D. Toghraie, A. Karimipour, M. Hajian, Modeling different structures in perturbed Poiseuille flow in a nanochannel by using of molecular dynamics simulation: Study the equilibrium, Phys. A: Statist. Mech. Appl. 515 (2019) 13–30. [22] M. Tohidi, D. Toghraie, The effect of geometrical parameters, roughness and the number of nanoparticles on the self-diffusion coefficient in Couette flow in a nanochannel by using of molecular dynamics simulation, Phys. B: Condensed Matter 518 (2017) 20–32. [23] D. Toghraie, M. Mokhtari, M. Afrand, Molecular dynamic simulation of copper and platinum nanoparticles Poiseuille flow in a nanochannels, Phys. E: LowDimensional Syst. Nanostruct. 84 (2016) 152–161. [24] Jabbarzadeh, et al., Rheological properties of thin liquid films by molecular dynamics simulations, J. Non-Newtonian Fluid Mech. 69 (1997) 169–193. [25] D. Toghraie, M.M.D. Abdollah, F. Pourfattah, O.A. Akbari, B. Ruhani, Numerical investigation of flow and heat transfer characteristics in smooth, sinusoidal and
[26] [27] [28] [29] [30]
14
zigzag-shaped microchannel with and without nanofluid, J. Therm. Anal. Calorim. 131 (2) (2018) 1757–1766. M. Parsaiemehr, F. Pourfattah, O.A. Akbari, D. Toghraie, G. Sheikhzadeh, Turbulent flow and heat transfer of Water/Al2O3 nanofluid inside a rectangular ribbed channel, Phys. E: Low-Dimensional Syst. Nanostruct. 96 (2018) 73–84. R. Sarlak, S. Yousefzadeh, O.A. Akbari, D. Toghraie, S. Sarlak, The investigation of simultaneous heat transfer of water/Al2O3 nanofluid in a close enclosure by applying homogeneous magnetic field, Int. J. Mech. Sci. 133 (2017) 674–688. A. Akhgar, D. Toghraie, An experimental study on the stability and thermal conductivity of water-ethylene glycol/TiO2-MWCNTs hybrid nanofluid: developing a new correlation, Powder Technol. 338 (2018) 806–818. M. Keyvani, M. Afrand, D. Toghraie, M. Reiszadeh, An experimental study on the thermal conductivity of cerium oxide/ethylene glycol nanofluid: developing a new correlation, J. Mol. Liq. 266 (2018) 211–217. H. Khodadadi, D. Toghraie, A. Karimipour, Effects of nanoparticles to present a statistical model for the viscosity of MgO-Water nanofluid, Powder Technol. 342 (2019) 166–180.