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# Molecule Dynamics Simulation of Heat Transfer Between Argon Flow and Parallel Copper PlatesOPEN ACCESS

[+] Author and Article Information
Yong Tang, Wei Yuan

Key Laboratory of Surface Functional Structure
Manufacturing of Guangdong High Education Institutes,
School of Mechanical and Automotive Engineering,
South China University of Technology,
Guangzhou 510640, China

Ting Fu

Key Laboratory of Surface Functional Structure
Manufacturing of Guangdong High Education Institutes,
School of Mechanical and Automotive Engineering,
South China University of Technology,
Guangzhou 510640, China
Department of Mechanical and Aerospace Engineering,
University of Missouri,
Columbia, MO 65211
e-mail: futing1234gh@163.com

Yijin Mao

Department of Mechanical and Aerospace Engineering,
University of Missouri,
Columbia, MO 65211

Yuwen Zhang

Department of Mechanical and Aerospace Engineering,
University of Missouri,
Columbia, MO 65211

1Corresponding author.

Manuscript received September 10, 2014; final manuscript received November 11, 2014; published online December 10, 2014. Assoc. Editor: Abraham Wang.

J. Nanotechnol. Eng. Med 5(3), 034501 (Aug 01, 2014) (4 pages) Paper No: NANO-14-1059; doi: 10.1115/1.4029158 History: Received September 10, 2014; Revised November 11, 2014; Online December 10, 2014

## Abstract

Molecular dynamics (MD) simulation aiming to investigate heat transfer between argon fluid flow and two parallel copper plates in the nanoscale is carried out by simultaneously control momentum and temperature of the simulation box. The top copper wall is kept at a constant velocity by adding an external force according to the velocity difference between on-the-fly and desired velocities. At the same time the top wall holds a higher temperature while the bottom wall is considered as physically stationary and has a lower temperature. A sample region is used in order to measure the heat flux flowing across the simulation box, and thus the heat transfer coefficient between the fluid and wall can be estimated through its definition. It is found that the heat transfer coefficient between argon fluid flow and copper plate in this scenario is lower but still in the same order magnitude in comparison with the one predicted based on the hypothesis in other reported work.

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## Introduction

Micro/nanoscale thermofluidic transport is remarkably different from the conventional flow [1], due to its much smaller spatial and temporal scales. From the perspective of convectional theory, the smaller size of the device is, the higher the efficiency of convective heat transfer can be achieved. In other words, micro/nanoscale heat transfer brought up a promising opportunity to dramatically improve efficiency of thermal control if the scale of a device is miniaturized significantly. This explains why micro/nanoscale heat transfer became such an attractive research topic across the entire engineering field in the past two decades. In order to have a better understanding of the mechanism of convective heat transfer in micro/nanoscale, a number of researches have already been conducted in the past years. With the purpose of result benchmark, study from a simple fluid flow problem that have analytical solution available is always a better choice. One of these fundamental fluid flow problems is Couette type flow where the fluid flows at the space between two parallel plates with one plate moving and the other stationary [1]. In fact, a variety of work deal with Couette type heat transfer problem can be found in the literature [2-9]. It is well known that the analytical solution for Couette flow problem can be easily obtained; however, it is also true that continuum assumption break down when the size of interested domain is approaching micro/nanoscale (depends on Knudsen number). MD simulation, which is able to describe the physical process from atomic level, is emerging as a powerful tool to provide detailed information on thermal properties at high shear rates or other extreme conditions for fluid flow problems in the micro/nanoscale. In fact, several similar simulation works have been reported to study these particular phenomena caused by the size effect. Jabbarzadeh et al. [10] investigated the boundary condition of the flow in molecularly thin liquid films of alkanes with roughness modeled by a sinusoidal wall using MD simulation. Soong et al. [1] investigated a Couette flow to study the effects of wall crystal–fluid interactions in nanochannels by MD simulation. Slip characteristics on absorbing surfaces under different conditions were explored by using MD simulation of thin films of hexadecane [11]. Khare et al. [12] studied the thermal resistance of a model solid–liquid interface using MD. It was found that the interfacial thermal resistance increased with the presence of velocity slip while it was not affected by the mass flow in the absence of the velocity slip. Nagayama and Cheng [13] carried out MD simulations to study the effect of the interface wettability on the pressure driven flow in a nanochannel. The results showed that the temperature and pressure profiles were distributed nonuniformly due to the effect of interface wettability. Among these similar works, the temperature and velocity for fluid film were controlled separately, which conflict with the fact that both momentum and temperature are controlled by the same molecular transport. Meanwhile, few researchers paid attention on estimation of heat transfer coefficient between fluid and solid wall during the fluid flow.

To improve the understanding of momentum and energy transports in Couette flow between two parallel flat plates, a more physically sound method which intent to control temperature and velocity simultaneously is applied to the same atoms through the entire MD simulation. In addition, the convective heat transfer coefficient between argon flow and two parallel copper plates is estimated in this work.

## Physical Model and Methods

Figure 1 shows the arrangement of the computational domain, where the liquid argon atoms are placed in the space between bottom and top solid walls which are modeled with copper atoms. The copper atoms located on the top and bottom are fixed while atoms between the top fixed copper plate and fluid argon represent the moving wall. The upper plate moves at a constant velocity relative to the bottom one along with the streamwise direction. The entire simulation box has dimensional size of 18.05 × 18.05 × 1.805 nm in the x-, y-, and z-direction. The height between the two parallel walls is set to be 10.83 nm. The copper wall is modeled with face-centered cubic (FCC) unit that has a lattice constant of 3.61 Å corresponding to its density of 8.69 × 103kg/m3. In order to apply a more physically sound thermostat, each copper wall is modeled with four layers of atoms. The two layers adjacent to fluid are considered to “phantom atoms” which serve as the heat source for each wall [14]. The other two layers are fixed in order to prevent atoms from penetration. Liquid argon that flows between parallel plates is also created by FCC type unit that has a lattice constant of 5.76 Å subject to its density at temperature of 90 K under the pressure of 0.1335 MPa. The total number of atoms in the simulation box is 26,068.

The pair interaction between fluid–fluid and fluid–solid atoms is given by Lennard-Jones (L-J) potentialDisplay Formula

(1)$U=4ɛ[(σr)12-(σr)6]$

where ε is the depth of the potential well, σ is the finite distance at which the intermolecular potential energy is zero, and r is the distance between each pair of the atoms. A cutoff distance of 14 Å is used in this simulation, beyond which the intermolecular force is considered to be zero.

For the interaction of copper–copper atoms, the embedded atom method (EAM) potential [5], which is able to account for the contribution caused by energy carrier of free electrons (interaction between electron and phonons) and more precisely depict the lattice thermal conductivity, is used to describe the interatomic interaction between metal atoms. For EAM type potential, the total potential energy Ei associated to one single atom i can be fitted into following the mathematical form [15] based on its experimental dataDisplay Formula

(2)$Ei=Fi(∑j≠iρi(rij))+12∑j≠iϕi(rij)$

where Ei is the embedding energy of atom i, which is a function of the atomic electron density ρ. In fact, the first term on the right-hand side accounts for the energy contributed from electrons due to interaction between electron gas and nuclei, while the second term φ is a short–range pair interaction between the atoms i and j.

In order to achieve a constant flow velocity at the top of the simulation box, an external force will be acting on the atoms that composing the moving wall. By following the existing work [16], this external force can be determined by:Display Formula

(3)$F=k(uT-1N∑j=1Nr·j)$

where k is the relaxation factor which typically range from 0 to 1, uT is the desired velocity and N is number atoms within desired control space. It can be seen that this force will vanish once the bulk velocity reach to the desired value.

In aspect to temperature control, the direct temperature control method is applied to the atoms associated with these two walls, such that a fixed gradient is applied to the simulation box. It should be pointed out that the desired temperature value in this thermostate should be estimated byDisplay Formula

(4)$T=23kBN[∑j=1N12mj(r·j-u)2]$

since an apparent bulk velocity of the simulation box exist. The thermal reservoir is realized by Langevin method [17]. The acceleration of atom i is estimated byDisplay Formula

(5)$r··i=α(r·i-1N∑j=1Nr·j)+fimi+Fmi$

where the first term on the right hand side denotes the thermal fluctuation. And in the second term, fi represents the force computed through potential function. In the third term, F is a random force vector that satisfies a uniform random. Specifically, the first term represent frictional drag force proportional to the particle’s thermal velocity. And α is a damping factor which is determined by—(m/dactor) where m is the mass of argon atom and dactor is the damping factor used to control the damping strength. In this work, the temperature of bottom copper wall is held at 90 K while the higher temperature of top copper wall is fixed to 190 K. Periodic boundary conditions are applied to the sides in both x- and z-directions. The entire simulation is performed with a time-step of 5 fs and the simulation runs for 2.75 ns in total, such that the flow in the computational domain is fully developed.

The simulations are carried out within the framework of open-source MD simulation package LAMMPS [18] with certain extension, while the data visualization in done with visual MD [19].

## Results and Discussions

In order to validate the implemented schemes, a three-dimensional case of Couette flow with heat transfer is created and the solution obtained here will be compared with the published analytical solution. The desired velocity of the top wall is (1, 0, 0) Å/ps, while the temperature difference between two solid copper walls is set to be 100 K. The final velocity and temperature profile in the y-direction will be extracted by chopping this computation domain into certain number of bins. In this case, 15 and 31 bins are used for temperature and velocity, respectively. In order to achieve a statistically meaningful result, eight similar cases that have almost the same configurations except for initial atomic positions and velocities are used. Figure 2(a) illustrates average velocity profile of the flow along the vertical direction (y-direction) with its error bar and fitting curve. It can be seen that the velocity profile is nearly parabolic, which agrees upon the slip boundary condition [5]. Figure 2(b) shows average temperature profile with its error bar and fitting curve. It also can be seen that the temperature profile along the vertical direction gradually developed to be a straight line, which also agree with the results presented in other researchers’ work [16].

According to the hypothesis initialized by Tuckerman and Pease [20], the value of heat transfer coefficient may vary significantly when the size approach to microscale or nanoscale from the perspective of Nusselt number which is defined as following:Display Formula

(6)$Nu=hD/k$

where h denotes convective heat transfer coefficient, D represents hydraulic diameter, and k is thermal conductivity. It is worth noticing that h may be scaled up to thousand or millions times when the hydraulic diameter D reduces to micro- or nano-size for a fully developed flow in the microchannel, while the definition of Nusselt number is still valid and thermal conductivity is still constant at such small scale. In order to validate this perspective, in this section, heat transfer coefficient h between argon flow and solid copper wall will be estimated with this new scheme through molecular dynamic approach.

In order to measure the heat flux flowing across the fluid when the flow is fully developed, a sample region between top and bottom wall is defined to record the fluctuation of heat flux flowing across the fluid when the flow reaches to the steady state which is considered as a state that both the velocity and temperature profile do no dramatically change along with time. Finally, the heat transfer coefficient between liquid argon flow and copper wall is estimated byDisplay Formula

(7)$h=QAΔtΔT$

where Q represents the total energy passes through copper wall to liquid argon, Δt stands for the time period that this energy pass through, A is the surface area of the wall that is exposed to the liquid argon flow and the ΔT is the temperature difference between the copper wall and the bulk temperature of argon flow. Moreover, from the atomic perspective, the heat flux can be calculated from Ref. [21]Display Formula

(8)$q=QAΔt=1V[∑ieivi-∑iSivi]$

where V is volume of this sample region, ei represents internal energy of atom i. vi is velocity of atom, S is the per-atom stress tensor which is calculated as following:Display Formula

(9)$∑iSivi=∑i

where fij is pairwise force between different atoms and xij is relative position vector between atoms i and j.

Figure 3 shows the fluctuation of heat flux passing through the sample region during the time period after the system reach to steady state. It can be seen that the recorded heat flux fluctuated within a certain range, which is a reflection of intuitive fluctuation of atoms. The heat flux trend shows that most of the values are negative and few of them are positive. This is mainly caused by relatively large noise and small temperature difference across the fluid domain. According to Eq. (8), the computational average heat flux value over the last period of 250 ps is 2.86 × 108W/m2. As a consequence, a simulation results of heat flux coefficient is 2.86 × 106W/m2 K according to Eq. (7). Meanwhile, based on the definition of Nusselt number, the predicted heat transfer coefficient can also be obtained. According to the derivation in Refs. [22,23], the Nu can be expressed as [5]

whereDisplay Formula

(11)$EcPr=U2(T2-T1)μk$

where in this case, U is the velocity in the x-direction which is 1 Å/ps, T2 and T1 are the temperature of top and bottom copper wall, respectively. In this case, the temperature difference is 100 K, the fluid argon viscosity μ is 63.62 × 10−6Pa · s and thermal conductivity k is 58.06 × 10−3 W/m K. In fact, a literature survey shows that the thermal conductivity of ultrathin liquid argon films confined between two plates spaced is smaller than the corresponding value of liquid argon at normal space by around 10% [24]. And Nusselt number will also reduce by around 20% due to the size effect [5]. Therefore, Nusselt number, which will be used to calculate heat flux coefficient according to Eq. (6), after correction is about 1.055. Finally, it is found that, the heat transfer coefficient is about 4.03 × 106W/m2 K. The MD simulation result is lower but still in the same order magnitude in comparison with the result predicted by convectional heat transfer theory.

## Conclusions

In this work, a more physically sound method aiming to control temperature and velocity simultaneously is applied to the same atoms. Based on this method, convective heat transfer between argon fluid and copper plate in Couette type flow is investigated by MD simulation. In order to achieve a constant flow velocity at the top of the simulation box, an external force is developed to compose the moving wall by following the existing work. It is found that both the average velocity profile and temperature profile along the vertical direction (y-direction) agree with the results presented in researchers’ work when the flow in the computational domain is fully developed and reaches to the steady state. In addition, the convective heat transfer coefficient between argon and two parallel copper plates is studied within this frame work. The measured heat flux flowing across the computational domain is recorded in order to compute heat transfer coefficient. It is found that the heat transfer coefficient is lower but still in same order magnitude with the one predicted based on the hypothesis in Ref. [23].

## Acknowledgements

The research was financially supported under the grant of the National Nature Science Foundation of China under Grant No. 51275180, the National Nature Science Foundation of China under Grant No. 51475172 and the U.S. National Science Foundation under Grant No. CBET-1066917. The authors also would like to acknowledge the Join-training Ph.D. Program (No. 201306150079) sponsored by the China Scholarship Council and hosted by the University of Missouri.

## References

Soong, C., Yen, T., and Tzeng, P., 2007, “Molecular Dynamics Simulation of Nanochannel Flows With Effects of Wall Lattice–Fluid Interactions,” Phys. Rev. E, 76(3), p. 036303.
Cahill, D. G., Ford, W. K., Goodson, K. E., Mahan, G. D., Majumdar, A., Maris, H. J., Merlin, R., and Phillpot, S. R., 2003, “Nanoscale Thermal Transport,” J. Appl. Phys., 93(2), pp. 793–818.
Chauhan, S., and Kumar, V., 2011, “Heat Transfer Effects in a Couette Flow Through a Composite Channel Partly Filled by a Porous Medium With a Transverse Sinusoidal Injection Velocity and Heat Source,” Therm. Sci., 15(Suppl. 2), pp. 175–186.
Das, S., Mohanty, M., Panda, J., and Sahoo, S., 2008, “Hydromagnetic Three Dimensional Couette Flow and Heat Transfer,” J. Nav. Archit. Mar. Eng., 5(1), pp. 1–10.
Faghri, A., Zhang, Y., and Howell, J. R., 2010, Advanced Heat and Mass Transfer, Global Digital Press, Columbia, MO.
Govindarajan, A., Ramamurthy, V., and Sundarammal, K., 2007, “3D Couette Flow of Dusty Fluid With Transpiration Cooling,” J. Zhejiang Univ., Sci., A, 8(2), pp. 313–322.
Jana, R. N., Datta, N., and Mazumder, B. S., 1977, “Magnetohydrodynamic Couette Flow and Heat Transfer in a Rotating System,” J. Phys. Soc. Jpn., 42(3), pp. 1034–1039.
Kuznetsov, A., 2000, “Fluid Flow and Heat Transfer Analysis of Couette Flow in a Composite Duct,” Acta Mech., 140(3–4), pp. 163–170.
Soundalgekar, V., Vighnesam, N., and Takhar, H., 1979, “Hall and Ion-Slip Effects in MHD Couette Flow With Heat Transfer,” IEEE Trans. Plasma Sci., 7(3), pp. 178–182.
Jabbarzadeh, A., Atkinson, J., and Tanner, R., 2000, “Effect of the Wall Roughness on Slip and Rheological Properties of Hexadecane in Molecular Dynamics Simulation of Couette Shear Flow Between Two Sinusoidal Walls,” Phys. Rev. E, 61(1), pp. 690–699.
Jabbarzadeh, A., Atkinson, J., and Tanner, R., 1999, “Wall Slip in the Molecular Dynamics Simulation of Thin Films of Hexadecane,” J. Chem. Phys., 110(5), pp. 2612–2620.
Khare, R., Keblinski, P., and Yethiraj, A., 2006, “Molecular Dynamics Simulations of Heat and Momentum Transfer at a Solid–Fluid Interface: Relationship Between Thermal and Velocity Slip,” Int. J. Heat Mass Transfer, 49(19), pp. 3401–3407.
Nagayama, G., and Cheng, P., 2004, “Effects of Interface Wettability on Microscale Flow by Molecular Dynamics Simulation,” Int. J. Heat Mass Transfer, 47(3), pp. 501–513.
Yi, P., Poulikakos, D., Walther, J., and Yadigaroglu, G., 2002, “Molecular Dynamics Simulation of Vaporization of an Ultra-Thin Liquid Argon Layer on a Surface,” Int. J. Heat Mass Transfer, 45(10), pp. 2087–2100.
Finnis, M., and Sinclair, J., 1984, “A Simple Empirical N-Body Potential for Transition Metals,” Philos. Mag. A, 50(1), pp. 45–55.
Wang, Y., and He, G., 2007, “A Dynamic Coupling Model for Hybrid Atomistic–Continuum Computations,” Chem. Eng. Sci., 62(13), pp. 3574–3579.
Schneider, T., and Stoll, E., 1978, “Molecular-Dynamics Study of a Three-Dimensional One-Component Model for Distortive Phase Transitions,” Phys. Rev. B, 17(3).
Plimpton, S., 1995, “Fast Parallel Algorithms for Short-Range Molecular Dynamics,” J. Comput. Phys., 117(1), pp. 1–19.
Humphrey, W., Dalke, A., and Schulten, K., 1996, “VMD: Visual Molecular Dynamics,” J. Mol. Graph., 14(1), pp. 33–38. [PubMed]
Tuckerman, D. B., and Pease, R., 1981, “High-Performance Heat Sinking for VLSI,” IEEE Electron Device Lett., 2(5), pp. 126–129.
Rapaport, D. C., 2004, The Art of Molecular Dynamics Simulation, Cambridge University, Cambridge, UK.
Macdonald, F., and Lide, D. R., 2003, “CRC Handbook of Chemistry and Physics: From Paper to Web,” CRC Press, London, UK.
Liu, C. W., Ko, H. S., and Gau, C., 2011, Heat Transfer—Theoretical Analysis, Experimental Investigations and Industrial Systems, In Tech, Taiwan.
Liu, Q.-X., Jiang, P.-X., and Xiang, H., 2010, “Molecular Dynamics Simulation of Thermal Conductivity of an Argon Liquid Layer Confined in Nanospace,” Mol. Simul., 36(13), pp. 1080–1085.
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## References

Soong, C., Yen, T., and Tzeng, P., 2007, “Molecular Dynamics Simulation of Nanochannel Flows With Effects of Wall Lattice–Fluid Interactions,” Phys. Rev. E, 76(3), p. 036303.
Cahill, D. G., Ford, W. K., Goodson, K. E., Mahan, G. D., Majumdar, A., Maris, H. J., Merlin, R., and Phillpot, S. R., 2003, “Nanoscale Thermal Transport,” J. Appl. Phys., 93(2), pp. 793–818.
Chauhan, S., and Kumar, V., 2011, “Heat Transfer Effects in a Couette Flow Through a Composite Channel Partly Filled by a Porous Medium With a Transverse Sinusoidal Injection Velocity and Heat Source,” Therm. Sci., 15(Suppl. 2), pp. 175–186.
Das, S., Mohanty, M., Panda, J., and Sahoo, S., 2008, “Hydromagnetic Three Dimensional Couette Flow and Heat Transfer,” J. Nav. Archit. Mar. Eng., 5(1), pp. 1–10.
Faghri, A., Zhang, Y., and Howell, J. R., 2010, Advanced Heat and Mass Transfer, Global Digital Press, Columbia, MO.
Govindarajan, A., Ramamurthy, V., and Sundarammal, K., 2007, “3D Couette Flow of Dusty Fluid With Transpiration Cooling,” J. Zhejiang Univ., Sci., A, 8(2), pp. 313–322.
Jana, R. N., Datta, N., and Mazumder, B. S., 1977, “Magnetohydrodynamic Couette Flow and Heat Transfer in a Rotating System,” J. Phys. Soc. Jpn., 42(3), pp. 1034–1039.
Kuznetsov, A., 2000, “Fluid Flow and Heat Transfer Analysis of Couette Flow in a Composite Duct,” Acta Mech., 140(3–4), pp. 163–170.
Soundalgekar, V., Vighnesam, N., and Takhar, H., 1979, “Hall and Ion-Slip Effects in MHD Couette Flow With Heat Transfer,” IEEE Trans. Plasma Sci., 7(3), pp. 178–182.
Jabbarzadeh, A., Atkinson, J., and Tanner, R., 2000, “Effect of the Wall Roughness on Slip and Rheological Properties of Hexadecane in Molecular Dynamics Simulation of Couette Shear Flow Between Two Sinusoidal Walls,” Phys. Rev. E, 61(1), pp. 690–699.
Jabbarzadeh, A., Atkinson, J., and Tanner, R., 1999, “Wall Slip in the Molecular Dynamics Simulation of Thin Films of Hexadecane,” J. Chem. Phys., 110(5), pp. 2612–2620.
Khare, R., Keblinski, P., and Yethiraj, A., 2006, “Molecular Dynamics Simulations of Heat and Momentum Transfer at a Solid–Fluid Interface: Relationship Between Thermal and Velocity Slip,” Int. J. Heat Mass Transfer, 49(19), pp. 3401–3407.
Nagayama, G., and Cheng, P., 2004, “Effects of Interface Wettability on Microscale Flow by Molecular Dynamics Simulation,” Int. J. Heat Mass Transfer, 47(3), pp. 501–513.
Yi, P., Poulikakos, D., Walther, J., and Yadigaroglu, G., 2002, “Molecular Dynamics Simulation of Vaporization of an Ultra-Thin Liquid Argon Layer on a Surface,” Int. J. Heat Mass Transfer, 45(10), pp. 2087–2100.
Finnis, M., and Sinclair, J., 1984, “A Simple Empirical N-Body Potential for Transition Metals,” Philos. Mag. A, 50(1), pp. 45–55.
Wang, Y., and He, G., 2007, “A Dynamic Coupling Model for Hybrid Atomistic–Continuum Computations,” Chem. Eng. Sci., 62(13), pp. 3574–3579.
Schneider, T., and Stoll, E., 1978, “Molecular-Dynamics Study of a Three-Dimensional One-Component Model for Distortive Phase Transitions,” Phys. Rev. B, 17(3).
Plimpton, S., 1995, “Fast Parallel Algorithms for Short-Range Molecular Dynamics,” J. Comput. Phys., 117(1), pp. 1–19.
Humphrey, W., Dalke, A., and Schulten, K., 1996, “VMD: Visual Molecular Dynamics,” J. Mol. Graph., 14(1), pp. 33–38. [PubMed]
Tuckerman, D. B., and Pease, R., 1981, “High-Performance Heat Sinking for VLSI,” IEEE Electron Device Lett., 2(5), pp. 126–129.
Rapaport, D. C., 2004, The Art of Molecular Dynamics Simulation, Cambridge University, Cambridge, UK.
Macdonald, F., and Lide, D. R., 2003, “CRC Handbook of Chemistry and Physics: From Paper to Web,” CRC Press, London, UK.
Liu, C. W., Ko, H. S., and Gau, C., 2011, Heat Transfer—Theoretical Analysis, Experimental Investigations and Industrial Systems, In Tech, Taiwan.
Liu, Q.-X., Jiang, P.-X., and Xiang, H., 2010, “Molecular Dynamics Simulation of Thermal Conductivity of an Argon Liquid Layer Confined in Nanospace,” Mol. Simul., 36(13), pp. 1080–1085.

## Figures

Fig. 1

Computational configuration, liquid argon atoms are placed between bottom and top solid copper atoms

Fig. 2

Final velocity and temperature profiles distribution along with y axis

Fig. 3

Heat flux variation along with simulation time-step

## Discussions

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