Non-Equilibrium Molecular Dynamics Simulation of Water Flow around a Carbon Nanotube

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1 Copyright c 27 Tech Science Press CMES, vol.22, no.1, pp.31-4, 27 Non-Equilibrium Molecular Dynamics Simulation of Water Flow aroun a Carbon Nanotube Wenzhong Tang 1 an Suresh G. Avani 1,2 Abstract: In this paper, non-equilibrium molecular ynamics (MD) simulations were performe to investigate water flow aroun a single-walle carbon nanotube. In the simulation, the nanotube was moele as a rigi cyliner of carbon atoms. Water molecules were escribe with the extene simple point charge (SPC/E) moel. The nanotube-water interactions were calculate with a Lennar-Jones potential between carbon-oxygen pairs. The water-water interactions comprise a Lennar-Jones potential between the oxygen-oxygen pairs an a Coulomb potential between all charge sites on interactive water molecules. It was shown that classical continuum mechanics oes not hol when the rag forces on the nanotube are consiere. In slow cross flows (perpenicular to the tube axis), the cross rag on a nanotube calculate from MD simulations were larger than those from the empirical equations, an the ifference increase as the flow velocity ecrease. It was also foun that in axial flow (in longituinal irection), ue to severe slippage on the nanotube surface, the axial rag on a nanotube from MD simulation was very small compare with the calculation from continuum mechanics. Keywor: Molecular ynamics simulation, water flow, nanotube, rag. 1 Introuction A circular cyliner in flui flow has been of theoretical an experimental interest for many years [Lamb (1945), White (1946), Finn (1953), Kaplun (1957), Broersma (196), Cox (197), 1 Department of Mechanical Engineering an Center for Composite materials, University of Delaware, Newark, DE , USA 2 Corresponing author; avani@uel.eu Batchelor (197), Happel an Brenner (1973), Ui, Hussey an Roger (1984)]. One main concern of this subject was the rag force on the cyliner in a flow fiel. Cross rag on a circular cyliner in cross (transverse) flow an axial rag in axial (parallel) flow form the basis for stuying the ynamics of flexible fibers in a suspension flow [Yamamoto an Matsuoka (1993), Ross an Klingenberg (1997), Joung, Phan-Thien an Fan (21), Tang an Avani (25)], which allows one to preict fiber orientation an istribution in the flowing suspension an its bulk rheological properties. In recent years, carbon nanotubes, a cylinrical fiber-like material with superior mechanical properties an large aspect ratios, have been extensively explore to reinforce various polymeric resins [Tang, Santare an Avani (23), Gong, Li, Bai, Zhao an Liang (24), Wu an Shaw (24), Gojny, Wichmann, Köpke, Fieler an Schulte (24)]. While consierable progress has been mae in the enhancement of polymernanotube composites, an a few viscosity measurements have been reporte on nanotube suspensions [Kinloch, Roberts an Winle (22), Pötschke, Fornes an Paul (22), Seo an Park (24)], little theoretical work has been one on the flow behavior an rheological properties of carbon nanotube suspensions. The major hurle to performing the theoretical stuy is lack of a practical evice to obtain the nano-level rag between the nanotube an the surrouning flui molecules. To our knowlege, currently no technique is available to measure the rag on a nanotube in a liqui flow. However, molecular ynamics simulation metho enables one to investigate nanotube behavior an liqui flows at the molecular level [Rapaport (1995), To (21), Wei, Srivastava an Cho (22)]. Due to the big ifference in length scale, continuum theories for

2 32 Copyright c 27 Tech Science Press CMES, vol.22, no.1, pp.31-4, 27 macroscale flow may not be applicable when consiering similar flows at the nanoscale. By nonequilibrium molecular ynamics simulations, Travis, To an Evans (1997) investigate Poiseuille flows of a flui compose of spherical particles. They foun that, in the simple flui case, classical behavior was approache when channel with was 1.2 times the molecular iameter of the flui particle, but when channel with reuce to 5.1 times the molecular iameter, Navier-Stokes theory began to break own. Walther, Werer, Jaffe an Koumoutsakos (24) investigate water flow past a single walle carbon nanotube using nonequilibriummolecular ynamics simulation. It was foun that slip length in the plane normal to nanotube axis was comparable to the van er Waals istance of the carbonwater potential whereas significant slip occurre along its axis. They also calculate the flui force acting on the nanotube an, base on a few ata, they reporte that the cross rag forces on the nanotubes were in reasonable agreement with the Stokes-Oseen solution for macroscale cyliners. In our previous work [Tang an Avani (26)], we stuie the cross rag on a nanotube in uniform argon flow. It was foun that the calculate rag forces by molecular ynamics simulation were ifferent from what one woul expect from continuum mechanics. In the range of velocities encountere in engineering applications, the rag on a nanotube is expecte to be much larger than the one calculate using continuum mechanics theory, which inirectly explains the increase in viscosity of the nanotube suspension even at small concentrations. However, one question arises from our previous stuy - Does our conclusion about the rag on a nanotube in liqui argon flow exten to other polyatomic fluis? In this paper, we focus on the cross an axial rag forces on a nanotube in water flow. 2 Simulation etails Water molecules flowing aroun a single walle zigzag (6, ) nanotube (with a iameter of =.4752nm) was investigate in this stuy. Figure 1 illustrates a unit cell of a carbon nanotube - water system use in our simulation. In the L W H cubic cell, the nanotube is fixe at the center of the system, with its axis along the z irection. Aroun the nanotube are water molecules at 25 o C, with a ensity of ρ = 996kg/m 3 an a ynamic viscosity of μ = kg/m s [Kunu (199)]. The water molecule was escribe with the extene simple point charge (SPC/E) moel as shown in Figure 2 [Berensen, Grigera an Straatsma (1987), Kusalik an Svishchev (1994)], where the bon length (r OH = m) an bon angle ( HOH = o ) are fixe; the oxygen carries a negative electric charge of q O =.8476e an each hyrogen atom carries a positive charge of q H =.4238e; an the oxygenoxygen well epth an van er Waals raius are ε OO =.651KJ/mol an σ OO = m respectively. The interactions between two water molecules consist of a Lennar-Jones potential between the oxygen atoms an a Coulomb potential comprising of all electrically interactive pairs. The Leonar-Jones potential between the oxygen pair is calculate as [Kotsalis, Walther an Koumoutsakos (24)] [ (σoo ) 12 E LJ (r OO )=4ε OO r OO ( σoo r OO ) ] 6, r OO r c_lj (1) where r OO is the istance between the two oxygen atoms an r c_lj = m is a cutoff istance beyon which the Lennar-Jones interaction is neglecte. The Coulomb potential between two electrically interactive sites is expresse as [Kotsalis, Walther an Koumoutsakos (24)] E El (r ij )= q iq j 4πε r ij, r ij r c_el (2) where q i an q j are the electric charges of the two sites, ε = F/m is the permittivity in vacuum, r ij is the istance between the two interaction sites, an r c_el is a cutoff istance beyon which the electrostatic interaction is neglecte. The truncation of the Coulomb potential has been shown to have little effect on the thermoynamic an structural properties of water for cutoff istances larger than m [Anrea, Swope an Anersen (1984)].

3 Non-Equilibrium Molecular Dynamics Simulation of Water Flow aroun a Carbon Nanotube 33 U y x W Carbon nanotube z Water molecules L H (a) (b) Figure 1: Unit cell of carbon nanotube water system: (a) top view; (b) sie view. σ OO H ( q H ) θ x' O ( q ) O z' y' H ( q H ) r OH Figure 2: SPCE water moel (Atoms are on the y z plane). In this work, we employe a cutoff istance of r c_el = m. A larger value i not make much ifference in the rag on the nanotube from our simulation. As in our previous work [Tang an Avani (26)] an in the work by Gorillo an Marti (23) an by Walther, Werer, Jaffe an Koumoutsakos (24), the carbon nanotube was moele as a rigi structure, in which the bons between carbon atoms are fixe (bon length r C C = m) an the carbon atoms o not move relative to each other. The rigi tube moel is use in our stuy since it has been verifie that the consieration of carbon mobility has little effect on the hyroynamic properties of water/carbon nanotube system [Werer, Walther, Jaffe, Halicioglu an Koumoutsakos (23)]. The nanotube-water interaction is escribe by the Lennar-Jones potential between the carbon an oxygen sites which is expresse as [ (σco ) 12 E LJ (r CO )=4ε CO r CO ( σco r CO ) ] 6, r CO r c_lj (3) where ε CO =.4389KJ/mol an σ CO = m are the carbon-oxygen well epth an carbon-oxygen van er Waals raius respectively [Bojan an Steele (1987), Werer, Walther, Jaffe, Halicioglu an Koumoutsakos (23)], r CO is the istance between the interaction sites, an r c_lj = m is the same cutoff istance as in Equation 1 for oxygen-oxygen pairs. With the above potentials, a coe base on the one by Rapaport (1995) for two-imensional liqui argon flow aroun a circular obstacle was evelope to simulate the water flow aroun a carbon nanotube. In this paper, two types of flow are investigate. One is cross (transverse) flow in which the flow irection is perpenicular to the axis of the nanotube, an the other is axial flow in

4 34 Copyright c 27 Tech Science Press CMES, vol.22, no.1, pp.31-4, 27 which the water molecules flow parallel to the tube axis. In both cases, the water molecules are initially evenly place in the omain, an all water molecules have ientical translational an rotational spees but move in ranom irections resulting in a system with zero resultant velocity. A cross (transverse) flow is initialize by superimposing a esire flow velocity in the x irection on all water molecules, an maintaine by first reefining ranom velocity of molecules within.25 L to the x = L/2 bounary every 4 time steps, an then superimposing the esire flow velocity on these molecules. This technique of maintaining uniform flow also removes excess heat, which is equivalent to aing a heat sink at the inlet [Rapaport (1995), Tang an Avani (26)]. An axial flow is generate in a similar way. It is initialize by superimposing a esire flow velocity in the z irection on all water molecules, an maintaine by superimposing the esire velocity in the axial (z-) irection on molecules within.25 L to the x = ±L/2 bounaries an within.25 W to the y = ±W/2 bounaries. Perioic bounary conition was use in all three irections. In this work, neighbor list metho was use in the calculation of interactions between atoms, an the equations of motion were integrate with preictor-corrector metho. In our calculations, non-imensional parameters were use by choosing σ OO, ε OO,anm H2 O (molecular mass of water) as the units of length, energy an mass respectively, leaing to a time unit of t = s. In this stuy, parameters will be given in the nonimensional units (or reuce units) unless otherwise state. Throughout this work, a time step of s was use. 3 Results an iscussion In this paper, the cross an axial rag forces on the nanotube were obtaine from cross (transverse) an axial (parallel) flow respectively. Drag forces from MD simulations were compare with classical results from empirical equations for flow aroun a circular cyliner. 3.1 Cross (transverse) flow Time-average cross rag To obtain the cross rag on the nanotube in the cross flow, simulations were performe until the system reache steay state an maintaine the state for a series of aitional time steps. The number of time steps neee for a run varie from 8, to 1,6,. When a lower flow spee U was applie, more time steps were neee for the system to reach steay state. Figure 3 shows typical results of total, kinetic an potential energy of the system as a function of time. In this case, the nanotube was place in a omain of L W H = nm 3 an a flow spee U = 95m/s (or.5 in reuce units) was applie. We observe that both kinetic an potential energy ecrease uring the inception stage. The kinetic energy ecrease because initially a flow velocity of U was superimpose on all water molecules. Due to resistance from the nanotube, the flow slowe own with time until the input of impose flow balance the resistance ue to the presence of the nanotube; thus the system reache steay state after t = 15. At each time step, the rag force on the nanotube (instantaneous rag) was obtaine by summing up forces on all carbon atoms. Figure 4a presents the instantaneous rag as a function of time. To characterize the rag on the nanotube, time-average values were obtaine by averaging the instantaneous rag over time. The time-average rag F avg (t) was calculate as F avg (t)= 1 t N t F (t), (4) t t where t was chosen as a fixe starting time when the system was at steay state, t is any time after t,ann t t is the number of sampling points from t to t. Figure 4b shows the time-average rag when the starting time t = 2. It is seen that the time-average rag converges with time, resulting in an average value of an a stanar eviation of.26 for t = In this work, the time-average value from t (vary from 2 to 4) to the en of each simulation t en (vary from 45 to 6) will be use to characterize the rag force on the nanotube.

5 Non-Equilibrium Molecular Dynamics Simulation of Water Flow aroun a Carbon Nanotube 35 Energy total kinetic potential Time Figure 3: Energy evelopment in a waternanotube flowing system (in reuce units) Instant rag. Time-average rag Time (a) Time (b) Figure 4: Typical simulate rag on a nanotube (in reuce units): (a) instantaneous rag; (b) time-average rag Effect of omain size on cross rag Due to the time consuming nature of molecular ynamic simulations, a small system is esirable but a system too small may provie inaccurate information. Our iea was to use a small cell thickness H to spee up our calculation. The simulation time is approximately proportional to the size of the system. When the cell thickness H is reuce, the number of molecules in the system reuces, requiring less calculation time. Table 1 lists the unit rag (rag force per unit length) on the nanotube for 4 ifferent values of cell thickness H while the other imensions were fixe (L W = nm 2 ) when a flow spee of U =.5(95m/s) was applie. It shoul be note that the unit cell is repeate in all the three irections (x, y an z) ue to the perioic bounary conition, so a molecule in the unit cell has its image in each of the repeating cells [Rapaport (1995)]. When the cell thickness H is small ( r c_el ), the istance between a molecule an its image in a repeating cell may fall within the interaction range, hence this interaction shoul also be inclue. It is seen when the cell thickness is larger than the cutoff istance (H/r c_el = 1.12 an 2.25), the ifference in the simulate unit rag force is very small (.28%). The ifference increases as the cell thickness ecreases. Using the thickest cell as a reference (H/r c_el = 2.25), the errors in the simulate rag are 3.4% an 68.1% for the thinner cells with H/r c_el =.75 an.38 respectively. Consiering both calculation time an accuracy, the cell thickness H =.85nm (H/r c_el =.75) was use in our parametric stuy. To examine the effect of omain size L W on cross rag, two flow spees U =.5 (95m/s) an 1. (19m/s) an five omain sizes (L/=11.2, 16.7, 22.3, 27.9 an 33.5, withl = W ) were consiere. Figure 5 shows that the unit rag on the nanotube ecreases an converges with growing omain size for both flow spees. The ifferences in rag force between the last two points on the curves are.99% an 3.4% for U =.5 an 1. respectively. We settle on a omain size of L/ = 27.9 (L W H = nm 3 ) to investigate how the cross rag changes with the flow spee. Unit rag U =.5 U = Domain size, L/D Figure 5: Convergence of rag force (in reuce units) with growing omain size.

6 36 Copyright c 27 Tech Science Press CMES, vol.22, no.1, pp.31-4, 27 Table 1: Cross rag on the nanotube for ifferent cell thicknesses H (L W = nm 2, flow spee U =95m/s). Cell thickness H, nm (H/r c_el ) (.38) (.75) (1.12) (2.25) Unit rag (error, %) (68.1) (3.4) (.28) (.) Cross rag as a function of flow spee To examine the effect of flow spee on the cross rag, simulations were performe on a omain size of L W H = nm 3. The system ha 4976 water molecules an 48 carbon atoms. The initial ranom spee of water molecules was U i = 3.38 (642.2m/s) at a temperature of T = 25 o C. The flow spee U was varie from (47.5 to 475m/s), corresponing to a Reynols number of Re = (base on the iameter of the nanotube). Table II lists the simulate rag per unit length F an rag coefficient C at ifferent flow spees. The rag coefficient was calculate as [Kunu (199)] F C = 1 2 ρu 2, (5) where ρ is the ensity of water, an is the iameter of the nanotube. It shoul be note that since perioic bounary conition is use in our MD simulation, the actual problem is flow aroun an array of carbon nanotubes. However, as illustrate in Figure 5, the simulate rag converges as omain size increases, implying that a converge rag from a large unit cell can also be approximate as unboune flow aroun a single nanotube. To compare our simulate rag with continuum theory, the rag an rag coefficient were also calculate using two empirical equations: Stokes-Oseen rag an Huner rag. The Stokes-Oseen rag is for flow past an array of two-imensional circular cyliners an the rag coefficient is calculate as [Probstein (1989), Walther, Werer, Jaffe an Koumoutsakos (24)] C Oseen = C cc 3 +2φ 5/ φ 1/ φ 5/3 3φ 2, (6) where φ = π 2 /4LW is the volume fraction of the nanotube in the unit cell, an C cc is the rag coefficient on a single circular cyliner given by C cc 8π = Reln(7.4/Re). (7) The Huner rag, which is base on experiments using macroscale cyliners, is expresse as [Huner an Hussey (1977)] F = { F Lamb 1.87ε [1 exp( Re)]ε 3}, (8) where F Lamb is the Lamb rag given by F Lamb = 4πμU ε, (9) where μ is flui viscosity, an ε is calculate as ε =[.5 γ ln(re/8)] 1, (1) in which γ = is Euler s constant an Re is Reynols number. Table 2 shows that, similar to what happens at the macroscale, the cross rag force on the nanotube from MD simulation increases with the flow spee U while the rag coefficient increases as the flow spee ecreases. Figures 6 compares the rag coefficients calculate from MD simulation an from the empirical equations. When a flow spee of U =.25 (47.5 m/s) is applie, the simulate rag (or rag coefficient) is 75% larger than the Huner an 54% larger than the Stokes-Oseen rag (or rag coefficient). As shown in Figure 6, the ifferences are expecte to increase as the flow spee ecreases. However, when the flow spee increases, the rag coefficient from MD reuces faster than that from the empirical equations. This ifference in cross rag force (or rag coefficient) between MD simulation an empirical equations is similar to what we observe in our previous simulation on nanotube in liqui argon flow [Tang an Avani

7 Non-Equilibrium Molecular Dynamics Simulation of Water Flow aroun a Carbon Nanotube 37 Table 2: Cross rag forces (reuce units) an rag coefficients from MD simulation an empirical equations. Drag force (Drag coefficient) Flow spee U Re F m (C m) (C Huner ) F Oseen (C Oseen ) (297.9) (17.6) (194.) (154.) (96.38) (11.5) (66.12) (55.28) 5.97 (64.14) (33.95) 72.2 (4.28) (47.22) (22.54) 12.7 (32.32) (38.24) (15.76) (27.32) 162. (32.62) Drag coefficient MD Huner Oseen Flow spee U Figure 6: Comparison of rag coefficients from molecular ynamics simulation an empirical equations erive from continuum mechanics. (26)]. It can be conclue that empirical equations for cross rag force on macroscale cyliners oes not translate to the scale of the carbon nanotubes. Figure 7 plots the flui velocity in the flow (x-) irection as a function of y on the yz plane through the nanotube axis (Figure 2) when flow spees are.75 an 1.5. The etails of how the velocity profiles are obtaine can be foun elsewhere [Tang an Avani (26)]. It is seen that there is nonzero velocity on the nanotube surface, implying that slippage occurs on the tube wall. For a higher flow spee, this nonzero wall velocity (slip velocity) is larger. This explains why the rag coefficient from MD simulations reuces faster than that from the empirical equations as the flow spee increases. 3.2 Axial (parallel) flow The axial rag was obtaine on a unit cell of L W H = nm 3, which containe 48 carbon atoms an 1776 water molecules. The simulation was performe in a similar way to the cross flow, except that the flow was in the lon- Normalize y U =.75 U = Flui velocity in flow (x-) irection Figure 7: Velocity profile on the yz-plane through tube axis illustrating slip velocity on the nanotube surface. gituinal (z) irection of the nanotube an maintaine by superimposing a esire flow velocity U on molecules within.25 L to the x = ±L/2 bounaries an within.25 W to the y = ±W/2 bounaries (shae regions in Figure 8a). The simulate axial rag is not sensitive to the increase in the omain size L W or the omain thickness H. For qualitative comparison, the unit rag on a circular cyliner of infinite length moving parallel to its axis in a coaxial cylinrical bounary (Figure 8b) was also calculate using Happel an Brenner s equation [Happel an H. Brenner (1973), Ui, Hussey an Roger (1984)] F Happel = 2πμU σ 2 3 +[4(lnσ)/(σ 2 1)] [(σ 2, +1)/(lnσ)] 1 (11) where μ is the ynamic viscosity of the flui, U is the moving spee of the cyliner, an σ = L/D

8 38 Copyright c 27 Tech Science Press CMES, vol.22, no.1, pp.31-4, 27.25L.25W y x Nanotube W Unit axial rag Axial-MD Axial-Happel Cross-MD L (a) Flow spee Figure 9: Drag force per unit length on the nanotube. Wall.6 Cyliner y D x Normalize y U =.5 U = L (b) Figure 8: Domain for axial flow: (a) MD simulation; (b) Happel equation Flui velocity in flow (z-) irection Figure 1: Velocity profile (axial flow) on the yzplane through tube axis illustrating slipage on the surface of the nanotube. is the ratio of the iameter of the flow fiel to the iameter of the moving cyliner. Figure 9 compares the axial rag on the nanotube from MD simulation an the above equation, in which the simulate cross rag is also plotte. It is seen the simulate axial rag forces are very small compare with the values calculate from the Happel an Brenner s equation. Figure 1 plots the velocity (flow irection) profile on the yz plane through the nanotube axis for flow spees of U =.5 (95 m/s) an 1. (19 m/s). It is shown that the magnitue of the flui velocity on the tube wall is of the orer of the applie flow spee, implying that significant slippage occurs on the surface of the nanotube in the axial flow. This explains the small axial rag forces on the nanotubes from MD simulation. Figure 9 also shows that the axial rag is negligible compare with the cross rag on the nanotube at the same flow spee. 4 Conclusions In this work, nonequilibrium MD simulations were performe to investigate water flow aroun a carbon nanotube. It was foun that the rag forces on the nanotube were ifferent from what one woul expect from continuum mechanics. The results show that when the flow spee was low, the cross rag coefficient from MD simulation was larger than the ones calculate from empirical equations base on continuum mechanics. The ifference increase as the flow spee ecrease. As the flow velocity was increase, the simulate rag from MD reuce faster an fell below the empirical results. This was attribute to slippage of the water molecules on the nanotube at large flow spees. Above conclusions about cross rag on a nanotube in water flow isagree with what was reporte by Walther, Werer, Jaffe an Koumoutsakos (24) while agree with the con-

9 Non-Equilibrium Molecular Dynamics Simulation of Water Flow aroun a Carbon Nanotube 39 clusions in our previous work on nanotube in liqui argon flow. It was also shown that, ue to significant slippage on the nanotube surface in axial flow, the axial rag on a nanotube from MD simulation was very small compare with the calculation from continuum mechanics. Results also show that the axial rag on the nanotube is negligible in comparison with the cross rag at the same flow spee. Acknowlegement: This work was financially supporte by the National Science Founation on Processing of Nanotubes Using Liqui Composite Moling Techniques uner grant no. DMI References Anrea, T. A.; Swope, W. C.; Anersen, H. C. (1984): The role of long range forces in etermining the structure an properties of liqui water. J. Chem. Phys, vol. 79, pp Batchelor, G. K. (197A): Slener-boy theory for particles of arbitrary cross-section in Stokes flow. J. Flui Mech, vol. 44, pp Berensen, H. J. C.; Grigera, J. R.; Straatsma, T. P. (1987): The missing term in effective pair potentials. J. Phys. Chem, vol. 91, pp Bojan, M. J.; Steele, W. A. (1987): Interactions of iatomic molecules with graphite. Langmuir, vol. 3, pp Broersma, S. (196): Viscous force constant for a close cyliner. J. Chem. Phys, vol. 32, pp Cox, R. B. (197): The motion of long slener boies in a viscous flui Part 1: General theory. J. Flui Mech, vol. 44, pp Finn, R. K. (1953): Determination of the rag on a cyliner at low Reynols numbers. J. Appl. Phys, vol. 24, pp Gojny, F. H.; Wichmann, M. H. G.; Köpke, U.; Fieler, B.; Schulte, K. (24): Carbon nanotube-reinforce epoxy-composites: enhance stiffness an fracture toughness at low nanotube content. Composites Sci. Tech, vol. 64, pp Gong, Q. M.; Li, Z.; Bai, X. D.; Li, D.; Zhao, Y.; Liang, J. (24): Thermal properties of aligne carbon nanotube/carbon nanocomposites. Mater. sci. eng. A Structural mater. properties microstructure & processing, vol. 384, pp Gorillo, M. C.; Marti, J. (23): Water on the outsie of carbon nanotube bunles. Phys. Rev. B, vol. 67, article no Happel, J.; Brenner H. (1973): Low Reynols number hyroynamics. Noorhoff International Publishing, Leyen, Netherlans. Huner, B.; Hussey, R. G. (1977): Cyliner rag at low Reynols number. Phys. Fluis, vol. 2, pp Joung, C. G.; Phan-Thien, N.; Fan, X. J. (21): Direct simulation of flexible fibers. J. Non-Newtonian Flui Mech, vol. 99, pp Kaplun, S. (1957): Low Reynols number flow past a circular cyliner. J. Math. Mech, vol. 6, pp Kinloch, I. A.; Roberts, S. A.; Winle, A. H. (22): A rheological stuy of concentrate aqueous nanotube ispersions. Polymer, vol. 43, pp Kotsalis, E. M.; Walther, J. H.; Koumoutsakos, P. (24): Multiphase water flow insie carbon nanotubes. Inter. J. Multiphase Flow, vol. 3, pp Kunu, P. K. (199): Flui Mechanics. Acaemic Press, California, USA. Kusalik, P. G.; Svishchev, I. M. (1994): The spatial structure in liqui water. Science, vol. 265, pp Lamb, S. H. (1993): Hyroynamics. Dover Publications, New York, USA. Pötschke,P.;Fornes,T.D.;Paul,D.R.(22): Rheological behavior of multiwalle carbon nanotube/polycarbonate composites. Polymer, vol. 43, pp Probstein, R. F. (1989): Physicochemical Hyroynamics. Butter-worths, Lonon. Rapaport, D. C. (1995): The Art of Molecuar Dy-

10 4 Copyright c 27 Tech Science Press CMES, vol.22, no.1, pp.31-4, 27 namics Simulation. Cambrige University Press, New York, USA. Ross, R. F.; Klingenberg, D. J. (1997): Dynamic simulation of flexible fibers compose of linke rigi boies. J. Chem. Phys, vol. 16, pp Seo, M. K.; Park, S. J. (24): Electrical resistivity an rheological behaviors of carbon nanotubes-fille polypropylene composites. Chem. Phys. Lett, vol. 395, pp Tang, W.; Avani, S. G. (25): Dynamic simulation of long flexible fibers in shear flow. CMES: Computer Moeling in Engineering & Sciences, vol. 8, pp Tang, W.; Avani, S. G. (26): Drag on a nanotube in uniform liqui argon flow. J. Chem. Phys, vol. 125, article no Tang, W.; Santare, M. H.; Avani, S. G. (23): Melt processing an mechanical property characterization of multi-walle carbon nanotube/high ensity polyethylene (MWNT/HDPE) composite films. Carbon, vol. 41, pp To, B. D. (21): Computer simulation of simple an complex atomistic fluis by nonequilibrium molecular ynamics techniques. Computer Phys. Communications, vol. 142, pp Travis,K.P.;To,B.D.;Evans,D.J.(1997): Poiseuille flow of molecular fluis. Physica A: Statistical & Theoretical Phys, vol. 24, pp Ui, T. J.; Hussey, R. G.; Roger, R. P. (1984): Stokes rag a cyliner in axial motion. Phys. Fluis, vol. 27, pp Walther, J. H.; Werer, T.; Jaffe, R. L.; Koumoutsakos, P. (24): Hyroynamic properties of carbon nanotubes. Phys. Rev. E, vol. 69, article no Wei, C. Y.; Srivastava, D.; Cho, K. (22): Molecular ynamics stuy of temperature epenent plastic collapse of carbon nanotubes uner axial compression. CMES - Computer Moeling in Engineering & Sciences, vol. 3, pp Werer, T.; Walther, J. H.; Jaffe, R. L.; Halicioglu, T.; Koumoutsakos, P. (23): On the water-carbon interaction for use in molecular ynamics simulations of graphite an carbon nanotubes. J. Phys. Chem. B, vol. 17, pp White, C. M. (1946): The rag of cyliners in fluis at slow spees. Proceeings Royal Soc. Lonon, Series A, Math. Phys. Sci, vol. 186, pp Wu, M.; Shaw, L. L. (24): On the improve properties of injection-mole, carbon nanotubefille PET/PVDF blens. J. Power Sources, vol. 136, pp Yamamoto, S.; Matsuoka, T. (1993): A metho for ynamic simulation of rigi an flexible fibers in a flow fiel. J. Chem. Phys, vol. 98, pp