Projektbericht Applikationsserver Phoenix

Transcription

1 Guido Fuchs Institut für Strömungsmechanik und Wärmeübertragung, Tu-Wien Projektbericht Applikationsserver Phoenix MOLECULAR DYNAMICS SIMULATION OF MARANGONI FLOW IN A CAVITY Guido Fuchs Institute for Fluid Mechanics and Heat Transfer, Vienna University of Technology, A-1040 Vienna, Austria guido.fuchs@tuwien.ac.at ABSTRACT The flow in two liquids inside a rectangular side-heated cavity, driven by a thermal-gradient-induced Marangoni effect along the common interface, is investigated numerically using molecular dynamics simulations. For this configuration the surface-tension gradient drives two counter-rotating vortices inside the cavity. The model fluids are simple Lennard-Jones type fluids, selected for varying liquid-solid contact angles. The velocity field is determined and compared to analytical results of the continuum creeping-flow approximation of this configuration. INTRODUCTION The Marangoni effect is a promising means to force a fluid flow on a microscopic scale, since surface forces dominate over volume forces on small length scales. But from a design point of view it is not straightforward how to model the fluid dynamics and thus thermocapillary effects at such scales. For small systems the validity of the continuum assumption, the linearity of the transport laws in the presence of high gradients, and the boundary conditions - like the no-slip condition, taken for granted in the macroscopic description, may be questioned (Gad-El-Hak 1999). Molecular dynamics (MD) simulation is an alternative way to model the properties of simple fluids and was adopted early to simulate fluid flows, like e. g. Rayleigh- Bérnard convection (Mareschal 1988). One important problem of particular interest is the validity of the no-slip condition at solid interfaces which has been investigated by Koplik et al. who applied MD simulations to Couette and Poiseuille flow (Koplik 1989). He found the no-slip condition to be valid for dense fluids in most cases, except for regions where very high shear gradients occur as, e. g.,near moving contact lines or in the vicinity of the singulaorner of an one-sided lid-driven cavity (Koplik 1989, 1995, and 200). In these cases slip occurs. Later on, these findings have been confirmed by other investigators (Qian 2003, Thompson 1997). Molecular dynamics was also used to simulate static properties of the liquid-solid interface like the contact angle of a liquid with a wall (Maruyama 1998). It was used, moreover, to simulate flows driven by surface-tension effects like, e. g., dewetting of a solid (Koplik 2000), imbibition of liquids into narrow pores (Martic 2002), oapillary rise (Seveno 2004). A problem similar to the moving contact line occurs at the static triple fluid-fluid-solid - contact line for Marangonidriven flows, since there is a discontinuity in the stress tensor and, therefore, an unresolved singularity of the vorticity at the triple contact line (Kuhlmann 1999). The aim of this work is to better understand the flow near the contact line. NOMENCLATURE Symbols H Aspect ratio of cavity k Parameter of molecular potential, ratio of the strength of attraction to the strength of repulsion m Molecular mass N Number of molecules r Distance between two atoms Cutoff radius of potential u U Velocity Potential energy function Greek Symbols Energy parameter of the Lennard-Jones potential Size parameter of the Lennard-Jones potential n Even Papkovich-Fadle functions 1 n Adjoint Papkovich-Fadle funcitons Stokes stream function Stream function based on the mass flux Density 1

2 MOLECULAR DYNAMICS SIMULATION In our simulations we consider the Marangoni flow in a closed cavity at a microscopic scale using the well-established tool of molecular dynamics simulation (Griebel 2004). As shown in Figure 1., the basic configuration used in all simulations is a cavity formed by crystalline solid walls, containing two immiscible fluids each of which being in contact with the heated sidewalls. The sidewalls are kept at different constant temperatures by thermostatted ghost particles. Hence, a temperature gradient tangential to the liquidliquid interface is developing, driving two counter-rotating vortices via the thermocapillary effect. Figure 1: Snapshot of a simulation volume containing two separated liquids inside crystalline, solid walls forming the cavity. The simulation volume has a height, width and depth of 2.2, 41.5, and 5.2, respectively, and contains 951 particles. The lower fluid is stronger wetting than the upper one. Ghost particles to the left and right act as thermostats. The basic interatomic force potential used in the simulations is a force-continuous Lennard-Jones type potential, similar to the potential proposed by Stoddard (1973). It is given by k 12 U R 4 r r c r 4 k 1 12 k r 2. (1) Where r is the distance between two particles, is the strength of the potential, is roughly the diameter of the modeled atom, is the cutoff radius and k is a dimensionless parameteontrolling the strength of attraction between atoms. The advantage of this potential stems from the fact, that it is simple and similar to the basic Lennard-Jones potential. But contrary to the Lennard-Jones potential both the potential energy and the force between atoms tend to zero continuously. This is an important property for simulations of capillary-driven flows. The mean force per atom in direction to the cold wall, resulting from the mean surface tension gradient and being caused by the rearrangement of the atoms by the thermal gradient along the interface, is rather small. This mean force is comparable in magnitude to the discontinuity of the force at the cutoff radius in the normal Lennard-Jones potential for the small cutoff radii commonly used in MD simulation to save computing time. All quantities used in the simulation are dimensionless. For simplicity, we chose the following potential parameters I, I II,II I, II I,W II,W W,W 3 1 k I, I k II,II k W,W 2 k I,II 1. (2),I, I,II,II, I, II,I,W, II,W 2,W,W 5 I,I II, II I, II I,W II,W W,W 1 The subscripts stand for fluid one (I), fluid two (II), and the wall (W). For example, I,W is the energy parameter for the interaction between liquid I and the wall. The ratios of attractions k I,W and k II,W are varied for different simulation runs. They are given in Table 1. The fluid properties are standard and chosen to realize two immiscible liquids with otherwise identical properties. The wall parameters were selected to create the lattice of a solid. The wall atoms are additionally tethered to bcc lattice sites with linear springs. The length of the lattice unit cell is The walls are thermostatted using ghost particles, similar to the ones used by Maruyama (2000). They are coupled to an Anderson thermostat with a coupling constant of 0.05 and a temperature of 1.2 at the hot side and 0.75 at the cold side. The ghost particles interact with each other and with the wall, thereby thermostatting the wall, using the Weeks-Chandler-Anderson potential, which consists only of a repulsive core. The integration was accomplished by the simple velocity- Verlet integrator using 500 time steps per unit time. Each simulation was equilibrated for two million time steps to make sure that the system is well temperated and quasi-stationary. Subsequently it is followed by a production phase of eight million time steps. 2

3 A total of six simulations have been carried out. The simulations denoted 1.1, 1.2, and 1.3, are foavities of increasing size and identical wall interactions, thus a contact angle of 90. The simulation runs denoted 2.1 and 2.2, also of increasing size, were made for slightly unsymmetrical wall interactions. Therefore, liquid II is wetting stronger than liquid I. The simulation 3.1 has again a symmetrical wall interaction, but the interaction with the wall is strongly repulsive, so that neither fluid tends to wet the solid. The data of all simulations are given in Table 1. The ratio of the liquid-wall attraction are specified as k I,W and k II,W. N I and N II give the number of fluid atoms in the cavity of size l x and l y as measured from the position of the solid atoms. The size of the simulation volume is given by the parameters B x, B y, and B z. Table 1: Parameters for the different simulation runs. Simulation k I,W k II,W N I N II B x B y B z l x l x DATA ANALYSIS The data acquired during the simulation runs are the mean averages of the momentum flux, kinetic energy, potential energy, the virial and the density. Also the standard deviations of these properties are calculated. All values are made dimensionless with respect to the parameters of the fluids' molecular potentials. For the averaging the simulation volume is divided into small bins of roughly the size 0.1, 0.1, and 5.18 in x, y, and z- direction. We are primarily interested in the momentum flux. Since it shows a poor signal to noise ratio (Figure 2), we are smoothing the velocity data. The strong noise results from the random atomic velocity (the system's temperature) which is orders of magnitude higher than the rather small thermocapillary drift velocity. For example, the calculated mean standard deviation for the momentum flux in x-direction, for a bin of simulation 3.1, is about Compared to a typical momentum flux of the value is rather high. This is expected, however, since the bin volume is small and contains, on time average, only about 0.04 particles. To smooth the results the mass-flux-based stream function, defined by u x y and u y x, is calculated from the noisy data. This is done by starting the integration several thousand times from a randomly chosen point inside the cavity and fitting the result from each integration to the previous integrations. The resulting stream function is then filtered in the frequency space using a perfect low-pass filter keeping only the first eight Fourier modes. This procedure turned out to be sufficient to keep the significant properties of the original stream function. Velocities are then derived from the smoothed stream function and the density field. RESULTS Figure 2 shows typical data from simulation run 2.1. The direction of view is the same as in Figure 1. The black, solid frame corresponds to the mean position of the first atomic layer of the solid wall. The inner dashed frame indicated a distance of about 0.5 (half a unit cell of the solid lattice) from the mean location of the first solid atoms. The plots of the momentum flux in vertical u y and horizontal u x directions clearly show the two counter-rotating vortices, which are driven by the surface tension gradient. Since the fluids have different wetting properties for this run, a meniscus is formed. The contact angle at the hot wall is about 57 and at the cold wall it is about 51. The velocity in the center is, as expected, directed towards the cold wall. Apart from the horizontal gradient the density is significantly reduced near the liquidliquid interface caused by the repulsive liquid-liquid interaction. Moreover, the density reflects a surface layering near the solid walls, generated by the lattice order of the solid. The density of both fluids is slightly larger at the cold wall as compared to the hot wall. The density profile is almost linear, as is the temperature field (see also Figure 3). The linear temperature variation indicates that the heat transport is mainly by conduction. Figure 3 shows additionally the temperature of the thermostatting ghost particles and the temperature of the wall. Convective temperature transport is negligibly small. Finally, the potential energy plot gives an idea of the surface tension gradient along the interface. Figures 4 and 5 displays the unsmoothed stream function as a color-coded plot for two different simulations. Smoothed streamlines are added together with velocity vectors. In both figures two counter-rotating vortices are visible. The velocity is highest inside the liquid-liquid interface, since both the momentum flux is highest and the density is lowest inside the interfacial region. In Figure 4 (simulation 2.1) the liquids have different wetting properties of the fluids, resulting in a meniscus. In Figure 5 (simulation 1.3), which was the biggest cavity simulated, the wall interaction is symmetric, thus the interface is flat. The MD simulations are now compared with the analytical result for a side-heated open cavity in the creeping-flow approximation, assuming a free surface and a linear dependence of the surface tension on the temperature. A sample result is 3

4 Figure 3: Temperature profile in horizontal direction from the hot to the cold wall (Simulation 2.1). y/σ Figure 2: Mean values of different properties from a simulation run coded by color (Simulation 2.1). Displayed are the velocity in x (horizontal) and y (vertical) direction, as well as the density, temperature, and kineticand potential energy. All units are dimensionless x/σ Figure 4: Stream function (in color), smoothed streamlines and velocity field from data of Figure 2 (Simulation 2.1). The maximum momentum flux is u max m 3. 4

5 shown in Figure. The free surface condition and the constant surface-tension gradient is imposed at y 0. The aspect ratio is H 0.9. Within the bounds of the limited accuracy the error bounds of the MD simulation, there is no significant discrepancy in the form of the velocity field between the MD results from simulations 1.1, 1.2, 1.3, 2.1, and 2.2, and the analytical solution, even though the continuum model is not y/σ able to predict the properties of the liquid-liquid, and liquidwall interfaces. The slight asymmetry of the MD results can easily be explained by the non-constant material properties like viscosity and density. The data of the MD simulations are not accurate enough to prove the existence of viscous corner eddies (which would be of size of an atom), as predicted by the continuum model. The present MD simulations do not provide any indication of velocity slip at the solid walls. However, our data are not accurate enough to rule slip out completely Table 2: Maximum momentum flux for different simulations, and the driving temperature difference between the fluid at the cold and at the fluid at the hot wall. Simulation u max T Finally, we present the data for a simulation with symmetrical, but strongly repulsive wall interactions of the liquids (Simulation 3.1) in Figure y/σ x/σ Figure 5: Stream function, smoothed streamlines and e velocity field for the biggest cavity simulated (Simulation 1.3). The maximum momentum flux is u max m y x 1 Figure : Stream function (color) and stream lines from an analytical solution of Marangoni flow in a side-heated open cavity. The stream function is normalized to x/σ Figure 7: Stream function, stream lines, and velocity field for two fluids with strong repulsion from the walls. The maximal momentum flux is u max m 3. Even though we only changed the wall interactions, and not the properties of the internal liquids, a completely different 5

6 velocity field arises. Apart from the two main vortices, additional smaller eddies appear near the walls. The driving thermal gradient is reduced by a factor of 1.5. The overall velocities are reduced by a factor of two as compared to simulation 1.2, but the absolute velocity gradients only by a factor of 1.5. Also, velocity slip at the walls is clearly present. Most likely, it is the slip which causes the different velocity field and enforces viscous dissipation to act mainly in the inner flow. CONCLUSIONS It has been shown, that MD simulation is nowadays fast enough to simulate Marangoni driven flow. In the simulations presented, that thermocapillar driven flow featuring two contra rotating vortices could be generated. Further work in this direction will include measurements of forces and material properties. ACKNOWLEDGMENTS We are indebted to the ZID of the Vienna University of Technology for providing computational recources. REFERENCES Gad-el-Hak M., 1999, The fluid mechanics of microdevices. The freeman scholar lecture, J. Fluids Eng. 121, pp 5-33 Griebel S., Knapek S., Zumbusch G., and Caglar A., 2004, Numerische Simulation in der Moleküldynamik, Springer- Verlag, Heidelberg Joseph D. D., 1977, The convergence of biorthogonal series for biharmonic and Stokes flow edge problems. Part I, SIAM J. Appl. Math. 33, pp Joseph D. D., and Struges, L., 1978; The convergence of biorthogonal series for biharmonic and Stokes flow edge problems. Part II, SIAM J. Appl. Math. 34, pp. 7-2 Koplik J., Banavar J. R., and Willemsen J. F., 1989, Molecular dynamics of fluid flow at solid surfaces, Phys. Fluids A 1 (5), pp Koplik J., Banavar J. R., 1995, Corner flow in the sliding plate flow, Phys. Fluids 7 (12), pp Koplik J., Banavar J. R., 2000, Molecular simulations of dewetting, Phys. Rev. Let. 84 (19), pp Koplik J., Banavar J. R., 200, Slip, immiscibility, and boundary conditions at the liquid-liquid interface, Phys. Rev. Let. 9, Kuhlmann H. C., 1999, Thermocapillary convection in models of crystal growth, Springer Verlag, Berlin Mareschal M, Malek, Mansour M., Puhl A., and E. Kestemont, 1988, Molecular dynamics versus hydrodynamics in a twodimensional Rayleigh-Bénard system, Phys. Rev. Let. 1 (22), pp Martic G., Gentner F., Seveno D., Coulon D., and De Coninck J., 2002, A molecular dynamics simulation of capillary imbibition, Langmuir 18, pp Maruyama S., Kurashige T., Matsumoto S., Yamaguchi Y. and Kimura T., 1998, Liquid droplet in contact with a solid surface, Microscale Thermophysical Eng. 2, 49-2 Maruyama S., 2000, Molecular dynamics method for microscale heat transfer, in Minkowycz W. J., and E. M. Sparrow (Eds), Advances in numerical heat transfer, vol. 2, Chap., Taylor & Francis, New York Qian T., Wang X.-P., Sheng P., 2003, Molecular scale contact line hydrodynamics of immiscible flows, Phys. Rev E 8, 0130 Seveno D., and De Coninck J., 2004, Possibility of different time scales in the capillary rise around a riber, Langmuir 20, pp Stoddard S. D., and Ford J., 1973, Numerical experiments on the stochastic behaviour of a Lennard-Jones gas system. Phys. Rev. A. 8 (3), pp Thompson P. A., and Troian S. M., 1997, A general boundary condition for liquid flow at solid surfaces, Nature 389, pp IMPLEMENTATION The molecular dynamics program was realized using well known space division techniques for reducing the n*n dependence of particle number to a n log(n) dependence. The parallelization was implemented using the library openmpi which implements the Message Passing Interface. The Code scales well linearily with the number of processes. Furthermore the GnuScientific Library was used for some minoalculations like Fast Fourier transformation or random numbers generation.