Capillary filling using Lattice Boltzmann Equations: the case of multi-component fluids
aa r X i v : . [ n li n . C G ] J a n EPJ manuscript No. (will be inserted by the editor)
Capillary filling for multicomponent fluid using thepseudo-potential Lattice Boltzmann method
S. Chibbaro , L. Biferale F. Diotallevi , and S. Succi Istituto per le Applicazioni del Calcolo CNR, Viale del Policlinico 137, 00161 Roma. Dept. of Physics and INFN, University of Tor Vergata, Via della Ricerca Scientifica 1, 00133 Roma, Italy.Received: date / Revised version: date
Abstract.
We present a systematic study of capillary filling for a binary fluid by using mesoscopic a latticeBoltzmann model describing a diffusive interface moving at a given contact angle with respect to thewalls. We compare the numerical results at changing the ratio the typical size of the capillary, H , andthe wettability of walls. Numerical results yield quantitative agreement with the theoretical Washburnlaw, provided that the channel height is sufficiently larger than the interface width and variations of thedynamic contact angle with the capillary number are taken into account. PACS.
The physics of capillary filling is an old problem, originat-ing with the pioneering works of Washburn [1] and Lucas[2]. It remains, however, an important subject of researchfor its relevance to microphysics and nanophysics [4,5,6]. Capillary filling is a typical “contact line” problem,where the subtle non-hydrodynamic effects taking placeat the contact point between liquid-gas and solid phaseallow the interface to move, pulled by capillary forces andcontrasted by viscous forces. As already remarked, Wash-burn in 1921 [1] described theoretically the dynamics ofcapillary rise. Considering also inertial effects, except the“vena contracta”, and two fluids with the same density( ρ a = 1 , ρ b = ρ a ) and the same viscosity ( µ a = µ b = µ ),the equation of motion of the moving front is [3]: d z ( t ) dt + 12 µH ρ dz ( t ) dt = 2 γcosθHρL (1)where H is the capillary height, L its length, γ the surfacetension and θ the contact angle. This model is obtainedunder the assumption that (i) the instantaneous bulk pro-file is given by the Poiseuille flow, (ii) the microscopic slipmechanism which allows the motion of the interface is notrelevant to bulk quantities (such as the overall positionof the interface inside the channel), (iii) inlet and out-let phenomena can be neglected (limit of infinitely longchannels). In the following, we will show to which extentthis phenomenon can described by a mesoscopic Lattice-Boltzmann equation for multicomponent. The model here Send offprint requests to : [email protected] used is a suitable adaptation of the Shan-Chen pseudo-potential LBE [7] with hydrophobic/hydrophilic bound-aries conditions, as developed in [8].
The relevant geometry is depicted in fig. (1). The bot-tom and top surfaces are coated only in the right half ofthe channel with a boundary condition imposing a givenstatic contact angle [8]; in the left half we impose peri-odic boundary conditions at top and bottom surfaces inorder to mimic an “infinite reservoir”. Periodic boundaryconditions are also imposed at the two lateral sides suchas to ensure total mass conservation inside the system. Atthe solid surface, bounce back boundary conditions for theparticle distributions were imposed. The conditions whichallow the wetting of the surfaces will be discussed in thefollowing.
Let us review the multicomponent LB model proposedby Shan and Chen [7]. This model allows for distributionfunctions of an arbitrary number of components with dif-ferent molecular mass: f ki ( x + c i ∆t, t + ∆t ) − f ki ( x , t ) = − ∆tτ k h f ki ( x , t ) − f k ( eq ) i ( x , t ) i (2)where f ki ( x , t ) is the kinetic probability density functionassociated with a mesoscopic velocity c i for the k th fluid, τ k is a mean collision time of the k th component (with ∆t S. Chibbaro et al.: Capillary filling for multicomponent fluid using the pseudo-potential Lattice Boltzmann method a time lapse), and f k ( eq ) i ( x , t ) the corresponding equilib-rium function. For a two-dimensional 9-speed LB model(D2Q9) f k ( eq ) i ( x , t ) takes the following form [9]: f k ( eq )0 = α k n k − n k u eqk · u eqk f k ( eq ) i = (1 − α k ) n k n k c i · u eqk + 12 n k ( c i · u eqk ) − n k u eqk · u eqk for i=1 . . . f k ( eq ) i = (1 − α k ) n k
20 + 112 n k c i · u eqk + 18 n k ( c i · u eqk ) − n k u eqk · u eqk for i=5 . . . c i ’s are discrete velocities, definedas follows c i = , i = 0 , (cid:16) cos ( i − π , sin ( i − π (cid:17) , i = 1 − √ (cid:16) cos [ ( i − π + π ] , sin [ ( i − π + π ] (cid:17) , i = 5 − α k is a free parameter related to the soundspeed of a region of pure k th component according to( c ks ) = (1 − α k ); n k = P i f ki is the number densityof the k th component. The mass density is defined as ρ k = m k n k , and the fluid velocity of the k th fluid u k is defined through ρ k u k = m k P i c i f ki , where m k is themolecular mass of the k th component. The equilibriumvelocity u eqk is determined by the relation ρ k u eqk = ρ k u ′ + τ k F k (6)where u ′ is the common velocity of the two components.To conserve momentum at each collision in the absence ofinteraction (i.e. in the case of F k = 0) u ′ has to satisfythe relation u ′ = s X i ρ k u k τ k ! / s X i ρ k τ k ! . (7)The interaction force between particles is the sum of abulk and a wall components. The bulk force is given by F k ( x ) = − Ψ k ( x ) X x ′ s X ¯ k =1 G k ¯ k Ψ ¯ k ( x ′ )( x ′ − x ) (8)where G k ¯ k is symmetric and Ψ k is a function of n k . In ourmodel, the interaction-matrix is given by G k ¯ k = g k ¯ k , | x ′ − x | = 1 ,g k ¯ k / , | x ′ − x | = √ , , otherwise . (9)where g k ¯ k is the strength of the interparticle potentialbetween components k and ¯ k . In this study, the effectivenumber density Ψ k ( n k ) is taken simply as Ψ k ( n k ) = n k . 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Periodic B.C. Periodic B.C.Fluid 2 WallsFluid 1ny=H nxPeriodic B.C.L L
Fig. 1.
Geometrical set-up of the numerical LBE. The two di-mensional geometry, with length 2 L and width H , is divided intwo parts. The left part has top and bottom periodic bound-ary conditions such as to support a perfectly flat gas-liquidinterface, mimicking a “infinite reservoir”. In the right half, oflength L , there is the true capillary: the top and bottom bound-ary conditions are those of a solid wall, with a given contactangle θ . Periodic boundary conditions are also imposed at thewest and east sides. Other choices would lead to a different equation of state(see below).At the fluid/solid interface, the wall is regarded as aphase with constant number density. The interaction forcebetween the fluid and wall is described as F k ( x ) = − n k ( x ) X x ′ g kw n w ( x ′ )( x ′ − x ) (10)where n w is the number density of the wall and g kw is theinteraction strength between component k and the wall.By adjusting g kw and n w , different wettabilities can beobtained. This approach allows the definition of a staticcontact angle θ , by introducing a suitable value for thewall density n w [8], which can span the range θ ∈ [0 o :180 o ]. In particular, we have chosen g w = 0 , g w = − g while n w is varied in order to adjust the wettability. In thesequel, we choose g = 0 . n w are associatedwith hydrophilicity.In a region of pure k th component, the pressure is givenby p k = ( c ks ) m k n k , where ( c ks ) = (1 − α k ). To simulatea multiple component fluid with different densities, we let( c ks ) m k = c , where c = 1 /
3. Then, the pressure of thewhole fluid is given by p = c P k n k + P k, ¯ k g k, ¯ k Ψ k Ψ ¯ k ,which represents a non-ideal gas law. The viscosity is givenby ν = ( P k β k τ k − ), where β k is the mass densityconcentration of the k th component.The Chapman-Enskog expansion [9] shows that thefluid mixture follows the Navier-Stokes equations for a sin-gle fluid: ∂ t ρ + ∇ · ( ρ u ) = 0 , (11) ρ [ ∂ t u + ( u · ∇ ) u ] = −∇ P + F + ∇ · ( νρ ( ∇ u + u ∇ ) . Chibbaro et al.: Capillary filling for multicomponent fluid using the pseudo-potential Lattice Boltzmann method 3 Time z ( t ) H=15H=30H=50H=100Analytical
Fig. 2.
Front displacement vs time for different channel height H = 15 , , ,
100 with their corresponding analytical solu-tions. The discrepancy from washburn’s law is stronger for thesmallest channel. The channel length is always L = 450 exceptfor H=100, for which L = 500. where ρ = P k ρ k is the total density of the fluid mix-ture and the whole fluid velocity u is defined by ρ u = P k ρ k u k + P k F k . All simulations were performed using the Shan-Chen modeldescribed above, setting ν l = ν g = 0 . ρ l = ρ g = 1, g = 0 . α = 4 /
9, that is c s = , and the interfacialtension is γ = 0 .
07. The channel length is chosen to be L = 450. By taking θ constant in time, a simple analyti-cal solution of equation (1) can be obtained: z ( t ) = V cap Hcosθ L t d [exp( − t/t d ) + t/t d −
1] + z , (12)where z is the starting point of the interface at the be-ginning of the simulation, t d = ρH µ is a typical transienttime and V cap = γµ is the capillary speed. This solutionhas been used to compare with simulations.The front displacement as a function of time is shownin Fig. 2 for different values of the channel height H =15 , , , n w = 1, for which static contact anglewas found to be θ ≈
5. As expected, the velocity of thefront grows with channel height. The analytical curves aregiven by the solution of Eq. (12), where the contact angleis the dynamic one computed from numerical data. Thecontact angles computed for the four heights 15 , , , ◦ , ◦ , ◦ , ◦ . The dynamic contact an-gle has been obtained directly as the slope of the contoursof near-wall density field, and independently through theLaplace’s law, ∆P = γcosθH . The latter has been chosenfor the comparison with analytical fitting curves, becausethe direct computation from density contours turns outto be less precise. Nevertheless, the values calculated inthe two ways are approximately consistent. For instance,the contact angle computed for the case H = 30 from the Time z ( t ) n w =1.0n w =0.75n w =0.5analytical Fig. 3.
Front dynamics for different n w (1 . , . , .
5) that isfor different degree of wettability. The configuration consideredis with H = 30. The case n w = 0 . θ = 0 .
78 and the case n w = 0 .
75 with θ = 0 . direct measurements of the pressure is θ ≈ ◦ againstthe value θ ≈ ◦ computed via density contours. Somecomments on the front dynamics are in order.. The case ofsmallest channel height does not follow the analytical so-lution, showing the finite size of the interface ( w/H / H = 100, where the transienttime-scale t d = ρH µ is sufficiently long to make the expo-nential term in the solution (12) important over a macro-scopic time span. The results show that the dynamic con-tact angles experience a strong dependence on the channelheight. In particular, in small channels, dynamic contactangles remain near their static values. On the other hand,for large ones the discrepancy is evident. This is due tothe increasing value of the capillary number ( Ca ∼ . H = 100), since it is known that there is a correc-tion of the dynamic contact angles due to finite capillarynumbers. This correction takes the form the general form cos ( θ d ) − cos ( θ s ) = g ( Ca ). Our results are best fitted by g ( Ca ) = 18 Ca . , which is in line with previous formsused in different LBE methods [11,12]Hereafter the configuration with H = 30 and n w =1 . n w =1 . , . , .
5. As expected, it is found that more hy-drophobic cases correspond to smaller velocities . The ana-lytical solutions which fit the numerical data are obtainedrespectively with θ = 22 ◦ , θ = 24 ◦ , θ = 40 ◦ . These an-gles are consistent with the values computed via Laplace’slaw directly from numerical data, that is θ = 0 . , θ =0 . , θ = 0 . t = 50000 at differ-ent positions are shown in fig. 4, for the standard case H = 30, n w = 1 .
0. Some comments are in order. Thevelocity profile is parabolic everywhere except very near
S. Chibbaro et al.: Capillary filling for multicomponent fluid using the pseudo-potential Lattice Boltzmann method y U y x=350x=220x=200x=50 xfront=210 Fig. 4.
Velocity profile U x ( y ) for different cuts taken at time t = 40000 with the front located at at x ≈ x = 50, another is far ahead at x = 350. For these cases, approximately the same Poiseuilleparabolic flow is found. The other two curves correspond tothe velocities just ahead and behind the interface. In thesecase, the velocity profile is necessarily distorted in order to letthe interface advance with an uniform velocity along y . Theinterface acts as an obstacle and the velocity shows a corre-sponding decrease (but not a recirculation) in the middle ofthe channel, giving rise to a two-humped profile. Fig. 5.
Velocity streamlines. The value of velocities are mag-nified by a factor 1000. The interface is located at x ≈ the interface. This is consistent with the assumption of aparabolic (Poiseuille) velocity profile. A small difference ispresent between the parabolic profile ahead and past theinterface. This is tentatively interpreted as due to the dif-ferent boundary conditions applied to the fluids ( n w = 1for the hydrophilic invading fluid 1, and n w = 0 for fluid2 ahead of the front). This difference were found to dis-appear by setting nearer values of n w for both fluids. Inother terms, boundary conditions are such that the fluidafter the interface is less slipping, with a velocity at thewall almost recovering no-slip condition.In fig. 5, velocity patterns are presented. Consistentlywith fig. 4, this figure shows that the flow is one-directionalfar from the interface, confirming the assumption of a Poiseuille flow. Moreover, although the flow appears tobe distorted near the interface to allow slippage, no recir-culation is observed at variance with other methods LBEs[13,12,11], spurious currents are negligible. The spikes infig. 4 reflect the existence of a hydrodynamic singularitynear the wall. A detailed understanding of the LB dynam-ics in the near vicinity of this singularity remains an openissue for future research. The present study shows that Lattice Boltzmann modelswith pseudo-potential energy interactions are capable ofreproducing the basic features of capillary filling for binaryfluids, as described within the Washburn approximation.Moreover, it has been shown that the method is able toreproduce the expected front dynamics for different de-gree of surface wettability, as well as the correct Poiseuillevelocity profile, in the whole domain, except for a thin re-gion near the interface. Quantitative agreement has beenobtained with a sufficiently thin interface, w/H < . ∆x (current values are about 5 ∆x ). Workalong these lines is underway. References
1. E.W. Washburn, Phys. Rev. (1921) 273.2. R. Lucas, Kooloid-Z (1918) 15.3. J. Szekelely, A:W. Neumann, and Y.K. Chuang Journal ofColl. and Int. Science, (1971) 273.4. P.G. de Gennes, Rev. Mod. Phys. (1985) 827.5. L.J. Yang, T.J. Yao and Y.C. Tai, J. Micromech. Microeng. (2004) 220.6. N.R. Tas et al., Appl. Phys. Lett. (2004) 3274.7. X. Shan, and H. Chen Phys Rev E , 1815 (1993).8. Kang, Zhang and Chen PHF (9) 3203, 20029. D.A. Wolf-Gladrow Lattice-gas Cellular Automata andLattice Boltzmann Models (Springer, Berlin, 2000).10. Hou, Shan, Zou, Doolen and Soll, JCP (2), 1997.11. M. Latva-Kokko, and D.H. Rothman Phys Rev Lett to bepublished.12. L. Dos Santos, F. Wolf, and P. Philippi J. Stat. Phys. ,197 (2005).13. F. Diotallevi, L. Biferale, S. Chibbaro, F. Toschi, and S.Succi EpjB submitted. r X i v : . [ n li n . C G ] J a n EPJ manuscript No. (will be inserted by the editor)
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