ISSN: 2180-1053 Vol. 2 No. 2 July-December 2010 Numerical Study of Droplet Dynamics on Solid Surface
93
or luid-solid interface is very simple. LBM automatically generate the interface region with no special treatment on the simulation program
meaning that no extra computational burdens are needed to track the interface. Application of LBM is expected to increase the eiciency,
accuracy and the capability of the current computer performance without sacriicing the need of the detailed behavior of luid particles
in the multiphase phenomena. There were several works have been done on multiphase low using
LBM, however, only few researches studied droplet dynamics on surface. Therefore, the objective of current study is to demonstrate the
capability of LBM in simulating droplet dynamics on solid surface. The structure of this paper is as follow. To begin, we show the formulation
of multiphase latice Boltzmann model from Landau free energy theory. Ater showing how the formulation of the particles interaction its in
to the framework of latice Boltzmann simulations, numerical results of droplet spreading on a solid surface are presented to highlight
the applicability of the approach. The inal section concludes current study.
2.0 multipHaSe lattice BoltZmaNN model
The starting points for the latice Boltzmann simulations is the evolution equation, discrete in space and time, for a set of distribution functions
f . If a two-dimensional nine-velocity model D2Q9 is used, then the
evolution equation for a given f takes the following form
f
i
x e
i
∆t,t ∆t − f
i
x , t
= − 1
f
i
x , t
− f
i eq
x , t
where ∆t is the time step, e is the particle’s veloci
∆ ∆
− = −
−
1 ∆
for the collision
where ∆t is the time step, e is the particle’s velocity, τ is the relaxation time for the collision and i = 0,1,…,8. Note that the term on the right
hand side of Equation S. L. Manzello and J. C. Yang, 2002, is the collision term where the BGK approximation has been applied P. L.
Bhatnagar, 1954. The discrete velocity is expressed as ;
e = 0, 0
e
1, 3, 5, 7
= cos i
−1 4
, sin i
−1 4
e
2, 4, 6, 8
= 2 sin i
−1 4
, cos i
−1 4
f
eq
is an equilibrium distribution func
=
= =
=
= −
∆
=
− =
= =
− −
= −
−
2
physics inherent
=
= =
=
= −
∆
=
− =
ISSN: 2180-1053 Vol. 2 No. 2 July-December 2010 Journal of Mechanical Engineering and Technology
94
f
i eq
is an equilibrium distribution function, the choice of which determines the physics inherent in the simulation. In the free-energy
two-phase latice Boltzmann model, the equilibrium distribution is writen in the following term
f
i eq
= A
i
B
i
e
i,
u C
i
e
i,
e
i,
u u Du
2
G e
i
e
i
where the summation, over repeated Car
=
=
=
= −
∆
=
− =
=
3 efficients A, B, C,
=
=
=
= −
∆
=
− =
where the summation, over repeated Cartesion indices, is understood. The coeicients A, B, C, D and
= −
−
=
where the summa and G are
conserves mass constrained by
=
=
=
= −
∆
=
− =
are determined by placing constraints on the moments of
f
i eq
. The collision term conserves mass and momentum, and therefore the irst and second moments of
f
i eq
are constrained by
= −
= −
−
=
f
i eq
i
=
=
=
= −
∆
=
− =
= −
= −
−
=
=
4
=
=
= −
∆
=
− =
= −
= −
−
=
=
e
i,
f
i eq
i
= u
The continuum macr
=
= −
∆
=
− =
= −
= −
−
=
= =
5 uation correctly
=
= −
∆
=
− =
The continuum macroscopic equations approximated by the evolution equation correctly describe the hydrodynamics of a one-component,
non-ideal luid by choosing the next moment of
f
i eq
. This gives
= =
− −
= −
−
=
= =
e
i,
e
i,
f
i eq
i
= P u
u u
u
= −
∆
=
− =
= =
− −
= −
−
=
=
=
=
6
= −
∆
=
− =
where
u
= t −1 2 ∆t
3
is the kinematic shear viscosity,
= =
− −
= −
−
=
= =
=
= −
∆
viscosity,
P
is th the coefficients
=
− =
is the pressure tensor, and r is the time relaxation. In order to fully constrain
the coeicients A, B, C, D and
= =
− −
= −
−
=
=
=
=
= −
∆
essure tensor, and ,
and G , a
=
− =
, a fourth condition is applied, which is
= =
− −
= −
−
=
=
=
=
= −
∆
e
i,
e
i,
e
i,
f
i eq
i
= c
2
3 u
u u
The thermodynamics of the fluid enters the lat
− =
= =
− −
= −
−
=
=
=
=
= −
∆
=
7 ia the pressure
− =
The thermodynamics of the luid enters the latice Boltzmann simulation via the pressure tensor
= =
− −
= −
−
=
= =
=
= −
∆
viscosity,
P
is th the coefficients
=
− =
. The equilibrium properties of a system can
ISSN: 2180-1053 Vol. 2 No. 2 July-December 2010 Numerical Study of Droplet Dynamics on Solid Surface
95
be described by a Landau free energy functional as follow [20]
− =
s x
dS T
dx
2
2 ,
− =
8
The right hand side terms of Equation 8 represent free energy density of the bulk phase, free energy from density gradient and contribution
from interaction between luid and solid [21] respectively. k is a constant related to the surface tension. Following Gennes [22] and Seppecher P.
Seppecher, 1996, we expand ted to the surface
we expand
s
as be sufficient for th
e energy functional can
− =
=
− −
=
= −
− −
=
= −
=
−
as a power series in Following Ge
series in
s
and id-solid interactio
rewritten as
− =
=
− −
=
= −
− −
=
= −
=
−
and keep only the irst order term since this turn out to be suicient for the
liquid-solid interaction scenarios that we want to consider. Therefore, free energy functional can be rewriten as
s x
T dx
1 2
2 ,
− =
It then follows that J. S. Rowlinson
=
− −
=
= −
− −
=
= −
=
−
− =
9
=
− −
=
= −
− −
=
= −
=
−
It then follows that J. S. Rowlinson and B. Widom, 1982
− =
It then follows that J. S. Rowlinson
=
x x
p P
with −
− =
= −
− −
=
= −
=
−
− =
=
10
− −
=
= −
− −
=
= −
=
−
with
− =
=
2
2 −
− =p
p x
= −
− −
=
= −
=
−
− =
=
− −
= 11
= −
− −
=
= −
=
−
where
− =
=
− −
= where
p =
T , − T,
is t The Cahn model is used to re
tangent plane to the droplet and th
− −
=
= −
=
−
is the equation of state of the luid. The Cahn model is used to relate
− =
=
− −
=
= −
is the equation of to relate
1
to , th and the substrate. Th
− −
=
= −
=
−
, the contact angle deined as the angle between the tangent plane to the droplet and the substrate.
This gives
− =
=
− −
=
= −
− −
= 3
cos 1
3 cos
2 sign
2 2
1 c
p
p
where
= −
,
=
−
and i
− =
=
− −
=
= −
− −
=
12
= −
=
−
− =
=
− −
=
= −
− −
=
where
p
= T
c
−T T
c
,
= cos
−1
sin
2
and is a constant typically equal to 0.1.
3.0 Results and Discussion
