Three-dimensional Finite Element Model of Three-phase Contact Line
Dynamics and Dynamic Contact Angle
KONSTANTIN A. CHEKHONIN, VICTOR D. VLASENKO
Computing Center of the Far Eastern Branch of the RAS,
65 Kim Yu Chena, Khabarovsk 680000,
RUSSIAN FEDERATION
Abstract: - An unconventional model of three-phase contact liny dynamics is suggested for the numerical
solution of the boundary value problem of dipping and spreading. The numerical modeling is conducted with
the use of the finite-element method in Lagrange variables. The mathematical model of the process is described
by the equation of motion, continuity, and natural boundary conditions on the free surface. To exclude the ity
of viscous stresses in the mathematical model on three-phase contact lines (TPCL) there was suggested a
gridded model of gliding that takes into consideration peculiarities of dissipative processes in the neighborhood
of TPCL at the microlevel. To reduce oscillations of pressure in the neighborhood of TPCL, a finite element is
used. The suggested method allows for natural monitoring of free surface and TPCL with an unconventional
model for dynamic contact micro-angle. A stable convergent algorithm is suggested that is not dependent on
the grid step size and that is tested through the example of a three-dimensional semispherical drop and a drop in
the form of a cube. The investigations obtained are compared to well-known experimental and analytical results
demonstrating a high efficiency of the suggested model of TPCL dynamics at small values of capillary number.
Key-Words: - Navier–Stokes equations, continuity equation, free surface, drop, gliding, adhesion, dynamic
boundary angle, three-phase contact line, finite element method.
Received: January 14, 2023. Revised: November 8, 2023. Accepted: December 6, 2023. Published: January 23, 2024.
1 Introduction
The processes of dripping and spreading of a drop
of liquid on solid and liquid surfaces are the initial
and most important stages of many physicochemical
phenomena accompanying modern technologies.
For example, the building of microelectronic
components, modeling of deformable biological
membranes, the process of forming powder
coatings, technology of settling micro-drops of ink
in the conditions of ink-jet printing.
In the processes of dripping and spreading, there
are hydrodynamical peculiarities: the presence of
interphase or free surface and three-phase contact
line (TPCL) that moves along a solid surface in a
tangent direction for instance, a solid body-liquid-
gas. The position of the interphase surface and
TPCL in the given area is unknown in advance and
is a part of the solution to a problem. The analytical
solution of such boundary problems can seldom be
obtained and the capabilities of an experimental
investigation are very limited. Therefore, one of the
basic tools of investigation of such type boundary
problems is their numerical solution. Herewith, in
the case of using mathematical model approaches
the main difficulties are related to obtaining a
solution in the TPCL area and building an effective
computational algorithm.
Beginning from the 60-ties of the XX century,
there has been completed a huge number of
theoretical and experimental investigations at the
presence of interphase boundary and TPCL. The
overview of such works is described in the works,
[1], [2]. However, despite much attention to the
problem and numerous practical applications of the
solution thereof, so far there has been no full
understanding of the mechanisms of interaction of
phase on the line of contact and related-to-them
formation of the interface front, peculiarities of the
contact line dynamics and liquid flow in its
neighborhood in the conditions of dripping and
spreading. The universal dependence between outer
volume flow and local dynamics in the
neighborhood of TPCL is absent, [3], [4].
When modeling static problems of the
hydrodynamics of dripping, TPCL stays unmovable
with a given static contact angle to a smooth surface
determined under Young-Dupré law, [1], [3]. In
dynamic problems, TPCL moves with a time-variant
rate and dynamic contact angle. The problem of
determining dynamic boundary angle (DBA) is
multiscale. In basic DBA models, the microscopic
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DOI: 10.37394/232013.2024.19.1
Konstantin A. Chekhonin, Victor D. Vlasenko
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Volume 19, 2024
angle is supposed to be determined by
intermolecular forces of short-range and to preserve
its equilibrium value. The Microscopic DBA equal
to the visually observed in experiments is
determined by the kinematics of flow, TPCL
moving rate, and properties of adhesion of liquid on
a solid surface. All theoretical models contain
unknown parameters, which need agreeing with the
experiment and which hide the physics of dynamic
processes in the neighborhood of TPCL. All
empirical models are only applicable to certain
conditions of flow and a limited circle of liquids and
surfaces. In numerical modeling, the algorithm of
implementing the selected model of DBA on the
grid area is also all-important. In the works, [1], [2],
[5], [6], [7], [8] it is noted that for obtaining
approximated convergence it is necessary to use a
grid dependence of the length of gliding with the
use of advanced approximation of initial functions
in the neighborhood of TPCL, which in its place
requires the use of special “approaches” of their
implementation. To a great degree, this impacts the
kinematics of flow near TPCL, and as a
consequence, the efficacy of modeling in
comparison with the experiment, stability, and
convergence of the numerical algorithm, [9], [10],
[11], [12], [13], [14] [15], [16], [17], [18], [19],
[20], [21].
The main requirements when building numerical
models of the TPCL dynamics in the conditions of
dripping and spreading are as follows: it is
necessary to establish a monotonous dependence of
microscopic and macroscopic dynamical contact
angles ensuring the kinematics of movements of
micro volumes in the neighborhood of TPCL in
form of swell; determining the length of gliding
from the rate of TPCL and volumetric parameters of
flow, for instance, of the Reynolds or the Bond
number; the rate of TPCL is a priori unknown and is
a solution to the problem in implementing kinematic
conditions. Therefore, additional assessments of its
determination are needed that exclude its non-physic
movements.
This work introduces a modeling of the flow of
viscous incompressible liquid with a free surface of
various conditions of dripping on solid walls with a
strong effect of surface tension in comparison with
gravitation and viscous forces. For the modeling, the
method of finite elements in Lagrange variables,
[22], [23] is applied. When applying the Laplace–
Beltrami operator in transforming boundary
conditions for the pressure jump on the free surface,
[24], the order of derivatives on the free surface was
reduced. This allowed for naturally including the
boundary conditions of gliding on TPCL, its rate,
and DBA. All the dynamic parameters of DBA flow
are computed naturally in the nodes belonging to
TPCL and do not require additional interpolation.
The use of setting up the problem in Lagrange
variables allows for direct monitoring of free
surface and TPCL, for excluding the nonlinearity of
convective member and implementing a completely
unexpressed algorithm of computation of the
position of free surface with the DBA model built
on equipoising Young’s dynamic forces by viscous
friction on TPCL with due consideration of the
effect of the dynamical microscopic contact angle.
All these are obtained in the frame of molecular-
kinetic theory, [25] and agree with molecular-
dynamic computation, [26], [27]. The algorithm is
tested on the example of the spreading of viscous
incompressible drop under the impact of surface
tensions and gravitation forces in static and dynamic
conditions of dripping and spreading on a smooth
solid surface.
2 Building the Model of TPCL
Movement
We shall consider the model of the movement of
TPCL on the example of a problem of a drop
spreading on a solid smooth base layer. Figure 1
introduces a computational area and details of the
geometry of the area with contact angle θ.
TPCL -----»
a)
b)
Fig. 1: a) Viscous drop with free surface Г1 lying on
solid base layer ГS, b) TPCL (red dot), revealed as
the circular curve of intersection of boundaries Г1
and ГS
The balance of forces affecting TPCL can be
presented as follows:
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+ cos cos ,
SS
σ n u ν
(1)
where, u – velocity vector,
= p +2μDIu
– stress
tensor, p – pressure, μ – dynamic coefficient of
viscosity, n – vector of normal to free boundary Г1,
I – unit tensor, γ – coefficient of surface tension at
the inter-phase surface of liquid-air,
– Dirac
delta function,
S
ν
– singular vector of the tangent to
solid wall,
ν
– singular vector of the tangent to free
surface.
If dynamic and static angles equal θ = θS, then
the tilting angle of the free surface on TPCL to the
solid wall in the process of spreading is constant and
equals the static contact angle.
The majority of investigations when modeling
exclude the consideration of dissipative processes at
the microlevel in the neighborhood of TPCL, and
only consider meso- and macro-levels applying the
condition of static contact angle at meso-level (fir.
1), for instance, work, [28]. It’s justified, since from
the point of mechanics of continuum, the
microregion is an area of the unavailability of the
solution. However, the investigations, [29], show
that the establishment of boundary conditions for
dynamic contact angle at the microlevel as a static is
incorrect. Moreover, the conducted molecular-
dynamical calculations also show various
mechanisms flowing in dissipative processes in the
neighborhood of TPCL on various scales.
The idea of building the TPCL dynamics model
consists of presenting the balance of forces on
TPCL (1) at the microlevel or, in other words,
building of correct boundary conditions for the
TPCL macro-model. In such cases, the contact
angles on TPCL are presented as microscopic.
According to molecular-dynamical calculations, we
shall express the coefficient of gliding in the form of
a sum of Navier friction-gliding and the coefficient
of friction on TPCL itself:
N CL
.
The coefficient of Navier friction-gliding
distributes in the neighborhood of TPCL on the
solid wall under exponential law from some given
N
with nearing of the gliding length
SN
l
to
zero with distancing from it, i.e. no-slip condition.
We shall compute the coefficient of friction
directly on TPCL from the relations molecular
kinetic theory, [25]
3
0
,
B
CL T
K
where, κB – Boltzmann constant, T – Kelvin
temperature, K0, λ – frequency and length of jumps
of molecules of liquid.
A special difficulty in modeling the dynamics of
TPCL appears in the case of hysteresis of contact
angle, [1], [2]. In this care, there differ the
advancing and the receding contact angles, which
are not equal. Their difference is the one that forms
the hysteresis. The difficulty is in the fact that TPCL
is immovable for all angles located in the interval of
the hysteresis and begins moving only at the angle
leaving the interval of hysteresis. In this case,
similar to the Signorini contact problems with
friction, we get to the variational setting in the form
of variational inequalities. To solve this problem,
we shall apply Lagrange multipliers.
On the basis of the mathematical model of the
considered problem we shall put Navier-Stokes
equations and equation of continuity, [24]:
0g, ,
uuu σu
t
(2)
where, ρ – density, g – gravitational acceleration.
We shall compute the system of equations (2)
with the use of the following boundary conditions:
– on the free surface moving with kinematic
condition:
d0
dt
xun
we shall set up boundary dynamic boundary
conditions consisting in the absence of tangent
stresses and equality of normal to the sum of
external and capillary pressures:
0nσ I nn
,
1κ
Ca
n n n
a
p
,
where, n – vector of normal to free surface, κ –
curvature of the free surface, Ca – capillary number,
pa – pressure over the free surface, which we accept
as equal to zero without losing the solidarity of
purpose.
At the lines of three-phase contact, we shall
establish boundary conditions of gliding and
impermeability, [23]
, 0,
s s s s s s
nσ I n n u I n n n u
where, β – nondimensional parameter of gliding
(β = 0 – complete gliding, β = ∞ – adhesion), ns –
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singular vector of normal to a solid wall in the
neighborhood of TPCL.
Projection-fine-difference equations are reported
in, [24], so we shall report here the peculiarities of
numerical implementation with consideration of the
hysteresis of the angle.
With due consideration of the angle hysteresis,
the smallness of the Reynolds number and the
condition (1) in the neighborhood of TPCL, the
projection-fine-difference equations of the problem
can be expressed as follows:
To find
11
( , )
nn
h h h h
p W Q
u
consistent with the
equation:
11
2 : d d
nn
h h h h
D D p
u w w
11
11
: d ( ) d
nn
hh
nn
h h S h
P w M w
1d,
n
hh
w
u
(3)
11
d d 0,
nn
h h h h h
q p q
u
(4)
1
[cos cos ( )]dS 0,
n
h
h S h S
uν
(5)
where,
,
SS
M
arccos M,
S
i.e.
cos S S S
– Young-Dupré law,
P I n n
(
ij ij i j
P = - n n
),
=c
h
h
h
u
–
parameter of the equation stability, λh, ηh – Lagrange
multipliers,
c
=0.5.
3 Results of Computations
We shall consider the spreading of the viscous drop
(Figure 1) with known initial θ0 and static contact
angles θS under the effect of the surface tension
forces. Density, viscosity, and surface tension
coefficient shall be taken to be equal to one. We
believe that the gravitation forces are sufficiently
small (the Bond number << 1). At the initial
moment, the liquid is moveless and occupies the
volume equal to V, which doesn’t change in the
process of spreading. We shall take Navier–Stokes
equations and equations of continuity as a basis of
mathematical description. For the
dVσ n w
member for normal stresses, we shall substitute the
right part of the equation (3) with the additional
condition (5). The introduction of Lagrange
multipliers is needed in the case of contact angle
hysteresis since they ease out the conditions in the
form of inequation at the ratio of the contact angle
and the rate of TPCL. We shall obtain the
projection-fine-difference equations in Lagrange
variables by way of adding to the left part of the
equation (3) a nonstationary member of Navier–
Stokes equations or consider the solution (3)-(5) as a
sequence of quasistatic problems with integrating of
kinematic conditions on Г1 free surface by the
implicit scheme.
We shall investigate the grid convergence of the
suggested algorithm based on the spreading of the
drop in the form of a semisphere with an initial
angle of 90 degrees and a static angle of 60 degrees.
The initial radius of the drop equals 0,5. For the
case, when the capillary number is substantially
smaller than one, the radius of spreading of the drop
in the function from the visual contact angle can be
expressed as follows
30
3( ),
dd
V
RФ
3
3
sin
( ) .
2 3cos cos
d
d
dd
Ф
(6)
After differentiating equation (6) by time, we
obtain a regular differential equation of evolution of
the radius of spreading in the function from the rate
of the rate of contact angle changing, which is
solved by the Runge–Kutta method. The spreading
radius of the drop to the static state can be
calculated from the relation (6) by substituting the
static contact angle with the condition of the volume
stationarity. In our case, for the static angle of 60
degrees, the relation of the static spreading radius to
the initial equals 1,276186.
We shall produce the numerical solution of the
problem (3)-(5) on the sequence of grids.
Figure 2 shows the grid convergence of the
suggested algorithm and the comparison of the
evolution of the spreading radius in the functions
from the time with an analytic solution at values of
the microscopic parameters of friction on TPCL
βN = 100, βCL = 1. From the calculation results there
follows a grid convergence of the considered
algorithm and a good correspondence of the
evolution of the spreading radius to the analytic
solution.
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Fig. 2: Evolution of spreading radius of
semispherical drop in time t to a static contact angle
of 60 degrees
Figure 3 shows the influence of the changing of
dynamical parameters of friction of TPCL on the
evolution of the radius of spread. The computational
experiment was conducted on the grid with minimal
step on TPCL equal to hmin = 1/64. It follows from
the calculation results that the growth coefficient of
Navier friction-gliding has a weak effect on the
evolution of the radius of spreading, but has a strong
effect on the grid convergence. The growth of the
friction coefficient βCL substantially slows down the
evolution of the radius of spreading and reduces the
effective length of gliding LS (Figure 3).
Fig. 3: Influence of changing in parameters of
friction on TPCL on the evolution of radius of
spreading
Now, therefore, the contribution of friction
coefficients on TPCL in the efficiency of the
considered algorithm becomes clear.
Now, we shall conduct a comparison of the
calculation results with the experiment of drop
spreading conducted in work, [30]. For the
calculations, there was taken the value of static
contact angle θS = 54°. The initial radius of the drop
was taken as equal to 0,5 with the center of mass
(0; 0.48) on a solid base layer. The liquid viscosity
was normalized and was taken as equal to 1. The
calculation results are stated in Figure 4.
Fig. 4: Evolution of radius of spreading of a drop in
time t at various values of friction coefficient with
hmin = 0.01, – experimental result, [30]
It follows from the calculation results that the
coefficient of friction on TPCL is an adjustable
parameter of the model of the TPCL dynamics (for
the correspondence of the numerical model to the
experiment). In our case, the best result is obtained
at friction coefficient βCL = 0.5. In addition, the
calculation analysis shows that the suggested model
of the TPCL dynamics does not correspond to the
initial period of spreading of the drop, which is
inertial. The model requires clarification by way of
dependence of the friction coefficient on TPCL
from the TPCL rate. It must be noted, that all the
above-stated investigations were conducted in a
three-dimensional set-up. Figure 5 illustrates an
expressed three-dimensional case of the spreading
of the drop of the initial form of a cube, initial
contact angle of 90 degrees and a static contact
angle of 60 degrees. The calculation results show a
satisfactory agreement with the drop form of a
constant volume in a static state. In all the
calculations, the drop volume loss didn’t exceed
0.01% at the static contact angle detection error, not
above 0,1 degree.
t = 0.0 s t = 0.5 s t = 1.0 s
Fig. 5: Spreading of a drop in the form of a cube
lying on the solid base layer
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The peculiarities of modeling a drop flow with a
hysteresis of the angle of dripping are considered in
the example of a drop lying on the oblique surface
(Figure 6).
Fig. 6: Determining parameters of drop on the
oblique surface
All these methods consist in the observance of
the following conditions for normal rate Ucl of
contact line:
0 if ,
0 if ,
0 if .
cl d R
cl R d A
cl A R
U
U
U
The oblique angle of surface αc in the beginning
of the movement can be obtained by watching the
balance of forces affecting the drop:
1
3
sin (cos cos ) ,
4
c R A Eo
a
where,
2/
Eo ga
– the Eötvös number, a – drop
diameter, so the critical contact angle changes as
1
(cos cos )
RA
Eo
and depends on the form of
the contact line. Figure 7 shows the evolution of the
linear rate of three-phased contact in the dependence
on the hysteresis of the angle of dripping, where
4(sin sin )
3
c
Ca Eo
C
.
Fig. 7: Effect of hysteresis on velocity of liquid fall
at α=50
Ca is illustrated as a function
cos cos
RA
,
□ – modeling, ······ – у = 0,0128 - 0,013x
4 Conclusion
As the investigation results there was suggested a
correct dynamic model of TPCL and a variational
formulation of a problem with variable dynamic
contact micro-angle. To exclude the singularity of
tangent stresses on TPCL, the coefficient of Navier
friction-gliding is applied. The friction coefficient
on TPCL is adjustable for the comparison with an
analytical solution or experiment. There was
suggested a stable, not-depending on a grid state,
convergent numerical algorithm that is tested on the
example of a three-dimensional semispherical drop
and a drop in the form of a cube. The investigations
obtained are compared with the known experimental
and analytical results demonstrating a high
efficiency of the suggested model of the TPCL
dynamics at small values of capillary number. These
investigations may be useful for solving and
forecasting such situations as, for instance, the
production of ink for ink-jet printing, applying of
liquid coatings and drainage in porous media,
spreading of pesticides on leaves, whole blood
dripping, spreading and drying of a blood serum
drop.
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WSEAS TRANSACTIONS on FLUID MECHANICS
DOI: 10.37394/232013.2024.19.1
Konstantin A. Chekhonin, Victor D. Vlasenko
E-ISSN: 2224-347X
8
Volume 19, 2024
