# Numerical Simulation of Three-Ph

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```					 12Numerical Simulation of Three-Phase Contact Lines
Ivan B. Bazhlekov, Patrick D. Anderson and Han E. H. Meijer
Eindhoven University of Technology, Department of Mechanical Engineering

Introduction                                                             Results
In most of the multiphase systems that involve more than two              An example of the liquid-ﬂuid-liquid case is so called ’drop
phases, contact regions appear. In the contact-line regions               engulﬁng’ process, ﬁgure 2, during which one of the phases
three phases are in mutual interaction, which can have a in-              can be partially or completely covered by the other.
ﬂuence on the hydrodynamics. Depending on the phases that                   1

0.8                                                                                           0.8
1                                                                                                      1

0.8
1

0.8
1

0.8

are in contact two general types of contact line exist: liquid-
0.6                                                                                           0.6                                                                                                        0.6                                                                                                       0.6                                                                                                                   0.6

0.4                                                                                           0.4                                                                                                        0.4                                                                                                       0.4                                                                                                                   0.4

0.2                                                                                           0.2                                                                                                        0.2                                                                                                       0.2                                                                                                                   0.2

0                                                                                                 0                                                                                                      0                                                                                                         0                                                                                                                     0

ﬂuid-liquid and liquid-ﬂuid-solid, see ﬁgure 1. The contact
−0.2                                                                                          −0.2                                                                                                       −0.2                                                                                                      −0.2                                                                                                                  −0.2

−0.4                                                                                          −0.4                                                                                                       −0.4                                                                                                      −0.4                                                                                                                  −0.4

−0.6                                                                                          −0.6                                                                                                       −0.6                                                                                                      −0.6                                                                                                                  −0.6

−0.8                                                                                          −0.8                                                                                                       −0.8                                                                                                      −0.8                                                                                                                  −0.8

−1

lines are usually modelled by ad hoc boundary condition for
−1                                                                                                                                                                                                       −1                                                                                                        −1                                                                                                                    −1

−1.5              −1             −0.5           0         0.5       1      1.5                  −1.5          −1       −0.5         0           0.5              1                 1.5                  −1.5           −1             −0.5       0              0.5            1               1.5                 −1.5          −1       −0.5                   0             0.5               1             1.5                 −1.5      −1              −0.5               0               0.5               1

the values of the dynamic contact angles, [1].                                    1
t = 0.0                                                  1
t = 0.007                                                 1
t = 0.028                                       1
t = 0.13                   1
t = 0.691

0.8                                                                            0.8                                                                            0.8                                                                                              0.8                                                                                        0.8                                                                                         0.8

0.6                                                                            0.6                                                                            0.6                                                                                              0.6                                                                                        0.6                                                                                         0.6

0.4                                                                            0.4                                                                            0.4                                                                                              0.4                                                                                        0.4                                                                                         0.4

0.2                                                                            0.2                                                                            0.2                                                                                              0.2                                                                                        0.2                                                                                         0.2

0                                                                          0                                                                                 0                                                                                             0                                                                                          0                                                                                           0

−0.2                                                                           −0.2                                                                           −0.2                                                                                          −0.2                                                                                       −0.2                                                                                        −0.2

−0.4                                                                           −0.4                                                                           −0.4                                                                                          −0.4                                                                                       −0.4                                                                                        −0.4

−0.6                                                                           −0.6                                                                           −0.6                                                                                          −0.6                                                                                       −0.6                                                                                        −0.6

−0.8                                                                           −0.8                                                                           −0.8                                                                                          −0.8                                                                                       −0.8                                                                                        −0.8

−1                                                                         −1                                                                             −1                                                                                               −1                                                                                         −1                                                                                          −1

−1           −0.5              0       0.5         1                     −1               −0.5        0          0.5       1                            −0.8   −0.6   −0.4   −0.2     0     0.2   0.4   0.6    0.8   1    1.2                        0   0.2    0.4   0.6   0.8     1   1.2     1.4     1.6   1.8   2                         0.8       1   1.2   1.4   1.6   1.8       2   2.2   2.4   2.6   2.8                         1.8   2      2.2   2.4   2.6   2.8   3         3.2   3.4   3.6   3.8

t = 1.4                                                            t = 2.17 t = 4.73 t = 9.24 t = 15.1 t = 29.8
Figure 2 The compound drop evolution at 0.5σ12 = σ13 = σ23 .
The liquid-ﬂuid-solid contact lines are important for pro-
Fig. 1 Schematic sketch of three-phase contact line: liquid-ﬂuid-         cesses as wetting and dewetting. Figure 3 shows detachment
liquid type (left); liquid-ﬂuid-solid type (right).                       of a pendant drop from a plane horizontal wall.
0                                                                      0                                                                            0                                                                                             0                                                                                          0                                                                                          0

Objective
−0.5                                                                           −0.5                                                                          −0.5                                                                                       −0.5                                                                                     −0.5                                                                                            −0.5

−1                                                                        −1                                                                            −1                                                                                            −1                                                                                     −1                                                                                             −1

To develop a computational model for a 3D simulation of mul-               −1.5

−2
−1.5

−2
−1.5

−2
−1.5

−2
−1.5

−2
−1.5

−2

tiphase ﬂows that involve three-phase contact lines.                       −2.5                                                                           −2.5                                                                          −2.5                                                                                       −2.5                                                                                     −2.5                                                                                            −2.5

−3                                                                        −3                                                                            −3                                                                                            −3                                                                                     −3                                                                                             −3

−3.5                                                                           −3.5                                                                          −3.5                                                                                       −3.5                                                                                     −3.5                                                                                            −3.5

Mathematical method                                                                        −1             −0.5             0           0.5             1            −1             −0.5               0          0.5           1                −1                    −0.5                    0                0.5              1            −1                −0.5                 0                 0.5                1              −1                  −0.5               0                  0.5                1              −1              −0.5                  0                    0.5               1

The mathematical model is based on the assumptions that                                   t = 0.0                                                                  t = 1.3                                                                         t = 2.4                                                                                        t = 3.0                                                                                    t = 3.4                                                                                 t = 3.7
inertia is negligible and interfaces are pure (no surfactant).            Figure 3 Time evolution of a pendant drop from a horizontal solid
The main elements of the model are:                                       wall at σ13 = σ23 and Bo = (∆ρR2 g)/σ12 = 2.
✷ Stokes equations in all liquid regions;
✷ continuity of the velocity across the interfaces;
✷ contact-line model based on a force balance in a
contact-line vicinity, [2], which is rewritten in terms of
capillary pressure, [3];
✷ slip condition on the solid wall in the case of liquid-
ﬂuid-solid case;
✷ normal stress balance on the interfaces which takes
t = 0.0                                                                                                   t = 1.0                                                                                                                                t = 2.1                                                                                                                                     t = 4.3
into account the capillary as well as disjoining pres-
sure;                                                                Figure 4 The evolution of a spreading drop on an inclined at 45o solid
✷ the evolution is governed by the kinematic condition.                wall for σ23 − σ13 = σ12 and Bo = (∆ρR2 g)/σ12 = 2.

An important feature of the present model is that the values              In the case when σ23 − σ13 ≥ σ12 one of the liquids com-
of the dynamic contact angles are not input parameters for                pletely wet the solid surface, then van der Waals forces play
the problem, but are part of the solution.                                an important role and so called ’precursor’ ﬁlm is formed. In
ﬁgure 4 an example of drop sliding due to gravity is shown.

Numerical method
The numerical method is based on a standard boundary in-                Conclusions
tegral formulation, extended with the following features, [3]:            A computational model is developed for simulation of dy-
namic contact-line problems. The contact line boundary con-
✷ non-singular contour integration of the singular layer               ditions express force balance on a vicinity of the contact line
potentials, which improves the accuracy;                             in terms of capillary pressure, which allows straightforward
✷ implementation of three-phase contact line boundary                  incorporation in the boundary-integral method.
conditions and the slip condition in the liquid-ﬂuid-              References:
solid case;                                                                          [1] Shikhmurzaev, Y.: J. Fluid Mech. 334 (1997) 211–249
✷ multiple step integration which improves the numerical                               [2] Bazhlekov, I. and Shopov, P.: J. Fluid Mech. 352 (1997) 113–133
stability and increases the performance.                                             [3] Bazhlekov, I.: Ph.D. thesis, Eindhoven University of Technology (2003)

/department of mechanical engineering                                                                                                                                                                                                   PO Box 513, 5600 MB Eindhoven, the Netherlands

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