5.1 The mesh refinement analysis of the velocity u, v, and the
surfactant concentration γ. . . 62 5.2 The mesh refinement analysis of the velocity u, v, and the
surfactant concentration γ. . . 63 5.3 The mesh refinement analysis of interface positions, the
con-tact angles, and the area of drop. . . 63
List of Figures
1.1 An unhappy molecule at the surface: it is missing half its attractive interaction. . . 2 1.2 (a) Two bugs stay on the water-air surface. (b) A
cross-sectional view of one leg of a bug. . . 3 1.3 The diagram of the alveoli. . . 5 1.4 Simplified diagram of the interface between two condensed
phases a and b. . . . 6 1.5 Diagrammatic representation of heptane-water interface with
adsorbed surfactant. . . 8 1.6 The effect of detergence in a water-tetradecane system. On
increasing the concentration of surfactant (SPAN 80), the sys-tem goes from a partial wetting regime to a total wetting regime of tetradecane on the substrate. The water droplet thus tends to detach itself from the substrate. . . 10 2.1 The diagram of nine components of stress. . . 17 2.2 The diagram of a bubble in a two-phase interfacial flow. . . 21 2.3 (a) The surface tension with which an element of the interface
acts on its boundary is normal to the boundary and tangential to the interface; it tries to minimize the area of this element.
(b) The resultant action of the surface tension on a surface element from a surrounding surface has a normal component if the interface in not flat, and a tangential component if the surface tension varies along the interface. . . 23 2.4 The diagram of contact lines and the problem setting. . . 26 3.1 (a) Hat function. (b) Cosine approximation. (c) Second-order
approximation with 4-point support. (d) Second-order ap-proximation with 3-point support. . . 43 4.1 A diagram of the staggered grid. . . 45
5.1 Comparison of stretching factor |Xα| with (UA = 0, dashed line) and (UA6= 0, solid line), where h = 1/128. . . 64 5.2 The time evolution of a bubble in a quiescent flow. The
dash-line is the configuration of the steady state which is a circle with re = 0.4243. (a) Relative error of area loss. (b) Total length of the bubble. . . 65 5.3 (a) A bubble with bulk surfactant in the second quadrant
moves to the left-top corner of the box. (b) The correspond-ing evolution of surfactant concentration of (a). (c) A bubble with bulk surfactant in the third quadrant moves to the left-bottom corner of the box. (d) The corresponding evolution of surfactant concentration of (c). . . 66 5.4 The time evolution of a drop in a shear flow with clean (η = 0,
’.’) and contaminated interface (η = 0.25, ’-.’, η = 0.5, ’-’). . . 68 5.5 The vorticity plot for the drop with surfactant near the left
and right tips (η = 0.5, T = 12). . . . 68 5.6 Distributions of the surfactant concentration (left) and the
corresponding surface tension (right). Notations and parame-ters are same as in Fig. 5.14. . . 70 5.7 The corresponding capillary force (left) and Marangoni force
(right). Notations and parameters are same as in Fig. 5.14. . 71 5.8 (a) Total mass of the surfactant. (b) Time plot of m(t)−m(0).
(c) Total area of the bubble. (d) Total length of the interface.
Notations and parameters are same as in Fig. 5.14. . . 72 5.9 The time evolution of a bubble under a shear flow with linear
(’.’) and nonlinear (’-’) equation of state. . . 73 5.10 Distributions of the surfactant concentration (left) and the
corresponding surface tension (right). Notations and parame-ters are same as in Fig. 5.9. . . 73 5.11 The corresponding capillary force (left) and Marangoni force
(right). Notations and parameters are same as in Fig. 5.9. . . 74 5.12 (a) Total mass of the surfactant. (b) Time plot of m(t)−m(0).
(c) Total area of the bubble. (d) Total length of the interface.
Notations and parameters are same as in Fig. 5.9. . . 74 5.13 The effect of capillary number Ca on the deformation of the
bubble. (Ca = 0.05 : ’.’, Ca = 0.25: ’-’, Ca = 0.5: ’-.’, Ca = 1.0: ’–’) . . . 75 5.14 The time evolution of a hydrophilic drop with clean (η = 0,
dashed line) and contaminated interface (η = 0.3, solid line). 76 5.15 The velocity field for the drop with surfactant near the left
and right contact lines (η = 0.3, T = 1.5625). . . 77
5.16 Distribution of the surfactant concentration (top) and the cor-responding surface tension (bottom). . . 78 5.17 (a) Left contact line speed of the drop. (b) Right contact line
speed of the drop. (c) Contact angle of the drop. (d) Total length of the drop. Notations and parameters are same as in Fig. 5.14. . . 79 5.18 The time evolution of a hydrophobic drop with clean (η = 0,
dashed line) and contaminated interface (η = 0.3, solid line). 80 5.19 The velocity field for the drop with surfactant near the left
and right contact lines (η = 0.3, T = 1.5625). . . . 81 5.20 (a) Left contact line speed of the drop. (b) Right contact line
speed of the drop. (c) Contact angles of the drop. (d) Total length of the drop. Notations and parameters are same as in Fig. 5.18. . . 82 5.21 Wettability and the initial drop set up. . . 83 5.22 The time evolution of a hydrophilic drop with clean (η = 0,
dashed line) and contaminated interface (η = 0.3, solid line). 83 5.23 The velocity field for the drop with surfactant near the left
and right contact lines at T = 1.0938, 3.5938. . . . 84 5.24 Distribution of the surfactant concentration (top) and the
cor-responding surface tension (bottom). . . 85 5.25 (a) Left contact line speed of the drop. (b) Right contact line
speed of the drop. (c) Left contact angle of the drop. (d) Right contact angle of the drop. . . 86 5.26 (a) A rising bubble with Eo = 0.1. (b) A rising bubble with
Eo = 10. . . 87
Chapter 1 Introduction
The real world is abundant in phenomena of free surfaces, interfaces and mov-ing boundaries (generally called interfaces), that interact with a surroundmov-ing substances, like gas, fluid, or solid. These interfaces separate one fluid from another, for instance air and water form the case of bubbles or free surface flows, and behave as boundaries between two materials of different physical properties. In some respects, the interface may be a rigid wall that moves with some specified time dependent motion, or an elastic membrane that deforms and stretches in response to the fluid motion. In addition, motion of interface may involve not only the dynamics of the liquid and surrounding air but also their interaction with adjacent solid surfaces. Many industrial processes, ranging from spin coating of microchips to de-icing of aircraft sur-faces, rely on the ability to control these interactions.
Fluid flows with moving interfaces play important roles in many scientific, biomedical, and engineering applications. The interaction of muscle tissue with blood in the heart and arteries, coating of solid substrates with liquids, film boiling and crystal growth, micro-organisms utilize for locomotion the anisotropic drag properties of their long flexible flagella, are part of interest-ing applications. If an incompressible fluid flow contains an interface and the interface is between fluid 1 and fluid 2, then the flow often refers to be a free surface flow. The position of the interface is determined by the capillary force (a force acts in the direction perpendicular to the tangent plane of an inter-face point), which results from the balance between the normal stress and the surface tension on the interface. Generally speaking, these problems are usu-ally described by the time-dependent incompressible Navier-Stokes equations together with interface jump conditions (can be viewed as a balance of forces on the interface). These types of problems are generally called free-boundary problems, multi-phase flow problems or interfacial flow problems.
1.1 Surface tension
In the microscopic sense of a matter, molecules attract one another all the time. When the attraction is stronger than thermal agitation, molecules switch from a gas phase to a more dense phase, so called a liquid. A molecule in the midst of a liquid interacts with all its neighbors and finds itself in a
”happy” state. By contrast, a molecule that floats at the surface loses half of its cohesive interactions, see Fig 1.1, and becomes ”unhappy”. That is
L iq u id
G a s
Figure 1.1: An unhappy molecule at the surface: it is missing half its attrac-tive interaction.
the fundamental reason that liquids adjust their shape in order to expose the smallest possible surface area.
In physics, a liquid molecule is in an unfavorable energy state when it moves to the surface. If the cohesion energy per molecule is U inside the liquid, a molecule sitting at the surface goes short of energy roughly U/2.
The surface tension is a direct measure of this energy shortfall per unit surface area. If r is the molecule’s size and r2 is its exposed area (like one face of a cube), the surface tension is of order σ ∼= U/(2r2). For most oils, for which the interactions are of the van der Waals type, we have U ∼= kT , which is the thermal energy. At a temperature of 25◦C, kT is equal to 1/40 eV , which gives σ = 20 mJ/m2. Because water involves hydrogen bonds, its surface tension is larger (σ ≈ 72 mJ/m2). For mercury, which is a strongly cohesive liquid metal, U ≈ 1 eV and σ ≈ 500 mJ/m2. Note that σ can
equivalently be expressed in units of mN/m. Similarly, the surface energy between two non-miscible liquids a and b is characterized by an interfacial tension σab. Table 1.1 [47] lists the surface tensions of some ordinary liquids (including those usually used in the experiments of related applications), as
Table 1.1: Surface tension of a few common liquids (at 20◦C unless otherwise noted) and interfacial tension of the water/oil system.
Liquid Helium (4K) Ethanol Acetone Cyclohexane Glycerol
σ(mNm ) 0.1 23 24 25 63
Liquid Water Water (100◦C) Molten glass Mercury Water/oil
σ(mNm ) 73 58 ∼ 300 485 ∼ 50
well as the interface tension between water and oil. Although its origin can be explained at the molecular level, the surface tension σ is a macroscopic
( a ) ( b )
Figure 1.2: (a) Two bugs stay on the water-air surface. (b) A cross-sectional view of one leg of a bug.
parameter defined on a macroscopic scale. In Fig. 1.2(a), two waterstriders can mat on the water mainly due to the effect of surface tension. One can think the surface tension as the force acting parallel to the water-air surface and perpendicular to the line (the long leg of the bug), see Fig. 1.2(b), fs
and fw reach a balance so that the leg is static on the water-air interface.
So far, one knows that supplying energy is necessary to create surfaces.
Suppose one wants to distort a liquid to increase its surface area by an amount dA. The work required is proportional to the number of molecules that must be brought up to the surface, i.e., to dA; and one can write:
δW = σ dA
where σ is the surface (or interfacial) tension. Dimensionally, [σ] = E/L2. The surface tension σ is thus expressed in units of mJ/m2. In words, σ is the energy that must be supplied to increase the surface area by one unit.