In this section, we aim to validate that our numerical method can be extend straightforward to more realistic interfacial flow problems, that is, the fluids inside and outside the interface have different fluid properties (density and viscosity), although we consider only constant fluid properties in all previous sections. Basically, we just need to consider an extra indicator function I(x, t) evaluated by solving a Poisson equation in Eq. (4.45) to determine where is fluid 1 and where is fluid 2 and use Eqs. (4.44) and (4.43) to update the density and viscosity, respectively. Since we now have different fluid properties, the fast direct solver we used to solve linear systems in the projection method is no longer available due to variable coefficient of the diffusion operator. Alternatively, an iterative solver, based on a semi-coarsening geometric multigrid method [3, 32], replaces to solve these linear systems. In this test, we consider the motion of a liquid bubble rising in a two-dimensional domain Ω = [−0.64, 0.64] × [−0.64, 1.92] due to gravity
0 10 20 30 10−4
10−3 10−2 10−1 100
0 10 20 30
10−4 10−3 10−2 10−1 100
0 10 20 30
0.35 0.4 0.45 0.5 0.55 0.6 0.65 0.7
0 10 20 30
0.35 0.4 0.45 0.5
unit of π
Figure 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.
g∞ = 1. The radius of the liquid bubble set to be R = 0.35. The boundary conditions are given as u(x; t)|∂Ω= 0 and the initial conditions are u(x; 0) = v(x; 0) = 0. The density and viscosity ratios ρ2/ρ1 = µ2/µ1 = 10, where subscripts 1 and 2 stand for values inside and outside the bubble, respectively.
In addition, we set ρ1 = 10 and µ1 = 0.01. The surface tension σ is varied as it determines the shape of the interface separating the two fluids. More precisely, we use a non-dimensional parameter called the Eotvos number Eo, which is the ratio of the gravity and the surface tension force.
Fig. 5.26(a) is the evolution of the interface which separates the two fluids at t = 0, t = 7, t = 14, and t = 20 with Eo = 0.1 while the shape of the rising bubble with Eo = 10 at the same time intervals is shown in Fig. 5.26(b).
In each case, the leakage of the area inside the bubble is within 0.5%. It can be seen that the shape of the bubble is clearly affected by the value of the Eotvos number. The shape of the liquid drop remains circular when the Eotvos number is small (corresponding to a large surface tension coefficient).
When the Eotvos number is large (corresponding to a small surface tension
−0.5 0 0.5
Figure 5.26: (a) A rising bubble with Eo = 0.1. (b) A rising bubble with Eo = 10.
coefficient), however, significant deformation can be observed. In another words, the surface tension affects the shape of the bubble. This is consistent with experimental observations.
Chapter 6
Summary and future work
In this dissertation, we introduced a mathematical model for the interfa-cial flow problems such as a bubble in a shear flow and the moving contact line problems. We have developed an immersed boundary method for two-dimensional interfacial flows with insoluble surfactant. The governing equa-tions (Navier-Stokes equaequa-tions) are formulated in a usual immersed bound-ary framework where a mixture of Eulerian fluid and Lagrangian interfacial variables are used, with the linkage between those two different variables is provided by the Dirac delta function. The immersed boundary force comes from the nonhomogeneous surface tension which is affected by the distribu-tion of surfactants along the interface. In addidistribu-tion, the unbalanced Young force should be applied at the contact lines to mimic the tendency to the equilibrium in the moving contact line problems (Navier-slip boundary con-ditions should be imposed on the solid substrate which contact lines adhere to). By tracking the interface in a Lagrangian manner, a simplified surfac-tant concentration equation can be obtained. The numerical method involves solving the Navier-Stokes equations on a staggered grid by a semi-implicit pressure increment projection method where the immersed interfacial forces are calculated at the beginning of each time step. Once the velocity values and interfacial configurations are obtained, a dynamical control of Lagrangian markers is introduced so that the physical spacing of the markers can be kept uniformly. Then the corresponding modified surfactant equation is solved in a new symmetric discretization such that the total mass of surfactant along the interface is conserved numerically. Numerical results include the con-vergence analysis, a bubble with surfactant in a shear flow, the effect of the surfactant for hydrophilic and hydrophobic drops, and a rising bubble in a flow with gravitational effect. These numerical results match mathematical predictions well, and show that one can trustfully use the method we pre-sented here to simulate related interfacial flow problems or extend the idea
of this work to predict phenomena with more complicated effects.
As a next step, we will generalize the present algorithm to simulate two phase flows with distinct densities and viscosities for the moving contact line problems. In particular, we plan to study the effect of soluble surfactant on drop detachment from a solid surface, i.e., a problem with moving contact points/lines. We also like to extend the idea of this work to first three-dimensional axi-symmetric flows, and then we plan to generalize the current work to general 3D simulations.
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