The effect of surfactant on the deformation of a drop is of considerable interest in polymer and emulsion industries. It is also a good theoretical model for illustrating subtle physics in viscous interfacial flow. In this section, the immersed boundary method is applied to study the effect of surfactant on the deformation of a bubble in Navier-Stokes flows.
Following the set up in [56], we consider a computational domain Ω = [−5, 5] × [−2, 2] where a circular bubble of radius one is initially located at the center of the domain. We apply a steady shear flow to the bubble;
that is, we set the boundary condition ub = (0.5 y, 0), for −2 ≤ y ≤ 2.
For comparison purposes, both clean (without surfactant) and contaminated (with surfactant) bubbles are used in these computations. Using the equation of state given by Eq. (2.37), η = 0 implies no contamination, in which case we do not need to solve the surfactant equation (2.36). Throughout this section, we set σc = 1 so the clean interface has a uniform surface tension σ = σc. For the contaminated case, the initial surfactant concentration is uniformly distributed along the interface such that γ(α, 0) = 1. Unless otherwise, we set the Reynolds number Re = 10, the capillary number Ca = 0.5, the surface Peclet number P es= 10, and the parameter η = 0.25.
5.3.1 Clean vs. contaminated interface
To examine the effect of the surfactant on interfacial dynamics, we com-pare a bubble with and without surfactant in a steady shear flow. When the surfactant are present in the interface, the surface tension can be reduced sig-nificantly, cf. equation of state (2.37). Throughout the rest of this section, we use a uniform Cartesian mesh h = ∆x = ∆y = 0.02, and a Lagrangian grid with size ∆s ≈ h/2. The mesh size of time is set to be ∆t = h/8.
Fig. 5.14 shows the time evolution plots of deformation of the bubble in a steady shear flow field. Here, we consider three different values of η in Eq. (2.37); namely, η = 0 (dotted, clean interface), η = 0.25 (dash-dotted), and η = 0.5 (solid). As expected, the magnitude of deformation of the bub-ble increases when the value of η increases, as in the case of Stokes flow [56].
Fig. 5.5 shows the vorticity plot for the bubble with surfactant near the left and the right tips. One can see that two vortices with positive and negative signs are generated near the tips of the bubble.
During the deformation of the bubble, the Lagrangian markers will grad-ually sweep into the tips and cause clustered distribution near the tips. If the
−3 −2 −1 0 1 2 3
−2
−1 0 1 2
T = 0
−3 −2 −1 0 1 2 3
−2
−1 0 1 2
T = 4
−3 −2 −1 0 1 2 3
−2
−1 0 1 2
T = 8
−3 −2 −1 0 1 2 3
−2
−1 0 1 2
T = 12
Figure 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, ’-’).
vorticity
−3.2 −3 −2.8 −2.6 −2.4 −2.2 −2
−1.2
−1
−0.8
−0.6
−0.4
−0.2 0
vorticity
2 2.5 3
0 0.2 0.4 0.6 0.8 1 1.2 1.4
Figure 5.5: The vorticity plot for the drop with surfactant near the left and right tips (η = 0.5, T = 12).
markers become too crowdedly or too coarsely distributed, it will affect the numerical accuracy. Thus, in order to maintain the numerical stability and accuracy, we need to apply the equi-distributed technique we performed in previous chapter, moreover, if necessary, we will double the number of grid when the bubble is over stretched. The detail is given as follows.
In each time step, we first apply the equi-distributed technique to get an artificial velocity and compute the resultant distance between two adjacent markers. If the distance is less than 0.75h, then we basically keep the original resolution. However, if the distance is greater than 0.75h, then we double the number of the grid. (Should be clarified. One important thing during the grid redistribution process is to keep the mass conservation of the surfactant.
This can be done in a local way. For instance, in the segment of adding more grid points, we simply distribute the surfactant mass into those points uni-formly. On the other hand, in the segment of removing grid points, we add up those surfactant mass to be a new surfactant concentration in the new combining segment. Thus, the overall surfactant mass is conserved exactly without any scaling.)
Plots of the corresponding surfactant concentration (left column) and sur-face tension (right column) versus arc-length are given in Fig. 5.16. For the surfactant concentration plot, we omit the case of clean interface since the concentration is zero everywhere on the interface. It can be seen from this figure, the bubble is elongated by the shear flow so that the total length of the interface is increased from the rest state. Since there is no surfactant transferred between the interface and the fluid, the surfactant concentration is diluted on a portion of the interface, partly due to the elongation of the in-terface, but mainly because it is swept to the tips of the bubble. As a result, the smallest surface tension occurs at the tips. One can also see that the value of η affects the surfactant concentration by shifting the distributions slightly along the length of the bubble. Once again, we confirm the same qualitative behavior as in [56].
In Fig. 5.7, the corresponding capillary (defined as σκ|∂X
∂α |/(ReCa), left column) and the Marangoni forces (defined as ∂σ∂α/(ReCa), right column) are plotted versus the arc-length for different cases of η. Since the capillary force depends on the curvature and surface tension, we see that the largest capil-lary force occurs at the tips of the bubble due to the high curvature there.
For clean interface, the Marangoni force is obviously zero.
In Fig. 5.17, we present four different plots; namely, (a) total mass of the surfactant, (b) the error of total mass, m(t) − m(0), (c) total area of
0 5 10
Figure 5.6: Distributions of the surfactant concentration (left) and the cor-responding surface tension (right). Notations and parameters are same as in Fig. 5.14.
the bubble, (d) total length of the interface. Clearly, the present method preserves the total surfactant mass and the errors reach machine precision.
However, there is a slight area losing or fluid leakage inside the bubble as shown in Fig. 5.17(c). It seems that the bubble without surfactant has a more serious leakage than the ones with surfactant. Here, the area loss is not that significant, thus no modification is applied. Once again, we can see from Fig. 5.17(d) that the bubble with surfactant has larger deformation than the one without surfactant due to the increase of total length of the interface.
5.3.2 Linear vs. nonlinear equation of state
In this test, we use the same set up as in the previous one except that a simplified form of nonlinear Langmuir equation of state σ(γ) = σc(1 + ln(1 − ηγ)) is used and compared with the results of the linear equation of the state.
In Fig. 5.9, the evolution of the bubble under steady shear flow is shown at different times using the linear (dotted) and nonlinear (solid) equations of state with η = 0.5. Once again, our results are consistent with those in [56],
0 5 10
Figure 5.7: The corresponding capillary force (left) and Marangoni force (right). Notations and parameters are same as in Fig. 5.14.
i.e., deformation of the bubble increases when the nonlinear equation of state is used. The corresponding surfactant concentrations and surface tensions are shown in Fig. 5.10. One can easily see that the nonlinear equation of state generates smaller surface tension at tips of the bubble which leads to a larger deformation. As shown in Fig. 5.11, the capillary forces are roughly similar but the Marangoni force for the nonlinear case is slightly larger at tips of the bubble. The four different plots for both linear and nonlinear cases including the total mass of the surfactant, the error of total mass, the total area of the bubble, and the total length of the bubble are shown in Fig. 5.12.
5.3.3 Effect of capillary number on drop deformation
As the last test, we perform the study on how different capillary numbers affect the deformation of the bubble. Here, we fix the Reynolds number Re = 10 and the surface Peclet number P es = 10. We vary the capillary number as Ca = 0.05, 0.25, 0.5, 1.0 and perform our runs up to time T = 4.
As confirmed in previous literature such as [30], a larger capillary number means a smaller surface tension (with the viscosity fixed) so the bubble
un-0 2 4 6 8 10 12
total mass of surfactant
0 2 4 6 8 10 12
error of surfactant mass
0 2 4 6 8 10 12
total area of the drop
0 2 4 6 8 10 12
total length of the drop
Figure 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.
der shear flow can deform more substantially. This is exactly what we see in our simulations as illustrated in Fig. 5.13. We also make runs by varying the different surface Peclet number while keeping the Reynolds and capillary numbers fixed. However, the effect of surface Peclet number is not as signif-icant as the effect of the capillary number on the deformation of the bubble, so we omit the results here.