四、 Numerical results
4.4 Effects of the drop deformation on surfactant … 22
There is different behavior for clean interface and contaminated interface, so we will study it in this subsection. We compare a clean drop with a contaminated drop in an extensional flow, ur= −0.5Gr and uz = Gz, where G = 1. By the comparison, we observe that the surface tension reduces when the surfactant is on the interface.
Here, we use the mesh h = ∆x = ∆y = 1/64, the Lagrangian grid with size ∆s ≈ h, and the time step size ∆t = h/8.
The problem is set up as a drop immersed in a fluid domain Ω = [0, 1] × [−3, 3]
with viscosity 1, which the drop is a circle with radius r0 = 0.25. We also fix the den-sity inside and outside the drop ρ1 = ρ2 = 1, the Reynolds number Re = 1, the Cap-illary number Ca = 0.6,and the surface Peclet number Pes = 1.25. The initial surfac-tant concentration is uniformly distributed along the interface such as Γ (0, s) = 0.5, and the parameters in E.q.2.23 are chosen as σc= 1 and β = 0.5,respectively. We ex-amine viscosity ratio λ = 1, 5 and 10 repeatedly and observe the distinction between the clean interface (no surfactant) and contaminated interface (with surfactant).
−1 0 1
−3
−2
−1 0 1 2 3
r
z
Time =0 clean surfactant
−1 0 1
−3
−2
−1 0 1 2 3
r
z
Time =1 clean surfactant
−1 0 1
−3
−2
−1 0 1 2 3
r
z
Time =1.5 clean surfactant
−1 0 1
−3
−2
−1 0 1 2 3
r
z
Time =2 clean surfactant
−1 0 1
Figure 4.4: The time evolution of a drop in an extensional flow with λ = 1.
From the Fig4.4, the tip points of the interface on the z-axis are about at the positions (r, z) ≈ (0, ±2.0376), then we obtain the distance of contaminated drop from the tip point to the center is larger than that of clean drop. We observe the contaminated drop deforms more intensely because surfactant decrease the surface tension, causing the drop more likely to deform.
−1 0 1
−1 0 1
Figure 4.5: The time evolution of a drop in an extensional flow with λ = 5.
From the Fig4.5, the tip points of the interface on the z-axis are about at the positions (r, z) ≈ (0, ±0.8187), then we also obtain the distance of contaminated drop from the tip point to the center is larger than that of clean drop. Then, We examine viscosity ratio λ = 10 and observe the distinction between the clean interface (no surfactant) and contaminated interface (with surfactant).
−1 0 1
−1 0 1
Figure 4.6: The time evolution of a drop in an extensional flow with λ = 10.
From the Fig4.6, the tip points of the interface on the z-axis are about at the positions (r, z) ≈ (0, ±0.4929), then we we also obtain the distance of contaminated drop from the tip point to the center is larger than that of clean drop.
0 0.5 1 1.5 2 2.5 3 3.5 4
Figure 4.7: This is the total mass of surfactant for P es = 1.25.
Even though surfactant would affect the drop deformation, viscosity is the main factor, and flow is more sticky as viscosity is higher, the shape of drop is more difficult to be affected by flow. Since the surfactant is insoluble, the total mass of the surfactant must be conserved In Fig.4.7, no matter which viscosity we choose, the total mass of surfactant for P es= 1.25 is conserved very well by our method.
4.5 Effects of the drop deformation on Peclet number
In this subsection, we compare the influence of different surface Peclet number P eson the elongation of contaminated drop, and we choose peclet number P es from 0 to 500, partitioned 50 choices at once. Here, we first put a contaminated drop in the extensional flow, ur = −0.5Gr and uz = Gz, where G = 1 and compute up to T = 4 in the domain Ω = [0, 1] × [−3, 3]. Then, we fix the Reynolds number Re= 1, the Capillary number Ca = 0.3,and Γ (0, s) = 0.5 with the parameters in E.q.2.23 chosen as σc= 1 and β = 0.5,respectively.
Figure 4.8: Drop in axisymmetric flow.
A convenient method for expressing the results of experiment is to measure L, the length of the drop in the direction of the r axis, and B, the breadth in the direction of the z axis. As noting in Fig.4.8, the degree of deformation can be characterized by the scalar Df that was originally introduced by Taylor [11].
Df = L − B
L + B (4.1)
0 50 100 150 200 250 300 350 400 450 500
0.2 0.22 0.24 0.26 0.28 0.3 0.32 0.34 0.36 0.38 0.4
Peclet number Pes degree of deformation Df
Figure 4.9: This is the deformation of drop with Peclet number P es at T = 4.
0 0.5 1 1.5 2 2.5 3 3.5 4 0.628
0.6281 0.6282 0.6283 0.6284 0.6285 0.6286 0.6287 0.6288 0.6289 0.629
Time
Mass
Figure 4.10: This is the total mass of surfactant for every Peclet number P es.
Since the underlying extensional drives the surfactant toward to the tip points of the drop, the surfactant concentration is higher and thus the surface tension is lower near the tips. When the Peclet number is lager, the surfactant gradient along the interface is lager too, therefore, the overall drop elongation is greater. From Fig.4.9 we noticed from Peclet number P es from 0 to 150 the larger the deformation the more obvious, but past 150 the larger the change will also be additional, but the higher the number the higher the tendency to plateau. By Fig.4.10 we also obtain the total mass of surfactant conserved no matter Peclet number which we choose.
5 Conclusion
In this paper, we used the immersed boundary method such that the interface is in the different phases with different densities and viscosities by an indicator function in the cylindrical coordinates to construct our formulation. Since the velocity field is in the axisymmetric coordinates, the velocity boundary conditions on r = 0 must be set ur = 0 and ∂uz/∂r = 0.We solve the Navier-Stokes equations by projection method, and update the new marker positions by imposing the artificial velocity in order to make the new positions uniformly. Controlling marker positions uni-formly can make the volume error smaller. Similarly, the surfactant equation must impose the artificial velocities to control the concentration locations. The surface tension is reduced by the surfactant, and Peclet number influences the diffusion of the surfactant.
In our future work, for a drop on a solid, controlling the equilibrium contact angle can produce the hydrophobic phenomenon or hydrophilic phenomenon. Imposing the surfactant on the interface affects the equilibrium contact angle in this mov-ing contact line case. Furthermore, we can develop the more general 3-dimension simulation.
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