Algorithm of The Method

We have given the details from Step1 to Step6 in above chapters and we just write the algorithm of Step7 since the surfactant equation is a little different as comparing with heat equation.

The surfactant equation can be rewritten as

Γ_t+ u · ∇Γ − n · (∇un)Γ = 1

P e(∆Γ − ∂²Γ

∂n² − κ∂Γ

∂n) (66)

and again we use semi-implicit Crank-Nicholson method to solve Eq.(64) which we have mentioned above. Note that Crank-Nicholson is just used in diffusion part which is ∆_sΓ and we treat the convection part with explicit discretization, then the discretization form is :

Γⁿ⁺¹− Γⁿ

We refer [21] to write the form of Eq.(66) and [21] suggested that in the discretization for spatial of Eq.(66), central difference is used in all term except this term u·∇Γ which is discretized by upwind-WENO-3 scheme which we have introduced. Note that we encounter the data structure problem again as in above chapter.

Note that σ_x and σ_y are discretized by central difference.

5.5 Numerical Results

The effect of surfactant is to decrease the surface tension coefficient and so the interface will deform much heavier than clean interface.

In this section we refer the example of [22] and we apply the steady shear flow with boundary condition u_b = (2y, 0) in the computational domain [−0.5, 0.5] × [−0.5, 0.5].The initial bubble is centered at (0, 0) with radius r = 0.15.

Example.1

We give our setting as follows.Initial surface tension coefficient σ₀ = 0.5, gravity g = 0, ρ₁ = ρ₂ = 1, µ₁ = 0.1, µ₂ = 1 and Peclet number P e = 1.

Note that our time step is as the example of above chapter, ∆t = ₄₀¹ ∆x and we consider three cases that are β = 0(clean), β = 0.25 and β = 0.5(See figure23).

As our expectation, if β is larger then the interface will deforms heavier since we know that

σ(Γ) = σ₀(1 − βΓ)

which means that concentration affects the surface tension coefficient.

Note that our inner area is conserved within 0.01% but the surfactant concentration which computed by Eq.(64) will not be conserved as we predict before.(See figure24)

Figure 23: Time evolution of a bubble in shear flow with β = 0(black), β = 0.25(blue) and β = 0.5(red)

6 Conclusion and Future Work

In this article we introduce some methods to simulate two-phase flow with or without surfactant. We also introduce level set method to solve heat equation on a circle and we can extend the level set method to 3D directly;

that is to solve heat equation on sphere φ(x, y, z) = px²+ y²+ z² − 1. We can also couple level set method and IIM(immerse interface method) to solve heat equation on the 2D domain with jump condition on the interface. The application of level set is very widely and it is really a valuable method.

The VOF method used in this article is to conserve the inner area and we get nice result in our numerical simulation. By coupling VOF and level set method, we are not only able to reconstruct the interface accurately but also to conserve the volume of inner area. Moreover, the method is also adapted to changing of interface topology such as two bubbles merging or one bubble braking into two parts.

Figure 24: (a) Total surfactant in each time step for β = 0.25. (b) Total area of drop in each time step for β = 0.25. (c) Total surfactant in each time step for β = 0.5. (d) Total area of drop in each time step for β = 0.5.

In our simulation of two-phase flow with surfactant on interfaces, the inner area is still conserved as we expect but the mass of surfactant is not conserved which does not make sense in our real life. So our future work is to develop another method to let the mass of surfactant to be conserved, that is we use other method to handle Eq.(64) and get feasible result in mass of surfactant.

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