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Diffuse interface surface tension models in an expanding flow

yhhuang 添加于 2011-1-5 15:21 | 3162 次阅读 | 0 个评论

作者
Wangyi Liu, Andrea L. Bertozzi, Theodore Kolokolnikov
摘要
We consider a diffusive interface surface tension model under compressible flow. The equation of interest is the Cahn-Hilliard or Allen-Cahn equation with advection by a non-divergence free velocity field. We prove that both model problems are well-posed. We are especially interested in the behavior of solutions with respect to droplet breakup phenomena. Numerical simulations of 1,2 and 3D all illustrate that the Cahn-Hilliard model is much more effective for droplet breakup. Using asymptotic methods we correctly predict the breakup condition for the Cahn-Hilliard model. Moreover, we prove that the Allen-Cahn model will not break up under certain circumstances due to a maximum principle.
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diffuse-interface reaction-diffusion
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Cahn-Hilliard and Allen-Cahn equation with advection.

Introduction
Basic Models for interfaces

Sharp interface
Level-set method
Diffusive interface

Both equation has the energy
$E(u)=\int \left(\epsilon|\nabla u|^2+\frac{1}{\epsilon}g(u)\right)$
where $g(u)=u^2(1-u^2)$ . The difference is that the Cahn-Hilliard equation is the $H^{-1}$ gradient flow for the Ginzbur-Landau free energy $E(u)$ , while the Allen-Cahn equation is the $L^2$ gradient flow. In other words, the Cahn Hilliad equation is
$u_t=\Delta \left(\frac{\delta E}{\delta u}\right)$
while the Allen-Cahn equation is
$u_t=-\frac{\delta E}{\delta u}$
where
$\frac{\delta E(u)}{\delta u} = -\epsilon \Delta u + \frac{1}{\epsilon}f(u)$ .

Model problems

Three equations are investigated in this paper

The advective Cahn-Hilliard equation
$u_t+\nabla\cdot(u\vec{V})=\Delta \left(\frac{\delta E}{\delta u}\right)$
The advective Allen-Cahn equation
$u_t+\nabla\cdot(u\vec{V})=-\frac{\delta E}{\delta u}$
The advective Allen-Cahn equation with mass conservation
$u_t+\nabla\cdot(u\vec{V})=-\frac{\delta E}{\delta u}+\lambda u,\qquad \lambda = \frac{\int_\Omega K(u)}{M}.$

Properties of the advective Cahn-Hilliard equation
If g is a polynormial of order 2p with leading order coefficient positive, there there exists a unique solution u belongs to
$C([0,T];L^2(\Omega)])\bigcap L^2(0,T;H_0^2(\Omega)\bigcap L^{2p}(0,T;L^{2p}(\Omega))$ .
We can get bounds on the norms and energy with suitable smoothness condition on the velocity V.
Properties of the advective Allen-Cahn equation
This equation is a standard semilinear parabolic equation, whose regularity results can be found from the references.
Numerical Results
The numerical results can be summarized as follow:

For the advective Cahn-Hilliard equation, the initial square may or may not break up, depending on V_0M^2
For the advective Allen-Cahn equation, depending on the velocity, there is a trivial or nontrivial (u=1) solution
For the nonlocal advective Allen-Cahn equation, there may be a nontrivial connected steady state or break up

Diffuse interface surface tension models in an expanding flow

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