examples.levelSet.surfactant.expandingCircle

This example represents an expanding circular interface with an initial coverage of surfactant. The rate of expansion is dependent on the coverage of surfactant, The governing equations are given by:

\[\begin{split}\dot{\theta} &= -\frac{\dot{r}}{r} \theta \\ \dot{r} &= k \theta\end{split}\]

The solution for these set of equations is given by:

\[\begin{split}r &= \sqrt{2 k r_0 \theta_0 t + r_0^2} \\ \theta &= \frac{r_0 \theta_0}{\sqrt{2 k r_0 \theta_0 t + r_0^2}}\end{split}\]

The following tests can be performed. First test for global conservation of surfactant:

>>> surfactantBefore = numerix.sum(surfactantVariable * mesh.cellVolumes)
>>> totalTime = 0
>>> steps = 5
>>> from builtins import range
>>> for step in range(steps):
...     velocity.setValue(surfactantVariable.interfaceVar * k)
...     distanceVariable.extendVariable(velocity)
...     timeStepDuration = cfl * dx / velocity.max()
...     distanceVariable.updateOld()
...     advectionEquation.solve(distanceVariable, dt = timeStepDuration)
...     surfactantEquation.solve(surfactantVariable, dt=1)
...     totalTime += timeStepDuration 
>>> surfactantEquation.solve(surfactantVariable, dt=1)
>>> surfactantAfter = numerix.sum(surfactantVariable * mesh.cellVolumes)
>>> print(surfactantBefore.allclose(surfactantAfter))
1

Next test for the correct local value of surfactant:

>>> finalRadius = numerix.sqrt(2 * k * initialRadius * initialSurfactantValue * totalTime + initialRadius**2)
>>> answer = initialSurfactantValue * initialRadius / finalRadius
>>> coverage = surfactantVariable.interfaceVar
>>> error = (coverage / answer - 1)**2 * (coverage > 1e-3)
>>> print(numerix.sqrt(numerix.sum(error) / numerix.sum(error > 0)) < 0.04)
1

Test for the correct position of the interface:

>>> x, y = mesh.cellCenters
>>> radius = numerix.sqrt((x - L / 2)**2 + (y - L / 2)**2)
>>> solution = radius - distanceVariable
>>> error = (solution / finalRadius - 1)**2 * (coverage > 1e-3)
>>> print(numerix.sqrt(numerix.sum(error) / numerix.sum(error > 0)) < 0.02) 
1

Last updated on Jun 26, 2024. Created using Sphinx 7.1.2.