examples.reactiveWetting.liquidVapor1D¶

Solve a single-component, liquid-vapor, van der Waals system.

This example solves a single-component, liquid-vapor, van der Waals system as described by Wheeler et al. [5]. The free energy for this system takes the form,

(1)¶ $f = - \frac{e \rho^2}{m^2} + \frac{R T}{m} \left( \ln \frac{\rho}{m - \bar{v} \rho} \right)$

where $\rho$ is the density. This free energy supports a two phase equilibrium with densities given by $\rho^l$ and $\rho^v$ in the liquid and vapor phases, respectively. The densities are determined by solving the following system of equations,

(2)¶ $P \left( \rho^l \right) = P \left( \rho^v \right)$

and

(3)¶ $\mu \left( \rho^l \right) = \mu \left( \rho^v \right)$

where $\mu$ is the chemical potential,

(4)¶ $\mu = \frac{\partial f}{\partial \rho}$

and $P$ is the pressure,

(5)¶ $P = \rho \mu - f$

One choice of thermodynamic parameters that yields a relatively physical two phase system is

>>> molarWeight = 0.118
>>> ee = -0.455971
>>> gasConstant = 8.314
>>> temperature = 650.
>>> vbar = 1.3e-05

with equilibrium density values of

>>> liquidDensity = 7354.3402662299995
>>> vaporDensity = 82.855803327810008

The equilibrium densities are verified by substitution into Eqs. (2) and (3). Firstly, Eqs. (1), (4) and (5) are defined as python functions,

>>> from fipy import CellVariable, Grid1D, TransientTerm, VanLeerConvectionTerm, DiffusionTerm, ImplicitSourceTerm, ConvectionTerm, CentralDifferenceConvectionTerm, Viewer
>>> from fipy.tools import numerix

>>> def f(rho):
...     return ee * rho**2 / molarWeight**2 + gasConstant * temperature * rho / molarWeight * \
...            numerix.log(rho / (molarWeight - vbar * rho))

>>> def mu(rho):
...     return 2 * ee * rho / molarWeight**2 + gasConstant * temperature / molarWeight * \
...            (numerix.log(rho / (molarWeight - vbar * rho)) + molarWeight / (molarWeight - vbar * rho))

>>> def P(rho):
...     return rho * mu(rho) - f(rho)

The equilibrium densities values are verified with

>>> print(numerix.allclose(mu(liquidDensity), mu(vaporDensity)))
True

and

>>> print(numerix.allclose(P(liquidDensity), P(vaporDensity)))
True

In order to derive governing equations, the free energy functional is defined.

$F = \int \left[ f + \frac{\epsilon T}{2} \left( \partial_j \rho \right)^2 \right] dV$

Using standard dissipation laws, we write the governing equations for mass and momentum conservation,

(6)¶ $\frac{\partial \rho}{\partial t} + \partial_j \left(\rho u_j \right) = 0$

and

(7)¶ $\frac{\partial \left( \rho u_i\right) }{\partial t} + \partial_j\left( \rho u_iu_j \right) = \partial_j\left( \nu \left[ \partial_j u_i + \partial_i u_j \right] \right) - \rho \partial_i \mu^{NC}$

where the non-classical potential, $\mu^{NC}$ , is given by,

(8)¶ $\mu^{NC} = \frac{\delta F}{\delta \rho} = \mu - \epsilon T \partial_j^2 \rho$

As usual, to proceed, we define a mesh

>>> Lx = 1e-6
>>> nx = 100
>>> dx = Lx / nx
>>> mesh = Grid1D(nx=nx, dx=dx)

and the independent variables.

>>> density = CellVariable(mesh=mesh, hasOld=True, name=r'$\rho$')
>>> velocity = CellVariable(mesh=mesh, hasOld=True, name=r'$u$')
>>> densityPrevious = density.copy()
>>> velocityPrevious = velocity.copy()

The system of equations is solved in a fully coupled manner using a block matrix. Defining $\mu^{NC}$ as an independent variable makes it easier to script the equations without using higher order terms.

>>> potentialNC = CellVariable(mesh=mesh, name=r'$\mu^{NC}$')

>>> epsilon = 1e-16
>>> freeEnergy = (f(density) + epsilon * temperature / 2 * density.grad.mag**2).cellVolumeAverage

In order to solve the equations numerically, an interpolation method is used to prevent the velocity and density fields decoupling. The following velocity correction equation (expressed in discretized form) prevents decoupling from occurring,

(9)¶ $u_{i,f}^c = \frac{A_f d_f}{\overline{d}_f} \left( \overline{ \rho \partial_i \mu^{NC} }_f - \overline{\rho}_f \partial_{i,f} \mu^{NC} \right)$

where $A_f$ is the face area, $d_f$ is the distance between the adjacent cell centers and $\overline{a}_f$ is the momentum conservation equation’s matrix diagonal. The overbar refers to an averaged value between the two adjacent cells to the face. The notation $\partial_{i,f}$ refers to a derivative evaluated directly at the face (not averaged). The variable $u_i^c$ is used to modify the velocity used in Eq. (6) such that,

(10)¶ $\frac{\partial \rho}{\partial t} + \partial_j \left(\rho \left[u_j + u_i^c \right] \right) = 0$

Equation (10) becomes

>>> matrixDiagonal = CellVariable(mesh=mesh, name=r'$a_f$', value=1e+20, hasOld=True)
>>> correctionCoeff = mesh._faceAreas * mesh._cellDistances / matrixDiagonal.faceValue
>>> massEqn = TransientTerm(var=density) \
...           + VanLeerConvectionTerm(coeff=velocity.faceValue + correctionCoeff \
...                                         * (density * potentialNC.grad).faceValue, \
...                                   var=density) \
...           - DiffusionTerm(coeff=correctionCoeff * density.faceValue**2, var=potentialNC)

where the first term on the LHS of Eq. (9) is calculated in an explicit manner in the VanLeerConvectionTerm and the second term is calculated implicitly as a DiffusionTerm with $\mu^{NC}$ as the independent variable.

In order to write Eq. (7) as a FiPy expression, the last term is rewritten such that,

$\rho \partial_i \mu^{NC} = \partial_i \left( \rho \mu^{NC} \right) - \mu^{NC} \partial_i \rho$

which results in

>>> viscosity = 1e-3
>>> ConvectionTerm = CentralDifferenceConvectionTerm
>>> momentumEqn = TransientTerm(coeff=density, var=velocity) \
...               + ConvectionTerm(coeff=[[1]] * density.faceValue * velocity.faceValue, var=velocity) \
...               == DiffusionTerm(coeff=2 * viscosity, var=velocity) \
...               - ConvectionTerm(coeff=density.faceValue * [[1]], var=potentialNC) \
...               + ImplicitSourceTerm(coeff=density.grad[0], var=potentialNC)

The only required boundary condition eliminates flow in or out of the domain.

>>> velocity.constrain(0, mesh.exteriorFaces)

As previously stated, the $\mu^{NC}$ variable will be solved implicitly. To do this the Eq. (8) is linearized in $\rho$ such that

(11)¶ $\mu^{NC} = \mu^* + \left( \frac{\partial \mu}{\partial \rho} \right)^* \left( \rho - \rho^* \right) - \epsilon T \partial_j^2 \rho$

The $^*$ superscript denotes the current held value. In FiPy, $\frac{\partial \mu}{\partial \rho}$ is written as,

>>> potentialDerivative = 2 * ee / molarWeight**2 + gasConstant * temperature * molarWeight / density / (molarWeight - vbar * density)**2

and $\mu^*$ is simply,

>>> potential = mu(density)

Eq. (11) can be scripted as

>>> potentialNCEqn = ImplicitSourceTerm(coeff=1, var=potentialNC) \
...                  == potential \
...                  + ImplicitSourceTerm(coeff=potentialDerivative, var=density) \
...                  - potentialDerivative * density \
...                  - DiffusionTerm(coeff=epsilon * temperature, var=density)

Due to a quirk in FiPy, the gradient of $\mu^{NC}$ needs to be constrained on the boundary. This is because ConvectionTerm’s will automatically assume a zero flux, which is not what we need in this case.

>>> potentialNC.faceGrad.constrain(value=[0], where=mesh.exteriorFaces)

All three equations are defined and an are combined together with

>>> coupledEqn = massEqn & momentumEqn & potentialNCEqn

The system will be solved as a phase separation problem with an initial density close to the average density, but with some small amplitude noise. Under these circumstances, the final condition should be two separate phases of roughly equal volume. The initial condition for the density is defined by

>>> numerix.random.seed(2011)
>>> density[:] = (liquidDensity + vaporDensity) / 2 * \
...    (1  + 0.01 * (2 * numerix.random.random(mesh.numberOfCells) - 1))

Viewers are also defined.

>>> from fipy import input
>>> if __name__ == '__main__':
...     viewers = Viewer(density), Viewer(velocity), Viewer(potentialNC)
...     for viewer in viewers:
...         viewer.plot()
...     input('Arrange viewers, then press <return> to proceed...')
...     for viewer in viewers:
...         viewer.plot()

The following section defines the required control parameters. The cfl parameter limits the size of the time step so that dt = cfl * dx / max(velocity).

>>> cfl = 0.1
>>> tolerance = 1e-1
>>> dt = 1e-14
>>> timestep = 0
>>> relaxation = 0.5
>>> if __name__ == '__main__':
...     totalSteps = 1e10
... else:
...     totalSteps = 10

In the following time stepping scheme a time step is recalculated if the residual increases between sweeps or the required tolerance is not attained within 20 sweeps. The major quirk in this scheme is the requirement of updating the matrixDiagonal using the entire coupled matrix. This could be achieved more elegantly by calling cacheMatrix() only on the necessary part of the equation. This currently doesn’t work properly in FiPy.

>>> while timestep < totalSteps:
...
...     sweep = 0
...     dt *= 1.1
...     residual = 1.
...     initialResidual = None
...
...     density.updateOld()
...     velocity.updateOld()
...     matrixDiagonal.updateOld()
...
...     while residual > tolerance:
...
...         densityPrevious[:] = density
...         velocityPrevious[:] = velocity
...         previousResidual = residual
...
...         dt = min(dt, dx / max(abs(velocity)) * cfl)
...
...         coupledEqn.cacheMatrix()
...         residual = coupledEqn.sweep(dt=dt)
...
...         if initialResidual is None:
...             initialResidual = residual
...
...         residual = residual / initialResidual
...
...         if residual > previousResidual * 1.1 or sweep > 20:
...             density[:] = density.old
...             velocity[:] = velocity.old
...             matrixDiagonal[:] = matrixDiagonal.old
...             dt = dt / 10.
...             if __name__ == '__main__':
...                 print('Recalculate the time step')
...             timestep -= 1
...             break
...         else:
...             matrixDiagonal[:] = coupledEqn.matrix.takeDiagonal()[mesh.numberOfCells:2 * mesh.numberOfCells]
...             density[:] = relaxation * density + (1 - relaxation) * densityPrevious
...             velocity[:] = relaxation * velocity + (1 - relaxation) * velocityPrevious
...
...         sweep += 1
...
...     if __name__ == '__main__' and timestep % 10 == 0:
...         print('timestep: %e / %e, dt: %1.5e, free energy: %1.5e' % (timestep, totalSteps, dt, freeEnergy))
...         for viewer in viewers:
...             viewer.plot()
...
...     timestep += 1

>>> from fipy import input
>>> if __name__ == '__main__':
...     input('finished')

>>> print(freeEnergy < 1.5e9)
True

Last updated on Jun 27, 2023. Created using Sphinx 6.2.1.