Drift terms

A drift term makes hydrogen move by something other than its own concentration gradient. All of them add a velocity to the flux, \(J = -D \nabla c + c \mathbf{v}\), and differ only in what sets \(\mathbf{v}\). See the theory guide for the equations.

They are passed to the problem as drift_terms.

Soret effect

Thermodiffusion along a temperature gradient, with the heat of transport \(Q^*\) in eV:

import festim as F

my_model.drift_terms = [
    F.SoretTerm(species=mobile_H, Q_star=0.2, subdomain=my_volume)
]

The term needs a temperature that varies in space, so give it as a function of x or couple an instance of festim.CoupledTransientHeatTransferHydrogenTransport. A uniform temperature gradient, and then the Soret has no effect. FESTIM warns when this happens.

Electromigration

Drift of a charged species in an electric potential, in volts. charge is the charge number \(z\):

my_model.drift_terms = [
    F.ElectromigrationTerm(
        species=hydroxyl,
        charge=1,
        potential=lambda x: 0.5 * (1 - x[0] / 5e-4),
        subdomain=membrane,
    )
]

FESTIM does not solve for the potential, so it must be passed in as an existing field. Give it as a float, a callable of x, t and/or T, or a fenics object. As with the Soret term, a spatially uniform potential makes the term do nothing, and FESTIM sends a warning.

Advection

Hydrogen carried by a moving fluid, with the velocity given directly as a dolfinx.fem.Function on a vector function space.

my_model.drift_terms = [
    F.AdvectionTerm(velocity=my_velocity_field, subdomain=my_volume, species=H)
]

The divergence form

Every drift term is assembled as \(\nabla \cdot (c\mathbf{v})\). There is no option to assemble \(\mathbf{v} \cdot \nabla c\) instead: the two differ by \(c \nabla \cdot \mathbf{v}\). Only the divergence form conserves the species.

The boundary term the divergence form leaves behind is the natural boundary condition, so flux boundary conditions constrain the total flux, drift included, and a boundary with no condition on it behaves like an impermeable wall (zero total flux, with the drift balanced by back-diffusion). If the boundary is intended to be an outlet, the OutflowBC is needed (see Letting the species out).

Note

FESTIM does not stabilise the advection-diffusion form. When drift dominates, i.e. when the cell Péclet number \(\mathrm{Pe} = |\mathbf{v}| h / D\) becomes large, the solution can oscillate. Refine the mesh where the drift dominates to mitigate this issue.

Warning

This changes results for existing advection models. Before FESTIM 2.2, festim.AdvectionTerm used \(\mathbf{v} \cdot \nabla c\). Your results are unchanged if both of the following hold:

• the velocity field is divergence-free, as an incompressible flow is;

• every boundary the flow crosses carries a boundary condition.

If the flow leaves through a boundary you did not tag, that boundary is now a closed end and the species backs up against it. Add a festim.OutflowBC there.

Letting the species out

festim.OutflowBC marks a surface the flow leaves through. It cancels the drift boundary term, so the natural condition there becomes zero diffusive flux, and the species is carried out at the rate the drift delivers it:

my_model.boundary_conditions = [
    F.FixedConcentrationBC(subdomain=inlet, value=1.0, species=H),
    F.OutflowBC(subdomain=outlet, species=H),
]

On a codimensional problem the surface may be the boundary of a manifold, e.g. the outlet of a 1D fluid running along a pipe wall.

festim.OutflowBC does nothing on a surface where no drift acts on the species.

Surface fluxes

festim.SurfaceFlux reports the total flux \((-D \nabla c + c\mathbf{v}) \cdot \mathbf{n}\), so drift contributes to it.