2.11. Force on the fluid

Forces on the fluid fall into a number of categories which are discussed in this section.

2.11.1. Body forces

The momentum equation may include a term which is a local body force density. The origin of such a force will depend on context, but may include uniform external forces, or local forces arising from the thermodynamic sector. An example of the former would be gravity, while an example of the latter would include bending forces at the interface arising from interfacial tension in the case of a symmetric binary fluid.

2.11.2. Constant body forces

If a constant body force density on the fluid is required, this may be included via:

force fx_fy_fz

where the three components are defined in the usual way. This force density is applied per fluid site per time step, so there must be some counter-balancing mechanism to prevent the fluid accelerating indefinitely. This counter-balancing force might be provided by, e.g., drag at external walls.

2.11.3. Local forces from the thermodynamic sector


As of version 0.19.0 the input key switches fd_force_divergence and fe_use_stress_divergence have been replaced by those described below. If the keys are present in the input, an error will occur; the old key value pairs should be replaced with the corresponding new single key fe_force_method.

The choice of free energy determines the form of chemical potential (and analogous quantities for more complex order parameters), and the form of the thermodynamic stress exerted locally on the fluid. There are a number of ways to implement how the force is computed and how it is applied to the fluid.

The list of available keys is as follows. Not all choices are relevant for all free energy selections. See the selections below for further details.

fe_force_method          no_force
fe_force_method          stress_divergence
fe_force_method          phi_gradmu
fe_force_method          phi_gradmu_correction
fe_force_method          relaxation_symmetric
fe_force_method          relaxation_antisymmetric Generic cases

All free energies define a “thermodynamic” stress \(P_{\alpha\beta}\) which acts on the fluid. There are a number of methods to implement the resulting force on the fluid. The first is to compute the divergence of the stress via a finite difference expression, which results in a local body force density

\[f_\alpha (\mathbf{r}; t) = \partial_\beta P_{\alpha\beta}.\]

The local body force density then enters the hydrodynamics at the lattice Boltzmann collision stage. This method has the advantage that it is by construction a divergence in the finite difference picture, and so momentum is conserved both locally, and therefore globally.

A second approach is to add the thermodynamic stress to the equilibrium stress \(S_{\alpha\beta}^{eq}\) at the lattice Boltzmann collision stage:

\[S_{\alpha\beta}^{eq} = u_\alpha u_\beta + P_{\alpha\beta}.\]

The current stress is then relaxed toward the equilibrium at a rate determined by the viscosity, and we refer to this as the relaxation method. It requires that the thermodynamic contribution to the stress is symmetric. The relaxation approach results in an effective divergence of a stress, which again conserves momentum.

These options are enabled by selections stress_divergence and relaxation_symmetric, respectively. If the option no_force is selected, no force from the thermodynamic sector will enter the hydrodynamics. This differs from setting no hydrodynamics at all in that with no_force, forces may still enter the hydrodynamics from elsewhere (e.g., from external gravity). Scalar order parameters and Cahn Hilliard

For scalar order parameters \(\phi(\mathbf{r}; t)\) with Cahn-Hilliard dynamics such as the composition in the symmetric free energy, it is possible to relate the local body force on the fluid to the gradient of the chemical potential \(\mu(\mathbf{r};t)\):

\[f_\alpha (\mathbf{r};t) = - \phi(\mathbf{r};t) \partial_\alpha\mu(\mathbf{r};t).\]

Formally, this expression is the same as the divergence of the corresponding stress (as seen above). However, in the finite difference picture, their discrete analogues do not behave in the same way. Specifically, the gradient of the chemical potential is not a divergence and therefore does not, by construction, result in a net force of zero and so does not conserve momentum.

However, the calculation does have the advantage that it admits an equilibrium. This is related to the fact that a consistent finite difference approach to the body force, and the diffusive fluxes in the Cahn-Hilliard equation \(-M\partial_\alpha\mu(\mathbf{r};t)\) where \(M\) is a uniform mobility, can be adopted. In this way, with a uniform chemical potential both the force in momentum equation and diffusive fluxes in the Cahn-Hilliard equation are simultaneously zero. This admits a state of no flow where the total flux in the Cahn-Hilliard equation

\[\phi u_\alpha + M \partial_\alpha \mu = 0.\]

This is in contrast to the computation of the force based on \(\partial_\beta P_{\alpha\beta}\), where a uniform chemical does not correspond to zero force in the finite difference picture. This often results in the generation of residual flows (“spurious currents”) in situations where an equilibrium might be expected.

For scalar order parameters one can choose

fe_force_method          phi_gradmu
fe_force_method          phi_gradmu_correction

The first option computes and applies the local force \(-\phi(\mathbf{r};t) \partial_\alpha \mu(\mathbf{r};t)\) without additional action. The second option computes the same force, but then applies a correction at every time step based on the current global force on the fluid. Any net force is subtracted uniformly from fluid sites to enforce global conservation of momentum before the force is applied. External chemical potential gradient

For Cahn-Hilliard dynamics, it is also possible to add a fixed external chemical potential gradient \(\partial_\alpha \mu^\textrm{ex}\) which generates an additional diffusive flux

\[\partial_t \phi + \partial_\alpha \phi u_\alpha = -\partial_\alpha (M\partial_\alpha \mu + M \partial_\alpha \mu^\textrm{ex}),\]

where \(M\) is the mobility. The addition chemical potential gradient also leads to a contribution to the body force on the fluid which is locally \(-\phi \partial_\alpha \mu^\textrm{ex}\). The external potential chemical gradient is a vector which may be defined in the input file as, e.g.,

grad_mu                  0.00001_0.0_0.0   # x-direction only

The additional force on the fluid may lead to a permanent acceleration of the fluid, so it may make sense to use this option only when walls are present, or some other mechanism is present to limit fluid motion.

Note this option is independent of the choice of fe_force_method. Vector order parameter

For vector order parameters involving the Leslie-Erickson equation, there are currently no options for fe_force_method: the stress divergence method is the default. Forces for tensor order parameter

The liquid crystal tensor order parameter \(Q_{\alpha\beta}\) gives rise to a stress which may be written in three parts:

\[P_{\alpha\beta} = P^\mathrm{symm}_{\alpha\beta} + P^\mathrm{anti}_{\alpha\beta} - \partial_\alpha Q_{\pi\nu} \frac{\delta \cal{F}}{\delta \partial_\beta Q_{\pi\nu}}.\]

Here the first two parts are related to the molecular field \(H_{\alpha\beta}\) and are symmetric and antisymmetric contributions, respectively. The third part, related to the functional derivative of the free energy density with respect to the order parameter gradient, is symmetric. In this case, the stress divergence may be computed to give a body force. However, as the stress relaxation requires a symmetric stress, only \(P^\mathrm{symm}_{\alpha\beta}\) and the third term may enter. The remaining antisymmetric must be treated via the divergence approach \(\partial_\beta P^\mathrm{anti}_{\alpha\beta}\).

For the bare liquid crystal free energy there is currently no implementation to compute the force via a term of the form

\[f_\gamma(\mathbf{r};t) = -H_{\alpha\beta}(\mathbf{r};t)\partial_\gamma Q_{\alpha\beta}(\mathbf{r};t),\]

which is the expression analogous to that involving the chemical potential in the scalar order parameter case.

Relevant choices of fe_force_method for the bare liquid crystal free energy are therefore:

fe_force_method          stress_divergence
fe_force_method          relaxation_antisymmetric

For the liquid crystal emulsion free energy there is both a chemical potential and a molecular field. Again, the stress from the thermodynamic sector may be decomposed into three parts

\[P_{\alpha\beta} = P^\mathrm{symm}_{\alpha\beta} + P^\mathrm{anti}_{\alpha\beta} + P^\mathrm{drop}_{\alpha\beta}\]

where the symmetric and antisymmetric parts are treated in the same way as the bare liquid crystal case (the same input keys are relevant).

The additional contribution \(P^\mathrm{drop}_{\alpha\beta}\) arising from the functional derivative of the free energy, gives rise to a force:

\[f_\alpha = - H_{\pi\nu} \partial_\alpha Q_{\pi\nu} - \phi \partial_\alpha \mu.\]

This is always computed separately and subject to a correction to conserve momentum. Forces in the electrokinetic picture

For free energies involving electrokinetics and the Nernst-Planck equation, there are two available options:

fe_force_method           stress_divergence
fe_force_method           phi_gradmu_correction

The details of the computation in this case a slightly different to other free energies (the phi in phi_gradmu_correction is a slight misnomer in the context of electrokinetics). Force near stationary solid objects (walls)

The following methods are available in the case of a static wall:

fe_force_method           no_force
fe_force_method           stress_divergence
fe_force_method           phi_gradmu
fe_force_method           phi_gradmu_correction

In the stress divergence approach, the stress acting on the wall does is accounted for as momentum lost by the fluid. If using the chemical potential, the approximation is made that the normal component of the gradient of the chemical potential at the wall is zero. Either the phi_gradmu or the phi_gramu_correction with no momentum conservation or with the correction to conserve momentum, respectively. Force near moving solid objects

For moving colloids, the following methods are available:

fe_force_method           no_force
fe_force_method           stress_divergence
fe_force_method           phi_gradmu_correction

For no_force there will be no force on the colloids from the thermodynamic sector. In the stress_divergence case, the net force on the colloid arising from the divergence of the thermodynamic stress is computed by accumulating the sum of the stress on the solid-fluid interfaces that make up the colloid at any given instant. This gives a body force on each colloid which goes into the dynamic update of their velocities. This conserves momentum.

In contrast, if the gradient of the chemical potential is used, there is no direct force on the colloids from the thermodynamic sector. Any influence comes indirectly through the hydrodynamic boundary conditions at the colloid surface in response to the local fluid velocities.