FilterDict assumes that the flow is stationary throughout a batch, i.e. no turbulences or oscillating behavior occurs. Therefore, the partial differential equations that need to be solved do not contain derivatives regarding time. Other assumptions are that the fluid is a Newtonian fluid and it is incompressible.

Navier Stokes Equations

If the geometry of the pores is resolved by the computational grid, all voxels are either pore space or solid material. The fluid flow in the pores is described by the stationary Navier-Stokes equations:

Here, is the fluid density, is the dynamic viscosity, is the velocity vector, is the pressure and is the force.

Stokes Equations

In a filter, the flow is often slow and laminar. In such cases, the Navier-Stokes equations can be simplified to the Stokes equations, which read as follows:

Here, is the dynamic viscosity, is the velocity vector, is the pressure and is the force.

Navier-Stokes-Brinkman Equations

If the geometry of the pores is not fully resolved by the computational grid, a voxel may describe pore space, solid material, or porous material. The Brinkman term allows to describe flow in an unresolved porous medium. In filter media simulations, this is the case when the filter becomes clogged with very small particles, i.e. the dust particle size is smaller than the voxel size. The Navier-Stokes-Brinkman equations are given as follows:

Here, is the velocity vector, is the pressure, is the force, and is the effective viscosity. The local viscous flow resistivity of a voxel is related to the local permeability by

where is the isotropic permeability of the voxel. In the pore space, the flow resistivity is zero and the Brinkman term disappears.

The initial structure as input for Filter Media simulations contains only solid material and pores. The porous domains appear when dust particles smaller than the voxel size are caught in the filter. Then, the number of regions where the Brinkman term is not zero becomes larger with each time step. In Filter Element or Complete Filter simulations, however, the pores and solids in the initial structure might be unresolved, and therefore a porous domain initially exists. Here, the Brinkman term describes porous voxels representing the porous media, as well as porous voxels containing deposited particles.

Stokes-Brinkman Equations

Again, under the assumption of a slow and laminar flow, the equations can be simplified and given as: