Dimensionless Numbers Important for Solver Stability
When solving a microfluidics problem numerically there are two critical dimensionless numbers to consider in terms of solver stability. These numbers are:
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The Reynolds number (Re)
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The Peclet number (Pe)
Each of these numbers is defined in Table 2-2. Both the Reynolds number and the Peclet number are associated with the relative importance of convective terms in the corresponding partial differential equation. The Peclet number describes the importance of convection in relation to diffusion (for either heat or mass transfer), and the Reynolds number describes the importance of the “convective” inertia term in relation to viscosity in the Navier–Stokes equations themselves.
Both the Reynolds number and the Peclet number can be defined on the “cell” or element level. As they are defined in COMSOL Multiphysics numerical instabilities can arise when the cell Reynolds or Peclet number is greater than one. These instabilities are usually manifested as spurious oscillations in the solution. Taking the Peclet number as an example, oscillations can occur when the cell Peclet number is greater than one in the following circumstances:
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A Dirichlet boundary condition can lead to a solution containing a steep gradient near the boundary, forming a boundary layer. If the mesh cannot resolve the boundary layer, this creates a local disturbance.
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A space-dependent initial condition that the mesh does not resolve can cause a local initial disturbance that propagates through the computational domain.
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A small initial diffusion term close to a nonconstant source term or a nonconstant Dirichlet boundary condition can result in a local disturbance.
In theory the grid can be refined to bring the cell Reynolds or Peclet number below one, although this is often impractical for many problems. Several stabilization techniques are included, which enable problems with larger cell Reynolds or Peclet numbers to be solved. At the crudest level additional numerical diffusion can be added to the problem to improve its stability. This is achieved by selecting Isotropic Diffusion under Inconsistent Stabilization for any physics interface.
The stabilization options are visible when the Show More Options button () is clicked and Stabilization is selected in the Show More Options dialog.
This method is termed “inconsistent” as a solution to the problem without numerical diffusion is not necessarily a solution to the problem with diffusion. COMSOL Multiphysics also has consistent stabilization options. A consistent stabilization technique reduces the numerical diffusion added to the problem as the solution approaches the exact solution. Both streamline diffusion and crosswind diffusion are available. Streamline diffusion adds numerical diffusion along the direction of the flow velocity (that is, the diffusion is parallel to the streamlines). Crosswind diffusion adds diffusion in the direction orthogonal to the velocity.
Generally it is best to use consistent stabilization where possible. If convergence problems are still encountered, inconsistent stabilization can be used with a parametric or time-dependent solver that slowly eliminates this term.
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Selection Information and Numerical Stabilization in the COMSOL Multiphysics Reference Manual
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Other Dimensionless Numbers
Table 2-2: Important dimensionless numbers related to the stability of various microfluidic problems.
Dimensionless Number
Symbol
Global Definition
Cell Definition
Notes
Reynolds number
 Re
spf.cellRe
Ratio of inertial forces to viscous forces.
Laminar flow applies:
Creeping flow applies:
Stabilization required:
Peclet number (Heat Flow)
 Pe
spf.cellPe
Ratio of heat convection to thermal diffusivity, .
Stabilization required:
Peclet number (Mass Transport)
 Pe
Ratio of concentration convection to diffusion.
Stabilization required:
Symbol definitions: v is the characteristic velocity or cell velocity, r is the fluid density, L is the characteristic length scale for the problem, μ is the fluid viscosity, h is the element size, cp is the heat capacity at constant pressure, κ is the thermal conductivity, and D is the diffusion constant. For anisotropic diffusion or conductivity an appropriate average is computed. The variables names assume the default name spf.