COMSOL 6.3 - Newtonian Formulation Without Inertial Terms

The Particle Tracing for Fluid Flow interface offers the formulation of the equations of motion, which can be selected from the list in the physics interface section. Use this formulation to model the motion of small particles in a fluid.

To use the formulation, a Drag Force must be exerted in every domain in the selection of the Particle Tracing for Fluid Flow interface. This could either be a single node with all domains selected, or multiple nodes with complementary selections.

The drag law is always used; other drag laws such as may not be selected.

Some other built-in forces can be used while the formulation is selected, but not as many forces as the or formulations. Forces such as the Gravity Force and Dielectrophoretic Force can be selected, but not the Lift Force or the Magnetic Force. As a general rule, a built-in force other than drag can be used with the formulation if the expression for that force does not include any explicit velocity dependence.

Consider first the equation of motion of a particle in the fluid, from Equation 5-1,

using the Stokes drag law from Equation 5-2,

Also recall the definition of the particle velocity response time τp from Equation 5-3,

To illustrate how a full Newtonian model of very small particles can give rise to numerically stiff problems, suppose that the particle starts from rest and that the only force acting on it is drag, where the background velocity is uniform. Under these simplifying assumptions, the equation of motion would be

Thus the velocity response time is the rate of the exponential decay by which the particle velocity begins to match the surrounding fluid velocity.

Consider a particle with diameter dp = 1 μm and density ρp = 2200 kg/m3 in water with dynamic viscosity μ = 8.9 × 10-4 Pa s. The velocity response time of such a particle would be approximately 1.4 × 10-7 s. Thus the time-dependent solver would be required to resolve exponential decay on a submicrosecond time scale. If the total simulation time is on the order of seconds or larger, this could result in the solver taking tens of millions of time steps, with a very long computation time.

The formulation uses a simplifying assumption that is valid when the velocity response time is much smaller than the time range of interest. The assumption is that each particle instantaneously changes its velocity so that the drag force exactly balances out all of the external forces on the particle,

If all forces other than drag are neglected, then for Stokes drag, this simplifies to

and particles will simply follow fluid velocity streamlines. Alternatively, considering only the drag and gravity forces (but including buoyancy), the particle motion follows the equation

or, solving for the particle velocity,

which gives the gravitational settling velocity of a small particle. Or more generally,