If the virtual mass (or added mass) term is applied in addition to the drag force, then the particle mass mp on the left-hand side is replaced by the virtual particle mass mv. For more information about the virtual mass term, see the Virtual Mass Force section.

In Equation 5-1, the Drag Force FD is defined as:

Strictly speaking, u is the value that the fluid velocity would have at the particle’s position if no particle were there, unless fluid-particle interactions are considered.

Many expressions for τp are available. The validity of a given drag law depends on the relative Reynolds number Rer (dimensionless) of particles in the flow,

Many expressions for the drag force, and the range of relative Reynolds numbers at which they are applicable, are given in Ref. 4.

In a creeping flow, where the relative Reynolds number is very low (), the Stokes drag law is applicable. This is the default behavior when adding a Drag Force node to the Particle Tracing for Fluid Flow interface. In the Stokes drag law, the velocity response time is defined as

By substituting Equation 5-3 into Equation 5-2, some other familiar expressions for the Stokes drag force can be obtained,

where rp (SI unit: m) is the particle radius.

To simplify the expressions for the velocity response time τp, a dimensionless drag coefficient CD is defined, such that

Comparing Equation 5-4 with Equation 5-3, the Stokes drag law can be stated in an alternative way, by defining the drag coefficient as

When Schiller-Naumann is selected from the Drag law list, the drag coefficient becomes

When Haider-Levenspiel is selected from the Drag law list, the drag coefficient is given by

where A, B, C, and D (all dimensionless) are empirical correlations of the particle sphericity. The sphericity is defined as the ratio of the surface area of a volume equivalent sphere to the surface area of the considered nonspherical particle

When Oseen correction is selected from the Drag law list, the drag coefficient is given by

Like Stokes drag law, the Oseen correction is applicable at low relative Reynolds numbers, typically Rer < 0.1. The Oseen correction is determined by simplifying, rather than neglecting, the inertia term in the Navier-Stokes equation, resulting in drag coefficients that are slightly greater than those based on the Stokes drag law.

The Hadamard-Rybczynski drag law is based on an analytical solution for creeping flow past a spherical liquid drop or gas bubble. The drag coefficient is defined as

where μp (SI unit: Pa·s) is the dynamic viscosity of the droplet or gas bubble. For arbitrarily large values of μp this expression asymptotically approaches the Stokes law. The Hadamard-Rybczynski drag law is only applicable if the droplets and the surrounding fluid are extremely pure and free from surface-active contaminants. If the fluids are not sufficiently pure, the drag force on the bubble or droplet is more likely to be accurately determined by the Stokes drag law.

The option Standard drag correlations defines the drag coefficient as a piecewise function of the relative Reynolds number:

where w = log Rer and the base 10 logarithm has been used.

The Standard drag correlations can be used when the relative Reynolds number is expected to change by several orders of magnitude during a simulation. It is also applicable at significantly higher relative Reynolds numbers than the Schiller-Naumann drag law.

At lower relative Reynolds numbers, this correlation agrees with the Oseen correction.

The drag coefficient as a function of the relative Reynolds number is shown in Figure 5-1.

When the Include wall corrections check box is selected in the settings for the Drag Force node, the drag force is given by Ref. 5 as

The first term in Equation 5-5 applies to the component of the relative particle velocity parallel to the nearest wall; the second term applies to the component of the relative velocity normal to this wall. The correction factors become more prominent as the wall distance becomes comparable to the particle radius.

The Gravity Force is given by: