Momentum Transfer

Consider a discrete particle traveling in a continuous fluid medium. The forces acting on the particle which affect the particle acceleration are due to the difference in velocity between the particle and fluid, as well as to the displacement of the fluid by the particle. The equation of motion for such a particle was derived by Basset, Boussinesq and Oseen for a rotating reference frame:

Equation 4.

which has the following forces on the right hand side:

: drag force acting on the particle.
: buoyancy force due to gravity.
: forces due to domain rotation (centripetal and Coriolis forces).
: virtual (or added) mass force. This is the force to accelerate the virtual mass of the fluid in the volume occupied by the particle. This term is important when the displaced fluid mass exceeds the particle mass, such as in the motion of bubbles.
: pressure gradient force. This is the force applied on the particle due to the pressure gradient in the fluid surrounding the particle caused by fluid acceleration. It is only significant when the fluid density is comparable to or greater than the particle density.
: Basset force or history term which accounts for the deviation in flow pattern from a steady state. This term is not implemented in ANSYS CFX.

The left hand side of Equation 4 can be modified due to the special form of the virtual mass term (see Virtual or Added Mass Force) which leads to the following form of the particle velocity:

Equation 5.

Only a part of the virtual mass term, , remains on the right hand side. The particle and fluid mass values are given by

Equation 6.

with the particle diameter as well as the fluid and particle densities and . The ratio of the original particle mass and the effective particle mass (due to the virtual mass term correction) is stored in

Equation 7.

Using , Equation 5 can be written as

Equation 8.

Each term on the right hand side of Equation 8 can potentially be linearized with respect to the particle velocity variable , leading to the following equation for each term:

Equation 9.

The following sections show the contribution of all terms to the right hand side values and the linearization coefficient .