For a particle of velocity v (SI unit: m/s) in a fluid of velocity
u (SI unit: m/s), the virtual mass term
Fvm (SI unit: N) is (
Ref. 19)
where mf (SI unit: kg) is the mass of the fluid displaced by the particle volume,
dp (SI unit: m) is the particle diameter, and
ρ (SI unit: kg/m
3) is the density of the fluid at the particle’s position.
The derivative term in Equation 5-12 is a total derivative or material derivative. Therefore it considers not only the time dependence of the velocity at a fixed point, but also the motion of the particle relative to the fluid. For an arbitrary scalar field
f, the material derivative is defined as
Since v is the velocity of a discrete particle, not a field variable, its gradient
vanishes, leaving only
The term involving the time derivative of particle velocity v can be moved to the left-hand side of the equation of motion, by manipulating the equation of motion of a particle as follows:
where mv (SI unit: kg) is the virtual mass,
To summarize, when including the virtual mass force, the particle mass mp on the left-hand side of the equation of motion is replaced with the virtual mass
mv. The contribution of the virtual mass force to the right-hand side then contains only the material derivative of the fluid velocity
u,