Biomedical Engineering Reference
In-Depth Information
the particle experience an increase in its effective mass by the addition of the fluid
mass that is moving with the particle. The virtual mass force for a spherical particle
is half of the volume of the sphere times the density of the fluid. The corresponding
effective inertial force is given as
ρ
p
m
du
f
du
p
dt
1
2
ρ
f
F
VM
=
dt
−
(6.35)
That is, the expression given by (6.35) needs to be added to the right-hand side of
Eq. (6.20). The virtual mass effects are particularly important when the fluid density
is greater than the particle density (
ρ
ρ
p
) such as for bubbles in a liquid.
The Basset history force describes the unsteady drag force that acts on an
accelerating particle and is given as
t
(
du
f
/dt
du
p
/dt
)
3
πμd
2
2
√
π v
−
|
t
1
F
Bass
=
√
t
dt
1
(6.36)
−
t
1
t
0
The Basset history force becomes important for high frequency unsteady motions.
Magnus Effect
A rotating spherical particle moving in fluid will experience a force
perpendicular to the direction of its velocity, which is known as the Magnus effect.
In Fig.
6.12
, a particle spins counter-clockwise, with an oncoming free stream fluid
velocity of
u
0
. The contribution by the rotating boundary layer will oppose the
velocity near the top of the particle, and assists the velocity flow in the front of the
particle (near the lower part). This means that
u
2
> u
0
> u
1
, and thus the pressure
near the upper surface of the particle becomes larger than that near the lower surface.
This pressure difference generates the aerodynamic lift that is referred to as the
Magnus force, and can be written as
1
2
ρ
f
u
f
AC
L
F
M
=
(6.37)
where
A
is the cross-sectional area of the particle and
C
L
is the coefficient of lift.
Fig. 6.12
Schematic of a
spinning particle creating the
Magnus effect. The velocities
u
0
,
u
1
, and
u
2
are the
upstream, particle front, and
particle rear velocities,
respectively.
F
represents the
resulting force perpendicular
to the fluid flow direction
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