That is, the external force is equal to the submerged weight of the grain.
The form drag can be written in terms of the shear velocity. For turbulent flow, the local
velocity, v is directly proportional to the shear velocity V*. Then, Equation 3.24 reduces to:
FD ~ ρD 2 V*2
(3.28)
s
The viscous drag is also related to the shear velocity, but it is the shear velocity for laminar
flow. For laminar flow:
dv
τ=
(3.29)
dy
Again, by replacing V with V* and y with Ds we can write
τ ≈ V* / D s
(3.30)
FV ~ D S V*
(3.31)
Now, consider the ratio of the form drag force FD to the viscous shear force Fv. According to
Equations 3.28 and 3.31:
FD ρD 2 V*2
s
~
D s V*
Fv
or
FD D s V*
(3.32)
~
υ
Fv
When the flow over the grain is turbulent, the form drag is predominant and the term DsV*/ν
is large. When the flow over the grain is laminar the viscous shear force is predominant and
the term DsV*/ν is small. Thus, the Reynolds' number of particle DsV*/ν is an indicator of the
characteristics of the flow in the vicinity of the grain.
As both the form drag and viscous shear are proportional to the shear velocity, the ratio of
the forces tending to move the grain to the forces resisting movement is:
ρD 2 V*2
τo
FD
s
=
(3.33)
~
Fn (ρs - ρ) gD 3 (γ s - γ)D s
s
3.35
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