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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