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Mud Hydraulics Fundamentals 31

Using these equations, the criterion for turbulent flow is the same as
for Newtonian fluids, with laminar flow occurring below a Reynolds
number of 2,100.

Power Law Fluids
The concept of apparent viscosity can also be used for Power Law fluids
for Reynolds number calculations. Equations (2.23) and (2.24), respec-
tively, become

μ = Kd ð1−nÞ 3 + 1/n n (2.29)
a
96v ð1−nÞ 0:0416

1−n n
μ = Kðd 2 − d 1 Þ 2 + 1/n (2.30)
a
144v ð1−nÞ 0:0208
If Dodge and Metzner’s (1959) correlation is used, the Reynolds
number for pipe flow and annular flow can be respectively expressed in
U.S. field units as

ρv 2−n 0:0416d n
N Re = 89,100 (2.31)
K 3 + 1/n
and

ρv 2−n 0:0208ðd 2 − d 1 Þ n
N Re = 109,000 (2.32)
K 2 + 1/n

When expressed in SI units, these two equations become
2−n n
1:638d
N Re = 743:5 (2.33)
ρð3:281vÞ
K 3 + 1/n
and
2−n n
N Re = 909:5 ρð3:281vÞ 0:819ðd 2 − d 1 Þ (2.34)
K 2 + 1/n
The turbulence criterion for Power Law fluids is based on a critical Rey-
nolds number (N Rec ) that depends on the value of the flow behavior
index. A simple correlation for estimating the critical Reynolds number
at the upper limit of laminar flow is

N Rec = 3,470 − 1370n (2.35)