Non-Stokesian drag regime Stokes number
ψ
{\displaystyle \psi }
describes non-stokesian drag correction factor spherical particle
ψ
(
re
o
)
=
24
re
o
∫
0
re
o
d
re
′
c
d
(
re
′
)
re
′
{\displaystyle \psi ({\text{re}}_{o})={\frac {24}{{\text{re}}_{o}}}\int _{0}^{{\text{re}}_{o}}{\frac {d{\text{re}}^{\prime }}{c_{d}({\text{re}}^{\prime }){\text{re}}^{\prime }}}}
considering limiting particle free-stream reynolds numbers,
re
o
→
0
{\displaystyle {\text{re}}_{o}\rightarrow 0}
c
d
(
re
o
)
→
24
/
re
o
{\displaystyle c_{d}({\text{re}}_{o})\rightarrow 24/{\text{re}}_{o}}
, therefore
ψ
→
1
{\displaystyle \psi \rightarrow 1}
. expected there correction factor unity in stokesian drag regime. wessel & righi evaluated
ψ
{\displaystyle \psi }
c
d
(
re
)
{\displaystyle c_{d}({\text{re}})}
empirical correlation drag on sphere schiller & naumann.
ψ
(
re
o
)
=
3
(
c
re
o
1
/
3
−
arctan
(
c
re
o
1
/
3
)
)
c
3
/
2
re
o
{\displaystyle \psi ({\text{re}}_{o})={\frac {3({\sqrt {c}}{\text{re}}_{o}^{1/3}-\arctan({\sqrt {c}}{\text{re}}_{o}^{1/3}))}{c^{3/2}{\text{re}}_{o}}}}
where constant
c
=
0.158
{\displaystyle c=0.158}
. conventional stokes number underestimate drag force large particle free-stream reynolds numbers. overestimating tendency particles depart fluid flow direction. lead errors in subsequent calculations or experimental comparisons.
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