
379Appendix
© 2010 Taylor & Francis Group, LLC
where
K
T
T
K
qf
qf
T
iii
iii
q
ji
ji
=+
′′
=+11
ψ
φ
φ
φ
()
;
()
;
γρ
βρ
"
KK
TT
T
iiiikiii ik
k
ϑ
ϑϑ
=+
′′′
+
′′ ′′′
+
′′
1
ργ ργ ρ
ω
ψ ()
iiiiik
iiiik
ji ji p
T
T
qf QV
τ
γ
βρ
β
′′′
=
ϑ
ϑ
;
/.
φ 1
(A.96)
Thus, a material ow and air it entrains is characterized, on the one hand, by a mac-
roscale process of the induced air translational movement and, on the other hand, by
a microscale pulsation of components—a mechanism of inter-component interaction
expanded in a theoretical description. An alternative method consists of develop-
ing approximations. However, a quite efcient theory of aerodynamic processes in
a bulk material gravity ow exists that ...