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44
Physics Formulary by ir. J.C.A. Wevers
with
D
=
±
2
κ
/
ω±
c
. At
x
=
π
D
the temperature variation is in antiphase with the surface. The one
dimensional solution at
Φ
=0
is
T
(
x, t
)=
1
2
√
π
at
exp
²

x
2
4
at
³
This is mathematical equivalent to the diffusion problem:
∂
n
∂
t
=
D
∇
2
n
+
P

A
where
P
is the production of and
A
the discharge of particles. The Fow density
J
=

D
∇
n
.
9.9
Turbulence
The time scale of turbulent velocity variations
τ
t
is of the order of:
τ
t
=
τ
√
Re
/
Ma
2
with
τ
the molecular
time scale. ±or the velocity of the particles holds:
v
(
t
)=
±
v
²
+
v
±
(
t
)
with
±
v
±
(
t
)
²
=0
. The NavierStokes
equation now becomes:
∂
±
²
v
²
∂
t
+(
±
²
v
²
·
∇
)
±
²
v
²
=

∇±
p
²
±
+
ν
∇
2
±
²
v
²
+
div
S
R
±
where
S
Rij
=

±
±
v
i
v
j
²
is the turbulent stress tensor. Boussinesq’s assumption is:
τ
ij
=

±
´
v
±
i
v
±
j
µ
. It is
stated that, analogous to Newtonian media:
S
R
=2
±ν
t
±
D
²
. Near a boundary holds:
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This note was uploaded on 01/30/2011 for the course PHYSICS 208 taught by Professor Ye during the Spring '10 term at Blinn College.
 Spring '10
 Ye
 Physics

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