Viscous Flow: Stress Strain Relationship
Objective:
Discuss assumptions which lead to the stressstrain relationship for
a Newtonian, linear viscous fluid:
j
i
k
ij
ij
j
i
k
k
k
u
u
u
x
x
x
u
u
v
w
V
x
x
y
z
τ
µ
δ λ
∂
∂
∂
=
+
+
∂
∂
∂
∂
∂
∂
∂
=
+
+
= ∇
∂
∂
∂
∂
K
i
where
=
µ
dynamic viscosity coefficient
λ
= bulk viscosity coefficient
Note:
0,
1,
ij
i
j
i
j
δ
≠
=
=
1
shear strain rate in
,
plane
2
j
i
ij
i
j
j
i
u
u
x
x
x
x
ε
∂
∂
=
+
≡
∂
∂
Thus, written in terms of the strain rates, the stress tensor is:
N
(
)
viscous stress
using indicial notation
due to shearing
this is
of a fluid element
viscous stress due to an
overall compression or
expansion of the fluid
element's volume
2
kk
ij
ij
ij
xx
yy
zz
ε
τ
µε
δ λ ε
ε
ε
=
+
+
+
±²²³²²´
±²²²³²²²´
This stressstrain relationship can be derived by the following two assumptions:
1. The shear stress is independent of a rotation of the coordinate system
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 Fall '03
 willcox
 Dynamics, Aerodynamics, Shear Stress, viscous stress, stressstrain relationship, τ ij

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