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StressStrain Relationship for a Newtonian Fluid
First, the notation for the viscous stresses are:
x
y
z
τ
xx
τ
xy
τ
xz
τ
zx
τ
zy
τ
zz
τ
yx
τ
yy
τ
yz
τ
ij
=
stress acting on the fluid element with a face whose normal is in
+
x
i
direction and the stress is in
+
x
j
direction.
Common assumption is that the net moment created by the viscous stresses are
zero.
⇒
ij
=
ji
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View Full Document StressStrain Relationship for a Newtonian Fluid
Let’s look at this in 2D:
dx
dy
x
y
τ
xx
τ
xy
τ
yx
τ
yy
τ
yy
τ
yx
τ
xy
τ
xx
M
z
dx
dy
τ
xy
2
dy
−
yx
dx
2
M
= Net moment about center=
z
dx
+
xy
dy
−
yx
dx
2
2
⇒
M
=
(
x
y
−
)
dxdy
z y
x
2
Thus, for
M
=
0,
x
=
z
y
Assumptions for Newtonian fluid stressstrain:
1)
ij
is at most a linear function of
ε
ij
.
2) The fluid is isotropic, thus its properties are independent of direction
⇒
stressstrain relationship cannot depend on choice of coordinate axes.
3) When the strain rates are zero, the viscous stresses must be zero.
16.100 2002
2
StressStrain Relationship for a Newtonian Fluid
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This note was uploaded on 11/06/2011 for the course AERO 100 taught by Professor Willcox during the Fall '03 term at MIT.
 Fall '03
 willcox
 Dynamics, Aerodynamics

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