Graphics Reference
In-Depth Information
8.3 Strain Rate and Newtonian Fluids
Viscosity is physically based on the fact that when molecules traveling at
different speeds collide or closely interact, some energy may be transferred
to vibrational or rotational modes in the molecule—i.e., heat —and thus the
difference in center-of-mass velocity between the two molecules is reduced.
At the continuum level, the net effect of this is that as a region of fluid
slips past another, momentum is transferred between them to reduce the
difference in velocity, and the fluids get hotter. The critical thing to note is
that this occurs when fluid moves
past
other fluid: in a rigid body rotation
there are differences in velocity, but the fluid moves together and there is no
viscous effect. Thus we really only care about how the fluid is deforming,
i.e., moving non-rigidly.
To measure differences in velocity locally, the natural quantity to con-
sider is the gradient of velocity:
u
. However, mixed up in the gradient is
information about the rigid rotation
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as well as the deformation induced
by the flow. We will want to separate out just the deformation part to
define viscous stress.
One way to characterize rigid motion is that the dot-product of any
two vectors remains constant. (If the two vectors are the same, this is just
saying lengths remains constant; for different vectors we're saying the angle
between them stays the same.) How much the dot-product between two
vectors changes is thus a measure of how fast the fluid is deforming. Let's
look at a point
x
:inasmalltimeintervalΔ
t
it moves to approximately
x
+Δ
tu
(
x
). Now look at two nearby points,
x
+Δ
x
and
x
+Δ
y
: linearizing
appropriately, they approximately move to
∇
x
+Δ
x
+Δ
t
(
u
(
x
)+
∇
u
Δ
x
)
x
+Δ
y
+Δ
t
(
u
(
x
)+
∇
u
Δ
y
)
,
and
respectively. The dot-product of the vectors from
x
to these points begins as
[(
x
+Δ
x
)
− x
]
·
[(
x
+Δ
y
)
−
x
]=Δ
x
·
Δ
y
and after the time interval is approximately
[(
x
+Δ
x
+Δ
t
(
u
(
x
)+
∇
u
Δ
x
))
−
(
x
+Δ
tu
(
x
))]
·
[(
x
+Δ
y
+Δ
t
(
u
(
x
)+
∇
u
Δ
y
))
−
(
x
+Δ
tu
(
x
))]
=[Δ
x
+Δ
t
∇
u
Δ
x
]
·
[Δ
y
+Δ
t
∇
u
Δ
y
]
.
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Later in the topic, we will take a look at the curl of velocity which is called vorticity,
ω
=
∇×u
; it measures precisely the rotational part of the velocity field.
