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Let's work through the simplification. If
η
is constant, then we can take
it out from under the divergence:
Du
Dt
+
1
p
=
η
u
T
)
ρ
∇
ρ
∇·
(
∇
u
+
∇
ρ
∇·∇
u
T
)
=
η
u
+
∇·
(
∇
⎡
⎣
⎛
⎝
⎞
⎠
⎤
⎦
∂u
∂x
+
∂
∂y
∂v
∂x
+
∂
∂z
∂w
∂x
∂
∂x
=
η
ρ
∂
∂x
∂u
∂y
+
∂
∂y
∂v
∂y
+
∂
∂z
∂w
∂y
∇·∇
u
+
∂
∂x
∂u
∂z
+
∂
∂y
∂v
∂z
+
∂
∂z
∂w
∂z
⎡
⎛
⎞
⎤
∂x
(
∂u
∂
∂x
+
∂v
∂y
+
∂w
∂z
)
⎣
⎝
⎠
⎦
=
η
ρ
∂y
(
∂u
∂
∂x
+
∂v
∂y
+
∂w
∇·∇
u
+
∂z
)
.
∂z
(
∂u
∂
∂x
+
∂v
∂y
+
∂w
∂z
)
In the last step we simply changed the order of partial derivatives in the
last term and regrouped. But now we see that the last term is simply the
gradient of
∇·
u
:
Du
Dt
+
1
p
=
η
ρ
∇
ρ
[
∇·∇
u
+
∇
(
∇·
u
)]
.
If the flow is incompressible, i.e.,
u
= 0, then the last term is zero and
we end up back at Equation (1.1). I emphasize that this only happens
when both the viscosity is constant through the flow and the velocity field
is incompressible. The second point becomes important numerically since
at intermediate stages in our time integration our velocity field may not be
discretely incompressible—and then this last term can't be blithely ignored.
Getting back to variable viscosity, some formula needs to be decided
for
η
. For a non-Newtonian fluid, it might be a function of the magnitude
of the strain rate, as we have seen. For regular Newtonian fluids it might
instead be a function of temperature—assuming we're tracking tempera-
ture in the simulation as we saw how to do with smoke and fire—which
is most important for liquids. Carlson et al. [Carlson et al. 02] suggest
that modeling melting and solidifying (freezing) can be emulated by mak-
ing the viscosity a low constant for temperatures above a transition zone
(centered on the melting point) and a high constant for temperatures below
the transition zone (thus giving near-rigid behavior), and smoothly varying
between the two in the narrow transition zone itself.
∇·
