Graphics Reference
In-Depth Information
Often a power-law is assumed:
n−
1
F
η
=
K
D
,
(8.2)
where
n
= 1 corresponds to a Newtonian fluid,
n>
1 a shear-thickening
fluid where apparent viscosity increases as you try to deform the fluid
faster (e.g., cornstarch suspended in water), and 0
<n<
1ashear-
thinning fluid where apparent viscosity decreases (e.g., paint). Granular
materials such as sand can even be modeled as the limit
n
=0ofshear-
thinning where the magnitude of “viscous” stress depends instead on pres-
sure, not the magnitude of the strain rate, making it more akin to dry
Coulomb friction; see Zhu and Bridson [Zhu and Bridson 05] for more on
this subject.
Getting back to simple Newtonian fluids, the relationship for incom-
pressible flow is
τ
=2
ηD
+
λ
tr(
D
)
δ,
(8.3)
where
η
is the
coecient of dynamic viscosity
. The second term, involv-
ing tr(
D
)=
u
, is of course zero for incompressible flow for any
λ
(which is termed the
second coecient of viscosity
). For compressible flow
λ
is often taken to be
∇·
2
3
η
, though theoretically this is only an ideal-
ization of monatomic gases. However, for an incompressible fluid we are
free to choose
λ
as we please,
6
−
and thus for simplicity's sake we'll set
λ
=0.
Plugging this into the momentum equation, we get
ρ
∇·
η
(
∇u
+
∇u
T
)
.
Du
Dt
+
1
1
ρ
∇p
=
(8.4)
You may notice that this isn't quite the same as our first statement of the
momentum equation, Equation (1.1). It turns out that for the common
case where
η
is constant, the correct equation (8.4) does in fact simplify to
Equation (1.1): but be aware, for simulations with variable viscosity, only
Equation (8.4) is correct. For example, Equation (1.1) doesn't conserve
angular momentum in the variable viscosity case.
2
3
η
, basic thermodynamics are violated, with viscosity
actually accelerating expansion or contraction, increasing the energy of the system. In a
numerical method, where divergence probably isn't exactly zero even for incompressible
flow, problems are bound to arise.
6
This isn't quite true: for
λ<−
