- applied to study certain hydrodynamics problems )
constant
=
ρ
Incompressible
0
=
τ
(as an equation of state)
Constitutive equation
Governing equations :
0
=
⋅
∇
u
d
p
dt
ρ
ρ
=
− ∇
u
f
(4 equations for 4 unknowns :
u
,
p
)
The energy equation is not required for solving
u
and p, and the
constitutive equation for
q
is irrelevant if we do not consider the
problem of heat transfer.
(Euler’s equation)

Fluid Mechanics (Spring 2019) – Chapter 2 - U. Lei (
李雨
)
Newtonian fluid (1)
(compressible and viscous fluid)
Real fluids
are compressible and have certain viscosity.
The simplest model for the constitutive equation of a real fluid
is the Newtonian fluid, which is a special case of the so-called
Stokesian fluid.
The Stokesian fluid satisfies
A Newtonian fluid is a special case of a Stokesian fluid in which
(1)
τ
is a linear function of the components of D, and
(2) there are no preferred direction properties (i.e., isotropy).
The most general linear form is
)
,
,
(
T
p
D
kl
ij
ij
τ
τ
=
kl
ijkl
ij
D
β
τ
=
where
is a fourth order tensor having 81 components, whose values depend
on the two chosen independent thermodynamic variables, say, for example, the
pressure,
p
, and the temperature,
T
.
ijkl
β

Fluid Mechanics (Spring 2019) – Chapter 2 - U. Lei (
李雨
)
Newtonian fluid (2)
ijkl
ij
kl
ik
jl
il
jk
β
αδ δ
βδ δ
γδ δ
=
+
+
where
α
,
β
, and
γ
are functions of thermodynamic state.
Theory of isotropic tensor
kl
jk
il
jl
ik
kl
ij
ij
D
)
(
δ
γδ
δ
βδ
δ
αδ
τ
+
+
=
Then
ji
ij
kk
ij
D
D
D
γ
β
αδ
+
+
=
ij
kk
ij
D
D
)
(
γ
β
αδ
+
+
=
2
ij
ij
D
λδ
μ
≡
∇ ⋅
+
u
ij
ji
D
D
=
(since
)
λ
: second viscosity ,
μ
: dynamic viscosity (to be determined experimentally)
D
u
I
τ
μ
λ
2
+
⋅
∇
=
Vector form

Fluid Mechanics (Spring 2019) – Chapter 2 - U. Lei (
李雨
)
Newtonian fluid (3)
ij
ij
ij
ij
D
p
T
μ
λδ
δ
2
+
⋅
∇
+
−
=
u
D
u
I
I
T
μ
λ
2
+
⋅
∇
+
−
=
p
We have
τ
I
T
+
−
=
p
Recall tensor decomposition :

Fluid Mechanics (Spring 2019) – Chapter 2 - U. Lei (
李雨
)
Newtonian fluid (4)
For isotropic fluid,
j
ij
i
x
T
K
q
∂
∂
=
ij
ij
k
K
δ
−
=
i
i
x
T
k
q
∂
∂
−
=
T
k
∇
−
=
q
Proposed that
q
is linearly proportional to the gradient of the
temperature field according to the experimental observation,
(Fourier’s law)
(
k
: thermal conductivity, a function of the thermodynamic state)
Then
or

Fluid Mechanics (Spring 2019) – Chapter 2 - U. Lei (
李雨
)
Newtonian fluid (5)
Check against the 2nd law of thermodynamics
0
2
≥
Φ
+
∇
⋅
∇
T
T
T
T
k
On substituting the constitutive laws into the entropy equation,
we have
D
D)
u
I
D
τ
τ
D
:
+
⋅
∇
=
:
=
:
≡
Φ
μ
λ
2
(
where the dissipation function :
(
2
(
2
λ
μ
λ
μ
=
∇ ⋅
:
+
:
=
∇ ⋅
+
:
2
u)I
D
D
D
u)
D
D
2
2
2
2
2
2
2
2
2
2
11
22
33
11
12
13
21
22
23
31
32
33
(
)
2
(
)
D
D
D
D
D
D
D
D
D
D
D
D
λ
μ
=
+
+
+
+
+
+
+
+
+
+
+
2
2
2
2
2
2
2
11
22
33
11
22
22
33
33
11
12
23
31
2
2
(
)(
)
[(
)
(
)
(
) ]
4
(
)
3
3
D
D
D
D
D
D
D
D
D
D
D
D
λ
μ
μ
μ
=
+
+
+
+
−
+
−
+
−
+
+
+
0,
k
≥
0,
μ
≥
0
3
2
≥
+
≡
μ
λ
κ
(bulk viscosity)

Fluid Mechanics (Spring 2019) – Chapter 2 - U. Lei (
李雨
)
Example 6 :
Interpretation of
μ
,
the viscosity coefficient
Consider simple shear flow :
(
),
u
u y
=
0,
v
=
0
=
w
The flow is incompressible since

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- Summer '15
- Tseng-Yuan Chen
- Fluid Dynamics, Fluid Mechanics