Continuum mechanics
IV. Thermodynamics
Aleš Janka
office Math 0.107
ales.janka@unifr.ch
http://perso.unifr.ch/ales.janka/mechanics
March 16, 2011, Université de Fribourg
Aleš Janka
III. Conservation laws and equilibria
0. Time-derivative of volume integral in Euler formulation
Scalar field Φ(y, t) (eg. density, concentration)
Volume integral: over current (deformed) domain Ωt :
Z
I=
Φ(y, t)dy
Ωt
Time-derivative of volume integral: (as we saw earlier)
Z Z i
DI
∂Φ
∂
∂Φ
∂Φ
∂v
=
+ i Φ v i dy =
+ vi
+ Φ i dy
i
Dt
∂t
∂y
∂t
∂y
∂y
Ωt
Ωt
Z
=
Ωt
DΦ
∂v i
+ Φ i dy
Dt
∂y
Application for Φ ≡ ρ: mass-conservation (continuity eqn.)
Dρ
∂v i
∂ρ
∂
0=
+ρ i =
+ i (ρv i )
Dt
∂y
∂t
∂y
Aleš Janka
III. Conservation laws and equilibria
1. Thermodynamical variables
Basic state variables:
temperature T : intensive quantity, ie. there is no “specific
temperature” or “temperature per unit mass”,
internal energy U and thermodynamical entropy S:
extensive quantities, one defines specific internal energy and
specific entropy η so that:
Z
Z
ρ η dy
U=
ρ dy
,
S=
Ωt
Ωt
Remark: the physical nature of these quantities is linked to
statistical mechanics beyond the scope of this lecture. We can
consider them as certain averaged characteristics of the particle
nature of the continuum.
Aleš Janka
III. Conservation laws and equilibria
2. First law of thermodynamics: conservation of energy
Theorem: The change of the kinetic and internal energy
δ(Ek + U) of a body Ωt is equal to the work δW of mechanical
forces and heat δQ.
DU
DW
DQ
DEk
+
=
+
Dt
Dt
Dt
Dt
Kinetic energy Ek :
1
Ek =
2
Z
1
ρ |v|2 dy =
2
Ωt
Internal energy U:
Z
Ωt
ρ v i vi dy
Z
U=
Ωt
ρ dy
Volume integrals in Euler formulation → apply formula for
time-derivatives.
Aleš Janka
III. Conservation laws and equilibria
2.1. Time-derivative of kinetic energy Ek
Time-derivative of Ek : (volume integral in Euler formulation):
DEk
Dt
=
1
2
=
1
2
=
1
2
=
1
2
Z
Ωt
Z
1
D(ρ v2 )
dy +
Dt
2
∂v i
ρv
dy
i
∂y
Ωt
Z
Z
Z
Z
D(v2 )
1
ρ
dy +
Dt
2
Ωt
2
Dρ
1
v2
dy +
Dt
2
Ωt
Z
∂v i
ρv
dy
i
∂y
Ωt
2
i
Dρ
∂v
v2
+ ρ i dy
Dt
∂y
Ωt
|
{z
}
= 0 by continuity eqn.
D(v2 )
1
ρ
dy +
Dt
2
Ωt
Z
D(v2 )
ρ
dy
Dt
Ωt
Aleš Janka
III. Conservation laws and equilibria
2.2. Time-derivative of internal energy U
Time-derivative of U: (volume integral in Euler formulation):
DU
Dt
Z
=
=
=
Ωt
D(ρ )
dy +
Dt
Z
Z
Z
Z
D
dy +
ρ
Dt
Ωt
D
ρ
dy +
Dt
Ωt
Z
=
Ωt
ρ
Z
∂v i
ρ i dy
∂y
Ωt
Dρ
dy +
Ωt Dt
Z
∂v i
ρ i dy
∂y
Ωt
Dρ
∂v i
+ ρ i dy
Dt
∂y
Ωt
|
{z
}
= 0 by continuity eqn.
D
dy
Dt
Aleš Janka
III. Conservation laws and equilibria
2.3. Time-derivative of heat
DQ
=−
Dt
Z
Z
q · n dΓ +
∂Ωt
Z
ρ r dy = −
Ωt
q i ni dΓ +
∂Ωt
Z
ρ r dy
Ωt
Here, q = q i gi is the heat flux [J m−2 s −1 ] and r is the specific
heat source intensity [J kg −1 s −1 ].
The minus sign appears because n is the external normal.
By the Divergence theorem:
Z
Z
∂q i
DQ
=−
dy + ρ r dy
Dt
∂y i
Ωt
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Ωt
III. Conservation laws and equilibria
2.4. Time-derivative of mechanical work = power
Power of surfacic traction forces and body-forces:
Z
Z
DW
=
τ ij nj vi dΓ + ρ f i vi dy
Dt
∂Ωt
Ωt
By the Divergence theorem:
Z
Z
∂(τ ij vi )
DW
=
dy + ρ f i vi dy
j
Dt
∂y
Ωt
Ωt
Aleš Janka
III. Conservation laws and equilibria
2. First law of thermodynamics: conservation of energy
DEk
DU
DW
DQ
+
=
+
Dt
Dt
Dt
Dt
For any ωt ⊂ Ωt , subdomain of the continuum:
Z Z D
ρ D(v2 )
∂q i
∂(τ ij vi )
+ρ
dy =
ρr − i +
+ ρ f i vi dy
j
2 Dt
Dt
∂y
∂y
ωt
ωt
Hence (pointwise formulation with worked-out derivatives):
D
∂q i
∂τ ij
Dv i
i
ij ∂vi
+ρ
= ρr − i +
v
+
τ
+
ρ
f
vi
ρ vi
i
Dt
Dt
∂y
∂y j
∂y j
And
D
∂q i
∂vi
ρ
= ρ r − i + τ ij j + vi
Dt
∂y
∂y
|
Aleš Janka
Dv i
∂τ ij
i
−ρ
+
+
ρ
f
Dt
∂y j
{z
}
= 0 by force-equilibrium
III. Conservation laws and equilibria
2. First law of thermodynamics: energy equation
Energy equation: in cartesian coordinates:
∂q i
∂vi
D
= ρ r − i + τ ij j
ρ
Dt
∂y
∂y
In curvilinear coordinates:
ρ
D
= ρ r − ∇i q i + τ ij ∇j vi
Dt
Aleš Janka
III. Conservation laws and equilibria
3. Second law of thermodynamics: entropy
Theorem: The change of total entropy in the body Ωt over time
is greater or equal to the sum of entropy flow over the boundary
∂Ωt from the exterior and the entropy produced by internal heat
sources on Ωt .
Total entropy: units [J/K ], defined up to a constant by
dS =
dQ
T
Clausius-Duhem inequality: mathematical form of the 2nd law:
Z
Z
ρr
1
DS
≥
dy −
q · n dΓ
Dt
T
T
Ωt
∂Ωt
For reversible processes we have the “=” sign,
for irreversible ones we have the “>” sign.
Aleš Janka
III. Conservation laws and equilibria
3.1. Clausius-Duhem inequality
Using the specific entropy η:


Z
Z
Z
D 
1
ρ
r
dy −
q · n dΓ
ρ η dy  ≥
Dt
T
T
Ωt
Ωt
∂Ωt
Time-derivative of an integral formula and the Divergence theorem:
i
Z
Z
Z
Z
Z
Dη
ρr
∂
q
Dρ
∂v i
ρ
dy + η
dy + ρ η i dy −
dy +
dy ≥ 0
Dt
∂y
Dt
T
∂y i T
Ωt
Ωt
Ωt
|
{z
}
= 0 by the continuity eqn.
Ωt
Ωt
Clausius-Duhem inequality in the integral form:
i
Z
Z
Z
Dη
ρr
∂
q
dy −
dy +
dy ≥ 0
ρ
Dt
T
∂y i T
Ωt
Ωt
Ωt
for any subdomain Ωt of the continuum.
Aleš Janka
III. Conservation laws and equilibria
3.1. Clausius-Duhem inequality
Local (pointwise) form of Clausius-Duhem inequality:
i
∂
ρr
q
+ i
≥0
ρ η̇ −
T
∂y
T
In cartesian coordinates
∂q i
q i ∂T
ρ T η̇ − ρ r + i −
≥0
∂y
T ∂y i
In curvilinear coordinates:
qi
ρ T η̇ − ρ r + ∇i q − ∇i T ≥ 0
T
i
Aleš Janka
III. Conservation laws and equilibria
3.2. Clausius-Duhem inequality with Helmholz free energy
Physical meaning of entropy: by itself S does not have any
meaning. However, dS T is the increase of the portion of internal
energy which cannot be used to do work.
Specific Helmholz free energy ϕ: the density of mechanically
exploitable internal energy:
ϕ=−T η
Derive ϕ in time
ϕ̇ = ˙ − Ṫ η − T η̇
and substitute into the Clausius-Duhem inequality.
Aleš Janka
III. Conservation laws and equilibria
3.2. Clausius-Duhem inequality with Helmholz free-energy
Express ∇i q i from the Energy equation:
∇i q i = −ρ ˙ + ρ r + τ ij ∇j vi
Plug it into Clausius-Duhem inequality:
qi
0 ≥ −ρ T η̇ + ρ r − ∇i q + ∇i T
T
i
qi
= −ρ T η̇ + ρ r + ρ ˙ − ρ r − τ ∇j vi + ∇i T
T
qi
ij
= ρ ϕ̇ + ρ Ṫ η − τ ∇j vi + ∇i T
T
Here we used ϕ̇ = ˙ − Ṫ η − T η̇.
ij
Dissipation inequality:
ij
ρ ϕ̇ + ρ Ṫ η − τ ∇j vi
|
{z
}
≡ −δ. . . internal dissipation
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qi
+ ∇i T ≤ 0
T
III. Conservation laws and equilibria