chapter 6. balance laws
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Continuum Mechanics
C. Agelet de SaracibarETS Ingenieros de Caminos, Canales y Puertos, Universidad Politécnica de Cataluña (UPC), Barcelona, Spain
International Center for Numerical Methods in Engineering (CIMNE), Barcelona, Spain
Chapter 6Balance Laws
October 8, 2013 Carlos Agelet de Saracibar 2
Chapter 6 · Balance Laws1. Introduction2. Conservation of mass3. Reynolds transport theorem4. Linear momentum balance5. Angular momentum balance6. Mechanical energy balance7. Assignments8. First law of thermodynamics9. Second law of thermodynamics10. Thermodynamic processes11. Governing equations
Balance Laws > Contents
Contents
October 8, 2013 Carlos Agelet de Saracibar 3
IntroductionThe fundamental laws of continuum mechanics are given by four
conservation/balance laws plus a restriction law.
The four conservation/balance laws are: Conservation of mass · Mass continuity equation Linear momentum balance · Cauchy first motion equation Angular momentum balance · Symmetry of Cauchy stress First law of thermodynamics · Energy balance equation
The restriction law is given by: Second law of thermodynamics · Clausius-Planck and heat
conduction inequalities
Balance Laws > Introduction
Introduction
October 8, 2013 Carlos Agelet de Saracibar 4
IntroductionThe mathematical expressions arising from the fundamental
laws will be given in: Global (or integral) form
o Global (or integral) spatial formo Global (or integral) material form
Local (or strong) formo Local (or strong) spatial formo Local (or strong) material form
Balance Laws > Introduction
Introduction
October 8, 2013 Carlos Agelet de Saracibar 5
Conservation of MassWe assume that during a motion there are neither mass sources
(reservoirs that supply mass), nor mass sinks (reservoirs thatabsorb mass), so the mass of a continuum body is a conserved
quantity.
Then, the mass is independent of the motion and, hence, thematerial time derivative of the mass of a continuum body (or a material volume) has to be zero,
Balance Laws > Conservation of Mass
Conservation of Mass
( ) ( )0 0m mΩ = Ω >
( ) 0d
mdt
Ω =
October 8, 2013 Carlos Agelet de Saracibar 6
Mass DensityThe mass at the material (or reference) configuration may be characterized by a continuous positive scalar field, denoted as
, which is a material property called material (orreference) mass density, such that,
The mass at the spatial (or current) configuration may be characterized by a continuous positive scalar field, denoted as
, which is called spatial (or current) mass density, such that,
Note that, taking t=0 as reference time,
Balance Laws > Conservation of Mass
Conservation of Mass
( ) ( )0 0dm dVρ= >X X
( )0 0 0ρ ρ= >X
( ) ( ), , 0dm t t dvρ= >x x
( ), 0tρ ρ= >x
( ) ( ) ( )0,0 ,0 0.ρ ρ ρ= = >x X X
October 8, 2013 Carlos Agelet de Saracibar 7
Conservation of Mass: Global Material FormThe mass of a continuum body (or a material volume) is a conserved quantity,
Using,
The global material form of the conservation of mass may be written as,
Balance Laws > Conservation of Mass
Conservation of Mass
( ) ( ) ( ) ( )0
0 0 , 0m dV t dv mρ ρΩ Ω
Ω = = = Ω >∫ ∫X x
( ), 0dv J t dV= >X
( ) ( )( ) ( )0 0
0 , , , 0dV t t J t dVρ ρΩ Ω
= >∫ ∫X X Xϕϕϕϕ
October 8, 2013 Carlos Agelet de Saracibar 8
Conservation of Mass: Local Material FormLet us consider the global material form of the conservation of mass given by,
Localizing the integral expression, the local material form of theconservation of mass reads,
Balance Laws > Conservation of Mass
Conservation of Mass
( ) ( )( ) ( )0 0
0 , , , 0dV t t J t dVρ ρΩ Ω
= >∫ ∫X X Xϕϕϕϕ
( ) ( )( ) ( )0 , , , 0t t J tρ ρ= >X X Xϕϕϕϕ
October 8, 2013 Carlos Agelet de Saracibar 9
Conservation of Mass: Global Material FormThe material time derivative of the mass of a continuum body(or a material volume) has to be zero,
Using,
The global material form of the conservation of mass may be written as,
Balance Laws > Conservation of Mass
Conservation of Mass
( ) ( ), 0d d
m t dvdt dt
ρΩ
Ω = =∫ x
( )( ) ( )0
, , , 0d
t t J t dVdt
ρΩ
=∫ X Xϕϕϕϕ
( ), 0dv J t dV= >X
October 8, 2013 Carlos Agelet de Saracibar 10
Conservation of Mass: Local Material FormLet us consider the global material form of the conservation of mass given by,
Localizing the integral expression, the local material form of theconservation of mass reads,
Balance Laws > Conservation of Mass
Conservation of Mass
( )( ) ( )
( )( ) ( )( )
0
0
, , , 0
, , , 0
dt t J t dV
dt
dt t J t dV
dt
ρ
ρ
Ω
Ω
=
=
∫
∫
X X
X X
ϕϕϕϕ
ϕϕϕϕ
( )( ) ( )( ), , , 0d
t t J tdt
ρ =X Xϕϕϕϕ
October 8, 2013 Carlos Agelet de Saracibar 11
Conservation of Mass: Global Spatial FormThe mass of a continuum body (or a material volume) is a conserved quantity,
Using,
The global spatial form of the conservation of mass may be written as,
Balance Laws > Conservation of Mass
Conservation of Mass
( )1 , 0dV J t dv−= >x
( )( ) ( ) ( )1 1
0 , , , 0t J t dv t dvρ ρ− −
Ω Ω= >∫ ∫x x xϕϕϕϕ
( ) ( ) ( ) ( )0
0 0 , 0m dV t dv mρ ρΩ Ω
Ω = = = Ω >∫ ∫X x
October 8, 2013 Carlos Agelet de Saracibar 12
Conservation of Mass: Local Spatial FormLet us consider the global spatial form of the conservation of mass given by,
Localizing the integral expression, the local spatial form of theconservation of mass reads,
Balance Laws > Conservation of Mass
Conservation of Mass
( )( ) ( ) ( )1 1
0 , , , 0t J t dv t dvρ ρ− −
Ω Ω= >∫ ∫x x xϕϕϕϕ
( )( ) ( ) ( )1 1
0 , , , 0t J t tρ ρ− − = >x x xϕϕϕϕ
October 8, 2013 Carlos Agelet de Saracibar 13
Conservation of Mass: Global Spatial FormThe material time derivative of the mass of a continuum body(or a material volume) has to be zero,
The global spatial form of the conservation of mass may be written as,
Balance Laws > Conservation of Mass
Conservation of Mass
( ), 0d
t dvdt
ρΩ
=∫ x
( ) ( ), 0d d
m t dvdt dt
ρΩ
Ω = =∫ x
October 8, 2013 Carlos Agelet de Saracibar 14
Conservation of Mass: Local Spatial FormLet us consider the global spatial form of the conservation of mass given by,
Localizing the integral expression, the local spatial form of theconservation of mass, or mass continuity equation, reads,
Balance Laws > Conservation of Mass
Conservation of Mass
( )
( ) ( )
( ) ( )
0 0 0
0
, 0
div div 0
dt dv
dt
d d ddv JdV J dV J J dV
dt dt dt
JdV dv
ρ
ρ ρ ρ ρ ρ
ρ ρ ρ ρ
Ω
Ω Ω Ω Ω
Ω Ω
=
= = = +
= + = + =
∫
∫ ∫ ∫ ∫
∫ ∫
x
v v
( ) ( ) ( ), , div , 0t t tρ ρ+ =x x v x
October 8, 2013 Carlos Agelet de Saracibar 15
Conservation of Mass: Local Spatial FormThe local spatial form of the conservation of mass, or masscontinuity equation, may be written as,
Using the following expressions,
The local spatial form of the conservation of mass, or masscontinuity equation, may be alternatively written as,
Balance Laws > Conservation of Mass
Conservation of Mass
( )( ) ( )( )
,div , , 0
tt t
t
ρρ
∂+ =
∂
xx v x
( ) ( ) ( )grad , div div gradt
ρρ ρ ρ ρ ρ
∂= + ⋅ = + ⋅
∂v v v v
( ) ( ) ( ), , div , 0t t tρ ρ+ =x x v x
Balance Laws > Conservation of Mass
Conservation of Mass
October 8, 2013 Carlos Agelet de Saracibar 16
Material Time DerivativeGiving the spatial description of an arbitrary property, the material time derivative of the property can be written as,Global and Local Spatial Forms
Global and Local Material Forms
( )
( )
div 0
div 0, div 0
ddv dv
dt
t
ρ ρ ρ
ρρ ρ ρ
Ω Ω= + =
∂+ = + =
∂
∫ ∫ v
v v
( )
( )
0 0
0
0
0, 0
d dJdV J dV
dt dt
dJ J
dt
ρ ρ
ρ ρ ρ
Ω Ω= =
= = >
∫ ∫
October 8, 2013 Carlos Agelet de Saracibar 17
Convective Flux of an Arbitrary PropertyConsider an arbitrary property of a continuum medium and let us denote as the spatial description of the amount of the property per unit of mass, the spatial density field
and the spatial velocity field.The convective flux of the property through a fixed spatial
surface with unit normal , i.e. the amount of the propertycrossing the spatial surface per unit of time due to theconvective flux, is given by,
Balance Laws > Reynolds Transport Theorem
Convective Flux
( ), tv x
( ), tn x
( ), tρ x
A
( ), tψ x
A
( ) ( ) ( ) ( ) ( ), , , ,s
t t t t t dsφ ρ ψ= ⋅∫ x x v x n xA
October 8, 2013 Carlos Agelet de Saracibar 18
Convective Flux of an Arbitrary PropertyThe net outcoming convective flux of the property through a fixed closed spatial surface with unit outward normal , i.e. the net amount of the property leaving the spatial volumeper unit of time due to the convective flux, i.e. outflow (+) plus inflow (-), is given by,
Balance Laws > Reynolds Transport Theorem
Convective Flux
( ), tn x
A
( ) ( ) ( ) ( ) ( ), , , ,v
t t t t t dsφ ρ ψ∂
= ⋅∫ x x v x n xA
inflowoutflow0⋅ ≤v n
0⋅ ≥v n
A
October 8, 2013 Carlos Agelet de Saracibar 19
Mass FluxGiven a spatial density field, denoted as and a spatial
velocity field, denoted as , the mass flux through a fixed
spatial surface with unit normal is given by,
Note that the mass flux may be viewed as a particular case of theconvective flux of an arbitrary property , setting .The net outcoming mass flux through a closed surface with unitoutward normal is given by,
Balance Laws > Reynolds Transport Theorem
Convective Flux
( ), tv x
( ), tn x
( ), tρ x
( ) ( ) ( ) ( ), , ,Ms
t t t t dsφ ρ= ⋅∫ x v x n x
A 1ψ =
( ) ( ) ( ) ( ), , ,Mv
t t t t dsφ ρ∂
= ⋅∫ x v x n x
( ), tn x
October 8, 2013 Carlos Agelet de Saracibar 20
Volume FluxGiven a spatial velocity field, denoted as , the volume flux through a fixed spatial surface with unit normal is givenby,
Note that the volume flux may be viewed as a particular case of the convective flux of an arbitrary property , setting . The net outcoming volume flux through a closed surface withunit outward normal is given by,
Balance Laws > Reynolds Transport Theorem
Convective Flux
( ), tv x
( ), tn x
( ) ( ) ( ), ,Vs
t t t dsφ = ⋅∫ v x n x
A 1ψ ρ −=
( ) ( ) ( ), ,Vv
t t t dsφ∂
= ⋅∫ v x n x
( ), tn x
October 8, 2013 Carlos Agelet de Saracibar 21
Reynolds LemmaConsider an arbitrary property of a continuum medium and let us denote as the spatial description of the amount of the property per unit of mass.The amount of the property at the current time t can be written as,
The material time derivative of the property can be writtenas,
Balance Laws > Reynolds Transport Theorem
Reynolds Transport Theorem
A
( ), tψ x
A
( )t dvρψΩ
= ∫A
A
( )d
t dvdt
ρψΩ
= ∫A
October 8, 2013 Carlos Agelet de Saracibar 22
Reynolds LemmaThe material time derivative of the property may be writtenas,
Balance Laws > Reynolds Transport Theorem
Reynolds Transport Theorem
A
( )
( )
( )
0 0
0 00 0
dt dv
dt
d dJdV J dV
dt dt
ddV dV
dt
dv
ρψ
ρψ ρψ
ρ ψ ρ ψ
ρψ
Ω
Ω Ω
Ω Ω
Ω
=
= =
= =
=
∫
∫ ∫
∫ ∫
∫
A
October 8, 2013 Carlos Agelet de Saracibar 23
Reynolds LemmaThe Reynolds Lemma for an arbitrary property takes theform,
Note that Reynolds lemma may be directly obtained using massconservation, taking into account that,
Balance Laws > Reynolds Transport Theorem
Reynolds Transport Theorem
A
ddv dv
dtρψ ρψ
Ω Ω=∫ ∫
( ) ( ), , 0dm t t dvρ= >x x
October 8, 2013 Carlos Agelet de Saracibar 24
Reynolds Transport TheoremThe following key expression holds,
Balance Laws > Reynolds Transport Theorem
Reynolds Transport Theorem
( )
( ) ( )
( ) ( )
( ) ( )
grad
grad div
div
d
dt
t
t
t
ρψ ρψ ρψ
ρψ ρψ ρψ
ρψ ρψ ρψ
ρψ ρψ
= −
∂= + ⋅ −
∂
∂= + ⋅ +
∂
∂= +
∂
v
v v
v
October 8, 2013 Carlos Agelet de Saracibar 25
Reynolds Transport TheoremThe Reynolds transport theorem for an arbitrary property
may be written as,
Balance Laws > Reynolds Transport Theorem
Reynolds Transport Theorem
A
( )
( ) ( )
( )
div
dt dv
dt
dv
dv dvt
dv dst
ρψ
ρψ
ρψ ρψ
ρψ ρψ
Ω
Ω
Ω Ω
Ω ∂Ω
=
=
∂= +
∂
∂= + ⋅
∂
∫
∫
∫ ∫
∫ ∫
v
v n
A
October 8, 2013 Carlos Agelet de Saracibar 26
Reynolds Transport Theorem: Mass ConservationTaking the mass as a particular case of an arbitrary property, theReynolds transport theorem yields a global spatial form of theconservation of mass and take the form,
Balance Laws > Reynolds Transport Theorem
Reynolds Transport Theorem
( ) ( )
( )
0
div
ddv
dt
dv dvt
dv dst
ρ
ρ ρ
ρ ρ
Ω
Ω Ω
Ω ∂Ω
=
∂= +
∂
∂= + ⋅
∂
∫
∫ ∫
∫ ∫
v
v n
Balance Laws > Reynolds Transport Theorem
Reynolds Transport Theorem
October 8, 2013 Carlos Agelet de Saracibar 27
Material Time DerivativeGiving the spatial description of an arbitrary property, the material time derivative of the property can be written as,Global Spatial Form
Local Spatial Form
( ) ( )
( )
div
ddv dv
dt
dv dvt
dv dst
ρψ ρψ
ρψ ρψ
ρψ ρψ
Ω Ω
Ω Ω
Ω ∂Ω
=
∂= +
∂
∂= + ⋅
∂
∫ ∫
∫ ∫
∫ ∫
v
v n
( ) ( )divt
ρψ ρψ ρψ∂
= +∂
v
October 8, 2013 Carlos Agelet de Saracibar 28
Assignment 6.1 [Classwork]Consider the spatial description of a velocity field given by,
The reference time is t=0. The mass density at the referenceconfiguration is constant. Obtain the spatial mass density and the mass flux through the open cylindrical surface S of the figure.
Balance Laws > Reynolds Transport Theorem
Assignment 6.1
, , 0t
x y zv ye v y v−= = =
A
A
October 8, 2013 Carlos Agelet de Saracibar 36
Balance Laws > Linear Momentum Balance
Linear Momentum Balance
Linear MomentumThe linear momentum of a material volume, denoted as , is defined as a vector-valued function given by,
( )LtM
( ) ( ) ( )
( ) ( )0
0
, ,
,
L t t t dv
t dV
ρ
ρ
Ω
Ω
=
=
∫
∫
M x v x
X V X
October 8, 2013 Carlos Agelet de Saracibar 37
Balance Laws > Linear Momentum Balance
Linear Momentum Balance
Linear MomentumUsing Reynolds Lemma, i.e. conservation of mass, the material
time derivative of the linear momentum takes the form,
( ) ( ) ( ) ( ) ( )
( ) ( ) ( ) ( )0 0
0 0
, , , ,
, ,
L
dt t t dv t t dv
dt
dt dV t dV
dt
ρ ρ
ρ ρ
Ω Ω
Ω Ω
= =
= =
∫ ∫
∫ ∫
M x v x x v x
X V X X V X
October 8, 2013 Carlos Agelet de Saracibar 38
Balance Laws > Linear Momentum Balance
Linear Momentum Balance
Resultant ForceThe resultant force acting on a material volume, denoted as is defined as a vector-valued function given by,
( ) ,tF
( ) ( ) ( ) ( )
( ) ( ) ( )0 0
0
, , ,
, ,
t t t dv t ds
t dV t dS
ρ
ρ
Ω ∂Ω
Ω ∂Ω
= +
= +
∫ ∫
∫ ∫
F x b x t x
X B X T X
October 8, 2013 Carlos Agelet de Saracibar 39
Balance Laws > Linear Momentum Balance
Linear Momentum Balance
Linear Momentum Balance LawThe linear momentum balance law states that the time-
variation of the linear momentum of a material volume is equalto the resultant force acting on that material volume.
If the continuum body is in equilibrium, the resultant force iszero and the linear momentum is a conserved quantity,
( ) ( ) ( )L L
dt t t
dt= =M M F
( ) ( )L L
dt t
dt= =M M 0
October 8, 2013 Carlos Agelet de Saracibar 40
Balance Laws > Linear Momentum Balance
Linear Momentum Balance
Linear Momentum Balance Law: Global Spatial FormThe global spatial form of the linear momentum balance lawcan be written as,
( ) ( ) ( ) ( )
( ) ( ) ( )
, , , ,
, , ,
dt t dv t t dv
dt
t t dv t ds
ρ ρ
ρ
Ω Ω
Ω ∂Ω
=
= +
∫ ∫
∫ ∫
x v x x v x
x b x t x
October 8, 2013 Carlos Agelet de Saracibar 41
Balance Laws > Linear Momentum Balance
Linear Momentum Balance
Linear Momentum Balance Law: Global Spatial FormThe surface forces can be written in spatial form as,
Substituting into the global spatial form yields,
divds ds dv∂Ω ∂Ω Ω
= =∫ ∫ ∫t nσ σσ σσ σσ σ
( ) ( ) ( ) ( )
( ) ( ) ( )
( ) ( ) ( )( )
, , , ,
, , div ,
, , div ,
dt t dv t t dv
dt
t t dv t dv
t t t dv
ρ ρ
ρ
ρ
Ω Ω
Ω Ω
Ω
=
= +
= +
∫ ∫
∫ ∫
∫
x v x x v x
x b x x
x b x x
σσσσ
σσσσ
October 8, 2013 Carlos Agelet de Saracibar 42
Balance Laws > Linear Momentum Balance
Linear Momentum Balance
Linear Momentum Balance Law: Local Spatial FormLocalizing, the local spatial form of the linear momentumbalance law, known as Cauchy’s first equation of motion, can be written as,
If the resultant force is zero, the local spatial form of the linear momentum balance law can be written as,
( ) ( ) ( ) ( ) ( ), , , , div ,t t t t tρ ρ= +x v x x b x x σσσσ
( ) ( ) ( ), , div ,t t tρ= +0 x b x xσσσσ
October 8, 2013 Carlos Agelet de Saracibar 43
Balance Laws > Linear Momentum Balance
Linear Momentum Balance
Linear Momentum Balance Law: Global Material FormThe global material form of the linear momentum balance lawcan be written as,
( ) ( ) ( ) ( )
( ) ( ) ( )
0 0
0 0
0 0
0
, ,
, ,
dt dV t dV
dt
t dV t dS
ρ ρ
ρ
Ω Ω
Ω ∂Ω
=
= +
∫ ∫
∫ ∫
X V X X V X
X B X T X
October 8, 2013 Carlos Agelet de Saracibar 44
Balance Laws > Linear Momentum Balance
Linear Momentum Balance
Linear Momentum Balance Law: Global Material FormThe surface forces can be written in material form as,
Substituting into the global material form yields,
0 0 0
DIVdS dS dV∂Ω ∂Ω Ω
= =∫ ∫ ∫T PN P
( ) ( ) ( ) ( )
( ) ( ) ( )
( ) ( ) ( )( )
0 0
0 0
0
0 0
0
0
, ,
, DIV ,
, DIV ,
dt dV t dV
dt
t dV t dV
t t dV
ρ ρ
ρ
ρ
Ω Ω
Ω Ω
Ω
=
= +
= +
∫ ∫
∫ ∫
∫
X V X X V X
X B X P X
X B X P X
October 8, 2013 Carlos Agelet de Saracibar 45
Balance Laws > Linear Momentum Balance
Linear Momentum Balance
Linear Momentum Balance Law: Local Material FormLocalizing, the local material form of the linear momentumbalance law can be written as,
If the resultant force is zero, the local material form of the linear momentum balance law can be written as,
( ) ( ) ( ) ( ) ( )0 0, , DIV ,t t tρ ρ= +X V X X B X P X
( ) ( ) ( )0 , DIV ,t tρ= +0 X B X P X
Balance Laws > Linear Momentum Balance
Linear Momentum Balance
October 8, 2013 Carlos Agelet de Saracibar 46
Material Time DerivativeGiving the spatial description of an arbitrary property, the material time derivative of the property can be written as,Global and Local Spatial Forms
Global and Local Material Forms
0 0 0 00 0 0
0 0 DIV
ddV dV dV dS
dtρ ρ ρ
ρ ρ
Ω Ω Ω ∂Ω= = +
= +
∫ ∫ ∫ ∫v v b T
v b P
div
ddv dv dv ds
dtρ ρ ρ
ρ ρ
Ω Ω Ω ∂Ω= = +
= +
∫ ∫ ∫ ∫v v b t
v b
σσσσ
October 8, 2013 Carlos Agelet de Saracibar 47
Balance Laws > Angular Momentum Balance
Angular Momentum Balance
Angular MomentumThe angular momentum of a material volume about a fixed
spatial point , denoted as , is defined as a vector-valuedfunction given by,
where the vector position is defined as,
( )AtM
( ) ( ) ( ) ( )
( ) ( ) ( )0
0
, , ,
, ,
A t t t t dv
t t dV
ρ
ρ
Ω
Ω
= ×
= ×
∫
∫
M r x x v x
R X X V X
0x
( ) ( ), ,t t= =r r x R X
( ) ( ) ( )0 0, , ,t t t= = − = − =r r x x x X x R Xϕϕϕϕ
October 8, 2013 Carlos Agelet de Saracibar 48
Balance Laws > Angular Momentum Balance
Angular Momentum Balance
Angular MomentumUsing Reynolds Lemma, i.e. conservation of mass, the material
time derivative of the angular momentum takes the form,
( ) ( ) ( ) ( )
( ) ( ) ( )
( ) ( ) ( )
( ) ( ) ( )
0
0
0
0
, , ,
, , ,
, ,
, ,
A
dt t t t dv
dt
t t t dv
dt t dV
dt
t t dV
ρ
ρ
ρ
ρ
Ω
Ω
Ω
Ω
= ×
= ×
= ×
= ×
∫
∫
∫
∫
M r x x v x
r x x v x
R X X V X
R X X V X
October 8, 2013 Carlos Agelet de Saracibar 49
Balance Laws > Angular Momentum Balance
Angular Momentum Balance
Resultant MomentThe resultant moment about a fixed spatial point , denoted as
is defined as a vector-valued function given by,( ) ,tM
( ) ( ) ( ) ( ) ( ) ( )
( ) ( ) ( ) ( ) ( )0 0
0
, , , , ,
, , , ,
t t t t dv t t ds
t t dV t t dS
ρ
ρ
Ω ∂Ω
Ω ∂Ω
= × + ×
= × + ×
∫ ∫
∫ ∫
M r x x b x r x t x
R X X B X R X T X
0x
October 8, 2013 Carlos Agelet de Saracibar 50
Balance Laws > Angular Momentum Balance
Angular Momentum Balance
Reference Configuration Current Configuration
0ΩΩ
time 0t = time t
1e
2e
3e
1 1,X x
2 2,X x
dA
N
dA
da
n
da
ϕϕϕϕ
X
x
F
0x
0= −r x x
t
T
T
P’
dv
ρb
0ρ B
PdV
0ρ B
0= −r x x
3 3,X x x
October 8, 2013 Carlos Agelet de Saracibar 51
Balance Laws > Angular Momentum Balance
Angular Momentum Balance
Angular Momentum Balance LawThe angular momentum balance law states that the time-
variation of the angular momentum of a material volume abouta fixed spatial point, is equal to the resultant moment about thisfixed spatial point, acting on that material volume.
If the resultant moment about the fixed spatial point is zero, then the angular momentum about this spatial point is a conserved quantity,
( ) ( ) ( )A A
dt t t
dt= =M M M
( ) ( )A A
dt t
dt= =M M 0
October 8, 2013 Carlos Agelet de Saracibar 52
Balance Laws > Angular Momentum Balance
Angular Momentum Balance
Angular Momentum Balance: Global Spatial FormThe global spatial form of the angular momentum balance lawcan be written as,
( ) ( ) ( ) ( ) ( ) ( )
( ) ( ) ( )
( ) ( )
, , , , , ,
, , ,
, ,
dt t t dV t t t dv
dt
t t t dv
t t ds
ρ ρ
ρ
Ω Ω
Ω
∂Ω
× = ×
= ×
+ ×
∫ ∫
∫
∫
r x x v x r x x v x
r x x b x
r x t x
October 8, 2013 Carlos Agelet de Saracibar 53
Balance Laws > Angular Momentum Balance
Angular Momentum Balance
Angular Momentum Balance: Global Spatial FormThe moment of the surface forces can be written in spatial form
as,
div abc cb ads ds dv dvε σ∂Ω ∂Ω Ω Ω
× = × = × +∫ ∫ ∫ ∫r t r n r eσ σσ σσ σσ σ
( )
( )
,
, ,
abc b c a abc b cd d a
abc b cd ad
abc b cd d a abc b d cd a
abc b a abc cb ac
r t ds r n ds
r dv
r dv r dv
r dv dv
ε ε σ
ε σ
ε σ ε σ
ε ε σ
∂Ω ∂Ω
Ω
Ω Ω
Ω Ω
=
=
= +
= ∇⋅ +
∫ ∫
∫
∫ ∫
∫ ∫
e e
e
e e
e eσσσσ
October 8, 2013 Carlos Agelet de Saracibar 54
Balance Laws > Angular Momentum Balance
Angular Momentum Balance
Angular Momentum Balance: Global Spatial FormSubstituting into the global spatial form yields,
( ) ( ) ( ) ( ) ( ) ( )
( ) ( ) ( )
( ) ( )
( )
, , , , , ,
, , ,
, div ,
,abc cb a
dt t t dV t t t dv
dt
t t t dv
t t dv
t dv
ρ ρ
ρ
ε σ
Ω Ω
Ω
Ω
Ω
× = ×
= ×
+ ×
+
∫ ∫
∫
∫
∫
r x x v x r x x v x
r x x b x
r x x
x e
σσσσ
October 8, 2013 Carlos Agelet de Saracibar 55
Balance Laws > Angular Momentum Balance
Angular Momentum Balance
Angular Momentum Balance: Local Spatial FormLocalizing, the local spatial form of the angular momentumbalance law yields,
Using the Cauchy’s first motion equation yields,
and the local spatial form of the angular momentum balance law can be written as,
( ) ( ), ,Tt t=x xσ σσ σσ σσ σ
( )div abc cb aρ ρ ε σ× = × + +r v r b e σσσσ
abc cb aε σ =e 0
October 8, 2013 Carlos Agelet de Saracibar 56
Balance Laws > Angular Momentum Balance
Angular Momentum Balance
Angular Momentum Balance: Global Material FormThe global material form of the angular momentum balance lawcan be written as,
( ) ( ) ( ) ( ) ( ) ( )
( ) ( ) ( )
( ) ( )
0 0
0
0
0 0
0
, , , ,
, ,
, ,
dt t dV t t dV
dt
t t dV
t t dS
ρ ρ
ρ
Ω Ω
Ω
∂Ω
× = ×
= ×
+ ×
∫ ∫
∫
∫
R X X V X R X X V X
R X X B X
R X T X
October 8, 2013 Carlos Agelet de Saracibar 57
Balance Laws > Angular Momentum Balance
Angular Momentum Balance
Angular Momentum Balance: Global Material FormThe moment of the surface forces can be written in material
form as,( )
0 0 0 0
DIVT
abc acbdS dS dV dVε
∂Ω ∂Ω Ω Ω× = × = × +∫ ∫ ∫ ∫r T r PN r P PF e
( )
( )
( )
0 0
0
0 0
0 0
0 0
,
, ,
DIV
abc b c a abc b cD D a
abc b cD aD
abc b cD D a abc b D cD a
abc b a abc bD cD ac
T
abc acb
r T dS r P N dS
r P dV
r P dV r P dV
r dV F P dV
dV dV
ε ε
ε
ε ε
ε ε
ε
∂Ω ∂Ω
Ω
Ω Ω
Ω Ω
Ω Ω
=
=
= +
= ∇⋅ +
= × +
∫ ∫
∫
∫ ∫
∫ ∫
∫ ∫
e e
e
e e
P e e
r P PF e
October 8, 2013 Carlos Agelet de Saracibar 58
Balance Laws > Angular Momentum Balance
Angular Momentum Balance
Angular Momentum Balance: Global Material FormSubstituting into the global material form yields,
( )
0 0
0 0
0
0 0
0 DIV
T
abc acb
ddV dV
dt
dV dV
dV
ρ ρ
ρ
ε
Ω Ω
Ω Ω
Ω
× = ×
= × + ×
+
∫ ∫
∫ ∫
∫
r v r v
r b r P
PF e
October 8, 2013 Carlos Agelet de Saracibar 59
Balance Laws > Angular Momentum Balance
Angular Momentum Balance
Angular Momentum Balance: Local Material FormLocalizing, the local material form of the angular momentumbalance law can be written as,
Using the Cauchy’s first motion equation yields,
and the local material form of the angular momentum balance law can be written as,
( ) ( ) ( ) ( ), , , ,T Tt t t t=P X F X F X P X
( )0 0 DIVT
abc acbρ ρ ε× = × + × +r v r b r P PF e
( )T
abc acbε =PF e 0
Balance Laws > Angular Momentum Balance
Angular Momentum Balance
October 8, 2013 Carlos Agelet de Saracibar 60
Material Time DerivativeGiving the spatial description of an arbitrary property, the material time derivative of the property can be written as,Global and Local Spatial Forms
Global and Local Material Forms
0 0
0 0
0 0
0
T T
ddV dV
dt
dV dS
ρ ρ
ρ
Ω Ω
Ω ∂Ω
× = ×
= × + ×
=
∫ ∫
∫ ∫
r v r v
r b r T
PF FP
T
ddV dv
dt
dv ds
ρ ρ
ρ
Ω Ω
Ω ∂Ω
× = ×
= × + ×
=
∫ ∫
∫ ∫
r v r v
r b r t
σ σσ σσ σσ σ
October 8, 2013 Carlos Agelet de Saracibar 61
Balance Laws > Mechanical Energy Balance
Kinetic Energy
Kinetic EnergyThe global spatial form of the kinetic energy of a continuum body, denoted as , takes the form,
The global material form of the kinetic energy of a continuum body, denoted as , takes the form,
( )tK
( ) ( ) ( ) ( ) ( ) ( )21 1
, , , , ,2 2
t t t dv t t t dvρ ρΩ Ω
= = ⋅∫ ∫x v x x v x v xK
( ) ( ) ( ) ( ) ( ) ( )0 0
2
0 0
1 1, ,
2 2, dV t t dVt tρ ρ
Ω Ω= ⋅= ∫ ∫X X V X V XV XK
( )tK
October 8, 2013 Carlos Agelet de Saracibar 62
Balance Laws > Mechanical Energy Balance
Internal Mechanical Power
Internal Mechanical PowerThe internal mechanical power per unit of spatial volume, is thework done by the stresses per unit of time and unit of spatialvolume, and can be written as,
( ) ( )
( ) ( )
( ) ( ) ( )
( ) ( )
1
1 1 1 1
1 1 1
1
1 1 1 1
1 1 1
: :
: : : :
: : :
T T
T T
ab ab ab ab
T T
aA Ab ab aA Ab ab aA ab bA aA aA
T T
aA AB Bb ab AB Aa ab bB AB AB
J
J J J J
J J J
d J d
J P F d J P F l J P l F J P F
J F S F d J S F d F J S E
σ τ
−
− − − −
− − −
−
− − − −
− − −
=
= = = =
= = =
=
= = = =
= = =
d d
PF d PF l P lF P F
FSF d S F dF S E
σ τσ τσ τσ τ
October 8, 2013 Carlos Agelet de Saracibar 63
Balance Laws > Mechanical Energy Balance
Internal Mechanical Power
Internal Mechanical PowerThe internal mechanical power in the continuum body, denotedas , i.e. work done per unit of time by the stresses, can be written as,
( )
0
0
0
1
1
1
:
: :
: :
: :
int t dv
J dv dV
J dv dV
J dv dV
Ω
−
Ω Ω
−
Ω Ω
−
Ω Ω
=
= =
= =
= =
∫
∫ ∫
∫ ∫
∫ ∫
d
d d
P F P F
S E S E
P σσσσ
τ ττ ττ ττ τ
( )int tP
Balance Laws > Mechanical Energy Balance
Internal Mechanical Power
October 8, 2013 Carlos Agelet de Saracibar 64
Material Time DerivativeGiving the spatial description of an arbitrary property, the material time derivative of the property can be written as,Global Spatial Form
Global Material Forms
( ) :int t dvΩ
= ∫ dσσσσP
( )0
0
0
:
:
:
int t dV
dV
dV
Ω
Ω
Ω
=
=
=
∫
∫
∫
d
P F
S E
ττττP
October 8, 2013 Carlos Agelet de Saracibar 65
Balance Laws > Mechanical Energy Balance
External Mechanical Power
External Mechanical PowerThe external mechanical power, denoted as , is the workdone per unit of time by the body forces and surface forces, and can be written in global spatial and material forms, respectively, as,
( ) ( ) ( ) ( ) ( ) ( )
( ) ( ) ( ) ( ) ( )0 0
0
, , , , ,
, , , ,
extt t t t dv t t ds
t t dV t t dS
ρ
ρ
Ω ∂Ω
Ω ∂Ω
= ⋅ + ⋅
= ⋅ + ⋅
∫ ∫
∫ ∫
x b x v x t x v x
X B X V X T X V X
P
( )exttP
October 8, 2013 Carlos Agelet de Saracibar 66
Balance Laws > Mechanical Energy Balance
Mechanical Energy Balance
Mechanical Energy BalanceThe external mechanical power, denoted as , i.e. workdone per unit of time by the body forces and surface forces, can be written in global spatial form as,
The external mechanical power of the surface forces can be written in global spatial form as,
( )extt dv dsρ
Ω ∂Ω= ⋅ + ⋅∫ ∫b v t vP
( )exttP
( )div div : grad
ds ds
dv dv dv
∂Ω ∂Ω
Ω Ω Ω
⋅ = ⋅
= ⋅ = ⋅ +
∫ ∫
∫ ∫ ∫
t v v n
v v v
σσσσ
σ σ σσ σ σσ σ σσ σ σ
October 8, 2013 Carlos Agelet de Saracibar 67
Balance Laws > Mechanical Energy Balance
Mechanical Energy Balance
Then, using the local spatial form of the linear momentum
balance equation, the external mechanical power can be writtenin global spatial form as,
( )
( )
( ) ( )
2
2
div : grad
div : grad
1: :
2
1:
2
ext
int
t dv ds
dv dv dv
dv dv
d ddv dv dv dv
dt dt
d ddv dv t t
dt dt
ρ
ρ
ρ
ρ ρ
ρ
Ω ∂Ω
Ω Ω Ω
Ω Ω
Ω Ω Ω Ω
Ω Ω
= ⋅ + ⋅
= ⋅ + ⋅ +
= + ⋅ +
= ⋅ + = +
= + = +
∫ ∫
∫ ∫ ∫
∫ ∫
∫ ∫ ∫ ∫
∫ ∫
b v t v
b v v v
b v v
vv l v d
v d
P
K P
σ σσ σσ σσ σ
σ σσ σσ σσ σ
σ σσ σσ σσ σ
σσσσ
October 8, 2013 Carlos Agelet de Saracibar 68
Balance Laws > Mechanical Energy Balance
Mechanical Energy Balance
Mechanical Energy BalanceThe mechanical energy balance states that the external
mechanical power supplied to the continuum body is spent in changing its kinetic energy and doing an internal mechanical
power.
( ) ( ) ( )ext int
dt t t
dt= +P K P
October 8, 2013 Carlos Agelet de Saracibar 69
Balance Laws > Mechanical Energy Balance
Mechanical Energy Balance
Mechanical Energy Balance: Global Spatial FormThe global spatial form of the mechanical energy balance can be written as,
( )
21:
2
ext t dv ds
ddv dv
dt
ρ
ρ
Ω ∂Ω
Ω Ω
= ⋅ + ⋅
= +
∫ ∫
∫ ∫
b v t v
v d
P
σσσσ
October 8, 2013 Carlos Agelet de Saracibar 70
Balance Laws > Mechanical Energy Balance
Mechanical Energy Balance
Mechanical Energy Balance: Global Material FormThe global material form of the mechanical energy balance can be written as,
( )0 0
0 0
0 0
0 0
0
2
0
2
0
2
0
1:
2
1:
2
1:
2
ext t dV dS
ddV dV
dt
ddV dV
dt
ddV dV
dt
ρ
ρ
ρ
ρ
Ω ∂Ω
Ω Ω
Ω Ω
Ω Ω
= ⋅ + ⋅
= +
= +
= +
∫ ∫
∫ ∫
∫ ∫
∫ ∫
b v T v
v d
v P F
v S E
P
ττττ
October 8, 2013 Carlos Agelet de Saracibar 71
Balance Laws > Mechanical Energy Balance
Mechanical Energy Balance
Quasistatic ProblemIf the material time derivative of the kinetic energy is zero (ornegligible), the problem is called quasistatic and the mechanical
energy balance reads,
( ) ( )ext intt t=P P
October 8, 2013 Carlos Agelet de Saracibar 72
Balance Laws > Mechanical Energy Balance
Mechanical Energy Balance
Free Vibration ProblemIf the external mechanical power is zero (or negligible), theproblem is called free vibration problem and the mechanical
energy balance reads,
( ) ( )0 int
dt t
dt= +K P
October 8, 2013 Carlos Agelet de Saracibar 73
Balance Laws > Mechanical Energy Balance
Mechanical Energy Balance
Conservative Mechanical SystemIf both the external and internal mechanical power derive from a potential, i.e. a potential energy for the external loading and a strain energy, such that,
the problem is said to be conservative and the mechanical
energy balance reads,
( ) ( ) ( ) ( ),ext ext int int
d dt t t t
dt dt= − Π = ΠP P
( ) ( ) ( )
( ) ( ) ( )( ) 0
ext int
int ext
d d dt t t
dt dt dt
dt t t
dt
− Π = + Π
+ Π + Π =
K
K
October 8, 2013 Carlos Agelet de Saracibar 74
Balance Laws > Mechanical Energy Balance
Mechanical Energy Balance
Conservative Mechanical SystemThe total potential energy, denoted as , is defined as thesum of the potential energy of the external loading, denoted as
and the strain energy, denoted as , yielding,
In a conservative mechanical system the sum of the kinetic
energy and the total potential energy is a conserved quantity,
( ) ( ) constantt t+ Π =K
( )tΠ
( )ext tΠ ( )int tΠ
( ) ( ) ( )( ) ( ) ( )( ) 0int ext
d dt t t t t
dt dt+ Π + Π = + Π =K K
October 8, 2013 Carlos Agelet de Saracibar 75
Balance Laws > Assignments
Assignment 6.2
Assignment 6.2A volume flux Q of an incompressible fluid flows in stationary
conditions through the pipeline of the figure. Velocity and pressure distributions at the sections A and B are uniform. Thepipeline is fixed through a rigid bar OP. The weights of thepipeline, rigid bar and fluid are neglected.
O
pA vA
pB
R
θ
P
SA
MF
eA
vB
SB
eB
October 8, 2013 Carlos Agelet de Saracibar 76
Balance Laws > Assignments
Assignment 6.2
1) Obtain the velocities at the sections A and B in terms of Q.2) Obtain the reaction force F and moment M at the point O.3) Obtain the values of the angle θ that maximize and minimize
the reaction at the point O.4) Obtain the external power needed to keep the volume flux Q
if the fluid is an incompressible ideal fluid with a sphericalstress state given byen by .p= − 1σσσσ
O
pA vA
pB
R
θ
P
SA
MF
eA
vB
SB
eB
October 8, 2013 Carlos Agelet de Saracibar 77
Balance Laws > Assignments
Assignment 6.2
The local spatial form of the conservation of mass or mass
continuity equation, plus the incompressibility condition yields,
The global spatial form of the conservation of mass for anincompressible medium reads,
Then the velocities at the sections A and B are given by,
div 0, 0 div 0ρ ρ ρ+ = = ⇒ =v v
div
0
A B
V V
S S
A A B B
dV dS
dS dS
v S v S
∂= ⋅
= ⋅ + ⋅
= − + =
∫ ∫
∫ ∫
v v n
v n v n
,A A B B A A B Bv S v S Q v Q S v Q S= = ⇒ = =
October 8, 2013 Carlos Agelet de Saracibar 78
Balance Laws > Assignments
Assignment 6.2
The global spatial form of the linear momentum balance for a stationary motion reads,
/ fV
V
ddV
dt
dVt
ρ
ρ
=
∂=
∂
∫
∫
R v
v ( ) ( )
( ) ( )2 2
2 2
A B
V V
S S
A A A B B B A B
A B
dS dS
dS dS
Q Qv S v S
S S
ρ ρ
ρ ρ
ρ ρ ρ ρ
∂ ∂+ ⋅ = ⋅
= ⋅ + ⋅
= − + = − +
∫ ∫
∫ ∫
v v n v v n
v v n v v n
e e e e
October 8, 2013 Carlos Agelet de Saracibar 79
Balance Laws > Assignments
Assignment 6.2
The resultant force acting on the volume V of fluid, taking intoaccount that the weight of the fluid is negligible, reads,
/ fV
dVρ= ∫R b
/
/
2 2
A B
A B
V V
wall S S
wall f p p
wall f A A A B B B
A B
A B
dS dS
dS dS dS
p S p S
Q Q
S Sρ ρ
∂ ∂+ =
= + +
= + +
= + −
= − +
∫ ∫
∫ ∫ ∫
t t
t t t
R R R
R e e
e e
October 8, 2013 Carlos Agelet de Saracibar 80
Balance Laws > Assignments
Assignment 6.2
The resultant force of the wall of the pipeline acting on the
volume V of fluid, reads,
Using the action-reaction principle, the resultant force of the
fluid acting on the wall of the pipeline, reads,
/ /
2 2
wall f f A A A B B B
A A A B B B
A B
p S p S
Q Qp S p S
S Sρ ρ
= − +
= − + + +
R R e e
e e
/ /
2 2
f wall wall f
A A A B B B
A B
Q Qp S p S
S Sρ ρ
= −
= + − +
R R
e e
October 8, 2013 Carlos Agelet de Saracibar 81
Balance Laws > Assignments
Assignment 6.2
The global spatial form of the angular momentum balance aboutthe point O, for a stationary motion, reads,
/
O
fV
V
ddV
dt
dVt
ρ
ρ
= ×
∂= ×
∂
∫
∫
M r v
r v ( ) ( )
( )A
V V
S
dS dS
dS
ρ ρ
ρ
∂ ∂+ × ⋅ = × ⋅
= × ⋅
∫ ∫
∫
r v v n r v v n
r v v n ( )
22
BS
B B z z
B
dS
Qv S R R
S
ρ
ρ ρ
+ × ⋅
= − = −
∫ r v v n
e e
October 8, 2013 Carlos Agelet de Saracibar 82
Balance Laws > Assignments
Assignment 6.2
The resultant moment about the point O acting on the volume V of fluid, taking into account that the weight of the fluid is
negligible, reads,
/
O
fV
dVρ= ×∫M r b
/
/
2
A B
A B
V V
wall S S
O O O
wall f p p
O
wall f B B z
z
B
dS dS
dS dS dS
p S R
QR
Sρ
∂ ∂+ × = ×
= × + × + ×
= + +
= +
= −
∫ ∫
∫ ∫ ∫
r t r t
r t r t r t
M M M
M e
e
October 8, 2013 Carlos Agelet de Saracibar 83
Balance Laws > Assignments
Assignment 6.2
The resultant moment of the wall of the pipeline acting on the
volume V of fluid about the point O, reads,
Using the action-reaction principle, the resultant moment of the
fluid acting on the wall of the pipeline about the point O, reads,
2
/ /
O O
wall f f B B z B B z
B
Qp S R p S R
Sρ
= − = − +
M M e e
/ /
2
O O
f wall wall f
B B z
B
Qp S R
Sρ
= −
= +
M M
e
October 8, 2013 Carlos Agelet de Saracibar 84
Balance Laws > Assignments
Assignment 6.2
Equilibrium of forces and moments at point O
aF
2
/
O
f wall B B
B
QM p S R
Sρ
= +
2
, /a f wall A A
A
QR p S
Sρ
= +
2
, /b f wall B B
B
QR p S
Sρ
= +
bF
M
O
October 8, 2013 Carlos Agelet de Saracibar 85
Balance Laws > Assignments
Assignment 6.2
The equilibrium of forces and moments about the point O on the
pipeline, taking into account that the weight of the pipeline and
the rigid bar are negligible, reads,
The reaction force F and moment M, at the point O, read,
/f wall+R W
/
O O
f wall W
+ =
+
F 0
M M + =M 0
( ) ( )2 2
/
2
/
f wall A A A A B B B B
O
f wall B B z
B
Q S p S Q S p S
Qp S R
S
ρ ρ
ρ
= − = − + + +
= − = − +
F R e e
M M e
October 8, 2013 Carlos Agelet de Saracibar 86
Balance Laws > Assignments
Assignment 6.2
The reactions force F at the point O, reads,
The norm of the reaction force F will take a maximum for anangle such that,
The norm of the reaction force F will take a minimum for anangle such that,
( ) ( )2 2
0 0
A A A A B B B BQ S p S Q S p Sρ ρ
> >
= − + + +F e e
3
2B A
πθ= − ⇒ =e e
2B A
πθ= ⇒ =e e
October 8, 2013 Carlos Agelet de Saracibar 87
Balance Laws > Assignments
Assignment 6.2
The external mechanical power needed to keep the volume flux Q, taking into account that the fluid is incompressible, stationary
and the stress state is spherical, is given by,
2
2
1:
2
1
2
ext intV V
V
d ddV dV
dt dt
dVt
ρ
ρ
= + = +
∂=
∂
∫ ∫
∫
v d
v
P K P σσσσ
21tr
2V VdS p dVρ
∂+ ⋅ + −∫ ∫v v n d
2 2
3
2 2
1 1
2 2
1 1 1
2
A BS S
B A
dS dS
QS S
ρ ρ
ρ
= ⋅ + ⋅
= −
∫ ∫v v n v v n
October 8, 2013 Carlos Agelet de Saracibar 88
Balance Laws > Assignments
Assignment 6.3
h
h
p = patm ≈ 0
C h/2
h/2v2v1
A
E
B
D
F
x
z
y
Assignment 6.3 [Classwork]An incompressible fluid flows in stationary conditions throughthe pipeline of the figure. Velocities and pressures distributionsare uniform at the sections AE and CD. Pressure on the walls isassumed to be uniform. There is a basculant barrier BC with a hinge on B. An horizontal force F, acting on the point C, iskeeping the barrier in vertical position. Body forces in the fluid are neglected. The weight of the barrier is also neglected.
h
h
p = patm ≈ 0
C h/2
h/2v2v1
A
E
B
D
F
x
z
y
October 8, 2013 Carlos Agelet de Saracibar 89
Balance Laws > Assignments
Assignment 6.3
1) Obtain the velocity v2 at the section CD in terms of thevelocity v1 at the section AE.
2) Obtain the resultant force and moment acting on the fluid at the point B.
3) Obtain the resultant force and moment of the fluid on the
barrier at the point B.4) Obtain the force F and the reaction at the point B.5) Obtain the external mechanical power needed, assuming the
stress tensor in the fluid is spherical,given by .p= − 1σσσσ
October 8, 2013 Carlos Agelet de Saracibar 104
Balance Laws > Assignments
Assignment 6.4
Assignment 6.4 [Homework]The figure shows the longitudinal section of a pump with a valveOA of weight W per unit of width (normal to the plane of thefigure). There is a hinge on O. The velocity of the pump is V.
v1
αaP2
b a
a/2 a/2
W
V
P1
v2
P2
B
A
O
B
A
Detail
O
αa
P1
The fluid is incompressibleand the motion stationary.Uniform pressure distribu-tions on the sections OB and AB are p1 and p2=0, respectively. Velocity dis-tributions at the sectionsOB and AB are uniform. Body forces in the fluid are negligible.
October 8, 2013 Carlos Agelet de Saracibar 105
Balance Laws > Assignments
Assignment 6.4
Assignment 6.4 [Homework]1) Obtain the uniform velocities v1 and v2 at the sections OB
and AB, respectively, in terms of the velocity v of thepumping tool
2) Obtain the resultant of the forces per unit of width given bythe fluid on the valve OA
3) Obtain the moment per unit of width at the point O of theforces given by the fluid on the valve OA
4) Obtain the weight W of the valve OA per unit of width. Environmental pressure p2 is neglected.
October 8, 2013 Carlos Agelet de Saracibar 119
Balance Laws > First Law of Thermodynamics
External Thermal Power
External Thermal Power: Global Spatial FormThe external thermal power is defined as the amount of heatthat enters into a material volume per unit of time. We will assume that heat may enter into a material volume dueto: Internal heat sources characterized by a scalar-valued
function, given in spatial description by , representingthe heat source per unit of mass.
Non-convective heat flux through the boundary, characterizedby a vector-valued function, given in spatial description by
, representing the non-convective outward heat flux
per unit of spatial surface.
( ),r tx
( ), tq x
October 8, 2013 Carlos Agelet de Saracibar 120
Balance Laws > First Law of Thermodynamics
External Thermal Power
External Thermal Power: Global Spatial FormThe global spatial form of the external thermal power is definedas,
where is the heat source per unit of mass and isthe non-convective outward heat flux per unit of spatial surface, both of them given in spatial description.
( ) ( ) ( ) ( )
( ) ( ) ( )
, , ,
, , div ,
extQ t t r t dv t ds
t r t dv t dv
ρ
ρ
Ω ∂Ω
Ω Ω
= − ⋅
= −
∫ ∫
∫ ∫
x x q x n
x x q x
( ),r tx ( ), tq x
October 8, 2013 Carlos Agelet de Saracibar 121
Balance Laws > First Law of Thermodynamics
External Thermal Power
External Thermal Power: Global Material FormUsing the conservation of mass yields,
Using Nanson’s formula, the heat flux through the spatialboundary of the continuum body may be written as,
where is the nominal heat flux, i.e. heat flux per unit of material surface, given by,
0 0
:Tds J dS dS−
∂Ω ∂Ω ∂Ω⋅ = ⋅ = ⋅∫ ∫ ∫q n q F N Q N
Q
1 1, A Aa aJ Q JF q− −= =Q F q
( ) ( ) ( ) ( )0
0, , ,t r t dv R t dVρ ρΩ Ω
=∫ ∫x x X X
October 8, 2013 Carlos Agelet de Saracibar 122
Balance Laws > First Law of Thermodynamics
External Thermal Power
External Thermal Power: Global Material FormThe global material form of the external thermal power takesthe form,
where is the heat source per unit of mass and isthe nominal heat flux or heat flux per unit of material surface, both of them given in material description.
( ) ( ) ( ) ( )
( ) ( ) ( )0 0
0 0
0
0
, ,
, DIV ,
extQ t R t dV t dS
R t dV t dV
ρ
ρ
Ω ∂Ω
Ω Ω
= − ⋅
= −
∫ ∫
∫ ∫
X X Q X N
X X Q X
( ),R tX ( ), tQ X
October 8, 2013 Carlos Agelet de Saracibar 123
Balance Laws > First Law of Thermodynamics
Total External Power
Total External Power: Global Spatial FormThe global material form of the total external power, i.e. mechanical plus thermal external power, can be written as,
( ) ( )
21:
2
ext extt Q t dv ds
r dv ds
ddv dv
dt
rdv ds
ρ
ρ
ρ
ρ
Ω ∂Ω
Ω ∂Ω
Ω Ω
Ω ∂Ω
+ = ⋅ + ⋅
+ − ⋅
= +
+ − ⋅
∫ ∫
∫ ∫
∫ ∫
∫ ∫
b v t v
q n
v d
q n
P
σσσσ
October 8, 2013 Carlos Agelet de Saracibar 124
Balance Laws > First Law of Thermodynamics
Total External Power
Total External Power: Global Material Form (I)The global material form of the total thermal power, i.e. mechanical plus thermal external power, can be written as,
( ) ( )0 0
0 0
0 0
0 0
0
0
2
0
0
1:
2
ext extt Q t dV dS
r dV dS
ddV dV
dt
rdV dS
ρ
ρ
ρ
ρ
Ω ∂Ω
Ω ∂Ω
Ω Ω
Ω ∂Ω
+ = ⋅ + ⋅
+ − ⋅
= +
+ − ⋅
∫ ∫
∫ ∫
∫ ∫
∫ ∫
b v T v
Q N
v d
Q N
P
ττττ
October 8, 2013 Carlos Agelet de Saracibar 125
Balance Laws > First Law of Thermodynamics
Total External Power
Total External Power: Global Material Form (II)The global material form of the total thermal power, i.e. mechanical plus thermal external power, can be written as,
( ) ( )0 0
0 0
0 0
0 0
0
0
2
0
0
1:
2
ext extt Q t dV dS
r dV dS
ddV dV
dt
rdV dS
ρ
ρ
ρ
ρ
Ω ∂Ω
Ω ∂Ω
Ω Ω
Ω ∂Ω
+ = ⋅ + ⋅
+ − ⋅
= +
+ − ⋅
∫ ∫
∫ ∫
∫ ∫
∫ ∫
b v T v
Q N
v P F
Q N
P
October 8, 2013 Carlos Agelet de Saracibar 126
Balance Laws > First Law of Thermodynamics
Total External Power
Total External Power: Global Material Form (III)The global material form of the total thermal power, i.e. mechanical plus thermal external power, can be written as,
( ) ( )0 0
0 0
0 0
0 0
0
0
2
0
0
1:
2
ext extt Q t dV dS
r dV dS
ddV dV
dt
rdV dS
ρ
ρ
ρ
ρ
Ω ∂Ω
Ω ∂Ω
Ω Ω
Ω ∂Ω
+ = ⋅ + ⋅
+ − ⋅
= +
+ − ⋅
∫ ∫
∫ ∫
∫ ∫
∫ ∫
b v T v
Q N
v S E
Q N
P
October 8, 2013 Carlos Agelet de Saracibar 127
Balance Laws > First Law of Thermodynamics
First Law of Thermodynamics
First Law of ThermodynamicsFirst Postulate. There exist a state function called total energy, denoted as , such that its material time derivative is equalto the total external power supplied to the system, i.e. theexternal mechanical plus thermal power,
( )tE
( ) ( ) ( ): ext ext
dt P t Q t
dt= +E
October 8, 2013 Carlos Agelet de Saracibar 128
Balance Laws > First Law of Thermodynamics
First Law of Thermodynamics
First Law of ThermodynamicsSecond Postulate. There exist a state function called internal
energy, denoted as , which is an extensive property, i.e. there exist a specific internal energy or internal energy per unit
of mass, denoted as , such that,
( )tU
( ) ( ), ,e e t E t= =x X
( ) ( ) ( ) ( ) ( )0
0: , , ,t t e t dv E t dVρ ρΩ Ω
= =∫ ∫x x X XU
October 8, 2013 Carlos Agelet de Saracibar 129
Balance Laws > First Law of Thermodynamics
First Law of Thermodynamics
First Law of ThermodynamicsTher first law of thermodynamics states that the material time derivative of the total energy is equal to sum of the material time derivative of the kinetic energy and the material time derivative of the internal energy,
( ) ( ) ( )d d d
t t tdt dt dt
= +E UK
October 8, 2013 Carlos Agelet de Saracibar 130
Balance Laws > First Law of Thermodynamics
First Law of Thermodynamics
Using the first postulate and the mechanical energy balance, thefirst law of thermodynamics reads,
yielding the following internal energy balance law,
( ) ( ) ( )
( ) ( )
( ) ( ) ( )
ext ext
int ext
d d dt t t
dt dt dt
P t Q t
dt t Q t
dt
= +
= +
= + +
E UK
K P
( ) ( ) ( )int ext
dt t Q t
dt= +U P
October 8, 2013 Carlos Agelet de Saracibar 131
Balance Laws > First Law of Thermodynamics
Energy Balance
Energy Balance Law: Global Spatial FormThe internal energy balance law in global spatial form can be written as,
( ) ( ) ( )
:
int ext
dt t Q t
dt
dedv edv
dt
dv r dv ds
ρ ρ
ρ
Ω Ω
Ω Ω ∂Ω
= +
=
= + − ⋅
∫ ∫
∫ ∫ ∫d q n
σσσσ
U P
October 8, 2013 Carlos Agelet de Saracibar 132
Balance Laws > First Law of Thermodynamics
Energy Balance
Energy Balance Law: Local Spatial FormUsing the divergence theorem, the internal energy balance lawin global spatial form can be written as,
Localizing, the local spatial form of the energy balance lawreads,
:
: div
dedv edv
dt
dv r dv ds
dv r dv dv
ρ ρ
ρ
ρ
Ω Ω
Ω Ω ∂Ω
Ω Ω Ω
=
= + − ⋅
= + −
∫ ∫
∫ ∫ ∫
∫ ∫ ∫
d q n
d q
σσσσ
σσσσ
: dive rρ ρ= + −d q σσσσ
October 8, 2013 Carlos Agelet de Saracibar 133
Balance Laws > First Law of Thermodynamics
Energy Balance
Energy Balance Law: Global Material FormThe internal energy balance law in global material form can be written as,
( ) ( ) ( )
0 0
0 0 0
0 0 0
0 0 0
0 0
0
0
0
:
:
:
int ext
dt t Q t
dt
dedV edV
dt
dV r dV dS
dV r dV dS
dV r dV dS
ρ ρ
ρ
ρ
ρ
Ω Ω
Ω Ω ∂Ω
Ω Ω ∂Ω
Ω Ω ∂Ω
= +
=
= + − ⋅
= + − ⋅
= + − ⋅
∫ ∫
∫ ∫ ∫
∫ ∫ ∫
∫ ∫ ∫
d Q N
P F Q N
S E Q N
ττττ
U P
October 8, 2013 Carlos Agelet de Saracibar 134
Balance Laws > First Law of Thermodynamics
Energy Balance
Energy Balance Law: Global Material FormUsing the divergence theorem, the internal energy balance lawin global material form can be written as,
0 0
0 0 0
0 0 0
0 0 0
0 0
0
0
0
: DIV
: DIV
: DIV
dedV edV
dt
dV r dV dV
dV r dV dV
dV r dV dV
ρ ρ
ρ
ρ
ρ
Ω Ω
Ω Ω Ω
Ω Ω Ω
Ω Ω Ω
=
= + −
= + −
= + −
∫ ∫
∫ ∫ ∫
∫ ∫ ∫
∫ ∫ ∫
d Q
P F Q
S E Q
ττττ
October 8, 2013 Carlos Agelet de Saracibar 135
Balance Laws > First Law of Thermodynamics
Energy Balance
Energy Balance Law: Local Material FormLocalizing, the internal energy balance law in local material form
can be written as,
Note that the local material form of the energy balance equationcould have been also obtained from the local spatial form using,
0 0
0
0
: DIV
: DIV
: DIV
e r
r
r
ρ ρ
ρ
ρ
= + −
= + −
= + −
d Q
P F Q
S E Q
ττττ
0 , DIV div , : : : :J J Jρ ρ= = = = =Q q d P F S E d τ στ στ στ σ
Balance Laws > First Law of Thermodynamics
Energy Balance
October 8, 2013 Carlos Agelet de Saracibar 136
Material Time DerivativeGiving the spatial description of an arbitrary property, the material time derivative of the property can be written as,Local Spatial Form
Local Material Forms
0 0
0
0
: DIV
: DIV
: DIV
e r
r
r
ρ ρ
ρ
ρ
= + −
= + −
= + −
d Q
P F Q
S E Q
ττττ
: dive rρ ρ= + −d q σσσσ
October 8, 2013 Carlos Agelet de Saracibar 137
Balance Laws > Second Law of Thermodynamics
Second Law of Thermodynamics
Second Law of ThermodynamicsFirst Postulate. There exist a state function called absolute
temperature, denoted as , which is always a positive scalar-valued function.
Second Postulate. There exist a state function called entropy, denoted as , which is an extensive property, i.e. there exista specific entropy or entropy per unit of mass, denoted as
, such that,
( ) ( ), ,t tθ θ= = Θx X
( )tH
( ) ( ), ,t tη η= = Ξx X
( ) ( ) ( ) ( ) ( )0
0, , ,t t t dv t dVρ η ρΩ Ω
= = Ξ∫ ∫x x X XH
( ) ( ), , 0t tθ θ= = Θ >x X
October 8, 2013 Carlos Agelet de Saracibar 138
Balance Laws > Second Law of Thermodynamics
Second Law of Thermodynamics
Second Law of ThermodynamicsThe second law of thermodynamics states that any admissible
thermodynamic process has to satisfy the following inequality, written in global spatial form as,
or in global material form as,
( )1 1d
t rdv dsdt
ρθ θΩ ∂Ω
≥ − ⋅∫ ∫ q nH
( )0 0
0
1 1dt rdV dS
dtρ
θ θΩ ∂Ω≥ − ⋅∫ ∫ Q NH
October 8, 2013 Carlos Agelet de Saracibar 139
Balance Laws > Second Law of Thermodynamics
Second Law of Thermodynamics
Second Law of ThermodynamicsAdmissible thermodynamic processes can be classified as reversible and irreversible processes.A thermodynamic process is said to be reversible if the followingcondition holds,
A thermodynamic process is said to be irreversible if thefollowing condition holds,
( )0 0
0
1 1 1 1dt rdv ds rdV dS
dtρ ρ
θ θ θ θΩ ∂Ω Ω ∂Ω> − ⋅ = − ⋅∫ ∫ ∫ ∫q n Q NH
( )0 0
0
1 1 1 1dt rdv ds rdV dS
dtρ ρ
θ θ θ θΩ ∂Ω Ω ∂Ω= − ⋅ = − ⋅∫ ∫ ∫ ∫q n Q NH
October 8, 2013 Carlos Agelet de Saracibar 140
Balance Laws > Second Law of Thermodynamics
Second Law of Thermodynamics
Second Law of Thermodynamics: Global Spatial FormThe global spatial form of the second law of thermodynamicscan be written as,
( )
1 1
d dt dv dv
dt dt
rdv ds
ρη ρη
ρθ θ
Ω Ω
Ω ∂Ω
= =
≥ − ⋅
∫ ∫
∫ ∫ q n
H
October 8, 2013 Carlos Agelet de Saracibar 141
Balance Laws > Second Law of Thermodynamics
Second Law of Thermodynamics
Second Law of Thermodynamics: Global Spatial FormUsing the divergence theorem, the global spatial form of thesecond law of thermodynamics can be written as,
2
1 1
1 1div
1 1 1div grad
ddv dv
dt
rdv ds
rdv dv
rdv dv dv
ρη ρη
ρθ θ
ρθ θ
ρ θθ θ θ
Ω Ω
Ω ∂Ω
Ω Ω
Ω Ω Ω
=
≥ − ⋅
= −
= − + ⋅
∫ ∫
∫ ∫
∫ ∫
∫ ∫ ∫
q n
q
q q
October 8, 2013 Carlos Agelet de Saracibar 142
Balance Laws > Second Law of Thermodynamics
Second Law of Thermodynamics
Second Law of Thermodynamics: Local Spatial FormLocalizing, the local spatial form of the second law of thermodynamics can be written as,
Multiplying by the absolute temperature yields,
2
1 1 1div gradrρη ρ θ
θ θ θ≥ − + ⋅q q
1div gradrρθη ρ θ
θ≥ − + ⋅q q
October 8, 2013 Carlos Agelet de Saracibar 143
Balance Laws > Second Law of Thermodynamics
Second Law of Thermodynamics
Clausius-Duhem Inequality: Local Spatial FormThe local spatial form of the Clausius-Duhem inequality statesthat the dissipation rate per unit of spatial volume is a non-negative scalar-valued quantity and it can be written as,
A stronger assumption is usually introduced, yielding,
1: div grad 0rρθη ρ θ
θ= − + − ⋅ ≥q qD
00
1: div grad 0
1: div 0, : grad 0
int
cond
int cond
int cond
r
r
ρθη ρ θθ
ρθη ρ θθ
≥≥
= + = − + − ⋅ ≥
= − + ≥ = − ⋅ ≥
q q
q q
DD
D D D
D D
October 8, 2013 Carlos Agelet de Saracibar 144
Balance Laws > Second Law of Thermodynamics
Second Law of Thermodynamics
Heat Conduction Inequality: Local Spatial FormThe local spatial form of the heat conduction inequality statesthat the projection of the heat flux per unit of spatial surface onthe direction of the spatial gradient of the temperature is a non-
positive scalar-valued quantity, i.e. heat flux takes place from thehot to the cold and not the other way around,
1: grad 0 grad 0
condθ θ
θ= − ⋅ ≥ ⇒ ⋅ ≤q qD
October 8, 2013 Carlos Agelet de Saracibar 145
Balance Laws > Second Law of Thermodynamics
Second Law of Thermodynamics
Heat Conduction Inequality: Local Spatial FormUsing Fourier law for heat conduction for an isotropic continuum medium, the second law of thermodynamics yields the followingrestriction on the admissible values of the spatial thermal
conductivity parameter,
1: grad 0 grad grad 0 0
condk kθ θ θ
θ= − ⋅ ≥ ⇒ ⋅ ≥ ⇒ ≥qD
October 8, 2013 Carlos Agelet de Saracibar 146
Balance Laws > Second Law of Thermodynamics
Second Law of Thermodynamics
Clausius-Planck Inequality: Local Spatial Form (I)The local spatial form of the Clausius-Planck inequality statesthat the internal dissipation rate per unit of spatial volume is a non-negative scalar-valued quantity and it can be written as,
The local spatial form of the Clausius-Planck inequality for a reversible process takes the form,
The local spatial form Clausius-Planck inequality for anirreversible process takes the form,
: div 0int
rρθη ρ= − + ≥qD
: div 0int
rρθη ρ= − + =qD
: div 0int rρθη ρ= − + >qD
October 8, 2013 Carlos Agelet de Saracibar 147
Balance Laws > Second Law of Thermodynamics
Second Law of Thermodynamics
Clausius-Planck Inequality: Local Spatial Form (II)Using the local spatial form of the internal energy balance
equation and the Clausius-Planck inequality given by,
the local spatial form of the Clausius-Planck inequality can be written in terms of the internal energy per unit of mass as,
: div
: div 0int
e r
r
ρ ρ
ρθη ρ
= + −
= − + ≥
d q
q
D
σσσσ
( ): : 0int
eρ θη= − − ≥d D σσσσ
div :r eρ ρ− = −q d σσσσ
October 8, 2013 Carlos Agelet de Saracibar 148
Balance Laws > Second Law of Thermodynamics
Second Law of Thermodynamics
Clausius-Planck Inequality: Local Spatial Form (III)Introducing the free energy per unit of mass, denoted as , defined as,
the local spatial form of the Clausius-Planck inequality can be written in terms of the free energy per unit of mass as,
( ): : 0int ρ ψ ηθ= − + ≥d D σσσσ
: eψ θη= −
ψ
October 8, 2013 Carlos Agelet de Saracibar 149
Balance Laws > Second Law of Thermodynamics
Second Law of Thermodynamics
Second Law of Thermodynamics: Global Material FormThe global material form of the second law of thermodynamicsstates that,
( )0
0 0
0 0
0
1 1
d dt dV dV
dt dt
r dV dS
ρ η ρ η
ρθ θ
Ω Ω
Ω ∂Ω
= =
≥ − ⋅
∫ ∫
∫ ∫ Q N
H
October 8, 2013 Carlos Agelet de Saracibar 150
Balance Laws > Second Law of Thermodynamics
Second Law of Thermodynamics
Second Law of Thermodynamics: Global Material FormUsing the divergence theorem, the global material form of thesecond law of thermodynamics can be written as,
0
0 0
0 0
0 0 0
0 0
0
0
0 2
1 1
1 1DIV
1 1 1DIV GRAD
ddV dV
dt
rdV dS
rdV dV
rdV dV dV
ρ η ρ η
ρθ θ
ρθ θ
ρ θθ θ θ
Ω Ω
Ω ∂Ω
Ω Ω
Ω Ω Ω
=
≥ − ⋅
= −
= − + ⋅
∫ ∫
∫ ∫
∫ ∫
∫ ∫ ∫
Q N
Q
Q Q
October 8, 2013 Carlos Agelet de Saracibar 151
Balance Laws > Second Law of Thermodynamics
Second Law of Thermodynamics
Second Law of Thermodynamics: Local Material FormLocalizing, the local material form of the second law of thermodynamics can be written as,
Multiplying by the absolute temperature yields,
0 0 2
1 1 1DIV GRADrρ η ρ θ
θ θ θ≥ − + ⋅Q Q
0 0
1DIV GRADrρ θη ρ θ
θ≥ − + ⋅Q Q
October 8, 2013 Carlos Agelet de Saracibar 152
Balance Laws > Second Law of Thermodynamics
Second Law of Thermodynamics
Clausius-Duhem Inequality: Local Material FormThe local material form of the Clausius-Duhem inequality statesthat the dissipation rate per unit of material volume is a non-negative scalar-valued quantity and it can be written as,
A stronger assumption is usually considered, yielding,
0 0 0
1: DIV GRAD 0rρ θη ρ θ
θ= − + − ⋅ ≥Q QD
0
0
0 0 0 0 0
00
0 0 0 0
1: DIV GRAD 0
1: DIV 0, : GRAD 0
int cond
int
cond
int cond
r
r
ρ θη ρ θθ
ρ θη ρ θθ
≥≥
= + = − + − ⋅ ≥
= − + ≥ = − ⋅ ≥
Q Q
Q Q
DD
D D D
D D
October 8, 2013 Carlos Agelet de Saracibar 153
Balance Laws > Second Law of Thermodynamics
Second Law of Thermodynamics
Clausius-Planck Inequality: Local Material Form (I)The local material form of the Clausius-Planck inequality statesthat the internal dissipation rate per unit of material volume is a non-negative scalar-valued quantity and it can be written as,
The local material form Clausius-Planck inequality for a reversible process takes the form,
The local material form Clausius-Planck inequality for anirreversible process takes the form,
0 0 0: DIV 0int
rρ θη ρ= − + ≥QD
0 0 0: DIV 0int
rρ θη ρ= − + =QD
0 0 0: DIV 0int
rρ θη ρ= − + >QD
October 8, 2013 Carlos Agelet de Saracibar 154
Balance Laws > Second Law of Thermodynamics
Second Law of Thermodynamics
Clausius-Planck Inequality: Local Material Form (II)Using the local material form of the internal energy balance
equation and the Clausius-Planck inequality yields,
0 0
0
0
0 0 0
: DIV
: DIV
: DIV
: DIV 0int
e r
r
r
r
ρ ρ
ρ
ρ
ρ θη ρ
= + −
= + −
= + − = − + ≥
d Q
P F Q
S E Q
Q
D
ττττ
0 0
0
0
DIV :
:
:
r e
e
e
ρ ρ
ρ
ρ
− = −
= −
= −
Q d
P F
S E
ττττ
October 8, 2013 Carlos Agelet de Saracibar 155
Balance Laws > Second Law of Thermodynamics
Second Law of Thermodynamics
Clausius-Planck Inequality: Local Material Form (II)The local material form of the Clausius-Planck inequality can be written in terms of the internal energy per unit of mass as,
( )
( )
( )
0 0
0
0
: :
:
: 0
inte
e
e
ρ θη
ρ θη
ρ θη
= − −
= − −
= − − ≥
d
P F
S E
D ττττ
October 8, 2013 Carlos Agelet de Saracibar 156
Balance Laws > Second Law of Thermodynamics
Second Law of Thermodynamics
Clausius-Planck Inequality: Local Material Form (III)Introducing the free energy per unit of mass, denoted as , defined as,
the local material form of the Clausius-Planck inequality can be written in terms of the free energy per unit of mass as,
: eψ θη= −
ψ
( )
( )
( )
0 0
0
0
: :
:
: 0
intρ ψ ηθ
ρ ψ ηθ
ρ ψ ηθ
= − +
= − +
= − + ≥
d
P F
S E
D ττττ
Balance Laws > Second Law of Thermodynamics
Clausius-Planck Inequality
October 8, 2013 Carlos Agelet de Saracibar 157
Material Time DerivativeGiving the spatial description of an arbitrary property, the material time derivative of the property can be written as,Local Spatial Forms
Local Material Forms
( ) ( )( ) ( )( ) ( )
0 0 0
0 0
0 0
: : :
: :
: : 0
inte
e
e
ρ θη ρ ψ ηθ
ρ θη ρ ψ ηθ
ρ θη ρ ψ ηθ
= − − = − +
= − − = − +
= − − = − + ≥
d d
P F P F
S E S E
D τ ττ ττ ττ τ
( ) ( ): : : 0int
eρ θη ρ ψ ηθ= − − = − + ≥d d D σ σσ σσ σσ σ
: div 0int rρθη ρ= − + ≥qD
0 0 0: DIV 0int
rρ θη ρ= − + ≥QD
October 8, 2013 Carlos Agelet de Saracibar 158
Balance Laws > Thermodynamic Processes
Adiabatic Process
Adiabatic Process: Local Spatial FormA thermodynamic process is said to be adiabatic if the net heat
transfer to or from the continuum body is zero.The internal dissipation rate per unit of spatial volume for anadiabatic process can be written as,
The stress power per unit of spatial volume for an adiabaticprocess is equal to the material time derivative of the internal
energy per unit of spatial volume,
0, div 0 :intr rρθη ρ= = ⇒ = −q D div+ q 0ρθη= ≥
( ): : 0 :int e eρθη ρ θη ρ= = − − ≥ ⇒ =d d D σ σσ σσ σσ σ
October 8, 2013 Carlos Agelet de Saracibar 159
Balance Laws > Thermodynamic Processes
Adiabatic Process
Adiabatic Process: Local Material Form (I)A thermodynamic process is said to be adiabatic if the net heat
transfer to or from the continuum body is zero.The internal dissipation rate per unit of material volume for anadiabatic process can be written as,
The stress power per unit of material volume for an adiabaticprocess is equal to the material time derivative of the internal
energy per unit of material volume,
0 0 00, DIV 0 :int
r rρ θη ρ= = ⇒ = −Q D DIV+ Q 0 0ρ θη= ≥
( )0 0 0 0: : 0 :int
e eρ θη ρ θη ρ= = − − ≥ ⇒ =d d D τ ττ ττ ττ τ
October 8, 2013 Carlos Agelet de Saracibar 160
Balance Laws > Thermodynamic Processes
Adiabatic Process
Adiabatic Process: Local Material Form (II)A thermodynamic process is said to be adiabatic if the net heat
transfer to or from the continuum body is zero.The internal dissipation rate per unit of material volume for anadiabatic process can be written as,
The stress power per unit of material volume for an adiabaticprocess is equal to the material time derivative of the internal
energy per unit of material volume,
0 0 00, DIV 0 :int
r rρ θη ρ= = ⇒ = −Q D DIV+ Q 0 0ρ θη= ≥
( )0 0 0 0: : 0 :int
e eρ θη ρ θη ρ= = − − ≥ ⇒ =P F P F D
October 8, 2013 Carlos Agelet de Saracibar 161
Balance Laws > Thermodynamic Processes
Adiabatic Process
Adiabatic Process: Local Material Form (III)A thermodynamic process is said to be adiabatic if the net heat
transfer to or from the continuum body is zero.The internal dissipation rate per unit of material volume for anadiabatic process can be written as,
The stress power per unit of material volume for an adiabaticprocess is equal to the material time derivative of the internal
energy per unit of material volume,
0 0 00, DIV 0 :int
r rρ θη ρ= = ⇒ = −Q D DIV+ Q 0 0ρ θη= ≥
( )0 0 0 0: : 0 :int
e eρ θη ρ θη ρ= = − − ≥ ⇒ =S E S E D
October 8, 2013 Carlos Agelet de Saracibar 162
Balance Laws > Thermodynamic Processes
Isentropic Process
Isentropic Process: Local Spatial FormA thermodynamic process is said to be isentropic if it takes place at constant entropy. The internal dissipation rate per unit of spatial volume for anisentropic process can be written as,
The stress power per unit of spatial volume for an isentropicprocess can be written as,
0 :intη ρθη= ⇒ = D div div 0r rρ ρ− + = − + ≥q q
: div :int r eρ ρ θη= − + = − −q d D σσσσ ( ) 0
: dive rρ ρ
≥ ⇒
= − +d qσσσσ
October 8, 2013 Carlos Agelet de Saracibar 163
Balance Laws > Thermodynamic Processes
Isentropic Process
Isentropic Process: Local Material Form (I)A thermodynamic process is said to be isentropic if it takes place at constant entropy. The internal dissipation rate per unit of material volume for anisentropic process can be written as,
The stress power per unit of material volume for an isentropicprocess can be written as,
0 00 :int
η ρ θη= ⇒ = D 0 0DIV DIV 0r rρ ρ− + = − + ≥Q Q
0 0 0: DIV :int
r eρ ρ θη= − + = − −Q d D ττττ ( )0 0
0
: DIVe rρ ρ
≥ ⇒
= − +d Qττττ
October 8, 2013 Carlos Agelet de Saracibar 164
Balance Laws > Thermodynamic Processes
Isentropic Process
Isentropic Process: Local Material Form (II)A thermodynamic process is said to be isentropic if it takes place at constant entropy. The internal dissipation rate per unit of material volume for anisentropic process can be written as,
The stress power per unit of material volume for an isentropicprocess can be written as,
0 00 :int
η ρ θη= ⇒ = D 0 0DIV DIV 0r rρ ρ− + = − + ≥Q Q
0 0 0: DIV :int
r eρ ρ θη= − + = − −Q P F D ( )0 0
0
: DIVe rρ ρ
≥ ⇒
= − +P F Q
October 8, 2013 Carlos Agelet de Saracibar 165
Balance Laws > Thermodynamic Processes
Isentropic Process
Isentropic Process: Local Material Form (III)A thermodynamic process is said to be isentropic if it takes place at constant entropy. The internal dissipation rate per unit of material volume for anisentropic process can be written as,
The stress power per unit of material volume for an isentropicprocess can be written as,
0 00 :int
η ρ θη= ⇒ = D 0 0DIV DIV 0r rρ ρ− + = − + ≥Q Q
0 0 0: DIV :int
r eρ ρ θη= − + = − −Q S E D ( )0 0
0
: DIVe rρ ρ
≥ ⇒
= − +S E Q
October 8, 2013 Carlos Agelet de Saracibar 166
Balance Laws > Thermodynamic Processes
Isentropic and Adiabatic Process
Isentropic + Adiabatic Process: Local Spatial FormA thermodynamic process is said to be isentropic and adiabaticif it takes place at constant entropy and the net heat flux to orfrom the continuum body is zero. The internal dissipation rate per unit of spatial volume for anisentropic and adiabatic process is zero and, then, the process isreversible.
The stress power per unit of spatial volume for an isentropic and adiabatic process is equal to the material time derivative of the
internal energy per unit of spatial volume,
0, 0, div 0 :intrη ρθη= = = ⇒ =q D rρ− div+ q 0=
: :int
eρ θη= − −d D σσσσ ( ) 0 : eρ= ⇒ =d σσσσ
October 8, 2013 Carlos Agelet de Saracibar 167
Balance Laws > Thermodynamic Processes
Isentropic/Adiabatic Reversible Process
Isentropic/Adiabatic Reversible ProcessIf a thermodynamic process is adiabatic and reversible, then theprocess is also isentropic,
If a thermodynamic process is isentropic and reversible, then theprocess is also adiabatic,
For reversible thermodynamic processes, any adiabatic processis also isentropic, and any isentropic process is also adiabatic .
0, 0 :int int
η ρθη= = ⇒ = D D div 0rρ− + =q
0, 0, div 0 :int int
r rρθη ρ= = = ⇒ = −q D D div+ q 0=
October 8, 2013 Carlos Agelet de Saracibar 168
Balance Laws > Thermodynamic Processes
Isothermal Process
Isothermal Process: Local Spatial FormA thermodynamic process is said to be isothermal if it takesplace at constant temperature. The internal dissipation rate per unit of spatial volume for anisothermal process is given by,
0 : :int
θ ρ ψ ηθ= ⇒ = − +d D σσσσ ( ) : 0ρψ= − ≥d σσσσ
October 8, 2013 Carlos Agelet de Saracibar 169
Balance Laws > Governing Equations
Governing Equations: Coupled TM Problem
Governing Equations: Spatial Form Conservation of mass. Mass continuity
❶ ❶❸ Balance of linear momentum. Cauchy first motion
❸ ❾ Balance of angular momentum. Symmetry of Cauchy stress
❸ Balance of energy
❶ ❶❸ Clausius-Planck and heat conduction inequalities
❶❶
div ρ ρ+ =b vσσσσ
T=σ σσ σσ σσ σ
: dive rρ ρ= + −d q σσσσ
div 0ρ ρ+ =v
: div 0, : grad 0int conrρθη ρ θ= − + ≥ = − ⋅ ≥q qD D
October 8, 2013 Carlos Agelet de Saracibar 170
Balance Laws > Governing Equations
Governing Equations: Coupled TM Problem
Constitutive Equations: Spatial Form Thermo-mechanical constitutive equations
❻ ❶
❶ Thermal constitutive equation. Fourier law
❸ State equations
❶
❶
( )
( )
,
, ,
θ π
η η θ π
=
=
v
v
σ σ ,σ σ ,σ σ ,σ σ ,
( ) ( ), , gradθ θ θ= = −q q v k v
( )
( )
,
,
e e θ
π π ρ θ
=
=
v
October 8, 2013 Carlos Agelet de Saracibar 171
Balance Laws > Governing Equations
Governing Equations: Coupled TM Problem
Governing Equations: Material Form (I) Conservation of mass. Mass continuity
❶ ❶ Balance of linear momentum. Cauchy first motion
❸ ❾❸ Balance of angular momentum. Symmetry restriction 1st P-K
❸ Balance of energy
❶ ❶❸ Clausius-Planck and heat conduction inequalities
❶❶
0 0DIV ρ ρ+ =P b v
T T=PF FP
0 0: DIVe rρ ρ= + −P F Q
0Jρ ρ=
0 0 0 0: DIV 0, : GRAD 0int cond
rρ θ η ρ θ= − + ≥ = − ⋅ ≥Q QD D
October 8, 2013 Carlos Agelet de Saracibar 172
Balance Laws > Governing Equations
Governing Equations: Coupled TM Problem
Constitutive Equations: Material Form (I) Thermo-mechanical constitutive equations
❻ ❶
❶ Thermal constitutive equation. Material Fourier law
❸ State equations
❶
❶
( )
( )
,
, ,
T θ π
η η θ π
=
=
PF v
v
τ ,τ ,τ ,τ ,
( ) ( ), , GRADθ θ θ= = −Q Q v K v
( )
( )
,
,
e e θ
π π ρ θ
=
=
v
October 8, 2013 Carlos Agelet de Saracibar 173
Balance Laws > Governing Equations
Governing Equations: Coupled TM Problem
Governing Equations: Material Form (II) Conservation of mass. Mass continuity
❶ ❶ Balance of linear momentum. Cauchy first motion
❸ ❾❸ Balance of angular momentum. Symmetry of Kirchhoff stress
❸ Balance of energy
❶ ❶❸ Clausius-Planck and heat conduction inequalities
❶❶
0 0: DIVe rρ ρ= + −d Q ττττ
0Jρ ρ=
0 0 0 0: DIV 0, : GRAD 0int cond
rρ θ η ρ θ= − + ≥ = − ⋅ ≥Q QD D
( ) 0 0DIV T ρ ρ− + =F b vττττ
T=τ ττ ττ ττ τ
October 8, 2013 Carlos Agelet de Saracibar 174
Balance Laws > Governing Equations
Governing Equations: Coupled TM Problem
Constitutive Equations: Material Form (II) Thermo-mechanical constitutive equations
❻ ❶
❶ Thermal constitutive equation. Material Fourier law
❸ State equations
❶
❶
( )
( )
,
, ,
θ π
η η θ π
=
=
v
v
τ τ ,τ τ ,τ τ ,τ τ ,
( ) ( ), , GRADθ θ θ= = −Q Q v K v
( )
( )
,
,
e e θ
π π ρ θ
=
=
v
October 8, 2013 Carlos Agelet de Saracibar 175
Balance Laws > Governing Equations
Governing Equations: Coupled TM Problem
Governing Equations: Material Form (III) Conservation of mass. Mass continuity
❶ ❶ Balance of linear momentum. Cauchy first motion
❸ ❾❸ Balance of angular momentum. Symmetry of 2nd P-K stress
❸ Balance of energy
❶ ❶❸ Clausius-Planck and heat conduction inequalities
❶❶
0 0: DIVe rρ ρ= + −S E Q
0Jρ ρ=
0 0 0 0: DIV 0, : GRAD 0int cond
rρ θ η ρ θ= − + ≥ = − ⋅ ≥Q QD D
( ) 0 0DIV ρ ρ+ =FS b v
T=S S
October 8, 2013 Carlos Agelet de Saracibar 176
Balance Laws > Governing Equations
Governing Equations: Coupled TM Problem
Constitutive Equations: Material Form (III) Thermo-mechanical constitutive equations
❻ ❶
❶ Thermal constitutive equation. Material Fourier law
❸ State equations
❶
❶
( )
( )
,
, ,
θ π
η η θ π
=
=
v
v
τ τ ,τ τ ,τ τ ,τ τ ,
( ) ( ), , GRADθ θ θ= = −Q Q v K v
( )
( )
,
,
e e θ
π π ρ θ
=
=
v
October 8, 2013 Carlos Agelet de Saracibar 177
Balance Laws > Governing Equations
Governing Equations: Mechanical Problem
Mechanical Problem: Spatial Form Conservation of mass. Mass continuity
❶ ❶❸ Balance of linear momentum. Cauchy first motion
❸ ❾ Balance of angular momentum. Symmetry of Cauchy stress
❸ Mechanical constitutive equation
❻
div ρ ρ+ =b vσσσσ
T=σ σσ σσ σσ σ
div 0ρ ρ+ =v
( )= vσ σσ σσ σσ σ
October 8, 2013 Carlos Agelet de Saracibar 178
Balance Laws > Governing Equations
Governing Equations: Mechanical Problem
Mechanical Problem: Spatial Form Conservation of mass. Mass continuity
❶ ❶❸ Balance of linear momentum. Cauchy first motion
❸ ❻ Mechanical constitutive equation
❻
div ρ ρ+ =b vσσσσ
div 0ρ ρ+ =v
( )= vσ σσ σσ σσ σ
October 8, 2013 Carlos Agelet de Saracibar 179
Balance Laws > Governing Equations
Governing Equations: Mechanical Problem
Mechanical Problem: Material Form (I) Conservation of mass. Mass continuity
❶ ❶ Balance of linear momentum. Cauchy first motion
❸ ❾❸ Balance of angular momentum. Symmetry of Kirchhoff stress
❸ Mechanical constitutive equation
❻
0 0DIV ρ ρ+ =P b v
T T=PF FP
0Jρ ρ=
( )T =PF vττττ
October 8, 2013 Carlos Agelet de Saracibar 180
Balance Laws > Governing Equations
Governing Equations: Mechanical Problem
Mechanical Problem: Material Form (II) Conservation of mass. Mass continuity
❶ ❶ Balance of linear momentum. Cauchy first motion
❸ ❾❸ Balance of angular momentum. Symmetry of Kirchhoff stress
❸ Mechanical constitutive equation
❻
( ) 0 0DIV T ρ ρ− + =F b vττττ
T=τ ττ ττ ττ τ
0Jρ ρ=
( )= vτ ττ ττ ττ τ
October 8, 2013 Carlos Agelet de Saracibar 181
Balance Laws > Governing Equations
Governing Equations: Mechanical Problem
Mechanical Problem: Material Form (II) Conservation of mass. Mass continuity
❶ ❶ Balance of linear momentum. Cauchy first motion
❸ ❻❸ Mechanical constitutive equation
❻
( ) 0 0DIV T ρ ρ− + =F b vττττ
0Jρ ρ=
( )= vτ ττ ττ ττ τ
October 8, 2013 Carlos Agelet de Saracibar 182
Balance Laws > Governing Equations
Governing Equations: Mechanical Problem
Mechanical Problem: Material Form (III) Conservation of mass. Mass continuity
❶ ❶ Balance of linear momentum. Cauchy first motion
❸ ❾❸ Balance of angular momentum. Symmetry of 2nd P-K stress
❸ Mechanical constitutive equation
❻
( ) 0 0DIV ρ ρ+ =FS b v
T=S S
0Jρ ρ=
( )=S S v
October 8, 2013 Carlos Agelet de Saracibar 183
Balance Laws > Governing Equations
Governing Equations: Mechanical Problem
Mechanical Problem: Material Form (III) Conservation of mass. Mass continuity
❶ ❶ Balance of linear momentum. Cauchy first motion
❸ ❻❸ Mechanical constitutive equation
❻
0Jρ ρ=
( )=S S v
( ) 0 0DIV ρ ρ+ =FS b v
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