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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 9Fluid Mechanics
October 10, 2013 Carlos Agelet de Saracibar 2
Chapter 9 · Fluid Mechanics1. Introduction2. Constitutive equations3. Governing equations
Fluid Mechanics > Contents
Contents
October 10, 2013 Carlos Agelet de Saracibar 3
Hydrostatic PressureThere exist experimental evidence that the stress state of a fluid
at rest is hydrostatic and it is characterized by a spherical stress
tensor given by,
where is a positive scalar-valued quantity denoted as hydrostatic pressure.
The traction vector for a fluid at rest, at a given spatial point, isthe same on any arbitrary plane with unit normal n, and is givenby a compression state along the unit normal,
Fluid Mechanics > Introduction
Introduction
0p= − 1σσσσ
0 0p >
0 0p p= = − = −t n 1n nσσσσ
October 10, 2013 Carlos Agelet de Saracibar 4
Mean PressureThe mean pressure, denoted as , is a scalar-valued quantitydefined as minus the mean stress,
For a fluid at rest, the mean pressure is equal to the hydrostatic
pressure,
Fluid Mechanics > Introduction
Introduction
1: tr
3mp σ= − = − σσσσ
p
( )0 0
1 1: tr tr
3 3mp p pσ= − = − = − − =1σσσσ
October 10, 2013 Carlos Agelet de Saracibar 5
Thermodynamic PressureThe thermodynamic pressure, denoted as , is a scalar-valuedquantity, that satisfies the following kinetic state equation,
For a fluid at rest, the hydrostatic pressure satisfies the kineticstate equation and, therefore, it is equal to the thermodynamic
pressure yielding,
For a fluid in motion the three pressures would be different,
Fluid Mechanics > Introduction
Introduction
( ), , 0F pρ θ =
p
0p p p= =
0 0, ,p p p p p p≠ ≠ ≠
October 10, 2013 Carlos Agelet de Saracibar 6
Barotropic FluidA fluid is said to be barotropic if the kinetic state equation doesnot depends on the temperature. The kinetic state equation for a barotropic fluid may be writtenas,
A particular case of barotropic fluid is the incompressible fluid.The kinetic state equation for an incompressible fluid may be written as,
Fluid Mechanics > Introduction
Introduction
( ) ( ), 0F p pρ ρ ρ= ⇒ =
( ) 00F ρ ρ ρ= ⇒ =
October 10, 2013 Carlos Agelet de Saracibar 7
Constitutive Equation for Stokes FluidsThe constitutive equation for a Stokes fluid may be written as,
� Ideal fluid:
� Newtonian fluid:
Fluid Mechanics > Constitutive Equations
Constitutive Equations
( ) ( ), , , , ,ab ab ab
p p p f pθ σ δ θ= − = −1+ f d + dσσσσ
( ), ,p θ =f d 0
( ) ( ) ( ) ( )0 1 1, ,, , p pp K I Kθ θθ = +f d d 1 d
October 10, 2013 Carlos Agelet de Saracibar 8
Constitutive Equation for Stokes Fluids� Quasi-Newtonian fluid:
� Reiner-Rivlin fluid:
Fluid Mechanics > Constitutive Equations
Constitutive Equations
( ) ( ) ( ) ( )( )
( ) ( ) ( )( )0 1 2 3
1 1 2 3
, , , ,
, , , ,
, , K I I I p
K I I I p
p θ
θ
θ =
+
d d d
d d d
f d 1
d
( ) ( ) ( ) ( )( )
( ) ( ) ( )( )
( ) ( ) ( )( )
0 1 2 3
1 1 2 3
1 2 32
, , , ,
, , , ,
, , , ,
, , K I I I p
K I I I p
I I I p
p
K
θ
θ
θ
θ =
+
+
d d d
d d d
d d d
f d 1
d
dd
October 10, 2013 Carlos Agelet de Saracibar 9
Constitutive Equation for Isotropic Newtonian FluidsThe constitutive equation for an isotropic Newtonian fluid maybe written as,
where are two scalar-valued functionsdenoted as dynamic viscosities.
Fluid Mechanics > Constitutive Equations
Constitutive Equations
( )( ) ( ), tr 2 ,p p pλ θ µ θ= − +1+ d 1 dσσσσ
( ) ( ), , , 0p pλ θ µ θ ≥
October 10, 2013 Carlos Agelet de Saracibar 10
Dynamic Viscosities for Isotropic Newtonian FluidsThe dynamic viscosity for an isotropic Newtonian fluid
may be written as,
where is a thermodynamic pressure-independent viscositycoefficient and is a constant, usually bigger enough such thatthe dynamic viscosity coefficient can be considered as thermodynamic pressure-independent.
Fluid Mechanics > Constitutive Equations
Constitutive Equations
( ) 0pµ ≥
( ) 0 exp 0p
pB
µ µ
= ≥
0 0µ ≥B
( ) 0pµ µ= ≥
October 10, 2013 Carlos Agelet de Saracibar 11
Dynamic Viscosities for Isotropic Newtonian FluidsThe dynamic viscosity for an isotropic Newtonian fluid
may be written as,
where is a thermodynamic pressure-independent viscositycoefficient, is an activation energy and is the universal constant of ideal gases.
Fluid Mechanics > Constitutive Equations
Constitutive Equations
( ) 0µ θ ≥
( ) 0 exp 0Q
Rµ θ µ
θ
= ≥
0 0µ ≥
Q R
October 10, 2013 Carlos Agelet de Saracibar 12
Dynamic Viscosities for Isotropic Newtonian FluidsThe dynamic viscosity for an isotropic Newtonian fluid
may be written as,
where the following parameters have been introduced as,
Fluid Mechanics > Constitutive Equations
Constitutive Equations
( ) 0µ θ ≥
( ) ( )( )0 0 0exp exp 0Q
Rµ θ µ µ α θ θ
θ
= = − − ≥
0 0 2
0 0
exp 0,Q Q
R Rµ µ α
θ θ
= ≥ = −
October 10, 2013 Carlos Agelet de Saracibar 13
Dynamic Viscosities for Quasi-Newtonian FluidsPower law model. The dynamic viscosity for a Quasi-
Newtonian fluid may be written as,
where is the consistency parameter and is the rate sensiti-vity coefficient.
Fluid Mechanics > Constitutive Equations
Constitutive Equations
( ) 0µ θ ≥
( )( ) ( )( )1
22 0 24
n
I K Iµ−
=d d
0K n
October 10, 2013 Carlos Agelet de Saracibar 14
Dynamic Viscosities for Quasi-Newtonian FluidsCarreau model. The dynamic viscosity for a Quasi-
Newtonian fluid may be written as,
where is the constant dynamic viscosity parameter, is a model parameter and is the rate sensitivity coefficient.
Fluid Mechanics > Constitutive Equations
Constitutive Equations
( ) 0µ θ ≥
0µn
( )( ) ( )( )1
2 22 0 21 4 , 0 1
n
I I nµ µ λ−
= + < <d d
λ
October 10, 2013 Carlos Agelet de Saracibar 15
Fluid Mechanics > Governing Equations
Governing Equations
Governing Equations� Conservation of mass. Mass continuity equation
❶ ❶❸� Balance of linear momentum. Cauchy first motion equation
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 v�σσσσ
T=σ σσ σσ σσ σ
: dive rρ ρ= + −d q� σσσσ
div 0ρ ρ+ =v�
: div 0, : grad 0int con
rρθη ρ θ= − + ≥ = − ⋅ ≥q q�D D
October 10, 2013 Carlos Agelet de Saracibar 16
Fluid Mechanics > Governing Equations
Governing Equations
Constitutive Equations� Thermo-mechanical constitutive equation for the stresses
❻ ❶
� Thermo-mechanical constitutive equation for the entropy❶
� Thermal constitutive equation. Fourier law❸
� Caloric state equation❶
� Kinetic state equation
❶
( ), ,p p θ= − 1+ f dσσσσ
( ) ( ), , gradθ θ θ= = −q q v k v
( ),e e ρ θ=
( ), ,pη η θ= d
( ),pρ ρ θ=
October 10, 2013 Carlos Agelet de Saracibar 17
Fluid Mechanics > Governing Equations
Governing Equations
Mechanical Problem� Conservation of mass. Mass continuity equation
❶ ❶❸� Balance of linear momentum. Cauchy first motion equation
❸ ❾� Balance of angular momentum. Symmetry of Cauchy stress
❸� Mechanical constitutive equation (temperature independent)
❻ ❶� Kinetic state equation for a barotropic fluid
❶
div ρ ρ+ =b v�σσσσ
T=σ σσ σσ σσ σ
div 0ρ ρ+ =v�
( ),p p= − 1+ f dσσσσ
( )pρ ρ=
October 10, 2013 Carlos Agelet de Saracibar 18
Fluid Mechanics > Governing Equations
Governing Equations
Mechanical Problem� Conservation of mass. Mass continuity equation
❶ ❶❸� Balance of linear momentum. Cauchy first motion equation
❸ ❻� Mechanical constitutive equation
❻ ❶
� Kinetic state equation for a barotropic fluid❶
div ρ ρ+ =b v�σσσσ
div 0ρ ρ+ =v�
( ),p p= − 1+ f dσσσσ
( )pρ ρ=
October 10, 2013 Carlos Agelet de Saracibar 19
Fluid Mechanics > Governing Equations
Governing Equations
Mechanical Problem� Conservation of mass. Mass continuity equation
❶ ❶❸� Balance of linear momentum. Cauchy first motion equation
❸ ❶� Kinetic state equation for a barotropic fluid
❶
div 0ρ ρ+ =v�
( )pρ ρ=
( )grad div ,p p ρ ρ− + + =f d b v�
October 10, 2013 Carlos Agelet de Saracibar 20
Fluid Mechanics > Governing Equations
Governing Equations
Incompressible Mechanical Problem� Conservation of mass. Mass continuity equation
❶ ❸� Balance of linear momentum. Cauchy first motion equation
❸ ❻� Mechanical constitutive equation
❻ ❶
0 0div ρ ρ+ =b v�σσσσ
div 0=v
( ),p p= − 1+ f dσσσσ
October 10, 2013 Carlos Agelet de Saracibar 21
Fluid Mechanics > Governing Equations
Governing Equations
Incompressible Mechanical Problem� Conservation of mass. Mass continuity equation
❶ ❸� Balance of linear momentum. Cauchy first motion equation
❸ ❶( ) 0 0grad div ,p p ρ ρ− + + =f d b v�
div 0=v
October 10, 2013 Carlos Agelet de Saracibar 22
Fluid Mechanics > Governing Equations
Governing Equations
Thermal Problem� Balance of energy
❶ ❶❸� Clausius-Planck and heat conduction inequalities
❶❶� Thermo-mechanical constitutive equation for the entropy
❶� Thermal constitutive equation. Fourier law
❸� Caloric state equation
❶
: dive rρ ρ= + −d q� σσσσ
: div 0, : grad 0int con
rρθ η ρ θ= − + ≥ = − ⋅ ≥q q�D D
( ), ,pη η θ= d
( ) ( ), , gradθ θ θ= = −q q v k v
( ),e e ρ θ=
October 10, 2013 Carlos Agelet de Saracibar 23
Fluid Mechanics > Governing Equations
Governing Equations
Thermal Problem� Balance of energy
❶ ❶❶� Clausius-Planck and heat conduction inequalities
❶
� Thermo-mechanical constitutive equation for the entropy❶
� Caloric state equation❶
( )( ): div , grade rρ ρ θ θ= + +d k v� σσσσ
( )( )( )
: div , grad 0,
: grad , grad 0
int
con
rρθ η ρ θ θ
θ θ θ
= − − ≥
= ⋅ ≥
k v
k v
�D
D
( ), ,pη η θ= d
( ),e e ρ θ=
October 10, 2013 Carlos Agelet de Saracibar 24
Fluid Mechanics > Governing Equations
Governing Equations
Incompressible Thermal Problem� Balance of energy
❶ ❶❶� Clausius-Planck and heat conduction inequalities
❶
� Thermo-mechanical constitutive equation for the entropy❶
� Caloric state equation❶
( )( )0 0: div , grade rρ ρ θ θ= + +d k v� σσσσ
( )( )( )
0 0: div , grad 0,
: grad , grad 0
int
con
rρ θ η ρ θ θ
θ θ θ
= − − ≥
= ⋅ ≥
k v
k v
�D
D
( ), ,pη η θ= d
( )e e θ=