Chapter 9. Fluid Mechanicsagelet.rmee.upc.edu/master/Chapter 9. Fluid Mechanics v1.0.pdfChapter 9·...

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


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


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,


Introduction

1: tr

3mp σ= − = − σσσσ

p

( )0 0

1 1: tr tr

3 3mp p pσ= − = − = − − =1σσσσ


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,


Introduction

( ), , 0F pρ θ =

p

0p p p= =

0 0, ,p p p p p p≠ ≠ ≠


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,


Introduction

( ) ( ), 0F p pρ ρ ρ= ⇒ =

( ) 00F ρ ρ ρ= ⇒ =


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


Constitutive Equation for Stokes Fluids� Quasi-Newtonian fluid:

� Reiner-Rivlin fluid:



( ) ( ) ( ) ( )( )

( ) ( ) ( )( )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


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.



( )( ) ( ), tr 2 ,p p pλ θ µ θ= − +1+ d 1 dσσσσ

( ) ( ), , , 0p pλ θ µ θ ≥


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.



( ) 0pµ ≥

( ) 0 exp 0p

pB

µ µ

= ≥

0 0µ ≥B

( ) 0pµ µ= ≥



may be written as,

where is a thermodynamic pressure-independent viscositycoefficient, is an activation energy and is the universal constant of ideal gases.



( ) 0µ θ ≥

( ) 0 exp 0Q

Rµ θ µ

θ

= ≥

0 0µ ≥

Q R



may be written as,

where the following parameters have been introduced as,



( ) 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

= ≥ = −


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.



( ) 0µ θ ≥

( )( ) ( )( )1

22 0 24

n

I K Iµ−

=d d

0K n


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.



( ) 0µ θ ≥

0µn

( )( ) ( )( )1

2 22 0 21 4 , 0 1

n

I I nµ µ λ−

= + < <d d

λ


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



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=



Governing Equations

Mechanical Problem� Conservation of mass. Mass continuity 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


T=σ σσ σσ σσ σ

div 0ρ ρ+ =v�

( ),p p= − 1+ f dσσσσ

( )pρ ρ=



Governing Equations



❸ ❻� Mechanical constitutive equation

❻ ❶

� Kinetic state equation for a barotropic fluid❶


div 0ρ ρ+ =v�

( ),p p= − 1+ f dσσσσ

( )pρ ρ=



Governing Equations



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�



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



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



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=



Governing Equations

Thermal Problem� Balance of energy

❶ ❶❶� Clausius-Planck and heat conduction inequalities

❶



( )( ): 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=



Governing Equations

Incompressible Thermal Problem� Balance of energy

❶ ❶❶� Clausius-Planck and heat conduction inequalities

❶



( )( )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=

Chapter 9. Fluid Mechanicsagelet.rmee.upc.edu/master/Chapter 9. Fluid Mechanics v1.0.pdfChapter 9·...

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