CH.9. CONSTITUTIVE EQUATIONS IN FLUIDSContinuum Mechanics Course (MMC) - ETSECCPB - UPC

Overview

Introduction Fluid Mechanics

What is a Fluid?

Constitutive Equations in Fluids

Fluid Models

Pressure and Pascal´s Law

Newtonian Fluids Constitutive Equations of Newtonian Fluids

Relationship between Thermodynamic and Mean Pressures

Components of the Constitutive Equation

Stress, Dissipative and Recoverable Power Dissipative and Recoverable Powers

Thermodynamic Considerations

Limitations in the Viscosity Values

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Ch.9. Constitutive Equations in Fluids

9.1 Introduction

A fluid is a continuum which cannot resist shearing forces (tangential stresses) while at rest. A fluid will continue to deform under applied stress and never reach

static equilibrium. A fluid has the ability to flow (will take the shape of the container it is

in). Fluids include liquids, gases and plasmas.

What is a fluid?

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Fluids can be classified into:

Ideal (inviscid) fluids: Also named perfect fluid. Only resists normal, compressive stresses (pressure). No resistance is encountered as the fluid moves.

What is a fluid?

Real (viscous) fluids: Viscous in nature and can be subjected to low

levels of shear stress. Certain amount of resistance is always offered

by these fluids as they move.

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Ch.9. Constitutive Equations in Fluids

9.2 Pressure and Pascal’s Law

Pascal´s Law

Pascal’s Law:In a confined fluid at rest, pressure acts equally in alldirections at a given point.

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In fluid at rest: there are no shear stresses only normal forces due to pressure are present.

The stress in a fluid at rest is isotropic and must be of the form:

Where is the hydrostatic pressure.

Consequences of Pascal´s Law

0

0 , 1,2,3ij ij

pp i j

1

0p

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Hydrostatic pressure, : normal compressive stress exerted on a fluid in equilibrium.

Mean pressure, : minus the mean stress.

Thermodynamic pressure, : Pressure variable used in the constitutive equations . It is related to density and temperature through the kinetic equation of state.

Pressure Concepts

0p

p

p

13mp Tr

, p, 0F REMARK In a fluid at rest,

0p p p

REMARK is an invariant,

thus, so are and . Tr

pm

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Barotropic fluid: pressure depends only on density.

Incompressible fluid: particular case of a barotropic fluid in which density is constant.

Pressure Concepts

, p 0F p f

, p, 0 .F k k const

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Ch.9. Constitutive Equations in Fluids

9.3 Constitutive Equations

Governing equations of the thermo-mechanical problem:

19 scalar unknowns: , , , , , , .

Conservation of Mass. Continuity Equation. 1 eqn.

Reminder – Governing Eqns.

0 v

Linear Momentum Balance. Cauchy’s Motion Equation. 3 eqns. b v

Angular Momentum Balance. Symmetry of Cauchy Stress Tensor. 3 eqns.T

Energy Balance. First Law of Thermodynamics. 1 eqn.:u r d q

Second Law of Thermodynamics.

2 restrictions 0u s :d

2

1 0

q

8 PDE + 2 restrictions

v u q s

Clausius-Planck Inequality.Heat flux Inequality.

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Constitutive equations of the thermo-mechanical problem:

The mechanical and thermal problem can be uncoupled if the temperature distribution is known a priori or does not intervene in the constitutive eqns. and if the constitutive eqns. involved do not introduce new thermodynamic variables.

Thermo-Mechanical Constitutive Equations. 6 eqns.

Reminder – Constitutive Eqns.

Thermal Constitutive Equation. Fourier’s Law of Conduction. 3 eqns.

State Equations. (1+p) eqns.

(19+p) PDE + (19+p) unknowns

, , v

, ,s s v 1 eqn.

K q q

, , 0 1,2,...,iF i p , , ,u f v

Kinetic

Caloric

Entropy Constitutive Equation.

set of new thermodynamic variables: . 1 2, ,..., p

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Constitutive equations Together with the remaining governing equations, they are used to

solve the thermo/mechanical problem.

In fluid mechanics, these are grouped into:

Constitutive Equations

, ,

, , , 1,2,3ij ij

p

p i j

f d

d

1

{ } g ,u

, 1, 2,3ii

q k i jx

q

{ }

k , ,s s d

Thermo-mechanical constitutive equations

Entropy constitutive equation

Fourier’s Law

Caloric equation of state

Kinetic equation of state

, p, 0F

REMARK sd v v

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General form of the thermo-mechanical constitutive equations:

In a moving fluid, this can be split into:

Depending on the nature of , fluids are classified into :1. Perfect fluid:

2. Newtonian fluid: f is a linear function of the strain rate

3. Stokesian fluid: f is a non-linear function of its arguments

Viscous Fluid Models

, ,

f , , , 1,2,3ij ij ij

p

p i j

f d

d

1

, , f d , , 0 p f d 1

, ,p f d 1

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Ch.9. Constitutive Equations in Fluids

9.4. Newtonian Fluids

Constitutive Equations of Newtonian Fluids

Mechanic constitutive equations:

where is the 4th-order constant (viscous) constitutive tensor.

Assuming: an isotropic medium the stress tensor is symmetrical

Substitution of into the constitutive equation gives:

22 , 1,2,3ij ij ll ij ij

p Trp d d i j

d d 1 1

C

, 1, 2,3ij ij ijkl kl

pp d i j

: dCC

1

C

2

, , , 1,2,3ijkl ij kl ik jl il jk

i j k l

C

C

1 1 I

REMARK and are not necessarily constant.

Both are a function of and .

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Relationship between Thermodynamic and Mean Pressures

Taking the mechanic constitutive equation,

Setting i=j, summing over the repeated index, and noting that, we obtain3ii

3 ( )

3 3 2 3ii ll

p Trp d p

d

2( )3

p p Tr p Tr d d

1( )3 iip

2 , 1,2,3ij ij ll ij ijp d d i j

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bulk viscosity

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Relationship between Thermodynamic and Mean Pressures

Considering the continuity equation,

And the relationship

10d ddt dt

v v

dp p pdt

v

p p Tr d vd

i

iii x

dTrv

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REMARK For a fluid at rest,

For an incompressible fluid,

For a fluid with ,

00 p p p v

0d p pdt

'0

Stokescondition

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p p

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Ch.9. Constitutive Equations in Fluids

9.5 Components of the Constitutive Equations

Components of the Constitutive Equation Given the Cauchy stress tensor, the following may be defined:

SPHERICAL PART – mean pressure

DEVIATORIC PART

p p p Tr v d

2p Tr d d 1 1 sph p 1

2p Tr p d d 1 1 1 2p p Tr d d 1 1

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2( ) 23

Tr Tr d d d 1 1

1( )3

Tr

d d d

d

1

p p Tr d

deviatoric part of the rate of strain tensor

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Components of the Constitutive Equation

Given the Cauchy stress tensor, the following may be defined: SPHERICAL PART – mean pressure

DEVIATORIC PART – deviator stress tensor

The stress tensor is then

p p p Tr v d

2 d

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Tr p 1 1

3p

Tr d

p

p

ijd

ij

2

from the definition of mean pressure

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Ch.9. Constitutive Equations in Fluids

9.6 Stress, Dissipative and Recoverable Powers

Mechanical Energy Balance:

Reminder – Stress Power

21 v 2

t

e V VV V V

dP t dV dS dV dVdt

b v t v :d

external mechanical power entering the medium

stress powerkinetic energy

edP t t Pdt K

REMARKThe stress power is the mechanical power entering the system which is not spent in changing the kinetic energy. It can be interpreted as the work per unit of time done by the stress in the deformation process of the medium.A rigid solid will have zero stress power.

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Dissipative and Recoverable Powers

Stress Power V

dV :d1 ( )3

Tr d d 1 d

p 1

1: :3

1 1: : : :3 3

:

p Tr

pTr p Tr

pTr

d d d

d d d d

d d

1 1

1 1 1 13 0Tr d

0Tr

p p Tr d2 d 2: 2 :pTr Tr d d d d d

RECOVERABLE POWER, .WR

DISSIPATIVE POWER, . 2WD

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Dissipative and Recoverable Parts of the Cauchy Stress Tensor

Associated to the concepts of recoverable and dissipative powers, the Cauchy stress tensor is split into:

And the recoverable and dissipative powers are rewritten as:

2p Tr d d 1 1

RECOVERABLE PART, .R

DISSIPATIVE PART, . D

22 :

R R

D D

W pTr p

W Tr

d :d :d

d d d :d

1

REMARK For an incompressible fluid,

W 0R p Tr d

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Work Energy Theorem

The mechanical energy balance can be re-written as follows

where The specific recoverable power is an exact differential. The dissipative power of the equation is necessarily non-negative.

: W 2Wde R DV V V

d dP dV dV dVdt dt

K K

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Thermodynamic considerations

Specific recoverable power is an exact differential,

Then, the recoverable work per unit mass in a closed cycle is zero:

This justifies the denomination

“recoverable power”.

1 1W : dR RdGdt

(exact differential)

1 1W 0B A B A B A

R R B A AA A A

dt dt dG G G

: d

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Thermodynamic Considerations

According to the 2nd Law of Thermodynamics, the dissipative power is necessarily non-negative for a fluid with and ,

In a closed cycle, the work done by the dissipative stress per unit mass will, in general, be different to zero:

This justifies the denomination “dissipative power”.

22W 0 2W : 0 0D D Tr d d d d

1 0B

DA

dt

: d

2W 0D

0 0

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Limitations in the Viscosity Values

The thermodynamic restriction,

introduces limitations in the values of the viscosity parameters and :

1. For a purely spherical deformation rate tensor:

2. For a purely deviatoric deformation rate tensor:

22W : 0D Tr d d d

,

2 03

2 2 : 2 0D ij ijW d d d d 0

22 0dDW Tr 0 d

0Tr d

0 d 0Tr d

0

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Ch.9. Constitutive Equations in Fluids

Summary

Summary

Constitutive equation for Newtonian fluids: Fluid at rest: For a moving fluid:

Pressure: For a fluid at rest,

For an incompressible fluid,

For a fluid with ,

Cauchy stress tensor:

Stress power

00 p p p v0d p pdt

0 2 3 p p

2p Tr d d 1 10p 1

2: 2 :pTr Tr d d d d d

RECOVERABLE POWER, .WR

DISSIPATIVE POWER, . 2WD

2p Tr d d 1 1R D

2sph

p

d

123

p p Tr d

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