ch.9. constitutive equations in fluidsmmc.rmee.upc.edu/documents/slides/ch9_v10.pdf · 2015. 12....
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CH.9. CONSTITUTIVE EQUATIONS IN FLUIDSContinuum Mechanics Course (MMC) - ETSECCPB - UPC
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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
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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
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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
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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
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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
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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
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
13
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
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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
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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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