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CH.9. CONSTITUTIVE EQUATIONS IN FLUIDS

Continuum Mechanics Course (MMC) - ETSECCPB - UPC Overview

 Introduction   What is a Fluid?  Constitutive Equations in Fluids  Fluid Models  and Pascal´s Law  Newtonian Fluids  Constitutive Equations of Newtonian Fluids  Relationship between Thermodynamic and Mean  Components of the Constitutive Equation  , Dissipative and Recoverable Power  Dissipative and Recoverable Powers  Thermodynamic Considerations  Limitations in the Values

2 9.1 Introduction

Ch.9. Constitutive Equations in Fluids

3 What is a fluid?

 A fluid is a continuum which cannot resist shearing (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.

4 What is a fluid?

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.

 Real (viscous) fluids:  Viscous in nature and can be subjected to low levels of .  Certain amount of resistance is always offered by these fluids as they move.

5 9.2 Pressure and Pascal’s Law

Ch.9. Constitutive Equations in Fluids

6 Pascal´s Law

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

7 Consequences of Pascal´s Law

 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:

  p01

ijpij0 ij ,1,2,3 

Where p 0 is the hydrostatic pressure.

8 Pressure Concepts

 Hydrostatic pressure, p 0 : normal compressive stress exerted on a fluid in equilibrium.

 Mean pressure, p : minus the mean stress.

1 REMARK pTr    Tr  is an invariant, m 3 thus, so are  m and p .

 Thermodynamic pressure, p : Pressure variable used in the constitutive equations . It is related to density and temperature through the kinetic equation of state. REMARK F ,p, 0 In a fluid at rest, ppp0  

9 Pressure Concepts

 Barotropic fluid: pressure depends only on density.

Fpf,p 0  

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

F ,p,    kkconst 0  .

10 9.3 Constitutive Equations

Ch.9. Constitutive Equations in Fluids

11 Reminder – Governing Eqns.

 Governing equations of the thermo-mechanical problem: Conservation of Mass.   v 0 1 eqn.  Continuity Equation.

Linear Momentum Balance.   bv   3 eqns.  Cauchy’s Motion Equation. Angular Momentum Balance. Symmetry 8 PDE + T 3 eqns.   of Cauchy Stress . 2 restrictions Energy Balance. First Law of  ur  :dq 1 eqn. Thermodynamics. Clausius-Planck us :d 0 Inequality. Second Law of 1 2 restrictions  q  0 Heat Thermodynamics. 2 Inequality.

 19 scalar unknowns:  , v ,  , u , q ,  , s .

12 Reminder – Constitutive Eqns.

 Constitutive equations of the thermo-mechanical problem: Thermo-Mechanical 6 eqns.   v,,  Constitutive Equations.

Entropy ss v,,  Constitutive Equation. 1 eqn. (19+p) PDE + (19+p) unknowns Thermal Constitutive Equation. Fourier’s qq  K  Law of Conduction. 3 eqns.

uf ,,,v  Caloric State Equations. (1+p) eqns. Kinetic Fi ,,  0ip 1,2,...,

set of new thermodynamic

variables:    12 , ,..., p  .  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.

13 Constitutive Equations

 Constitutive equations  Together with the remaining governing equations, they are used to solve the thermo/mechanical problem.

 In fluid mechanics, these are grouped into: Thermo-mechanical constitutive equations Caloric equation of state   p1 fd ,, u  g,    {}  ijpij ij d,, ,  1,2,3 constitutive equation Kinetic equation of state ss d,,   F ,p, 0 q k   REMARK Fourier’s Law    s qk ij,1,2,3 {} dv   v  i  xi 14 Viscous Fluid Models

 General form of the thermo-mechanical constitutive equations:

p1 fd ,, 

ijpij ij f,,, ij d  1,2,3

 In a moving fluid, this can be split into:  p1 fd ,,  

 Depending on the nature of fd  ,,    , fluids are classified into : 1. Perfect fluid: fd ,, 0  p1 2. : f is a linear function of the 3. Stokesian fluid: f is a non-linear function of its arguments

15 9.4. Newtonian Fluids

Ch.9. Constitutive Equations in Fluids

16 Constitutive Equations of Newtonian Fluids

 Mechanic constitutive equations:  p1 C :d

ijpdij ij C ijkl kl ,1,2,3  where C is the 4th-order constant (viscous) constitutive tensor.

C 112 I  Assuming:         an isotropic medium Cijkl ij kl ik jl il jk  the stress tensor is symmetrical ijkl,,, 1,2,3  Substitution of C into the constitutive equation gives: pTr11 dd 2 REMARK  and  are not necessarily constant. pd  2,1,2,3 dij  ij ij ll ij ij Both are a function of  and  .

17 Relationship between Thermodynamic and Mean Pressures

 Taking the mechanic constitutive equation,

ijpd ij  ll ij 2,1,2,3 dij ij  

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

ii  3 , we obtain 1 332p   dp  3()p   ii ll ii 3p Tr()d 3

bulk viscosity  2 2 pp()  Tr dd  p Tr   3 3

18 Relationship between Thermodynamic

and Mean Pressures

 Considering the continuity equation, dd 1  vv0  dt dt

 And the relationship pp Trd v  Trd  d  i    v ii x i  REMARK  d ppv p For a fluid at rest, v 0 ppp0  d dt For an incompressible fluid,  0 pp dt 2 For a fluid with ,   0  p p Stokes ' 3 condition

19 9.5 Components of the Constitutive Equations Ch.9. Constitutive Equations in Fluids

20 Components of the Constitutive Equation

 Given the Cauchy stress tensor, the following may be defined:  pTr11  dd 2  sph    p1

 SPHERICAL PART – mean pressure ppvd p Tr

 DEVIATORIC PART pTr11dd 2  p 1  pp11 Trdd 2

pp Trd 2 2    () Trddd11  Tr   2 3 3 1 deviatoric part of the rate of  ()ddTr 1  d strain tensor 3  d 21 Components of the Constitutive Equation

 Given the Cauchy stress tensor, the following may be defined:  SPHERICAL PART – mean pressure p p ppvd p Tr 

Tr d  DEVIATORIC PART – deviator stress tensor ij     2d 2  The stress tensor is then dij 1   Tr11 p  3  3p from the definition of mean pressure

22 9.6 Stress, Dissipative and Recoverable Powers Ch.9. Constitutive Equations in Fluids

23 Reminder – Stress Power

 Mechanical Energy Balance: d 1 P tdVdSdVdVbv tvv2  :d e VV dt 2 VVt  V

external mechanical power kinetic energy stress power entering the medium d Pt K  t P e dt  REMARK The 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 process of the medium. A rigid solid will have zero stress power.

24 Dissipative and Recoverable Powers

 Stress Power   :d dV  1 V dd1d Tr()  3   p1  1 ::dddpTr11    3  3  Tr d 0 11 pTrdddd11::   p 1 :   Tr  : 1  33 Tr  0 pTrdd :

 2d  :2:dddddpTr   Tr 2      pp Trd RECOVERABLE DISSIPATIVE POWER, 2W . POWER, W R . D

25 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:

 pTr11  dd 2 RECOVERABLE DISSIPATIVE PART,  R . PART,  D .

 And the recoverable and dissipative powers are rewritten as:

WpTrpRRd:d:d 1  2  2WTrDD ddd:d: REMARK For an incompressible fluid,

W0R  pTrd 

26 Work Energy Theorem

 The mechanical energy balance can be re-written as follows

ddKK P  :W2Wd dV  dV  dV eRD dtVVV dt

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

27 Thermodynamic considerations

 Specific recoverable power is an exact differential, 11 dG W  :d (exact differential) RRdt Then, the recoverable work per unit mass in a closed cycle is zero: BA11 BA BA  W dt  :d dt dG G G 0 RR  BAA AA A

 This justifies the denomination “recoverable power”.

28 Thermodynamic Considerations

 According to the 2nd Law of Thermodynamics, the dissipative power is necessarily non-negative for a fluid with   00 and   ,

2 2WDD 0 2W Tr ddd  : 0 d 0

In a closed cycle, the work done by the dissipative stress per unit mass will, in general, be different to zero: B 1  :d dt  0  D A 

2WD  0

 This justifies the denomination “dissipative power”.

29 18/12/2015 Limitations in the Viscosity Values

 The thermodynamic restriction, 2 2WD   Tr ddd : 0 introduces limitations in the values of the viscosity parameters , and :

1. For a purely spherical deformation rate tensor: Tr d  0 2 20WTr  2 d    0 d  0 D 3

2. For a purely deviatoric deformation rate tensor: Tr d  0 22:2Wdd dd  0   0 d  0 Dijij  0

30 18/12/2015 Summary

Ch.9. Constitutive Equations in Fluids

31 Summary

 Constitutive equation for Newtonian fluids:

 Fluid at rest:    p01  For a moving fluid:  pTr11  dd 2  Pressure:

 For a fluid at rest, v  0 ppp0 d  For an incompressible fluid,   0 pp dt  For a fluid with   023 , p p  Cauchy stress tensor:  R D   sph   pp Trd 2  pTr11  dd 2  p1 2d   3  Stress power RECOVERABLE POWER, W R . DISSIPATIVE  :2:dddddpTr   Tr 2      POWER, 2W D .

32