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Ch.9. Constitutive Equations in Fluids

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Ch.9. Constitutive Equations in Fluids CH.9. CONSTITUTIVE EQUATIONS IN FLUIDS Multimedia Course on Continuum Mechanics Overview Introduction Fluid Mechanics Lecture 1 What is a Fluid? Pressure and Pascal´s Law Lecture 3 Constitutive Equations in Fluids Lecture 2 Fluid Models Newtonian Fluids Constitutive Equations of Newtonian Fluids Lecture 4 Relationship between Thermodynamic and Mean Pressures Components of the Constitutive Equation Lecture 5 Stress, Dissipative and Recoverable Power Dissipative and Recoverable Powers Lecture 6 Thermodynamic Considerations Limitations in the Viscosity Values 2 9.1 Introduction Ch.9. Constitutive Equations in Fluids 3 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 shear stress. 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: σ = − p01 σδij =−∈p0 ij 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. REMARK 1 p = −σ= − Tr () σ Tr () σ is an invariant, m 3 thus, so are σ 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 ()ρθ,p,= 0 In a fluid at rest, p0 = pp = 9 Pressure Concepts Barotropic fluid: pressure depends only on density. F(ρρ,p) = 0 pf= ( ) Incompressible fluid: particular case of a barotropic fluid in which density is constant. F(ρθ,p, ) ≡ F( ρ) = ρ −= k 0 ρ==k const. 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. ρρ+ ∇⋅ = 1 eqn. v 0 Continuity Equation. Linear Momentum Balance. ∇⋅σ +ρρbv = 3 eqns. Cauchy’s Motion Equation. Angular Momentum Balance. Symmetry 8 PDE + T 3 eqns. σσ= of Cauchy Stress Tensor. 2 restrictions Energy Balance. First Law of ρρur =σ :dq + −∇⋅ 1 eqn. Thermodynamics. −−+ρθ()usσ :d ≥0 Clausius-Planck Inequality. Second Law of 1 2 restrictions −q ⋅≥∇θ 0 Heat flux Thermodynamics. ρθ 2 Inequality. 19 scalar unknowns: ρ , v , σ , u , q , θ , s . 12 Reminder – Constitutive Eqns. Constitutive equations of the thermo-mechanical problem: Thermo-Mechanical 6 eqns. σσ= ()v,,θ ζ Constitutive Equations. Entropy ss= ()v,,θ ζ Constitutive Equation. 1 eqn. (19+p) PDE + (19+p) unknowns Thermal Constitutive Equation. Fourier’s qq=()θθ =−∇K Law of Conduction. 3 eqns. uf= ()ρθ,,,v ζ Caloric State Equations. (1+p) eqns. Kinetic Fi (){}ρθ, ,ζ = 0 ip ∈ 1,2,..., set of new thermodynamic variables: ζ = {} ζζ 12 , ,..., ζ 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 σ =−+p1 fd(),,ρθ u = g,()ρθ σ=−+ δ ρθ ∈ ijp ijf ij ()d , ,ij , 1, 2, 3 Entropy constitutive equation Kinetic equation of state ss= ()d,,ρθ F ()ρθ,p,= 0 q =−⋅k ∇θ REMARK Fourier’s Law ∂θ s =−∈ dv() = ∇ v qi k ij, 1, 2, 3 ∂xi 14 Viscous Fluid Models General form of the thermo-mechanical constitutive equations: σ =−+p1 fd(),,ρθ σij=−+p δ ijf ij (){}d , ρθ ,ij , ∈ 1, 2, 3 Depending on the nature of fd () ,, ρθ , fluids are classified into : 1. Perfect fluid: fd(),,ρθ =⇒=− 0 σ p1 2. Newtonian fluid: f is a linear function of the strain rate 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: σ =−+p1 C :d σδij=−+p ijC ijkl d kl ij, ∈{} 1, 2, 3 where C is the 4th-order constant (viscous) constitutive tensor. C =λµ11 ⊗+2 I Assuming: =++λδ δ µ δ δ δ δ an isotropic medium Cijkl ij kl( ik jl il jk ) the stress tensor is symmetrical i, jkl , ,∈{} 1, 2, 3 Substitution of C into the constitutive equation gives: σ =−+p11λµ Tr ()dd +2 REMARK λµ σ=−++p δ λδ d2 µ d ij , ∈{} 1, 2, 3 and are not necessarily constant. ij ij ll ij ij Both are a function of ρθ and . 17 Relationship between Thermodynamic and Mean Pressures Taking the mechanic constitutive equation, σij=−++p δ ij λδ d ll ij2 µ d ij ij , ∈{} 1, 2, 3 Setting i=j, summing over the repeated index, and noting that δ ii = 3 , we obtain 1 σ=−++3p() 32 λµ dp =− 3 = − σ ii ll ()p ii −3p Tr()d 3 bulk viscosity κ 2 2 p=++ p()λµ Tr()dd =+ p κ Tr () κλ= + µ 3 3 18 Relationship between Thermodynamic and Mean Pressures Considering the continuity equation, ddρρ1 +ρ ∇⋅vv =0 ∇⋅ =− dt ρ dt And the relationship p= p +κ Tr (d) ∂vi Tr(d) = dii = = ∇ ⋅ v ∂xi REMARK κρd For a fluid at rest, v =0 ppp = = pp= +κ ∇⋅v = p − 0 ρ dρ dt For an incompressible fluid, =0 pp = dt 2 For a fluid with , κ = 0 λµ=−=pp 3 Stokes' 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: σσ= + σ′ σ =−+p11λµ Tr ()dd +2 sph = − p1 SPHERICAL PART – mean pressure p= p −κκ ∇⋅vd = p − Tr () DEVIATORIC PART −+p11λµ Tr ()dd +2 =−+p 1σ′ σ=′ ()()p−+ p11λµ Tr dd +2 p= p −κ Tr ()d 2 2 κλ= + µ σ=′ −+()λµTr ()()d11 + λTr dd +2 µ 3 3 1 deviatoric part of the rate of σ=2′′µµ()dd−=Tr ()1 2 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 p= p −κκ ∇⋅vd = p − Tr ( ) −κ DEVIATORIC PART – deviator stress tensor σ ij′ Tr (d) σ′′= 2µ d 2µ The stress tensor is then dij′ 1 σ=Tr ( σσ)11 +=−+′′p σ 3 REMARK = −3p from the definition Note that κ is not a of mean pressure function of d, while µµ = ( d ) . 22 9.6 Stress, Dissipative and Recoverable Powers Ch.9. Constitutive Equations in Fluids 23 Reminder – Stress Power Mechanical Energy Balance: d 1 P() t=ρρb ⋅ v dV +⋅ t v dS =v2dV +σ :d dV e ∫∫VV∂ dt ∫∫2 VVt ≡ V external mechanical power kinetic energy stress power entering the medium d Pt() =() t + P e dt K σ 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 deformation process of the medium. A rigid solid will have zero stress power. 24 Dissipative and Recoverable Powers Stress Power = σ :d dV ∫ 1 V d=Tr() d1 + d′ 3 σσ=−+p1 ′ 1 σσ::d=−+( p11′′) Tr (dd) + = 3 ′ = 3 =Tr (d ) = 0 11 =−pTr (d) 11:: +−σσ′′ ddp 1 : ′ + Tr ( d) ′ :1 = 33=Tr (σ′) = 0 =−+pTr (dd) σ′′: σ′′= 2µd σ :d=−+pTr( d) κµ Tr 2 ( d) +2: dd′′ p= p −κ Tr (d) RECOVERABLE STRESS DISSIPATIVE STRESS POWER, . POWER, . WR 2WD 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: σ =−+p11λµ Tr ()dd +2 RECOVERABLE DISSIPATIVE PART, . PART, . σ R σ D And the recoverable and dissipative powers are rewritten as: WRR=−=−= pTr()d p1 :dσ :d 2 2WDD=κµ Tr ()d +=2σ d′′: d :d REMARK For an incompressible fluid, W0R =−=pTr ()d 26 Thermodynamic considerations Specific recoverable stress power is an exact differential, 11 dG W =σ :d = → (exact differential) ρρRRdt Then, the recoverable stress work per unit mass in a closed cycle is zero: BA≡≡11BA BA ≡ =σ = = −= ∫∫WR dt R:d dt ∫ dG GBA≡ G A 0 AAρρ A This justifies the denomination “recoverable stress power”. 28 Thermodynamic Considerations According to the 2nd Law of Thermodynamics, the dissipative power is necessarily non-negative, 2 2WDD≥ 0 2W =κµ Tr (d) +=2 dd′′: 0 d = 0 In a closed cycle, the work done by the dissipative stress per unit mass will, in general, be different to zero: BB≡ 1 σ D :d dt > 0 ∫ A ρ 2WD > 0 This justifies the denomination “dissipative power”. 29 Limitations in the Viscosity Values The thermodynamic restriction, 2 2WD =κµ Tr ()d +≥2 dd′′: 0 introduces limitations in the values of the viscosity parameters κλ , and µ : 1. For a purely spherical deformation rate tensor: Tr ()d ≠ 0 2 20W=κ Tr 2 ()d ≥ κλ=+≥ µ0 d′ = 0 D 3 2. For a purely deviatoric deformation rate tensor: Tr ()d = 0 2W = 2:µµdd′ ′ = 2dd ′′ ≥ 0 ≥µ 0 d′ ≠ 0 D ij ij > 0 30 Chapter 9 Constitutive Equations in Fluids 9.1 Concept of Pressure Several concepts of pressure are used in continuum mechanics (hydrostatic pres- sure, mean pressure and thermodynamic pressure) which, in general, do not co- incide.
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