Computational Rheology (4K430)
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Computational Rheology (4K430) dr.ir. M.A. Hulsen m.a.hulsen@tue.nl Website: http://www.mate.tue.nl/~hulsen under link ‘Computational Rheology’. – Section Polymer Technology (PT) / Materials Technology (MaTe) – Introduction Computational Rheology important for: B Polymer processing B Rheology & Material science B Turbulent flow (drag reduction phenomena) B Food processing B Biological flows B ... Introduction (Polymer Processing) Analysis of viscoelastic phenomena essential for predicting B Flow induced crystallization kinetics B Flow instabilities during processing B Free surface flows (e.g.extrudate swell) B Secondary flows B Dimensional stability of injection moulded products B Prediction of mechanical and optical properties Introduction (Surface Defects on Injection Molded Parts) Alternating dull bands perpendicular to flow direction with high surface roughness (M. Bulters & A. Schepens, DSM-Research). Introduction (Flow Marks, Two Color Polypropylene) Flow Mark Side view Top view Bottom view M. Bulters & A. Schepens, DSM-Research Introduction (Simulation flow front) 1 0.5 Steady Perturbed H 2y 0 ___ −0.5 −1 0 0.5 1 ___2x H Introduction (Rheology & Material Science) Simulation essential for understanding and predicting material properties: B Polymer blends (morphology, viscosity, normal stresses) B Particle filled viscoelastic fluids (suspensions) B Polymer architecture macroscopic properties (Brownian dynamics (BD), molecular dynamics (MD),⇒ Monte Carlo, . ) Multi-scale. ⇒ Introduction (Solid particles in a viscoelastic fluid) B Microstructure (polymer, particles) B Bulk rheology B Flow induced crystallization Introduction (Multiple particles in a viscoelastic fluid) Introduction (Flow induced crystallization) Introduction (Multi-phase flows) Goal and contents of the course Goal: Introduction of the basic numerical techniques used in Computational Rheology using the Finite Element Method (FEM). Contents (tentatively): B Basic equations from Continuum Mechanics and Rheology B Introduction of basic FEM techniques: Galerkin method, mixed methods, Petrov-Galerkin: SUPG, DG, time discretization. B FEM for flow problems. Navier-Stokes, Mixed Stokes. Viscoelastic. B Stabilization techniques for viscoelastic flows. B Benchmarks B Micro-macro methods B Integral models B Suspensions B ... Configurations reference configuration time τ region Vτ P present configuration time t ~ dX region Vt path of P particle P ~e X~ 3 d~x ~x ~e2 ~e r 1 O Deformation (1) material description (Lagrangian): T = Tm(X,~ t) spatial description (Eulerian): T = Ts(~x,t) mapping deformation: ≡ ~x = ~x(X,~ t) X~ = X~ (~x,t) ⇔ deformation gradient (local deformation): ∂~x ∂xi d~x = F dX,~ F = ,Fij = · ∂X~ ∂Xj deformation of local volume: dV J = det F = > 0 dVτ Deformation (2) Deformation tensors: C = F T F “Green” · B = F F T “Finger” · Rates (1) material derivative: DT ∂T (X,~ t) ∂T = m = = T˙ Dt ∂t ∂t X~ =constant local derivative: ∂T ∂T (~x,t) ∂T = s = ∂t ∂t ∂t ~x=constant velocity: D~x ~u = = ~x˙ Dt acceleration: D2~x ~a = ~u˙ = = ~x¨ Dt2 DT ∂T = + ~u T Dt ∂t · ∇ Rates (2) velocity gradient tensor: D 1 T ∂ui (d~x) = L d~x with L = F˙ F − = ( ~u) ,Lij = Dt · · ∇ ∂xj rate-of-deformation tensor: 1 D = (L + LT ) 2 vorticity tensor: 1 W = (L LT ) 2 − volume-rate-of-deformation: J˙ = ~u = divergence of velocity ~u J ∇ · Balance (conservation) laws in Eulerian frame (1) Conservation of mass ρ˙ + ρ ~u = 0 ∇ · for constant density fluids: ρ˙ = 0: ~u = 0 ∇ · Linear momentum balance Cauchy stress tensor σ gives the ‘traction’ on surface with normal ~n: ~t = σT ~n = ~n σ · · ρ~u˙ = σ + ρ~b ∇ · constant density fluids: σ = pI + t, p : hydrostatic pressure, t : extra-stress tensor, − Balance (conservation) laws in Eulerian frame (2) angular momentum balance σT = σ symmetric energy balance ρε˙ = ~q + σ : D + ρr ∇ · with ε internal energy per unit mass ~q heat flux vector: amount of energy flowing through a surface with a normal ~n per unit area q = ~q ~n by conduction · r body heat source. CEs for the stress tensor Constant density fluids: σ = pI + t − Newtonian fluids: t = 2ηD with viscosity η a constant. Viscoelastic fluids: for example the Oldroyd-B fluid t = 2ηsD + τ with λ τ5 +τ = 2ηD where τ5= τ˙ L τ τ LT − · − · η = 0 upper-convected Maxwell (UCM) fluid s ⇒ Linear viscoelastic fluid γ τ τ G0 γ 1 step strain modulus G(t) t t Linear response theory (Boltzmann superposition): t dG τ(t) = M(t t0)[γ(t) γ(t0)] dt0,M(t) = (t) − − − dt Z−∞ Elastic reponse at t = 0+: 0 + + + τ(0 ) = M( t0) dt0 γ(0 ) = G γ(0 ) − 0 Z−∞ Non-linear viscoelastic fluid (integral model) Neo-Hookean elastic model τ = G(B I) − Viscoelastic (Lodge rubber like liquid) t τ (t) = M(t t0)[B (t) I] dt0 − t0 − Z−∞ Spectrum with single relaxation time t/λ G0 t/λ G(t) = G e− ,M(t) = e− 0 λ and t G (t t0)/λ τ (t) = e− − [B (t) I] dt0 λ t0 − Z−∞ G G 0 → Non-linear viscoelastic fluid (differential model) Differentiating to time t t t G (t t0)/λ G (t t0)/λ τ˙ = 0 e− − [B (t) I] dt0 + e− − B˙ (t) dt0 − λ2 t0 − λ t0 upper boundary Z−∞ Z−∞ |{z} T With B = F F from t0 to t and F˙ = L F · · T B˙ = F˙ F T + F F˙ = L B + B LT · · · · we get t τ T G (t t0)/λ T τ˙ = + L τ + τ L + e− − dt0(L + L ) −λ · · λ Z−∞ τ = + L τ + τ LT + 2GD −λ · · Non-linear viscoelastic fluid (Oldroyd-B, UCM) ⇒ λ τ5 +τ = 2ηD where τ5 = τ˙ L τ τ LT − · − · η = Gλ properties: B constant steady state viscosity η B single relaxation time λ 2 B steady state first-normal stress difference N1 = 2ηλγ˙ 1 B no steady state elongation viscosity for ˙ > 2λ B second-normal stress difference N2 = 0. Exercise 1 The linear viscoelastic properties of a particular fluid can be modeled by t/λ1 t/λ2 G(t) = G1e− + G2e− Apply the procedure we used to derive the Lodge integral model to propose a non- linear model suitable for large deformation of the fluid. Derive the corresponding differential model. CEs for ε and ~q We assume (in this course): ε˙ = cT˙ , c : specific heat (constant),T : temperature ~q = k T, k : thermal conductivity (constant) − ∇ and: σ does not depend on the thermal history decoupling ⇒ Set of equations (summary) Conservation of mass ~u = 0 ∇ · (Linear and angular) Momentum equation ρ~u˙ = σ + ρ~b, with σ = σT ∇ · CE for the stress tensor Newtonian: σ = pI + τ , τ = 2ηD − Viscoelastic: (Oldroyd-B/UCM) σ = pI + 2η D + τ , λ τ5 +τ = 2ηD, τ5= τ˙ L τ τ LT − s − · − · Energy equation ρcT˙ = (k T ) + σ : D + ρr ∇ · ∇ Boundary and initial conditions Convection-diffusion-reaction equation Γ D u(~x,t = 0) = u0(~x) in Ω ΓN u = uD on ΓD ∂u ~n Ω A = A~n u = h on Γ − ∂n − · ∇ N ~n: outside normal ∂u + ~a u (A u) + bu = f ∂t · ∇ − ∇ · ∇ Energy equation: k 1 r u = T, ~a = ~u, A = with A 0, b = 0, f = σ : D + ρc ≥ ρc c Set of equations (flow of a visco-elastic fluid) (1) Conservation of mass ~u = 0 ∇ · (Linear and angular) Momentum equation ρ~u˙ = σ + ρ~b, with σ = σT ∇ · Viscoelastic fluid model: (Oldroyd-B/UCM) σ = pI + 2η D + τ , λ τ5 +τ = 2ηD, τ5= τ˙ L τ τ LT − s − · − · Boundary and initial conditions Set of equations (flow of a visco-elastic fluid) (2) Rewrite: Find (~u,p, τ ) such that, ∂~u ρ( + ~u ~u) (2η D) + p τ = ρ~b, in Ω ∂t · ∇ − ∇ · s ∇ − ∇ · ~u = 0, in Ω ∇ · ∂τ λ( + ~u τ L τ τ LT ) + τ = 2ηD, in Ω ∂t · ∇ − · − · Boundary and initial conditions Convection-diffusion-reaction equation ΓD u(~x,t = 0) = u0(~x) in Ω ΓN u = uD on ΓD ~n Ω ∂u A = A~n u = h on ΓN ~n: outside normal − ∂n − · ∇ ∂u + ~a u (A u) + bu = f ∂t · ∇ − ∇ · ∇ Finite Element Method (FEM) Approximation method: d du (A ) = f Ku = f −dx dx ⇒ ¯ ˜ ˜ where u: an approximate solution using a finite number of unknowns N. ˜ For N :“u u” → ∞ ˜ → B quite general distribution of ‘elements’ without losing accuracy B local refinements near large gradients B quite general geometries in multiple dimensions Linear spaces (1) u V linear space V v u V, v V, w V ∈ ∈ ∈ w λ R, µ R ∈ ∈ B u + v V λu V ∈ B ∈ B (u + v) + w = u + (v + w) B λ(u + v) = λu + λv 0 such that u + 0 = u λ(µu) = (λµ)u B ∃ B B u such that u + ( u) = 0 1 u = u ∃ − − B · Linear spaces (2) Elements of a linear space V : linear combination of independent base vectors. With N base vectors: N-dimensional space. N u = uiei i=1 X Independence: N α e = 0 α = 0 i i ⇒ i i=1 X Examples (1) B 3D (N = 3) physical space Any 3 non-zero vectors not in a plane can act as a base. All periodic functions on ( π, π) B − f π π − g Fourier expansion: 1 ∞ f(x) = a + a cos(kx) + b sin(kx) 2 0 k k k=1 X Examples (2) with 1 π 1 π ak = f(x) cos(kx) dx, bk = f(x) sin(kx) dx π π π π Z− Z− Base: 1, cos x, sin x, cos 2x, sin 2x, . N = ∞ 0 B All continuous function on (a, b): C (a, b). 2 B All square integrable functions on (a, b): L (a, b): b f L2(a, b) then f 2 dx < ∈ ∞ Za Note: δ(x) L2(a, b): 6∈ Examples (3) h f(x) x 2/h f(x) dx = 1 Z 2h f 2(x) dx = for h 3 → ∞ → ∞ Z Inner product and norm Inner product (u, v): (u, v) = (v, u) for all u, v V B ∈ (αu + βv, w) = (αu, w) + (βv, w) for all u, v, w V, α, β R B ∈ ∈ (u, u) 0 for all u, v V B ≥ ∈ B (u, u) = 0 implies u = 0 u and v are orthogonal is (u, v) = 0.