On models for viscoelastic fluid-like materials that are mechanically incompressible and thermally compressible or expansible and their Oberbeck Boussinesq type approximations Vít Průša (joint work with K.R. Rajagopal) prusv@karlin.mff.cuni.cz Mathematical Institute, Charles University 30 October 2013
Buoyancy driven flows
Oberbeck Boussinesq approximation Original papers: J. Boussinesq. Théorie analytique de la chaleur. Gauthier-Villars, Paris, 1903 A. Oberbeck. Über die Wärmeleitung der Flüssigkeiten bei Berücksichtigung der Strömungen infolge von Temperaturdifferenzen. Ann. Phys. Chem., 1:271, 1879
Oberbeck Boussinesq approximation Governing equations: divv = 0 ρ ref dv dt = m+µ ref v+ρ ref (1+α(θ θ ref ))b ρ ref c m,ref dθ dt = κ θ Rayleigh number: Ra = def ρ ref gα ref θ diff l 3 char kη 1,ref
Full system of governing equations Governing equations: Constitutive relations: dρ dt +ρdivv = 0 ρ dv dt = div T+ρb ρ de dt = T : D divq T = pi+2µd q = κ θ p = p(ρ,θ) e = e(...)
Oberbeck Boussinesq approximation E. A. Spiegel and G. Veronis. On the Boussinesq approximation for a compressible fluid. Astrophys. J., 131:442 447, 1960 In equation (19) we have retained the term gε(ρ / ρ 0 )k even through it contains ε as a factor. John M. Mihaljan. A rigorous exposition of the Boussinesq approximations applicable to a thin layer of fluid. Astrophys. J., 136:1126 1133, 1962 K. R. Rajagopal, M. Růžička, and A. R. Srinivasa. On the Oberbeck Boussinesq approximation. Math. Models Methods Appl. Sci., 6(8):1157 1167, 1996
Oberbeck Boussinesq system as an approximation of an exact system Mechanically and thermally compressible/expansible material (Ma ε, Fr ε). Eduard Feireisl and Antonin Novotný. The Oberbeck Boussinesq approximation as a singular limit of the full Navier Stokes Fourier system. J. Math. Fluid Mech., 11:274 302, 2009 Mechanically incompressible and thermally compressible/expansible material. K. R. Rajagopal, M. Růžička, and A. R. Srinivasa. On the Oberbeck Boussinesq approximation. Math. Models Methods Appl. Sci., 6(8):1157 1167, 1996
Buoyancy driven flows in viscoelastic fluids Trench "SLAB PULL" Ridge LithosphereTrench Source: Wikipedia
Viscoelastic fluids Maxwell model l l d l s µ m κ m F Mechanical analogue: Spring energy storage. Dashpot energy dissipation. Problems: Distribution of the deformation between the elements (spring, dashpot). Three dimensional model. (Galilean invariance.) Meets laws of thermodynamics. (No perpetual motion.)
Upper convected Maxwell model A three-dimensional incompressible viscoelastic rate type model: T = pi+s τs+s = 2µD divv = 0 1 D = def ( v+( v) ) 2 L = def v da A = def dt LA AL
Oberbeck Boussinesq approximation for viscoelastic fluids Governing equations: divv = 0 ρ ref dv dt = div T+ρ ref(1+α(θ θ ref ))b ρ ref c m,ref dθ dt = κ θ Naive approach is to replace Navier Stokes fluid model by Maxwell model T = pi+2µd T = pi+s τ S+S = 2µD divv = 0
Full model Requirements: Mechanically incompressible but thermally compressible/expansible materials. (Liquids.) Different nature of the pressure. (No equation of state. Liquids.) Viscoelastic fluid like materials. (Internal energy has a non-thermal contribution.) Interplay between all the effects. Simplification: Let us consider only constant material coefficients. (Temperature independent viscosity.)
Plan Viscoelasticity: K. R. Rajagopal and A. R. Srinivasa. A thermodynamic frame work for rate type fluid models. J. Non-Newton. Fluid Mech., 88(3):207 227, 2000 Mechanical incompressibility and thermal expansivity/compressibility: K. R. Rajagopal, M. Růžička, and A. R. Srinivasa. On the Oberbeck Boussinesq approximation. Math. Models Methods Appl. Sci., 6(8):1157 1167, 1996
Example viscous fluids Internal energy ansatz: e(ρ,η) = g(ρ)η +h(ρ) Time derivative: de dt = g dρ ρ dt η +gdη dt + h dρ ρ dt Entropy production: ρ dη dt +div q θ = 1 θ Second law satisfied if: ( q θ+t δ : D δ + ( αρ 2 h ρη αm ρ ) ) dθ dt q = def κ θ, T δ = def 2µD δ αρ 2 h ρ ρη αm = def 0
Viscous fluids result Γ 4 Γ 2 ρ c m dϑ Γ 1 dt = div v ρ dv dt = Γ 2 ( v + 1 3 (div v ) dϑ dt = Γ 3Γ 2 ϑ +2Γ 2 D δ : D δ + ) + 1 ( m +ρ b ) Γ 1 ( ϑ + θ ) ref dm θ diff dt
Parameter values Experiment Water Water Water Glycerol Mercury Γ 1 2.40 10 5 7.64 10 4 1.55 10 2 1.68 10 2 1.82 10 3 Γ 2 2.17 10 1 3.84 10 2 1.39 10 5 7.48 10 3 2.70 10 5 Γ 3 6.68 10 7 6.68 10 7 1.01 10 6 1.15 10 3 3.16 10 7 Γ 4 2.19 10 9 1.23 10 10 3.18 10 11 3.91 10 8 2.89 10 10 Ra 1.5 10 2 4.8 10 3 2.3 10 10 4.5 10 7 3.4 10 7 Pr 7.1 7.1 4.4 2.5 10 3 2.5 10 2 Re 4.6 2.6 10 1 7.2 10 4 1.3 10 2 3.7 10 4 Table: Dimensionless parameters in some experiments with viscous fluids.
Viscoelastic fluids Maxwell model l l d l s µ m κ m F Mechanical analogue: Spring energy storage. Dashpot energy dissipation. Problems: Distribution of the deformation between the elements (spring, dashpot). Three dimensional model. (Galilean invariance.) Meets laws of thermodynamics. (No perpetual motion.)
Notion of natural configuration reference configuration R F = F RC C current configuration dissipative response F RN F NC elasic response TrD RN = 0 ζ = 2η 1 F NCD RN 2 N natural configuration h = c 1(TrB NC 1)+H(detF NC) Problem: Distribution of the deformation between the dissipative and elastic response. (Condition F = F NC F RN is not enough restrictive.)
Evolution equation for the elastic response Evolution of the elastic response: db NC dt Oldroyd upper convected derivative: = LB NC + B NC L 2F NC D RN F NC da A = def dt LA AL. Evolution equation for the elastic response: B NC = 2F NC D RN F NC
Evolution equation for the internal energy Internal energy ansatz: e = g(ρ)η +h(ĩ 1 (B NC ),Ĩ 2 (B NC ),det F NC ) Evolution equation for the internal energy: de dt = g dρ ρ dt η+gdη dt + h dĩ1 Ĩ1 dt + h dĩ2 Ĩ2 dt + h d det F NC dt (det F NC) Useful identities: d dt (det F NC) = det F NC (Tr D Tr D RN ) h Tr db NC = 2 h ( ( )) B NC : D Tr F NC D RN F NC Ĩ1 dt Ĩ1
Evolution equation for the entropy Evolution equation for the entropy: ρθ dη ( dt = divq+ T δ 2ρ h ( + αρ ) (B NC ) δ : D δ Ĩ1 +2ρ h ( ) Tr F NC D RN F NC Ĩ1 h det F NC ρη αm 2 det F NC 3 ρα h Ĩ1 h Tr B NC ) dθ dt +ρ (det F NC )Tr D RN det F NC
Refinement of constitutive assumptions Solid part is a compressible neo-hookean elastic solid: ) h(ĩ 1,Ĩ 2,det F NC ) = def c 1 (Ĩ1 3 +H(det F NC ) Fluid part is incompressible: Tr D RN = def 0 Fourier s law: Cauchy stress tensor: Entropy: q = def κ θ T δ = def 2ρ h (B NC ) δ Ĩ1 αρ 2 h ρ ρη αm 2 ρα h Tr B NC = def 0 3 Ĩ1
Entropy production Entropy production: ς = 2ρ h ( Tr F NC D RN F NC )+κ θ 2 Ĩ 1 θ
Entropy production Entropy production: Constitutive assumption: ς = 2ρ h ( Tr F NC D RN F NC )+κ θ 2 Ĩ 1 θ Maximization procedure leads to: ς = def 2η 1 F NC D RN 2 F NC D RN F NC = ρ η 1 h Ĩ 1 ( BNC λi ) Recall: B NC = 2F NC D RN F NC
Generalized Maxwell model Introduce notation: ( S = def ρµ 1 BNC λi ) ( ) 1 p = def m+ρµ 1 3 Tr B NC λ Generalized Maxwell model: η 1 S+ ρµ 1 ( 1+ η 1 ρµ 1 divv T = pi+s ) S = 2η 1 λd δ + 2 3 η dλ 1λ(divv) I η 1 dt I
Viscoelastic fluids result Λ 1 dϑ dt = div v 1 Ra 2 3 ρ dv Pr Ra 1 6 dt = div S δ + Ra 1 3Ra 1 6 crit ( m +ρ b ) Λ 1 crit Λ 2 S +(ρ +Λ 2 div v )S = 2ρ λ (D δ + 13 ) (divv) I ρ dλ dt I ρ Ra2 3 dϑ Ra 1 6 dt = ϑ + Λ 3 ρ Tr S Λ 2 2 crit + 2 ( 3 Λ 3Λ 1 ϑ + θ ref )[D : S + ρ θ diff Λ 2 +Λ 3 Ra 1 3 Ra 1 6 crit (λdiv v 12 Tr S )] ( ϑ + θ ) ref dm θ diff dt
Parameter values Parameter Definition Value Pr Ra η 1,ref ρ ref k ref 4.0 10 23 ρ ref gα ref θ diff l 3 char k ref η 1,ref 5.6 10 4 Λ 1 α ref θ diff 1.4 10 2 Λ 2 Λ 3 τ ref t char η 1,ref l 2 char κ ref θ diff t 2 char 1.5 10 4 5.1 10 1 Table: Dimensionless parameters for convection in the Earth s mantle.
Conclusion Thermodynamically consistent model for mechanically incompressible Maxwell type fluids that are thermally compressible or expansible. Identification of the full system of the governing equations. (No Oberbeck Boussinesq type approximation.) Discussion of the validity of the Oberbeck Boussinesq type approximation in experiments/real problems involving viscoelastic fluids. V. Průša and K. R. Rajagopal. On models for viscoelastic materials that are mechanically incompressible and thermally compressible or expansible and their Oberbeck Boussinesq type approximations. Math. Models Meth. Appl. Sci., 23(10):1761 1794, 2013