A constitutive model for non-reacting binary mixtures

A constitutive model for non-reacting binary mixtures Ondřej Souček ondrej.soucek@mff.cuni.cz Joint work with Vít Průša Mathematical Institute Charles University 31 March 2012 Ondřej Souček Charles University) Non-reacting binary mixtures 31 March 2012 1 / 21

Double diffusive convection Huppert & Turner 1981) Ondr ej Souc ek Charles University) Non-reacting binary mixtures 31 March 2012 2 / 21

Mixture theory Truesdell Idea of co-occupancy: v s v w ρ s ρ w θ s, e s θ w, e w velocity velocity density density temperature/internal energy temperature/internal energy Table: Unknowns in problems for binary mixtures. Eckart1940), Truesdell 1957), Samohýl 1987), Rajagopal & Tao 1995) Ondřej Souček Charles University) Non-reacting binary mixtures 31 March 2012 3 / 21

Governing equations Governing equations for components - non-reacting binary mixture of non-polar constituents d s ρ s = ρ s div v s, d w ρ w = ρ w div v w, ρ s d s v s = div T s + ρ s b + π, d w v w ρ w = div T w + ρ w b π d s ρ s e s + 12 ) v s 2 = div T s v s ) div q s + q s + e i + ρ s b s v s d w ρ w e w + 12 ) v w 2 = div T w v w ) div q w + q w e i + ρ w b w v w Ondřej Souček Charles University) Non-reacting binary mixtures 31 March 2012 4 / 21

Governing equations Governing equations for components - non-reacting binary mixture of non-polar constituents d s ρ s d w ρ w ρ s d s v s ρ w d w v w ρ s d s e s ρ w d w e w = ρ s div v s, = ρ w div v w, = div T s + ρ s b + π, = div T w + ρ w b π = T s : v s div q s + q s + e i π v s + ρ s b s v s = T w : v w div q w + q w e i + π v w + ρ w b w v w Ondřej Souček Charles University) Non-reacting binary mixtures 31 March 2012 4 / 21

Single-component form We formulate balance laws for the mixture as a whole, rewrite them in a single-component form in terms of mixture properties and identify additional balances for complementary quantities. Mixture components properties ρ s, ρ w, v s, v w, e s, e w Mixture properties ρ := ρ s + ρ w ρv := ρ s v s + ρ w v w ρe := ρ s e s + ρ w e w + 1 2 ρ su s u s + 1 2 ρ wu w u w where u s := v s v, u w := v w v. Complementary properties only c and, same temperature e s e w not needed) c := ρ s ρ := ρ s v s v) Ondřej Souček Charles University) Non-reacting binary mixtures 31 March 2012 5 / 21

Governing equations in new variables dρ ρ dc ρ dv d ρ de = ρ div v = div = div T + ρb 1 1 = v div v) I) div ρ c 1 ) ) 1 c + div 1 c) T s ct w ) + T c + π = div q + T : Dv) + q Ondřej Souček Charles University) Non-reacting binary mixtures 31 March 2012 6 / 21

Governing equations in new variables with dρ ρ dc ρ dv d ρ de = ρ div v = div = div T + ρb 1 1 = v div v) I) div ρ c 1 ) ) 1 c + div 1 c) T s ct w ) + T c + π = div q + T : Dv) + q T := T s + T w ρ s u s u s ρ w u w u w Ondřej Souček Charles University) Non-reacting binary mixtures 31 March 2012 6 / 21

Governing equations in new variables dρ ρ dc ρ dv d ρ de = ρ div v = div = div T + ρb 1 1 = v div v) I) div ρ c 1 ) ) 1 c + div 1 c) T s ct w ) + T c + π = div q + T : Dv) + q with T := T s + T w 1 ρc 1 c) Ondřej Souček Charles University) Non-reacting binary mixtures 31 March 2012 6 / 21

Governing equations in new variables ρ de = div q + T : Dv) + q where q := J e + J ek + J q + J T J e := ρ s e s u s + ρ w e w u w J ek := 1 2 ρ s u s 2 u s + 1 2 ρ w u w 2 u w J q := q s + q w J T := T s u s T w u w q := q s + q w Ondřej Souček Charles University) Non-reacting binary mixtures 31 March 2012 7 / 21

Governing equations in new variables where ρ de = div q + T : Dv) + q q := J e + J ek + J q + J T J e := e s e w ) J ek := 1 1 2c 2 ρ 2 c1 c)) 2 ) J q := q s + q w ) J T := 1 c)t s ct w ) q := q s + q w Ondřej Souček Charles University) Non-reacting binary mixtures 31 March 2012 7 / 21

Entropy balance ρ s d s η s + div J ηs = ξ s + ζ ρ w d w η w + div J ηw = ξ w ζ ρ dη + div J η = ξ provided we define ρη := ρ s η s + ρ w η w J η := J ηs + J ηw + η s η w ) ξ := ξ s + ξ w. Ondřej Souček Charles University) Non-reacting binary mixtures 31 March 2012 8 / 21

The minimal constitutive model From the energy and entropy balance ρe = ρ s e s + ρ w e w + 1 2 ρ su s u s + 1 2 ρ wu w u w ρη := ρ s η s + ρ w η w Ondřej Souček Charles University) Non-reacting binary mixtures 31 March 2012 9 / 21

The minimal constitutive model From the energy and entropy balance ρe = ρce s + ρ1 c)e w + 1 2 2 ρη := ρcη s + ρ1 c)η w Ondřej Souček Charles University) Non-reacting binary mixtures 31 March 2012 9 / 21

The minimal constitutive model From the energy and entropy balance ρe = ρce s + ρ1 c)e w + 1 2 2 ρη := ρcη s + ρ1 c)η w Constitutive assumption: ρeρ, c, η, ) = ρce s ρ, c, η) + ρ1 c)e w ρ, c, η) + 1 2 2 ρηρ, c, e) := ρ s η s ρ, c, e) + ρ w η w ρ, c, e) Ondřej Souček Charles University) Non-reacting binary mixtures 31 March 2012 9 / 21

The minimal constitutive model - Helmholz potential Defining absolute temperature ϑ := ê η Defining the Helmholz potential our constitutive assumptions imply c,ρ,js Ψ s := e s ϑη s Ψ w := e w ϑη w Ψ := e ϑη Ψρ, c, ϑ, ) = cψ s ρ, c, ϑ) + 1 c)ψ w ρ, c, ϑ) + 1 2 2 ρ 2 c1 c) This is one of the results of theory of fluid mixtures of Müller 1968). Ondřej Souček Charles University) Non-reacting binary mixtures 31 March 2012 10 / 21

The minimal constitutive model Using µ := Ψ c ρ,ϑ,js ) p := ρ 2 Ψ ρ c,ϑ,js ) Ψ ϑ ρ,c,js ) = η We may now express ρϑ dη = ρ de Ψ ρ dρ ρ + Ψ c dc + Ψ dj ) s = div q + T : Dv) + p div v + µ div d, Ondřej Souček Charles University) Non-reacting binary mixtures 31 March 2012 11 / 21

ρϑ dη = p + 4 2 3 + 1 ) ) ) δ 3 trt div v + T δ Js + : D δ v) + 1 c)t s ct w ) J s 1 ρ c 1 )) ) : 1 c ) π + µ + T c) div J q + J e µ J ek ) Ondřej Souček Charles University) Non-reacting binary mixtures 31 March 2012 12 / 21

ρϑ dη = p + 4 2 3 + 1 ) ) ) δ 3 trt div v + T δ Js + : D δ v) + 1 c)t s ct w ) J s 1 ρ c 1 )) ) : 1 c ) π + µ + T c) div J q + J e µ J ek ) ) ) ) µ µ = µ ρ ρ + µ c,ϑ, Js c c + ρc1 c) ρ,ϑ, Js ϑ ϑ ρc1 c) c,ρ, Js ρc1 c) ) + µ J s ρc1 c) c,ρ,ϑ Ondřej Souček Charles University) Non-reacting binary mixtures 31 March 2012 12 / 21

ρϑ dη = p + 4 2 3 + 1 ) ) ) δ 3 trt div v + T δ Js + : D δ v) + 1 c)t s ct w ) J s 1 ρ c 1 )) ) : 1 c ) π + µ + T c) div J q + J e µ J ek ) ) ) ) µ µ = µ ρ ρ + µ c,ϑ, Js c c + ρc1 c) ρ,ϑ, Js ϑ ϑ ρc1 c) c,ρ, Js ρc1 c) ) 1 2c Ondřej Souček Charles University) Non-reacting binary mixtures 31 March 2012 12 / 21

Our model thus suggests to identify ϑξ = p + 4 2 3 + 1 ) 3 trt div v + T δ + ) + 1 c)t s ct w ) : D ) µ µ π + ρ + ρ c c + T c ) Jq + J e µ J ek ) ϑ, ϑ ) ) δ Js : D δ v) ) ) and J η = J q + J e µ J ek ϑ. Ondřej Souček Charles University) Non-reacting binary mixtures 31 March 2012 13 / 21

Our model thus suggests to identify ϑξ = p + 4 2 3 + 1 ) 3 trt div v + T s + T w ) δ : D δ v) + 1 c)t s ct w ) : Du s u w ) ) ) µ µ π + ρ + ρ c c + T c u s u w ) Jq + J e µ J ek ) + µ ) ϑ, ϑ ϑ and J η = J q + J e µ J ek ϑ. Ondřej Souček Charles University) Non-reacting binary mixtures 31 March 2012 13 / 21

Our model thus suggests to identify ϑξ = p + 4 2 3 + 1 ) 3 trt div v + T s + T w ) δ : D δ v) ) + 1 c)t s ct w ) : D ) ) ) µ µ π + ρ + ρ c c + T c Jq + J e µ J ek ) + µ ) ϑ, ϑ ϑ and J η = J q + J e µ J ek ϑ. Ondřej Souček Charles University) Non-reacting binary mixtures 31 March 2012 13 / 21

Our model thus suggests to identify ϑξ = p + 4 2 3 + 1 ) 3 trt div v + T s + T w ) δ : D δ v) ) + 1 c)t s ct w ) : D ) µ µ µ π + ρ + c + ρ c ϑ ϑ ) Jq + J e µ J ek ) ϑ, ϑ ) + T c ) and J η = J q + J e µ J ek ϑ. Ondřej Souček Charles University) Non-reacting binary mixtures 31 March 2012 13 / 21

Our model thus suggests to identify ϑξ = p + 4 2 3 + 1 ) 3 trt div v + T s + T w ) δ : D δ v) ) + 1 c)t s ct w ) : D ) ) µ µ π + ρ + c + α µ ρ c ϑ ϑ + T c Jq + J e µ J ek ) + 1 α) µ ϑ ϑ ) ϑ, ) and J η = J q + J e µ J ek ϑ. Ondřej Souček Charles University) Non-reacting binary mixtures 31 March 2012 13 / 21

Let us postulate the rate of entropy production as 2ν + 3λ ϑξ = div v) 2 3 ) + 2νD d v) : D d v) + 2 νd ) + k for ν, ν 0, 2ν + 3λ 0, k 0, κ 0. : D ) + κ ϑ ϑ 0 ϑ Let us employ the postulate of maximization of rate of entropy production with respect to the thermodynamic ) affinities see ) e.g. Rajagopal & Srinivasa 2004)) div v, D d J v), D s J ρc1 c), s ρc1 c), ϑ ) Ondřej Souček Charles University) Non-reacting binary mixtures 31 March 2012 14 / 21

Final constitutive relations µ π + ρ J q +J e µ J ek ) ϑ p + 4 2 3 + 1 2ν + 3λ trt = div v 3 3 ) δ T δ Js + = 2νD δ v) ) 1 c)t s ct w ) = 2 νd ) + T c = k µ ρ + c + α µ c ϑ ϑ + 1 α) µ ϑ = κ ϑ ϑ Ondřej Souček Charles University) Non-reacting binary mixtures 31 March 2012 15 / 21

Final constitutive relations µ π + ρ J q +J e µ J ek ) ϑ p + 4 2 3 + 1 2ν + 3λ trt = 3 3 µ ρ + c + α µ c ϑ ϑ T δ s + T δ w = 2νD δ v) div v 1 c)t s ct w ) ) = 2 νd u s u w ) + T c = k + 1 α) µ ϑ = κ ϑ ϑ Ondřej Souček Charles University) Non-reacting binary mixtures 31 March 2012 15 / 21

Heat and entropy fluxes Let us inspect the final form of heat flux q and entropy flux J η q = κ ϑ + J e + J ek + J T + c e s c + 1 c) e w c αϑ ) c cη s + 1 c)η w ) J η = 1 ϑ q 2J e k J T µ ) = κ ϑ 1 α) µ ϑ ϑ Ondřej Souček Charles University) Non-reacting binary mixtures 31 March 2012 16 / 21

Heat and entropy fluxes Let us inspect the final form of heat flux q and entropy flux J η q = κ ϑ + J e + J ek + J T + c e s c + 1 c) e w c αϑ ) c cη s + 1 c)η w ) J η = 1 ϑ q 2J e k J T µ ) = κ ϑ ϑ + 1 α) c cη s + 1 c)η w ) Ondřej Souček Charles University) Non-reacting binary mixtures 31 March 2012 16 / 21

Heat and entropy fluxes Let us inspect the final form of heat flux q and entropy flux J η q = κ ϑ + J e + J ek + J T + c e s c + 1 c) e w c αϑ ) c cη s + 1 c)η w ) J η = 1 ϑ q 2J e k J T µ ) = κ ϑ ϑ + 1 α) c cη s + 1 c)η w ) Choice of α?: α = 0 Ondřej Souček Charles University) Non-reacting binary mixtures 31 March 2012 16 / 21

Heat and entropy fluxes Let us inspect the final form of heat flux q and entropy flux J η q = κ ϑ + J e + J ek + J T + c e ) s c + 1 c) e w c J η = 1 ϑ q 2J e k J T µ ) = κ ϑ ϑ + η s η w ) + c η ) s c + 1 c) η w c Choice of α?: α = 0 satisfactory - no entropy contribution in heat flux q + correct diffusive term present in entropy flux J η Ondřej Souček Charles University) Non-reacting binary mixtures 31 March 2012 16 / 21

Heat and entropy fluxes Let us inspect the final form of heat flux q and entropy flux J η q = κ ϑ + J e + J ek + J T + c e ) s c + 1 c) e w c J η = 1 ϑ q 2J e k J T µ ) = κ ϑ ϑ + η s η w ) + c η ) s c + 1 c) η w c Of particular interest is the quantity ϑj η q Ondřej Souček Charles University) Non-reacting binary mixtures 31 March 2012 16 / 21

Comparison ϑj η q Our result ϑj η q = J ek J T c cψ s + 1 c)ψ w ) Fluid Mixture theory of Bowen 1976) entropy flux postulated) ϑj η q = J ek J T Ψ s Ψ w ) Müller 1984) theory for mixtures of non-viscous fluids entropy flux derived) ϑj η q = J ek J T Ψ s Ψ w ) Ondřej Souček Charles University) Non-reacting binary mixtures 31 March 2012 17 / 21

Evolution equation for d 1 1 = v div v) I) div ρ c 1 ) ) 1 c + div 1 c) T s ct w ) + T c + π Ondřej Souček Charles University) Non-reacting binary mixtures 31 March 2012 18 / 21

Evolution equation for d 1 1 = v div v) I) div ρ c 1 ) ) 1 c + div 1 c) T s ct w ) + T c + π Using the constitutive relations for π + T c and 1 c)t s ct w : d 1 1 = v div v) I) + div ρ c 1 ) ) 1 c )) ) µ µ + div 2 ν sym ρ + ρ c c k Ondřej Souček Charles University) Non-reacting binary mixtures 31 March 2012 18 / 21

Evolution equation for d 1 1 = v div v) I) div ρ c 1 ) ) 1 c + div 1 c) T s ct w ) + T c + π Using the constitutive relations for π + T c and 1 c)t s ct w : d 1 1 = v div v) I) + div ρ c 1 ) ) 1 c )) ) µ µ + div 2 ν sym ρ + ρ c c k Provided fast relaxation djs 0 and neglecting non-linear terms in v,, and assuming ν 0, one obtains ) µ µ k ρ + ρ c c Ondřej Souček Charles University) Non-reacting binary mixtures 31 March 2012 18 / 21

Conclusions Key features of the presented model: reformulation in traditional variables ρ, c, v, ϑ and also, for all of them evolution equations are available - equivalent to 2 mass balances, 2 momentum balances and single energy balance class II mixtures Hutter)) Minimal scenario for Helmholz potential implied by balance eq. Ψρ, c, ϑ, ) = cψ s ρ, c, ϑ) + 1 c)ψ w ρ, c, ϑ) + 1 2 2 ρ 2 c1 c) fits well the framework of earlier mixture theories. Choice of rate of entropy production - satisfaction of 2nd law of thermodynamics Maximization of r.e.p. constitutive equations Ondřej Souček Charles University) Non-reacting binary mixtures 31 March 2012 19 / 21

Conclusions II Classical viscous-fluid relations for fluid part T s + T w Dv) Additional relation for 1 c)t s ct w Du s u w ) Interaction force π contains all relevant for diffusion) mechanisms. Energy and entropy flux consistent extensions of theories of Müller and Bowen. Physical limit of evolution eq. for diff. flux yields classical Fick s diffusion. Ondřej Souček Charles University) Non-reacting binary mixtures 31 March 2012 20 / 21

Thank you for your attention! Ondřej Souček Charles University) Non-reacting binary mixtures 31 March 2012 21 / 21