Flow of the glass melt through a die: Stationary flow with fixed boundaries of mechanically incompressible, but thermally expansible, viscous fluids

Lecture at the CIME-EMS School Mathematical models in the manufacturing of glass, polymers and textiles, Montecatini Terme, Italy, September 14, 2008 p. 1/3 Flow of the glass melt through a die: Stationary flow with fixed boundaries of mechanically incompressible, but thermally expansible, viscous fluids Andro Mikelić Institut Camille Jordan, UFR Mathématiques, Université Claude Bernard Lyon 1, Lyon, FRANCE

Lecture at the CIME-EMS School Mathematical models in the manufacturing of glass, polymers and textiles, Montecatini Terme, Italy, September 14, 2008 p. 2/3 Thanks: The results which I will present today are obtained in collaboration with A. Farina and A. Fasano (Dipartimento di Matematica, Universita degli Studi di Firenze, Italy). Detailed proofs are in A. Farina, A. Fasano, A. Mikelić : On the equations governing the flow of mechanically incompressible, but thermally expansible, viscous fluids, to appear in M 3 AS : Math. Models Methods Appl. Sci., Vol. 18, no. 6 (2008).

Lecture at the CIME-EMS School Mathematical models in the manufacturing of glass, polymers and textiles, Montecatini Terme, Italy, September 14, 2008 p. 3/3 Introduction In this talk, I ll present the study of the system div v = α v ϑ, ρ(ϑ) = 1 αϑ (1) ρ(ϑ) ρ(ϑ)( v ) v = Ar ϑ e g P + 1 Re Div { 2µ(ϑ)D( v) } 2 3Re ( µ(ϑ) div v ) (2) ρ(ϑ)c p1 (ϑ) v ϑ = 1 Pe div ( λ(ϑ) ϑ ), (3)

Lecture at the CIME-EMS School Mathematical models in the manufacturing of glass, polymers and textiles, Montecatini Terme, Italy, September 14, 2008 p. 3/3 Introduction In this talk, I ll present the study of the system div v = α v ϑ, ρ(ϑ) = 1 αϑ (4) ρ(ϑ) ρ(ϑ)( v ) v = Ar ϑ e g P + 1 Re Div { 2µ(ϑ)D( v) } 2 3Re ( µ(ϑ) div v ) (5) ρ(ϑ)c p1 (ϑ) v ϑ = 1 Pe div ( λ(ϑ) ϑ ), (6) in a cylindrical domain Ω = {r < R(x 3,φ) 1} [0, 2π] (0, 1) in IR 3. R : [0, 1] [0, 2π] (0, 1] is a C -map.

Lecture at the CIME-EMS School Mathematical models in the manufacturing of glass, polymers and textiles, Montecatini Terme, Italy, September 14, 2008 p. 4/3 I1 The boundary of Ω contains 3 distinct parts: the lateral boundary Γ lat = {r = R(x 3,φ}, the inlet boundary Γ in = {x 3 = 1 and r R(1,φ), φ [0, 2π]} and the outlet boundary Γ out = {x 3 = 0 and r R(0,φ), φ [0, 2π]}. α 0 is a parameter (the "expansivity"). System (1)-(3) describes a dimensionless model for the stationary motion of a mechanically incompressible, but thermally expansible viscous fluid. It is widely used in industrial simulations of flows of hot melted glasses, polymers etc. We ll consider the system (10)-(12) in the realistic situation, when the parameter α = K ρ (T w 1) (the " expansivity or the thermal expansion coefficient ) is small.

I1a Lecture at the CIME-EMS School Mathematical models in the manufacturing of glass, polymers and textiles, Montecatini Terme, Italy, September 14, 2008 p. 5/3

M10 Lecture at the CIME-EMS School Mathematical models in the manufacturing of glass, polymers and textiles, Montecatini Terme, Italy, September 14, 2008 p. 6/3

M10 Lecture at the CIME-EMS School Mathematical models in the manufacturing of glass, polymers and textiles, Montecatini Terme, Italy, September 14, 2008 p. 7/3

Lecture at the CIME-EMS School Mathematical models in the manufacturing of glass, polymers and textiles, Montecatini Terme, Italy, September 14, 2008 p. 8/3 M11 Our plan is the following: We prove that the system (1) (3), with suitable boundary conditions, has a solution. We establish uniqueness of the solution for small data. We prove that, in the limit α 0, our system reduces to the Oberbeck-Boussinesq system. Furthermore, we calculate the first correction term in this asymptotic limit. We recall the Oberbeck-Boussinesq system: D v OB Dt div v OB = 0 (7) = Ar ϑ OB e g p OB mot + 1 Re Div { 2µ(ϑ OB )D( v OB ) } (8)

Lecture at the CIME-EMS School Mathematical models in the manufacturing of glass, polymers and textiles, Montecatini Terme, Italy, September 14, 2008 p. 9/3 M12 c p1 (ϑ OB ) DϑOB Dt = 1 Pe div ( λ(ϑ OB ) ϑ OB). (9) In the situation which is of interest for us, the quantities from (??) are small but non-zero. Our goal is to study a model which generalizes the Oberbeck-Boussinesq approximation and reduces to it in the limit.

M12 c p1 (ϑ OB ) DϑOB Dt = 1 Pe div ( λ(ϑ OB ) ϑ OB). (13) In the situation which is of interest for us, the quantities from (??) are small but non-zero. Our goal is to study a model which generalizes the Oberbeck-Boussinesq approximation and reduces to it in the limit. NB: The full non-stationary system: div v = K ρ ρ(ϑ) (T w 1) Dϑ (14) Dt ρ(ϑ) D v Dt = Ar ϑ e g (P + 2 3Re µ(ϑ) div v) + 1 Div { 2µ(ϑ)D( v) Re (15) ρ(ϑ)c p1 (ϑ) Dϑ Dt = 1 Pe div ( λ(ϑ) ϑ ), (16) Lecture at the CIME-EMS School Mathematical models in the manufacturing of glass, polymers and textiles, Montecatini Terme, Italy, September 14, 2008 p. 9/3

Lecture at the CIME-EMS School Mathematical models in the manufacturing of glass, polymers and textiles, Montecatini Terme, Italy, September 14, 2008 p. 10/3 Existence for the stationary problem Problem A : We consider the system (1) (3) in Ω, with the boundary conditions v = v 1 e g, ϑ = 1 on Γ in (17) v = v 2 e g, ϑ = 0 on Γ out (18) v = 0, 1 Pe λ(ϑ) ϑ n = q 0ϑ + S on Γ lat (19) v 1 C 1 0 (Γ in) C ( Γ in ), v 2 C 1 0 (Γ out) C ( Γ out ), Γ in ρ(1)v 1 rdrdφ = S C ( Γ lat ), S 0 (20) (1 α)v 1 rdrdφ = v 2 ρ(0) rdrdφ Γ in Γ out (21)

Lecture at the CIME-EMS School Mathematical models in the manufacturing of glass, polymers and textiles, Montecatini Terme, Italy, September 14, 2008 p. 11/3 E1 We start by studying the energy equation for a given w = ρ v H(Ω), where H(Ω) = { z L 3 (Ω) 3 div z = 0 in Ω, z n = 0 on Γ lat, z n Γin = ρ(1)v 1, z n Γout = ρ(0)v 2 }. Our nonlinearities c p1 and λ are defined only for ϑ such that the density is not negative. We extend it on IR by setting c p1 (ϑ) = { cp1 (1)/ϑ 2 for ϑ > 1 ( S L (Γ lat ) q 0 ϑ ) 2 c p1 ( S L (Γ lat ) q 0 ) for ϑ < S L (Γ lat ) q 0, (24) λ(ϑ) = { λ(1) for ϑ > 1 λ( S L (Γ lat ) q 0 ) for ϑ < S L (Γ lat ) q 0. (25)

Lecture at the CIME-EMS School Mathematical models in the manufacturing of glass, polymers and textiles, Montecatini Terme, Italy, September 14, 2008 p. 12/3 E2 We will prove that ϑ [ S L (Γ lat ) q 0, 1] Now for a given { w H(Ω), S L (Γ lat ), S 0 λ,c p1 W 1, (IR), λ λ 0 and constants q 0, Pe 0, we consider the problem Problem Θ: Find ϑ H 1 (Ω) L (Ω) such that (30) c p1 (ϑ) w ϑ = 1 Pe div ( λ(ϑ) ϑ ), (31) ϑ = 1 on Γ in and ϑ = 0 on Γ out (32) 1 Pe λ(ϑ) ϑ n = q 0ϑ + S on Γ lat (33)

Lecture at the CIME-EMS School Mathematical models in the manufacturing of glass, polymers and textiles, Montecatini Terme, Italy, September 14, 2008 p. 13/3 E3 Proposition 2 Under the stated assumptions, Problem Θ has at least one variational solution in H 1 (Ω), satisfying the estimate λ 0 2Pe ϑ 2 L 2 (Ω) + q 0 3 4 ϑ 2 L 2 (Γ lat ) A 0 + B 0 w L1 (Ω) 3, (34) where B 0 = c p1 max { 1, S } L (Γ lat ), q 0 A 0 = qo + 4 4q 0 S L2 (Γ lat ) + 2 q 0 Γ lat + + λ 2Peλ 0 Ω + 3 2 ρ c p1 v 1 L2 (Γ in ).

Lecture at the CIME-EMS School Mathematical models in the manufacturing of glass, polymers and textiles, Montecatini Terme, Italy, September 14, 2008 p. 14/3 E4 Corollary 1 Under the assumptions of Proposition 2, we have the following estimate { Pe ϑ 2 L 2 (Ω) 8 max, 1 } (A0 ) x3 R λ o q + B 0 w L1(Ω). 3 o Lemma 1 Any variational solution ϑ H 1 (Ω) to Problem Θ, satisfying the a priori estimate (34), satisfies also (35) S L (Γ lat ) q 0 ϑ 1 a.e. on Ω. (36)

Lecture at the CIME-EMS School Mathematical models in the manufacturing of glass, polymers and textiles, Montecatini Terme, Italy, September 14, 2008 p. 15/3 E4A Remark 4 Under the condition that the interior angle between the lateral boundary r = R(x 3,φ) C 2 and the upper and lower surfaces is π/2, we can apply the elliptic regularity. This can be seen directly by extending the equation for x 3 > 1 and x 3 < 0. Let l 3 (Ω) be the elliptic regularity constant in estimating the W 1,3 (Ω) norm using elliptic potentials, i.e. the solution θ for the mixed problem div ( θ f) = 0 in Ω (37) θ n = g on Γ lat and θ = 0 on Γ out Γ in (38) is estimated as } θ L3 (Ω) l 3(Ω) { g 3 L3(Γlat) + f L3(Ω) 3. (39)

Lecture at the CIME-EMS School Mathematical models in the manufacturing of glass, polymers and textiles, Montecatini Terme, Italy, September 14, 2008 p. 16/3 E5 Then we have where E 00 and ϑ L3 (Ω) 3 E 00 + E 01 w L3 (Ω), (40) = Γ lat 1/3 l 3 (Ω)Pe λ 0 + λ ( Ω 1/3 + l 3 (Ω) Γ lat 1/3), λ 0 ( { S L (Γ lat ) + q o max 1, S } L (Γ lat ) q 0

Lecture at the CIME-EMS School Mathematical models in the manufacturing of glass, polymers and textiles, Montecatini Terme, Italy, September 14, 2008 p. 16/3 E5 Then we have where E 00 ϑ L3 (Ω) 3 E 00 + E 01 w L3 (Ω), (43) = Γ lat 1/3 l 3 (Ω)Pe λ 0 + λ ( Ω 1/3 + l 3 (Ω) Γ lat 1/3), λ 0 ( { S L (Γ lat ) + q o max 1, S } L (Γ lat ) q 0 and E 01 = l { 3(Ω) Pe c p1 max 1, S } L (Γ lat ). (45) λ 0 q 0

Lecture at the CIME-EMS School Mathematical models in the manufacturing of glass, polymers and textiles, Montecatini Terme, Italy, September 14, 2008 p. 17/3 E6 Now we turn to the continuity and momentum equations, determining the velocity field w = ρ v and the pressure p. Our system reads For given ϑ determine { w,p} satisfying div w = 0 in Ω (46) ( w ) w ρ = Arϑ e g (p + 2µ 3Re div w ρ )+ 1 Re Div ( 2µ(ϑ)D( w ρ )) in Ω (47) w = 0 on Γ lat, w = (1 α)v 1 e g on Γ in and w = v 2 e g on Γ out, (48)

Lecture at the CIME-EMS School Mathematical models in the manufacturing of glass, polymers and textiles, Montecatini Terme, Italy, September 14, 2008 p. 17/3 E6 Now we turn to the continuity and momentum equations, determining the velocity field w = ρ v and the pressure p. Our system reads For given ϑ determine { w,p} satisfying div w = 0 in Ω (49) ( w ) w ρ = Arϑ e g (p + 2µ 3Re div w ρ )+ 1 Re Div ( 2µ(ϑ)D( w ρ )) in Ω (50) w = 0 on Γ lat, w = (1 α)v 1 e g on Γ in and w = v 2 e g on Γ out, (51) Next, there exists

Lecture at the CIME-EMS School Mathematical models in the manufacturing of glass, polymers and textiles, Montecatini Terme, Italy, September 14, 2008 p. 18/3 E7 ζ H 2 (Ω) 3, ζ L 3 (Ω) 9, with curl ζ satisfying (48) (55) Now we adapt the well-known Hopf construction to our non-standard nonlinearities. we have Proposition 2 Let us suppose that 1/ρ L (Ω), 1/ρ 1/ρ min. Then, for every γ > 0 there is a ξ, depending on γ and ρ min, such that ξ H 1 (Ω) 3, div ξ = 0 in Ω, ξ = ζ on Ω and (56) 1 ρ ( φ ) φ ξ dx γ φ 2 H 1 (Ω), φ H0(Ω) 1 3 3 (57) Ω Let us introduce now some useful constants:

Lecture at the CIME-EMS School Mathematical models in the manufacturing of glass, polymers and textiles, Montecatini Terme, Italy, September 14, 2008 p. 19/3 E8 γ = 2µ min ρ max Re, A 00 = 4 6 ρ 2 min, B 00 = µ min ρ max, C 01 = 2 6 1/6 Ω 5/6 B 01 = 2 6 1/3 1 ρ 2 ξ L6 (Ω) 3, B 02 = 4µ max6 1/6 min ρ 3 min C 00 = 1 ( ξ 2 ρ L6 (Ω) + 2µ max 3 min Re D( ) ξ) L2 (Ω). 9 (61) Theorem 1 Let us suppose that B = 1 Re( B00 αb 02 ϑ L3 (Ω) 3 ) αb01 ϑ L2 (Ω) 3 > 0 (62) and det = 2 B 4A 00α ϑ L2 (Ω) 3(C 00 + ArC 01 ϑ L2 (Ω) 3) > 0, (63)

Lecture at the CIME-EMS School Mathematical models in the manufacturing of glass, polymers and textiles, Montecatini Terme, Italy, September 14, 2008 p. 20/3 E9 Then there is a solution w H 1 (Ω) 3 for the problem (46)-(48), satisfying the estimate D( w) L2 (Ω) 9 D( ξ) L2 (Ω) 9 + C 00 + ArC 01 ϑ L2 (Ω) 3 det (65) Next we define our iterative procedure: Let γ = µ min ρ max Re and ξ be the corresponding vector valued function from Hopf s construction. For a given w m = W m + ξ, such that W m B R = { z H 1 0 (Ω)3 : div z = 0 in Ω and D( W m ) L2 (Ω) 9 R}, we calculate ϑm, a solution to (27)-(29).

Lecture at the CIME-EMS School Mathematical models in the manufacturing of glass, polymers and textiles, Montecatini Terme, Italy, September 14, 2008 p. 21/3 E9A Next, with this ϑ m, we determine a solution w m+1 = W m+1 + ξ for the problem (46)-(48), satisfying the estimate (64). The natural question arising in the iterative process is if W m+1 remains in B R? We have the following result Proposition 3 Let the constants A 0 and B 0 be given as in Proposition 1 and let E 00 and E 01 be given by Lemma 2. Let the constants B 00,B 01,B 02,A 00,C 00 and C 01 be given by formula (58). Let ξ be generalized Hopf s lift, given by Proposition 2 and corresponding to γ. Let R be given by

Lecture at the CIME-EMS School Mathematical models in the manufacturing of glass, polymers and textiles, Montecatini Terme, Italy, September 14, 2008 p. 22/3 E10 R = 2Re ρ ( max 2Ar Ω 5/6 6 1/6 max{1, S L (Γ lat ) } + µ min q 0 1 ( ξ 2 ρ L6 (Ω) + 2µ ) max 3 min Re D( ) ξ) L2 (Ω) 9 Then for all α > 0 such that. (66) 1 = 1 ( ( B00 αb 02 E 00 + E 01 ( ξ L3 (Ω) Re 3 ) ) +48 1/12 Ω 1/6 Pe ( R) αb01 2A0 + λ 0 ( 2B0 ξ 1/2 + Ω 1/4 HR 1/2)) > 0 and (67) L 1 (Ω) 3

Lecture at the CIME-EMS School Mathematical models in the manufacturing of glass, polymers and textiles, Montecatini Terme, Italy, September 14, 2008 p. 23/3 E11 2 = 2 1 4A 00 α 2Pe [ C 00 + ArC 01 max{1, S L (Γ lat ) q 0 } λ 0 [ A0 + B 0 ( ξ 1/2 L 1 (Ω) 3 + Ω 1/4 R )] ] > B2 00 2Re2, (70) W m B R implies W m+1 B R. Theorem 2 There is a weak solution {ϑ, v} W 1,3 (Ω) H 1 (Ω) 3 for the Problem A, such that ϑ 2Pe L2 (Ω) 3 λ 0 ( A 0 + B 0 ξ L1 (Ω) + ) B 0 Ω 1/2 R, S L (Γ lat ) q 0 ϑ 1, and D( v ξ) L2 (Ω) R, 9 (71)

Lecture at the CIME-EMS School Mathematical models in the manufacturing of glass, polymers and textiles, Montecatini Terme, Italy, September 14, 2008 p. 24/3 E12 where ξ is given by Proposition 2 with γ = 2µ min /(ρ max Re). Now we are in position to pass to the limit when the expansivity parameter α tends to zero. First we remark that the a priori estimates from the previous section are independent of α, α α 0, where α 0 is the maximal positive α satisfying (67)-(68). Consequently we have

Lecture at the CIME-EMS School Mathematical models in the manufacturing of glass, polymers and textiles, Montecatini Terme, Italy, September 14, 2008 p. 25/3 E13 Theorem 3 Let {ϑ(α), v(α)}, α (0,α 0 ), be a sequence of weak solutions to Problem A, satisfying the bounds (69). Then there exists {ϑ OB, v OB } W 1,3 (Ω) H 1 (Ω) 3 and a subsequence {ϑ(α k ), v(α k )} such that ϑ(α k ) ϑ OB, uniformly on Ω ϑ(α k ) ϑ OB, weakly in W 1,3 (Ω) v(α) v OB, weakly in H 1 (Ω) 3. Furthermore, {ϑ OB, v OB } is a weak solution for the equations (7) (9), satisfying the boundary conditions (17)-(21) and the bounds (69).

Uniqueness Lecture at the CIME-EMS School Mathematical models in the manufacturing of glass, polymers and textiles, Montecatini Terme, Italy, September 14, 2008 p. 26/3

Lecture at the CIME-EMS School Mathematical models in the manufacturing of glass, polymers and textiles, Montecatini Terme, Italy, September 14, 2008 p. 26/3 Uniqueness The uniqueness Quite technical. For small data there is a unique weak solution { v,ϑ} H 1 (Ω) 3 (W 1,3 (Ω) C( Ω)) for Problem A, satisfying the bounds (69).

Lecture at the CIME-EMS School Mathematical models in the manufacturing of glass, polymers and textiles, Montecatini Terme, Italy, September 14, 2008 p. 27/3 The regularity of solutions Lemma 22 Let c p0 and λ C (IR). Furthermore let Γ lat C and S C ( Γ lat ). Then ϑ W 2,6 (Ω) C 1,1/2 ( Ω). Lemma 23 Let ϑ W 2,6 (Ω), let Γ lat C and let v j C, j = 1, 2 satisfy (20). Then { w,p} W 2,q (Ω) W 1,q (Ω), q < +. Theorem 24 Let the assumptions on the data from Lemmas 22 and 23 hold true. Then every weak solution { v,p,ϑ} H 1 (Ω) 3 L 2 (Ω) (H 1 (Ω) L (Ω)) for Problem A is an element of W 2,q (Ω) 3 W 1,q (Ω) W 2,q (Ω), q <. Furthermore { v,p,ϑ} C (Ω) 5.

Lecture at the CIME-EMS School Mathematical models in the manufacturing of glass, polymers and textiles, Montecatini Terme, Italy, September 14, 2008 p. 28/3 The Boussinesq limit In this section we reconsider the limit when the expansivity parameter α tends to zero. We saw at the end of the section on existence of a weak solution that the obtained a priori estimates allow passing to the Boussinesq limit. Having justified the Oberbeck-Boussinesq system as the limit equations when the expansivity parameter α tends to zero, the next question is: What is the accuracy of the approximation?. The answer relies on the uniqueness and regularity results from the previous sections.

(75) Lecture at the CIME-EMS School Mathematical models in the manufacturing of glass, polymers and textiles, Montecatini Terme, Italy, September 14, 2008 p. 29/3 BL1 Proposition 25 Let us suppose that the conditions (67)-(68), and (smallness of data), ensuring existence, uniqueness and regularity of a solution lying inside the ball defined by the bounds (69). Furthermore let the solution { v, p, ϑ} satisfies the inequalities N = λ 0 Pe C 6(Ω) (1 + H ( ) c p1 2 L (IR) v L3 (Ω) 3ρ max+ 2 2 Re λ ) L (IR) ϑ L3 (Ω) > 0 3 (74) Pe µ min > 4( 3 ρ max 2 )1/4 H D( v) L2 (Ω) 9 + HL {2α Θ Re µ max ρ min v L (Ω) 3 +Ar H2 2 + 2 Re µ L (IR) D( v) L3 (Ω) 9C 6(Ω) + αh v 2 L (Ω) 3 },

Lecture at the CIME-EMS School Mathematical models in the manufacturing of glass, polymers and textiles, Montecatini Terme, Italy, September 14, 2008 p. 30/3 BL2 where L Θ = H c p1 L (IR) ϑ L (Ω) 2N (77) Then derivatives of the solution, with respect to α, exist at all orders as continuous functions of α. With this result, we are ready to state the error estimate for Boussinesq s limit. First, we write the 1st order correction, i.e. the system defining the first derivatives { w 0, π 0,θ 0 } = d dα { v, p,ϑ} α=0:

Lecture at the CIME-EMS School Mathematical models in the manufacturing of glass, polymers and textiles, Montecatini Terme, Italy, September 14, 2008 p. 31/3 BL3 div { w 0 ϑ OB v OB} = 0 in Ω (78) ϑ OB ( v OB ) v OB + { ( w 0 ) v OB + ( v OB ) w 0} = Ar θ 0 e g π 0 + 2 Re Div { µ(ϑ OB )D( w 0 ) + µ (ϑ OB )θ 0 D( v OB ) } in Ω (79) div { λ(ϑob ) θ 0 + ( v OB c p (ϑ OB ) λ (ϑ OB ) ϑ OB )θ 0 Pe Pe ϑ OB v OB C p (ϑ OB ) + w 0 C p (ϑ OB ) } = 0 in Ω (80) θ 0 = 0, w 0 = 0 on Γ out ; θ 0 = 0, w 0 = V 2 e g on Γ in (81) w 0 = 0 and 1 Pe (λ(ϑob ) θ 0 + λ (ϑ OB )θ 0 ϑ OB ) n = q 0 θ 0 on Γ lat, (82)

Lecture at the CIME-EMS School Mathematical models in the manufacturing of glass, polymers and textiles, Montecatini Terme, Italy, September 14, 2008 p. 32/3 BL4 Under the conditions of the preceding Proposition, with α = 0, the system (78)-(82) has a unique smooth solution. Hence we have established rigorously the O(α 2 ) approximation for Problem A. Clearly, one could continue to any order. The result is given by the following theorem, which is a straightforward corollary of Proposition 25. Theorem 26 Let us suppose the assumptions of Proposition 25. Then we have v v OB α w 0 W 1, (Ω) 3 + ϑ ϑob αθ 0 W 1, (Ω) Cα 2 (85) inf C IR p pob αp 0 + C L (Ω) 3 Cα2 (86) where p 0 = π 0 2µ(ϑ OB ) div w 0 /(3 Re).

References [1] S.N. Antontsev, A.V. Kazhikhov, V.N. Monakhov, Boundary Value Problems in Mechanics of Nonhomogeneous Fluids, North-Holland, Amsterdam, 1990. [2] S.E. Bechtel, M. G. Forest, F.J. Rooney, Q. Wang, Thermal expansion models of viscous fluids based on limits of free energy, Physics of Fluids, Vol. 15, 2681-2693 (2003). [3] S.E. Bechtel, F.J. Rooney, M. G. Forest, Internal constraint theories for the thermal expansion of viscous fluids, International Journal of Engineering Science, Vol. 42, 43-64 (2004). [4] H. Beirao da Veiga, An L p -Theory for the n Dimensional, stationary, compressible Navier-Stokes Equations, and the Incompressible Limit for Compressible Limit for Compressible Fluids. The Equilibrium Solutions, Commun. Math. Phys., Vol. 109, 229-248 (1987). [5] J.I. Diaz, G. Galiano, Existence and uniqueness of solutions of the Boussinesq system with nonlinear thermal diffusion, Topol. Methods Nonlinear Anal. 11, 59 82 (1998). [6] G.P. Galdi, An Introduction to the Mathematical Theory of the Navier-Stokes Equations, Vol. I: Linearised Steady 32-1

Problems, Vol. II: Nonlinear Steady Problems, Springer- Verlag, New York, 1994. [7] G. Gallavotti, Foundations of Fluid Dynamics, Springer- Verlag, Berlin, 2002. [8] D. Gilbarg, N.S. Trudinger, Elliptic Partial Differential Equations of Second Order, 2nd edition, Springer-Verlag, Berlin, 1983. [9] R.H. Hills, P.H. Roberts, On the motion of a fluid that is incompressible in a generalized sense and its relationship to the Boussinesq Approximation, Stability Appl. Anal. Continua, Vol. 3, 205-212 (1991). [10] Y.Kagei, M. R užička, G. Thäter, Natural Convection with Dissipative Heating, Commun. Math. Phys., Vol. 214, 287-313 (2000). [11] O.A. Ladyzhenskaya and N.N. Uralceva, The mathematical theory of viscous incompressible flow, Gordon and Breach Science Publishers, New York, 1987. [12] O.A. Ladyzhenskaya and N.N. Uralceva, Linear and quasilinear elliptic equations, Academic Press, New York, 1968. 32-2

[13] L. Landau, E. Lifchitz, Mécanique des fluides, 2ème edition revue et complété, Editions Mir, Moscow, 1989. [14] K.R. Rajagopal, M. R užička, A.R. Srinivasa, On the Oberbeck-Boussinesq approximation, Math. Models Methods Appl. Sci, Vol. 6, 1157-1167 (1996). [15] H. Schlichting, K. Gersten, Boundary-Layer Theory, 8th edition, Springer, Heidelberg, 2000. [16] J.E. Shelby, Introduction to Glass Science and Technology, 2nd edition, RSCP Publishing, London, 2005. [17] R. Temam, Navier-Stokes equations, Revised edition, Elsevier Science Publishers, Amsterdam, 1985. [18] R. Temam, Infinite-Dimensional Dynamical systems in Mechanics and Physics, Springer-Verlag, New York, 1988. [19] R.Kh. Zeytounian, The Bénard problem for deep convection: rigorous derivation of approximate equations, Int. J. Engng. Sci., Vol. 27, 1361-1366 (1989). [20] R.Kh. Zeytounian, The Bénard-Marangoni thermocapillary instability problem: On the role of the buoyancy, Int. J. Engng. Sci., Vol. 35, 455-466 (1997). 32-3

[21] R.Kh. Zeytounian, The Bénard-Marangoni thermocapillary instability problem, Phys.-Uspekhi, Vol. 41, 241-267 (1998). [22] R.Kh. Zeytounian, Joseph Boussinesq and his approximation: a contemporary view, C.R. Mécanique, Vol. 331, 575-586 (2003). 32-4