Asymptotic analysis of the glass fibres shape

Transcription

1 Lecture at the CIME-EMS School Mathematical models in the manufacturing of glass, polymers and textiles, Montecatini Terme, Italy, September 10, 2008 p. 1/5 Asymptotic analysis of the glass fibres shape Andro Mikelić Institut Camille Jordan, UFR Mathématiques, Université Claude Bernard Lyon 1, Lyon, FRANCE

2 Lecture at the CIME-EMS School Mathematical models in the manufacturing of glass, polymers and textiles, Montecatini Terme, Italy, September 10, 2008 p. 2/5 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) and T. clopeau (UFr Mathématiques, Université Lyon 1). Detailed proofs are in T. Clopeau, A. Farina, A. Fasano, A. Mikelić : Asymptotic equations for the terminal phase of glass fiber drawing and their analysis, under revision in Nonlinear Analysis TMA: Real World Applications, 2008.

3 Lecture at the CIME-EMS School Mathematical models in the manufacturing of glass, polymers and textiles, Montecatini Terme, Italy, September 10, 2008 p. 3/5 Free Jet 1 In the process of glass fibers drawing, molten glass flows from a reservoir into a die and then it feeds a filament which is rapidly cooled to acquire a sufficiently high viscosity and it is pulled by a spinning device. During the whole process the glass can be modelled as a thermally expansible, but mechanically incompressible Newtonian fluid. We may distinguish four stages: (a) The flow of molten glass at high temperature in the reservoir, feeding the fiber production system. (b) The non-isothermal flow through the die, with rigid lateral boundaries. (c) The viscous jet flow with rapidly changing physical parameters, owing to the fast cooling, up to

4 Lecture at the CIME-EMS School Mathematical models in the manufacturing of glass, polymers and textiles, Montecatini Terme, Italy, September 10, 2008 p. 4/5 Free Jet 1a the formation of a "fiber" (high viscosity, small variation of the axial velocity and very small radial velocity). (d) The motion of the glass fiber, drawn down by a device called spinner (or spool). From (b) to (d) axial velocity changes by several orders of magnitude. Let us discus briefly the (b) and (c), that is the flow through the die and the subsequent free boundary flow in the air. Our goal is to derive a mathematical model, making some simplifying assumptions, in order to discuss the influence of some physical parameters involved.

5 Lecture at the CIME-EMS School Mathematical models in the manufacturing of glass, polymers and textiles, Montecatini Terme, Italy, September 10, 2008 p. 5/5 Free Jet 2 In stages (b) and (c) one meets several difficulties. The first is that we deal with the non-isothermal flow of a thermally expansible fluid. Next, when the fluid leaves the die it experiences strong cooling effects. It is observed that, in a suitable range of temperature and pulling velocity, defining the so called "cold breakdown", the jet detaches from the solid wall inside the die. Of course, phenomena of wetting of the outer surface of the die are also observed if temperature is large enough, making the molten glass less viscous, and/or if the pulling velocity is sufficiently small.

6 Lecture at the CIME-EMS School Mathematical models in the manufacturing of glass, polymers and textiles, Montecatini Terme, Italy, September 10, 2008 p. 6/5 Free Jet 3 Our approach could describe the flow in stages (b) and (c) for a general geometry of the die. Nevertheless, since the main difficulties are not really linked with the form of the die and in order to simplify our exposition, we consider cylindrical geometry with azimuthal symmetry. In particular, x will denote the axial coordinate, r the radial one and the symmetry axis coincides with the x axis. The dependence of the fluid viscosity µ, on the temperature T plays a major role in the process. The temperature dependence of the fluid density ρ and surface tension σ will also be taken into the account.

7 Lecture at the CIME-EMS School Mathematical models in the manufacturing of glass, polymers and textiles, Montecatini Terme, Italy, September 10, 2008 p. 7/5 Free Jet 4 The fluid constitutive model is T = (p + 2µ 3 div v)i + 2µD, with T Cauchy ( stress, p pressure and D = 1/2 v + ( v ) T). In particular, the continuity equation entails (ρ(t)v 1 ) x + 1 r (r (ρ(t)v 2 )) r = 0, (1) where v 1 is the fluid axial velocity and v 2 the radial one, that is v = v 1 (x,r,t) e x + v 2 (x,r,t) e r.

8 Lecture at the CIME-EMS School Mathematical models in the manufacturing of glass, polymers and textiles, Montecatini Terme, Italy, September 10, 2008 p. 8/5 Free Jet 5 We consider a die which is a cylinder of radius R and hight L 1. After entering the die the molten glass wets only a part of the die walls and then produces a viscous jet. We confine our analysis to the case in which the jet starts at some point x T (0,L 1 ) and then we have a free boundary S, r = h(x,t), for x T < x < L, separating the fluid from the surrounding air. The fiber is cooled down by a stream of air, whose interaction with the fibers is also important, but we will not deal with this aspect here.

9 Lecture at the CIME-EMS School Mathematical models in the manufacturing of glass, polymers and textiles, Montecatini Terme, Italy, September 10, 2008 p. 9/5 Free Jet 6 The flow domain is the set Ω die Ω = Ω die Ω jet, with = {0 < x < x T, 0 < r < R}, Ω jet = {x T < x < L, 0 < r < h(x,t)}, where, as mentioned, r = h(x,t) is the unknown boundary of the jet. The boundary of Ω contains four distinct parts Inlet boundary Γ in = {x = 0, 0 r R}. Wall boundary Γ wall = {0 < x x T, r = R}. Free boundary S = {r = h(x,t), x (x T,L)}. Outlet boundary Γ out = {x = L, 0 r R L < R}, where R L = h(l).

10 Free Jet 7 We note that the outlet boundary x = L is an artificial mathematical boundary, because the jet continues. We confine our attention to the portion of jet in the interval x T < x < L, defining L as a location at which the fiber is cooled down to a point such that the fluid viscosity is large enough to prevent any further important fiber radius variations. On the other hand, the temperature should still be larger than the glass transition temperature. We take L = 12R, which is the empirical choice from the literature. This value of L should be checked a posteriori and some iterations may be needed. oncerning the boundary conditions to be imposed on the jet boundary S, we first assume a purely kinematic condition: the fluid velocity is tangent to the free surface, namely v n = w S n = 0. (2) Lecture at the CIME-EMS School Mathematical models in the manufacturing of glass, polymers and textiles, Montecatini Terme, Italy, September 10, 2008 p. 10/5

11 Lecture at the CIME-EMS School Mathematical models in the manufacturing of glass, polymers and textiles, Montecatini Terme, Italy, September 10, 2008 p. 11/5 Free Jet 8 Next, we impose a dynamic condition: equilibrium of the forces acting on S, that is (p + 2µ div v) + 2µ (D n) n = σκ, 3 2µ (D n) τ = ( τ )σ, (3) where: κ = h 1 (1 + (h ) 2) 1/2 h (1 + (h ) 2) 3/2. The air pressure, p (assumed to be constant), has been rescaled to 0. In other words, in place of p we consider p p.

12 Lecture at the CIME-EMS School Mathematical models in the manufacturing of glass, polymers and textiles, Montecatini Terme, Italy, September 10, 2008 p. 12/5 Free Jet 9 Concerning the boundary conditions on Γ in and Γ wall, we have: On Γ in, we prescribe pressure, temperature and no radial velocity p (0,r) = P in, 0 r R, T (0,r) = T in, 0 r R, v 2 (0,r) = 0, 0 r R. On Γ wall, we assume no slip v (x,r) = 0, 0 < x < x T. Of course, some slipping occurs near to x T, but it is generally admitted that the slipping length l s is extremely small.

13 Lecture at the CIME-EMS School Mathematical models in the manufacturing of glass, polymers and textiles, Montecatini Terme, Italy, September 10, 2008 p. 13/5 Free Jet 10 The abscissa of the triple point, x T, is not a priori given and has to be determined as a part of problem (as well as R L ). In most of the references, one imposes the contact angle to be equal to 0 or to π and uses this additional condition for having a totally determined problem. The boundary conditions that are specified on Γ out required coupling with the stage (d). We observe that at x = L we are well inside the stage (d). MODEL FOR THE STAGE (d)?

14 Free Jet 11 Lecture at the CIME-EMS School Mathematical models in the manufacturing of glass, polymers and textiles, Montecatini Terme, Italy, September 10, 2008 p. 14/5

15 Free Jet 12 Lecture at the CIME-EMS School Mathematical models in the manufacturing of glass, polymers and textiles, Montecatini Terme, Italy, September 10, 2008 p. 15/5

16 Lecture at the CIME-EMS School Mathematical models in the manufacturing of glass, polymers and textiles, Montecatini Terme, Italy, September 10, 2008 p. 16/5 terminal phase 1 It is considered that the stage (d) starts when viscosity is sufficiently large (larger than 10 5 Pa sec) and the fiber radius is already rather small (smaller than hundred micrometers). Contrary to the stage (c), where one should treat the 3D Navier-Stokes system, with the free boundary, coupled with the nonlinear conduction of heat, here we have a long filament (several meters) of molten glass. In the stage (c) one needs an industrial code to solve the corresponding partial differential equations. Here an alternative is possible. Fundamental equations, describing the stage (c), are the temperature dependent incompressible Navier-Stokes equations with free boundary coupled with the energy equations. Their derivation from the first principles is in

17 Lecture at the CIME-EMS School Mathematical models in the manufacturing of glass, polymers and textiles, Montecatini Terme, Italy, September 10, 2008 p. 17/5 terminal phase 2 the article [1], where also the Oberbeck-Boussinesq approximation we use is justified rigorously, by passing to the singular limit when expansivity parameter goes to zero. In the engineering literature on the fiber drawing, this system is frequently approximated by a quasi 1D approximation for viscous flows, in which the radius of the free boundary r = R(z,α,t), axial speed w and the temperature ϑ are independent of the radial variable and depend only on the axial coordinate z and of the time t. That is possible in the situations where the fibre of liquid glass is long and thin, so that the principal flow is directed along the axis z and the velocity is basically one dimensional. In particular, this property to be of "low thickness" leads to the "approximation of lubrication" of Reynolds. The strategy of the "lubrication approximation" is

18 Lecture at the CIME-EMS School Mathematical models in the manufacturing of glass, polymers and textiles, Montecatini Terme, Italy, September 10, 2008 p. 18/5 terminal phase 3 to expand the velocity field with respect ε, being the ratio of the characteristic thickness R E in the radial direction with the characteristic axial length of fibre L. The terms of order zero should be sufficient to describe the movement. The strategy of the "lubrication approximation" is to expand the velocity field with respect ε, being the ratio of the characteristic thickness R E in the radial direction with the characteristic axial length of fibre L. The terms of order zero should be sufficient to describe the movement. Effective equations are then derived starting from the coefficients of expansion with respect to ε, which depend only on axial variable z (and of t). This idea is traditionally applied to flows through thin domains. Treating the flows with a free boundary is much more complicated. Initially, the radius describes the position of the free boundary and

19 Lecture at the CIME-EMS School Mathematical models in the manufacturing of glass, polymers and textiles, Montecatini Terme, Italy, September 10, 2008 p. 19/5 terminal phase 4 the smallness of the expansion parameter ε will depend on the solution itself. In the second place, it is not obvious which forces in the equations must be taken into account (for example torsion can be important or negligible). The historical references in the subject are papers by Kase and Matsuo and Matovich and Pearson, where the "equations of Matovich-Pearson" are formally obtained. They describe the axially symmetrical movement of the viscous filaments, and read A t + (wa) z = 0; z (3µ(T)A w z ) + (σ(t) A) z = 0, (4)

20 Lecture at the CIME-EMS School Mathematical models in the manufacturing of glass, polymers and textiles, Montecatini Terme, Italy, September 10, 2008 p. 20/5 terminal phase 5 where A = A(z,t) is the area of fibre section, w = w(z, t) is effective axial velocity, 3µ is Trouton s viscosity and σ is the surface tension. µ and σ depend on the temperature, and it is necessary to add the equation for the temperature T = T(z,t). They were obtained under the assumptions (H) i) The viscous forces dominate inertia ii) Effects of the surface tension are in balance with the normal stress at the free boundary. iii) The heat conduction is small compared with the heat convection in the fiber. iv) All the phenomena are axially symmetrical and the fiber is nearly straight.

21 Lecture at the CIME-EMS School Mathematical models in the manufacturing of glass, polymers and textiles, Montecatini Terme, Italy, September 10, 2008 p. 21/5 terminal phase 6 Derivation of the model of Matovich-Pearson for the thermal case Our unknowns are the following: velocity is v = v z e z + v r e r ; hydrodynamic pressure is p; temperature is T; fiber radius (being the distance from the symmetry axis ) is R = R(t,z); specific heat is c p = c p (T); density is ρ = ρ(t); surface tension is σ = σ(t); viscosity is µ = µ(t); thermal conductivity is λ = λ(t). For the heat transfer coefficient we adopt the formula of Kase-Matsuo h = λ R(z,t) C( 2ρ v z (z,t)r(z,t)) m, (5) µ where the subscript denotes the parameters of the

22 Lecture at the CIME-EMS School Mathematical models in the manufacturing of glass, polymers and textiles, Montecatini Terme, Italy, September 10, 2008 p. 22/5 terminal phase 7 surrounding air and C > 0 and m, 0 < m < 0.4, are constants determined from the experimental data. Concerning characteristic length, one takes L = ρ ER E c pe v E 2h E (1 T g /T E ). Other possibility is to take the distance between the spooler and extrusion dye. We will restrict our considerations to the stationary case, even if the generalization to the non-stationary case is straightforward. Our first difficulty is that viscosity changes over several orders of magnitude. This motivates us to write the Oberbeck-Boussinesq equations in Ω = {z (0,L); 0 r < R(z)}, in the following form:

23 Lecture at the CIME-EMS School Mathematical models in the manufacturing of glass, polymers and textiles, Montecatini Terme, Italy, September 10, 2008 p. 23/5 terminal phase 8 ( p µ(t) div v) 2 v + ( p ( µ(t) div v)i 2D(v) ) log µ(t) div v = ρ(t) µ(t) ge z ρ(t) (v )v µ(t) in Ω (6) div (ρ(t)v) = 0 in Ω (7) ρ(t)vc p (T) T = div (λ(t) T) in Ω. (8) The stress tensor is now ( p Σ = µ(t) ( µ(t) ) div v)i 2D(v). (9)

24 Lecture at the CIME-EMS School Mathematical models in the manufacturing of glass, polymers and textiles, Montecatini Terme, Italy, September 10, 2008 p. 24/5 terminal phase 9 The natural small parameter is ε = R E L following dimensionless quantities:. We introduce the r = R E r, t = L T T t, v z = v E ṽ z, v r = εv E ṽ r, = v E T T, p µ = µ E µ,σ = σ E σ, h = h E h, µ(t) div v = v E p L, λ = λ E λ. The axially symmetric generalized Oberbeck-Boussinesq system takes the following form: v r r + v r r + v z z = v z z log ρ(t) v r r log ρ(t), in Ω; (10)

25 Lecture at the CIME-EMS School Mathematical models in the manufacturing of glass, polymers and textiles, Montecatini Terme, Italy, September 10, 2008 p. 25/5 terminal phase 10 ε 2 Re ρ(t) µ(t) (v r ) ε 2 2 v r z 2 ε 2 Re ρ(t) µ(t) (v r ε 2 log µ z v r r + v v r z z ) = p r + ( 2 v r r r v r r v r r 2+ log µ (p 2 v r r r ) + log µ (ε 2 v r z z + v z r ) r (v z z log ρ(t) + v r r log ρ(t)), in Ω; (11) ( v z r + v v z 2 z z ) = ε2 p z + v z r v z r r + v z ε2 2 z 2 (p 2 v z z ) + log µ r (ε 2 v r z + v z Re ρ(t) ) + ε2 r Fr 2 µ(t) z (v z z log ρ(t) + v r r log ρ(t)), in Ω; (12) )

26 Lecture at the CIME-EMS School Mathematical models in the manufacturing of glass, polymers and textiles, Montecatini Terme, Italy, September 10, 2008 p. 26/5 terminal phase 11 ε 2 T Peρ(T)c p (T)(v r r + v T z z ) = 1 r Next we have ε 2 z (λ(t) T z r (rλ(t) T r )+ ), in Ω. (13) v z R(z) z = v r on r = R(z) (the kinematic condition) (14) and the dynamic conditions at the free boundary read: ( µ(t)ca 2ε 2 R(z) z ( v r r v z z ) + (1 ε2 ( R z )2 )(ε 2 v r z + v ) z r ) =

27 Lecture at the CIME-EMS School Mathematical models in the manufacturing of glass, polymers and textiles, Montecatini Terme, Italy, September 10, 2008 p. 27/5 terminal phase 12 ε( R(z) z σ(t) r + σ(t) ) 1 + ε z 2 ( R(z) z )2 on r = R(z); ( εcaµ(t) (1 + ε 2 ( R(z) z )2 )p 2 ( v r r + ε2 ( R(z) z )2 v z z ) ε 2 R(z) z v r z R(z) z ε 2 σ(t) 2 R(z) z 2 / v z r ) = σ(t) R(z) 1 + ε 2 ( R(z) z )2 (15) 1 + ε 2 ( R(z) z )2 on r = R(z). (16) Finally, the heat transfer to the environment is given by

28 Lecture at the CIME-EMS School Mathematical models in the manufacturing of glass, polymers and textiles, Montecatini Terme, Italy, September 10, 2008 p. 28/5 terminal phase 13 T r ε2 R z T z = Bi h λ(t) T 1 + ε 2 ( R(z) z )2 on r = R(z); (17) Lubrification approximation We study long and very viscous fibers. Therefore the parameter ε is small and earlier mentioned hypothesis (H) holds true. As we will see in the section with simulations, the experimental data imply that Re ε, Bo 1, Ca 1 ε, Pe 1, Bi ε as ε 0. ε (18)

29 Lecture at the CIME-EMS School Mathematical models in the manufacturing of glass, polymers and textiles, Montecatini Terme, Italy, September 10, 2008 p. 29/5 terminal phase 14 In order to perform the asymptotic analysis of our equations, we expand all unknown functions with respect to ε, i.e. for an arbitrary function f = f(r,z) we set f = f (0) + ε 2 Pe f (1) + ε 2 f (2) +... We note that contrary to the viscosity µ, the non-dimensional log µ is of order 1. Let Ω 0 = { 0 r < R (0) (z), 0 < z < 1 }. After inserting the expansions for unknowns into the system (10)-(17), we find out that the zero order terms in the system (10)-(13) are: v (0) r r + v(0) r r + v(0) z z = v (0) z z log ρ(t (0) ) v (0) r r log ρ(t (0) ) in Ω (19)

30 terminal phase 15 r (v(0) z z log ρ(t (0) ) + v r (0) r log ρ(t (0) )) = p(0) r + 2 v r (0) r 2 v(0) r r 2 + log µ(t (0) ) r (2 v(0) r r p) + log µ(t (0) ) z v (0) z r + 1 r in Ω 0 ; (20) v 0 = 2 v z (0) r r 0 = 1 r v (0) z r T (0) r with the boundary conditions v r (0) = v z (0) R (0) z + log µ(t (0) ) r v (0) z r in Ω 0 ; (21) + 2 T (0) r 2 in Ω 0, (22) on r = R (0) (z); (23) Lecture at the CIME-EMS School Mathematical models in the manufacturing of glass, polymers and textiles, Montecatini Terme, Italy, September 10, 2008 p. 30/5

31 Lecture at the CIME-EMS School Mathematical models in the manufacturing of glass, polymers and textiles, Montecatini Terme, Italy, September 10, 2008 p. 31/5 terminal phase 15 p (0) = 1 εca v z (0) = 0 on r = R (0) (z); (24) r σ(t (0) ) 2( v(0) µ(t (0) )R (0) + r z R (0) v(0) z r r ) on r = R(0) (z); (25) T (0) r = 0 on r = R (0) (z); (26) Using (22) we obtain T (0) = T (0) (z). Next, equation (21) yields v z (0) = v z (0) (z). Radial component of the velocity is calculated using the equation (19). We integrate it and get

32 Lecture at the CIME-EMS School Mathematical models in the manufacturing of glass, polymers and textiles, Montecatini Terme, Italy, September 10, 2008 p. 32/5 terminal phase 16 v (0) z r = 0 = T (0) r, v(0) r (r,z) = r 2 ( v(0) z z + z log ρ(t (0) )v (0) z ) (27) Consequently, equation (20) reads p(0) r = 0 and, after using the boundary value condition (25), we obtain p (0) = 1 εca σ(t (0) ) µ(t (0) )R 2 v(0) (0) + r r = 1 εca σ(t (0) ) µ(t (0) )R (0) v (0) z z z log ρ(t (0) )v (0) z. (28) The kinematic condition (23) now transforms to

33 Lecture at the CIME-EMS School Mathematical models in the manufacturing of glass, polymers and textiles, Montecatini Terme, Italy, September 10, 2008 p. 33/5 terminal phase 17 0 = z ( ρ(t (0) )v (0) z (R (0) ) 2). (29) At the order O(ε 2 Pe ), we have the following boundary value problem for the temperature correction T (1) : ρ(t (0) )c p (T (0) )v (0) z ε 2 Peλ(T (0) ) T (1) r T (0) z = 1 r + Bi (v(0) z R (0) ) m T (0) ε 2 R (0) r (rλ(t (0) T (1) ) r ) in Ω 0, (30) = 0, on r = R (0) (z), where m is the exponent from Kase-Matsuo s formula. Using Fredholm s alternative, we get the following effective energy equation: (31)

34 Lecture at the CIME-EMS School Mathematical models in the manufacturing of glass, polymers and textiles, Montecatini Terme, Italy, September 10, 2008 p. 34/5 terminal phase 18 ρ(t (0) )(R (0) ) 2 v z (0) c p (T (0) T (0) ) z + 2Bi ε 2 Pe (v(0) z R (0) ) m T (0) = 0 on (0, (32) Next, at the order O(ε 2 ), equation (12) and condition (15) give Re 2 ρ 2 (T (0) ) µ(t (0) ) z (v (0) z ) 2 ρ(t (0) ) + p(0) 2 v z (0) z v z (2) r 2 z + log µ(t (0) ) z + 1 r v (2) z r + Bo ρ(t (0) ) µ(t (0) ) (p (0) 2 v(0) z z ) = z (v(0) z z log ρ(t (0) )) in Ω 0. (33)

35 Lecture at the CIME-EMS School Mathematical models in the manufacturing of glass, polymers and textiles, Montecatini Terme, Italy, September 10, 2008 p. 35/5 terminal phase 19 2 R(0) (z) z 1 Now (27) yields ( v(0) r εcaµ(t (0) ) z (p 0 z v 0 z) = z ( r v(0) z z ) + v(0) r z + v(2) z r σ(t (0) ) z 1 εca σ(t (0) ) µ(t (0) )R = on r = R (0) (z). (34) z 2 v(0) (0) and integration of the equation (33) yields z ) z z log ρ(t (0) )v (0) (35)

36 Lecture at the CIME-EMS School Mathematical models in the manufacturing of glass, polymers and textiles, Montecatini Terme, Italy, September 10, 2008 p. 36/5 terminal phase 20 r v z (2) r=r (0) = R(0) (Re 2 2 log µ(t (0) ) z (3 v(0) z z 1 εca z ( 1 εca ρ 2 (T (0) ) µ(t (0) ) z (v (0) z ) 2 ρ(t (0) ) Bo ρ(t (0) ) µ(t (0) ) σ(t (0) ) µ(t (0) )R (0) + z log ρ(t (0) )v (0) z )+ σ(t (0) ) µ(t (0) )R z 2 v(0) (0) z )). (36) On the other hand, after inserting (27) into the boundary condition (34) we get r v (2) z r=r (0) = 1 εcaµ(t (0) ) σ(t (0) ) z + R(0) 2 ( zz v 0 z+

37 Lecture at the CIME-EMS School Mathematical models in the manufacturing of glass, polymers and textiles, Montecatini Terme, Italy, September 10, 2008 p. 37/5 terminal phase 21 z (vz 0 z log ρ(t (0) ))) + z R (0) (3 v(0) z z + v0 z z log ρ(t (0) )). (37) After comparing equations (36)-(37), we obtain the effective momentum equation: (3µ(T (0) )(R (0) ) 2 v(0) z z z + µ(t (0) )(R (0) ) 2 v z (0) z log ρ(t (0) )+ ) 1 εca σ(t (0) )R (0) = Re 2 (ρ(t (0) )R (0) ) 2 (v z (0) ) 2 z ρ(t (0) ) Bo ρ(t (0) )(R (0) As conclusion we summarize our results in dimensional form: (38)

38 Lecture at the CIME-EMS School Mathematical models in the manufacturing of glass, polymers and textiles, Montecatini Terme, Italy, September 10, 2008 p. 38/5 terminal phase 22 Proposition 1 Let v eff = v E v z (0) be the effective axial velocity, R eff = R E R (0) the effective fiber radius and T eff = T E T (0) the effective temperature. Let us suppose that the quantities Q 0 (the mass flow), R f (the final fiber radius), V f (the pulling velocity), F L (the traction force) and T E (the extrusion temperature) are given positive constants. Then all other relevant physical quantities are determined by {v eff,r eff,t eff } and given by: effective radial velocity: v eff r (r,z) = r 2( z v eff (z) + v eff z log ρ(t eff (z)) (39)

39 Lecture at the CIME-EMS School Mathematical models in the manufacturing of glass, polymers and textiles, Montecatini Terme, Italy, September 10, 2008 p. 39/5 terminal phase 22a effective pression: p eff (z) = σ(t eff(z)) R eff (z) µ(t eff (z)) z v eff (z) µ(t eff(z)) v eff (z) z log ρ(t eff (z)); (40) 3 effective axial stress: Σ eff (r,z)e z = µ(t eff (z)) ( 3 z v eff (z)+ v eff (z) z log ρ(t eff (z)) ) e z σ(t eff(z)) R eff (z) e z + µ(t eff (z)) ( 3 z v eff (z) +v eff (z) z log ρ(t eff (z)) ) r zr eff (z) R eff (z) e r + r zσ(t eff (z)) e r. R eff (z) (41)

40 Lecture at the CIME-EMS School Mathematical models in the manufacturing of glass, polymers and textiles, Montecatini Terme, Italy, September 10, 2008 p. 40/5 terminal phase 23 effective traction : F eff = πr2 eff g ( µ(t eff (z)) ( 3 z v eff (z) +v eff (z) z log ρ(t eff (z)) ) σ(t eff(z)) R eff (z) Functions {v eff,r eff,t eff } are given by the Cauchy problem ). (42) v eff (z)reff 2 (z)ρ(t eff(z)) = Q 0 π = V frf 2 ρ(t g), 0 < z < L; (43) ( 3µ(Teff (z))(r eff (z)) 2 v eff + σ(t eff (z))r eff (z)+ z z µ(t eff (z))(r eff (z)) 2 v eff z log ρ(t eff (z)) ) =

41 terminal phase 24 (ρ(t eff (z)) 2 Q 0 c p (T eff ) πcλ z T eff z Finally, we have v 2 eff ρ(t eff (z)) g) ρ(t eff (z))(r eff (z)) 2, 0 < z < L; (44) + ( 2ρ v eff R eff µ ) m(teff T ) = 0, 0 < z < L; (45) R eff (L) = R f, v eff (L) = V f, T eff (L) = T E. (46) v z (r,z) = v eff (z) + O(ε 2 Pe ); v r (r,z) = v eff r (r,z) + O(ε 2 Pe ); R(r,z) = R eff + O(ε 2 Pe ); p(r,z) = p eff (z) + O(ε 2 Pe ); Σ(r,z) = Σ eff (r,z) + O(ε 2 Pe ). Lecture at the CIME-EMS School Mathematical models in the manufacturing of glass, polymers and textiles, Montecatini Terme, Italy, September 10, 2008 p. 41/5

42 terminal phase 25 Solvability of the boundary value problems for the effective equations Clearly, the values at the extrusion boundary could be replaced by the values at the interface S E between the stages (c) and (d) of the fiber drawing process. In the industrial simulations it makes sense to solve the full 3D Navier-Stokes system in the stage (c), to solve the equations (43)-(45) corresponding to the stage (d) and to couple them at the interface S E. Coupling at the interface requires construction of the boundary layer. Conditions which clearly hold true are continuities of the temperature field and of the axial component of the velocity. After Renardy, a typical iterative procedure for the Navier-Stokes equations with free boundary is the following: for a given free boundary we solve the Navier-Stokes equations Lecture at the CIME-EMS School Mathematical models in the manufacturing of glass, polymers and textiles, Montecatini Terme, Italy, September 10, 2008 p. 42/5

43 Lecture at the CIME-EMS School Mathematical models in the manufacturing of glass, polymers and textiles, Montecatini Terme, Italy, September 10, 2008 p. 43/5 terminal phase 26 with normal stress given at the lateral boundary. Then we update position of the free boundary using the kinematic free boundary condition. Iterations are repeated until the stabilization. Such procedure requires solving the equations (43)-(45) with the boundary conditions (46) replaced by v eff (L) = V f, v eff (0) = v E, T eff (0) = T E. (47) In this section we study the boundary value problem (43)-(45), (47). We simply drop the indice eff and set Q = Q 0 /π. In the absence of the gravity and inertia effects, with constant density and with the heat transfer coefficient depending only on the temperature, the problem was solved by T. Hagen in 2002.

44 terminal phase 27 Using the temperature as variable, it was possible to write an explicit solution for the radius and prove existence and uniqueness. In the general situation, the approach by Hagen is not possible any more. Nevertheless, their change of the unknown function will be useful in our existence proof. We prove an existence result under the following physical properties on the coefficients: (H1) Functions µ, ρ and σ ρ1/3(t) are defined on R, bounded from above and from below by positive constants and decreasing. We suppose them infinitely derivable. ρ f = min T T T E ρ(t). (H2) 0 < v E = v z=0 < V f = v z=l and T E > T. (H3) c p is infinitely derivable strictly positive function with c p,min = min T T T E c p (T). Lecture at the CIME-EMS School Mathematical models in the manufacturing of glass, polymers and textiles, Montecatini Terme, Italy, September 10, 2008 p. 44/5

45 Lecture at the CIME-EMS School Mathematical models in the manufacturing of glass, polymers and textiles, Montecatini Terme, Italy, September 10, 2008 p. 45/5 terminal phase 28 we introduce the new unknown w by w = log V fρ 1/3 f vρ 1/3 (T). (48) Let G = V f ρ 1/3 g and C 1 = Cλ Q (2ρ G ) m. Then the µ boundary value problem (43)-(45), (47) transforms to ( 3 µ(t) w z ρ(t) z + 1 ) σ(t) QG ρ 1/3 (T) ew/2 = g G ρ1/3 (T)e w Gρ 1/3 e w z w 5G 6 ρ 4/3 e w z ρ, 0 < z < L; (49)

46 Lecture at the CIME-EMS School Mathematical models in the manufacturing of glass, polymers and textiles, Montecatini Terme, Italy, September 10, 2008 p. 46/5 terminal phase 29 c p (T) T z + C 1ρ 2m/3 (T)e mw/2 (T T ) = 0, 0 < z < L; (50) w(0) = w 0 = log V fρ 1/3 f v E ρ 1/3 > 0, w(l) = 0, T(0) = T E. (51) We will obtain existence of C -solutions to problem (49)-(51), such that w w 0. Then the velocity v is given by v = V f ( ρ f ρ(t) )1/3 e w. It is a C -function and satisfies v(z) v E on [0,L]. We note as well that ρ f = ρ(t(l)) is not given. For simplicity, we suppose w 0 known. Otherwise, we should do one more fixed point calculation for ρ f, which does not pose problems.

47 terminal phase 30 Definition The corresponding variational formulation for problem (49)-(51) is: Find functions w H 1 (0,L) and T H 1 (0,L), z T 0, such that the boundary conditions (51) are satisfied and we have L 0 L 0 3 µ(t) ρ(t) w z ϕ z dz L g G ρ1/3 (T)e min{w,w0} ϕ dz + L 0 1 σ(t) QG ρ 1/3 (T) emin{w,w 0}/2 ϕ z dz+ L 0 Gρ 1/3 (T)e w w z ϕ dz 5G ρ 4/3 (T)e w z ρϕ dz = 0, ϕ H0(0,L). 1 (52) T z = C 1 c p (T) ρ 2m/3 (T)e mmin{w,w0}/2 (T T ), 0 < z < L. Lecture at the CIME-EMS School Mathematical models in the manufacturing of glass, polymers and textiles, Montecatini Terme, Italy, September(53) 10, 2008 p. 47/5

48 Lecture at the CIME-EMS School Mathematical models in the manufacturing of glass, polymers and textiles, Montecatini Terme, Italy, September 10, 2008 p. 48/5 terminal phase 31 Proposition 2 Let µ, σ and ρ satisfy (H1)-(H2) and let {w,t } be a variational solution to (51), (52) and (53). Then we have w w 0 and T E T T. Corollary Under hypothesis (H1)-(H2), any variational solution {w, T } to (51), (52) and (53) solves equations (49)-(50). Theorem Let us suppose hypothesis (H1)-(H3). Let T be the solution for T z = C 1 c p ( T) ρ 2m/3 ( T)e mw 0/2 ( T T ), 0 < z < L; T(0) = TE. (54)

49 Lecture at the CIME-EMS School Mathematical models in the manufacturing of glass, polymers and textiles, Montecatini Terme, Italy, September 10, 2008 p. 49/5 terminal phase 32 Let κ = max T T T E T ρ(t) and let A = L dz 0. Then µ( T(z)) there is δ 0 > 0 such that for κa < δ 0, problem (49)-(51) admits a solution {w,t } C [0,L] 2, such that z T 0 and w(z) w 0. Remark We see that in the case of constant density, the necessary condition from the Theorem is always fulfilled. Furthermore, since viscosity takes large values with temperature decrease, A is a very small quantity. Consequently, the Theorem covers all situations of practical interest. Proof uses Brouwer s fixed point theorem. Numerical simulations of the effective boundary value problem

50 Lecture at the CIME-EMS School Mathematical models in the manufacturing of glass, polymers and textiles, Montecatini Terme, Italy, September 10, 2008 p. 50/5 terminal phase 33 Celcius Temperature Set Value 1 Set Value z

51 Lecture at the CIME-EMS School Mathematical models in the manufacturing of glass, polymers and textiles, Montecatini Terme, Italy, September 10, 2008 p. 51/5 Free Jet 34 m/s Velocity Set Value 1 Set Value z

52 Lecture at the CIME-EMS School Mathematical models in the manufacturing of glass, polymers and textiles, Montecatini Terme, Italy, September 10, 2008 p. 52/5 Free Jet 35 m Radius 5 10 Set Value 1 Set Value z

53 Lecture at the CIME-EMS School Mathematical models in the manufacturing of glass, polymers and textiles, Montecatini Terme, Italy, September 10, 2008 p. 53/5 Free Jet 36 N minus Traction Force Set Value 1 Set Value z

54 Conclusion In the 3rd and 4th lecture we discussed briefly the full problem and then present the solution for the terminal phase of the fiber drawing. The questions often posed by the industry and which concern formation of the free jet (second phase) are the following: Evaluation of the radiation in the fiber Integration of the viscoelastic nature of the glass into the model? Modeling kinematic conditions and stresses at the free boundary of the jet Adaptability of the solution methods, temperature gradient Exchange of heat at the free boundary Influence of the cooling wings. Lecture at the CIME-EMS School Mathematical models in the manufacturing of glass, polymers and textiles, Montecatini Terme, Italy, September 10, 2008 p. 54/5