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Global-in-time solutions for the isothermal Matovich-Pearson equations

In this paper we study the Matovich-Pearson equations describing the process of glass fiber drawing. These equations may be viewed as a 1D-reduction of the incompressible Navier-Stokes equations including free boundary, valid for the drawing of a long and thin glass fiber. We concentrate on the isothermal case without surface tension. Then the Matovich-Pearson equations represent a nonlinearly coupled system of an elliptic equation for the axial velocity and a hyperbolic transport equation for the fluid cross-sectional area. We first prove existence of a local solution, and, after constructing appropriate barrier functions, we deduce that the fluid radius is always strictly positive and that the local solution remains in the same regularity class. To the best of our knowledge, this is the first global existence and uniqueness result for this important system of equations.


Introduction
The drawing of continuous glass fibers is a widely used procedure. Industrial glass fibers are manufactured by a bushing with more than thousand nozzles. Bushings are supplied with a molten glass from a melting furnace. Its temperature ranges from 1300K to 1800K. In order to understand the glass fiber forming process, it is important to study the drawing of a single glass fiber. This is, of course, a significant simplification because we disregard interaction between fibers and between fibers and the surrounding air. For a single glass fiber, the hot glass melt is forced by gravity to flow through a die into air. After leaving the die, the molten glass forms a free liquid jet. It is cooled and attenuated as it proceeds through the air. Finally, the cold fiber is collected on a rotating drum.
The molten glass can be considered as a Newtonian fluid and the process is described by the non-isothermal Navier-Stokes equations for a thermally dilatable but isochoric fluid. Since we deal with a free liquid jet, the problem is posed as a free boundary problem for the Navier-Stokes equations, coupled with the energy equation. We refer to [5] for detailed modeling and analysis of the equations describing the stationary flow inside the die. There are several models proposed to describe the various stages of the flow of a molten glass from the furnace to the winding spool: the slow flow in the die (the "first phase" of the drawing), the jet formation under rapid cooling (the "second phase"), and the terminal fiber profile (the "third phase") (see [6]).
Since we consider long (their length is approximately 10 m) and thin (their diameter varies from 1 mm to 10 µm) fibers, it is reasonable to apply the lubrication approximation to the model equation. This approach yields good results, at least for flows far from the die exit and in the so-called "third phase" of the fiber drawing.
A standard engineering model for the isothermal glass fiber drawing in the "third phase" is represented by the Matovich-Pearson equations. For an axially symmetric fiber with a straight central line, they read where A = A(t, x) is the cross-area of the fiber section, v = v(t, x) is the effective axial velocity, 3µ is Trouton's viscosity, and σ denotes the surface tension. As the coefficients µ and σ depend on the temperature, it is necessary to take into account an equation for the temperature T = T (t, x). The original derivation of the system (1) is purely heuristic and obtained under the assumptions that: (i) the viscous forces dominate the inertial ones; (ii) the effect of the surface tension is balanced with the normal stress at the free boundary; (iii) the heat conduction is small compared with the heat convection in the fiber; (iv) the fiber is almost straight, and all quantities are axially symmetric. We refer to the classical papers by Kase & Matsuo [16,17], and Matovich & Pearson [19] for more details concerning the model.
Another derivation of the model based on a lubrication type asymptotic expansion can be found in the work by Schultz et al. [9,23], Dewynne et al. [3,4], and Hagen [12], with more emphasis on the mathematical aspects of the problem. The (formal) asymptotic expansion is developed with respect to a small parameter ε, proportional to the ratio of the characteristic thickness R E in the radial direction and the characteristic axial length of the fiber L. The fact that the viscosity changes over several orders of magnitude is surprisingly ignored in these studies. As a matter of fact, the viscosity coefficient depends effectively on the temperature, with values varying from 10 to 10 12 Pa sec, while in the above mentioned asymptotic expansions it is considered simply of order one. A correct formal derivation was given in [1], and it is in full agreement with the model announced in [12]. Finally, a full non-stationary model of a thermally dilatable molten glass, with density depending on the temperature, was derived in [6].
A mathematical analysis of generalized stationary Matovich-Pearson equations is performed in [1] (see also [2]). The non-stationary case, without surface tension and with advection equation for the temperature, is studied by Hagen & Renardy [10]. They prove a local-in-time existence result in the class of smooth solutions. Their approach is based on a precise analysis of the dependence of the solution of the mass conservation equation on the velocity. This method requires controlling higher order Sobolev norms in the construction of solutions by means of an iterative procedure and works only for short time intervals. Hagen et al. [11,13,14] have also undertaken a detailed study of the linearized equations of forced elongation. Despite this considerable effort, global-in-time solvability of the Matovich-Pearson equations was left open.
The ratio σ/µ is small, and, furthermore, the inertia and gravity effects are negligible in most applications. Accordingly, we consider the Matovich-Pearson equations (1) with σ = 0, meaning, the isothermal drawing with constant positive viscosity and in the absence of surface tension. For a prescribed velocity at the fiber end points, the important parameter is the draw ratio, being the ratio between the outlet and inlet fluid velocities. It is well known that the instability known as a draw resonance occurs at draw ratios in excess of about 20.2. Linear stability analysis was rigorously undertaken by Renardy [22]. Moreover, in [26], it was established that the cross section, given by the Matovich-Pearson equations with σ = 0, may vary chaotically at a draw ratio higher than 30, under the condition of periodic variations of the input cross section. There are also numerous articles devoted to numerical simulations confirming such a conclusion. Fairly complete simulations can be found in the papers by Gregory Forest & Zhou [8,25]. Their simulations predict various aspects of the physical process, like a linearized stability principle, bounds on the domain of convergence for linearly stable solutions, and transition to instability. Their analysis completes that of [7].
The above mentioned simplification of system (1) is briefly discussed in [15], however, without any rigorous proofs. Our idea is to use the particular structure of the system with σ = 0, and to prove short-time existence of smooth solutions satisfying good uniform estimates. Then, performing a qualitative analysis of the solutions and constructing appropriate barrier functions, we show that the cross-section area remains bounded below away from zero. This observation allows us to deduce existence as well as uniqueness of global-in-time solutions.
2 Isothermal fiber drawing without surface tension We study the system of equations supplemented with the boundary and initial conditions Here v is the axial velocity and A denotes the cross section, L, T are given positive numbers, and 3µ > 0 denotes Trouton's viscosity assumed to be constant. The data satisfy Moreover, the functions v in , v L , S 0 , S 1 belong to certain regularity classes specified below.

A priori bounds
Our construction of global-in-time solutions is based on certain a priori estimates that hold, formally, for any smooth solution of problem (2) -(5), with the cross-section area A > 0. The crucial observation is that, as a direct consequence of (3), where χ is a function of the time variable only. Moreover, as A is positive and the axial velocity satisfies the boundary conditions (6), we deduce that which in turn implies Next, we rewrite equation (2) in the form Integrating (7) over (0, L) and using (5) and (10) give rise to the uniform bound In order to deduce a lower bound for the cross section area A, we first observer that A and ∂ x A satisfy the same transport equation, namely, In particular, Evaluating the boundary values of ∂ x log(A) with (4), namely, where, in accordance with (2), (4), (5), (7), and (11) We deduce easily the desired lower bound on A where the constant A m is determined solely in terms of v m , V M , S m , S M , and the first derivatives of S 0 , S 1 . The a priori bounds derived in (7) -(15) form a suitable platform for the existence theory developed in the remaining part of this paper.

Short time existence of regularized strong solutions
In addition to (6), we shall assume that where the symbol W k,p denotes the standard Sobolev space of functions having k−derivatives L p −integrable, and H 2 ≡ W 2,2 . For further use, we introduce the notation Let us begin with a list of definitions: v satisfies (5) pointwise.
Similarly, the radius R is defined by Definition 3. For R given by (24) and 0 < t 0 ≤ T , we denote by S(t 0 , R) the convex set of nonnegative functions h defined on Q t 0 such that Definition 4. For R given by (24), we denote t * ∈ (0, T ] a positive time satisfying In this section, for δ > 0 small enough, we construct a family of approximate solutions (A δ , v δ ) solving the following initial-boundary value problem: Specifically, we prove the following result.
The rest of this section is devoted to the proof of Theorem 1. The basic and rather standard idea is to construct a sequence approaching a fixed point of a suitable nonlinear mapping. The proof is carried over by means of several steps. We fix t 0 ∈ (0, t * ) and δ ∈ (0, min {1, t 0 }) such that (34) is satisfied.
At this stage we follow the ideas of [10] and [15] to show that this nonlinear operator has a fixed point for all t 0 < t * . They study the nonisothermal fiber spinning and their system of equations is different. Hence we are obliged to give an independent proof of short time existence, but the result remains close to their considerations.
The natural approach is to apply Schauder's fixed point theorem. To use it we have to prove that S(t 0 , R) is a relatively compact convex set invariant under our nonlinear mapping. Finally, we should establish the continuity of the mapping (49).

STEP 2
We establish uniform L ∞ − bounds for the function u. We have Lemma 1. Let R be given by Definition 3 and let t 0 ≤ t * . Then the solution u of problem (47)-(48) satisfies the estimate Proof. Let s m ∈ W 2,∞ (0, t 0 ) be the solution to the Cauchy problem Then s m clearly solves (47) and it follows from (6) and (48) that while (6), (34), the nonnegativity of Q 0 , (48), and (51) ensure that The comparison principle then entails that u(t, x) ≥ s m (t) for (t, x) ∈ Q t 0 , whence, by (28), (34), and (46), This proves the lower bound. Proving the upper bound is analogous. The comparison function is now given by the solution to the Cauchy problem Applying again the comparison principle gives u(t, x) ≤ s M (t) for (t, x) ∈ Q t 0 , which completes the proof since (34) guarantees that This proves the Lemma. Now we use equation (47) to calculate G(t) = ∂ x u(t, 0) ∈ C([0, t 0 ]) getting: Due to the assumptions on the data, G ∈ W 1,∞ (0, t 0 ) and we have Next, we take the derivative of equation (47) with respect to the x variable. This yields that S = ∂ x u solves the function v 0 being still given by (37) with A 0 instead of h.
Multiplying equation (47) by u, equation (55) by S, integrating both equations on (0, t) × (0, x), (t, x) ∈ Q t 0 , and adding the obtained identities, we deduce: Now we use (6), (46) with (54) to get The third term on the left hand side of (57) is estimated with the help of (42) and (50) as while the fourth term obeys thanks to (46) and (50). Now, setting we may insert (58)-(60) into (57) and use (24) and (28) to get the following estimate: Recalling (6) and (42), we thus conclude that Next, we use equation (47) and estimate (61) to control ∂ t u, obtaining Therefore the image of a nonnegative function from S(t 0 , R) remains in S(t 0 , R) and satisfies the L ∞ − bound (50). Therefore our nonlinear map (49) maps the convex set into itself.
Denoting the corresponding velocity field by v δ , we have and since t → ∂ x v δ (t, 0) ∈ W 1,∞ (0, t 0 ), we may apply Theorem 4 in Appendix to conclude that, in fact, It remains to prove uniqueness.

STEP 4
With the obtained smoothness, uniqueness is easy to establish. It suffices to notice that L 2 (0, t 0 ; H 1 (0, L)) perturbation of v δ is controlled by the L 2perturbation of A δ in x and t. Then we use this observation, regularity of A δ , and Gronwall's lemma to obtain uniqueness.
The proof of Theorem 1 is now complete.

Global existence of regularized strong solutions
Now we suppose that the regularity of the solution or/and the strict positivity of A δ , stated in (50), breaks down at the time t p . Our goal is to prove that t p = T .

Theorem 2. Under the hypotheses of Theorem 1, the initial-boundary value problem (29)-(33) has a unique solution
The remaining part of this section is devoted to the proof of Theorem 2.

STEP 1
First we recall that, owing to (31) and (33), v δ is given by Next, by (6) and we put We point out that the upper bound in (50) is independent of t * . Indeed, we prove now that it is valid regardless the length of the time interval. To this end, we introduce the solution S δ M ∈ W 2,∞ (0, t 0 ) to the Cauchy problem Owing to the positivity (68) of Q δ , S δ M is a supersolution to (29)-(30) with This proves the upper bound and, in addition, the estimate is independent of the length of the time interval.
Therefore, by Jensen's inequality, we may infer that for any t < t p , where t p is the critical time at which A δ (t, x) attains zero for some x.

STEP 2
Having established that ∂ x A δ (t, 0) is bounded in L ∞ (0, t) independently of t, we now look for a strictly positive lower bound for A δ .

Proposition 1.
There are constants C 1 and C 2 , independent of the length of the time interval and of δ, such that we have Proof. We notice that y = ∂ x log A δ satisfies the equation and The function F (t, x) = (Q δ − Q cut,δ )(t)/A δ (t, x) vanishes for t ≥ δ and, due to (50), satisfies the following bound for every t since δ ≤ t 0 . We next introduce the solution y m to the ordinary differential equation with initial condition with Owing to (73), (76), (78), and (79), y m is nonpositive and thus a subsolution to (75), and satisfies y m (t) ≤ y m (0) ≤ y(t, 0) for t ∈ (0, t p ) and y m (0) ≤ y(0, x) for x ∈ (0, L). The comparison principle then entails that Since y m (0) ≤ 0, we deduce from (77) that y(t, x) ≥ C 1 for some positive constant C 1 , independent of δ and t 0 . Similarly, let Y M be the solution to the ordinary differential equation with initial condition the constant G M being defined in (80). It follows from (73), (76), (81), and (82) that Y M is a supersolution to (75) which satisfies Y M (t) ≥ Y M (0) ≥ y(t, 0) for t ∈ (0, t p ) and Y M (0) ≥ y(0, x) for x ∈ (0, L). Using once more the comparison principle and (77), we conclude that the constant C 2 being independent on δ and t.
Since log A δ (t, 0) = log S 0 (t) ≥ log S m > −∞ by (6) and we infer from (74) that Therefore A δ is strictly positive on Q t for all t ≤ t p , and bounded from below and from above by a constant which is independent of both t and δ.

STEP 3
Having established (83) we easily obtain the following estimates: where C is again independent of t and δ.
We conclude that, for all t, A δ is bounded from below by a positive constant, independent of t, and the norm of A δ in C 1 ([0, t]; H 1 (0, L)) ∩ C([0, t]; H 2 (0, L)) remains bounded by a constant, also independent of t, that may, however, depend on δ. The maximal solution therefore extends to [0, T ], and, in fact, we have established the existence of a unique strictly positive solution A δ on (0, T )×(0, L). The corresponding velocity v δ is given by (37), with h = A δ .
This completes the proof of Theorem 2 .

Existence of a unique strong solution
At this stage, we are ready to establish the main result of the paper.
Proof. We just recall the estimates obtained in the proof of Theorem 2, valid independently of δ: Since L ∞ (Q T ) is the dual space of the separable Banach space L 1 (Q T ), Alaoglu's weak * compactness theorem gives weak * sequential compactness. Therefore there exist A and v such that ∂ t A δ ⇀ ∂ t A weakly * in L ∞ (Q T ), as δ → 0, uniformly in Q T , as δ → 0. (91)