Research Article | Open Access

# Existence and Stability of the Solution of a Nonlinear Boundary Value Problem

**Academic Editor:**Carlos Lizama

#### Abstract

The purpose is to find conditions assuring the existence of solutions for a nonlinear, boundary value problem in case of the axis-symmetric Young-Laplace differential equation. The equation describes the capillary surface between two static fluids. Necessary or sufficient conditions are found for the existence of a solution. The static stability of the obtained solution is also analyzed and stability or instability results are revealed. For the NdYAG microfiber growth, by the pulling-down method, numerical illustrations are given.

#### 1. Motivation of the Mathematical Considerations

The near one-dimensional single crystal fibers have attracted some attention on the applications of optical and electronic devices [1–3]. There are several methods for growing fiber crystals [4], such as the edge-defined film-fed growth (EFG) and floating zone (pedestal growth) methods. The micro-pulling-down (*μ*-PD) process, a variant of the inverse EFG, developed by Fukuda’s laboratory in Japan [5–9], has been shown promising in producing single crystal fibers with good diameter control and concentration uniformity. Several oxide [5–7] and semiconductor fibers [8, 9] have been grown. The grown diameter ranges are 10–1500 *μ*m for oxides and about 300–900 *μ*m for semiconductors. According to [10], for this process some simple theoretical analyses for the operation limit [11] and solute distribution [12, 13] have been performed, and no detailed modeling has been conducted. In the previous reports the melt convection was ignored, and heat and mass transfer coupled with the capillary shaping and solidification have not been analyzed.

Since the system is small, the measurements of flow, dopant, and temperature distributions are difficult. In [10] a detailed numerical simulation is conducted in a self-consistent manner to analyze the *μ*-PD process for fibers; . The system of partial differential equations describing the melt flow, heat, and mass transfer, the crystallization front and meniscus, and the grown fiber diameter are solved numerically in stationary case for various process parameters. The model formulation and analysis are very similar to those developed in [14, 15] and appropriate for the analysis of the radial segregation. Since the model presented in [10] is a stationary model, it cannot be used for the analysis of the axial segregation, which can be relevant in *μ*-PD process, as is reported in [16].

In this paper the meniscus shape and size, appearing in *μ*-PD, are evaluated in function of a parameter , which incorporates the hydrostatic pressure of the melt column situated behind the meniscus, the hydrodynamic pressure associated to the thermoconvection in the meniscus, the Marangoni pressure associated to the Marangoni convection, and the gas pressure in front of the free surface of the meniscus. The obtained results can be useful for defining the geometry in case of the simulation of the growth process.

#### 2. Formulation of the Mathematical Problem

For single crystal fiber growth by *μ*-PD method, the free surface of the meniscus at each moment of time is described by the Young-Laplace equation [17, 18]:
Here is the melt surface tension; denote the main normal curvatures of the free surface at a point of the free surface; is the pressure in front of the free surface; is the pressure behind the free surface (Figure 1).

The pressure in front of the free surface is equal to the pressure of the gas introduced in the furnace and thereafter is denoted by (.

The pressure behind the free surface is the sum of the hydrodynamic pressure in the meniscus melt (due to the thermal convection), the Marangoni pressure due to the Marangoni convection, and the hydrostatic pressure of the melt column behind the free surface equal to (see Figure 1). Here denotes the melt density; is the gravity acceleration; is the coordinate of with respect to the *Oz *axis, directed vertically downwards; denotes the melt column height between the horizontal crucible melt level and the shaper bottom level (see Figure 1).

The pressure difference across the free surface is denoted usually by and, according to the above explanations, The part given by: of the pressure difference is considered to be known. That is because the major part of is known, and in general is small with respect to and is mainly determined by the thermal conditions. What is important to retain is that does not depend on spatial coordinate, but it can depend on the moment of time .

In the above terms the Young-Laplace equation (2.1) can be written as To calculate the meniscus-free surface shape and size are convenient to employ the Young-Laplace equation (2.4) in its differential form. This form can be obtained also as a necessary condition for the minimum of the free energy of the melt column [17, 18].

The differential equation of the meridian curve for axis-symmetric meniscus-free surface is If the solution of (2.5) is the meridian curve of the free surface of a meniscus on the interval , then it verifies the following boundary conditions:

The first condition in (2.6) expresses that at the three-phase point , where the thermal conditions for solidification have to be realized, the angle between the tangent line to the meridian curve of the free surface and the vertical is equal to the growth angle. If this condition is satisfied during the whole process, then the tangent to the fiber surface at the three-phase point is vertical; that is, the fiber diameter is constant.

The second condition in (2.6) expresses that the coordinate of the crystallization front is equal to .

The first condition in (2.7) expresses that, at the three-phase point , where the meridian curve touches the outer edge of the shaper, the catching angle is equal to .

The second condition in (2.7) expresses that at the three-phase point the meridian curve is fixed to the outer edge of the shaper.

Condition (2.8) expresses the fact that the meniscus shape is relatively simple.

The determination of for which (2.5) and (2.6) hold is a nonlinear boundary value problem (NLBVP).

Equation (2.5) is the Euler equation of the free energy functional of the melt column: The nonlinear boundary value problem (NLBVP), (2.5) and (2.6), is equivalent to the NLBVP:

#### 3. Existence of the Solution

Consider the differential equation: And such that .

*Definition 3.1. *A solution of (3.1) is a globally convex solution of the NLBVP (2.5) and (2.6) on the interval if it possesses the following properties: (a) , (b) , (c) and is strictly decreasing on , (d) , A solution of (3.1) is a globally-concave solution of the NLBVP (2.5), (2.6) on the interval if verifies (a) (b) (c) and in addition the following inequality holds: (e) ].

*Remark 3.2. *If , then for a globally concave solution of (3.1), which satisfies (b) and (c), there is no such that (a) holds. Therefore, if and we are interested in solutions of (3.1) which satisfy (b) and (c) and are not globally concave.

If , then for a globally convex solution of (3.1), which satisfies (b) and (c), and there is no such that a. holds. Therefore, if , we are interested in solutions of (3.1) which satisfy (b) and (c) and are not globally convex.

Theorem 3.3 (necessary condition for the existence of a globally convex solution). * If and there exists a globally convex solution of the NLBVP (2.5) and (2.6), on , then the following inequalities hold:
*

*Proof. * Let be a globally convex solution of the problem (2.5) and (2.6) on and . The function verifies the following quation:
and the boundary conditions:
Hence, according to the Lagrange mean value theorem, there exists such that the following equality holds:
Since on , is strictly increasing, and is strictly decreasing on . Therefore, the following inequalities hold:
Equality (3.5) and inequalities (3.6) imply (3.2).

Corollary 3.4. *In terms of the ratio the inequalities (3.2) can be written as
*

Corollary 3.5. *If , then and (3.7) become
**
If , then and .*

Theorem 3.6 (necessary condition for the existence of a globally concave solution). *If , and there exists a globally concave solution of the NLBVP (2.5) and (2.6) on , then the following inequalities hold:
*

*Proof. * Since on , is strictly decreasing and is strictly increasing on . Therefore the following inequalities hold:
for every .

Equality (3.5) and inequalities (3.10) imply inequalities (3.9).

Corollary 3.7. *In terms of the ratio the inequalities (3.9) can be written as
*

Corollary 3.8. *If , then and (3.11) become
**
If , then and .*

Theorem 3.9 (necessary condition for the existence of a convex-concave solution). * If for and and there exists a solution of the NLBVP (2.5) and (2.6) on the interval and then for the following inequalities hold:
*

*Proof. * Since , we have and hence . It follows that, for , close to , the following inequality: holds. Since , it follows that there exists such that is the point where is minimum). The equality implies that the following equality holds:
Taking into account the fact that is a decreasing function on , we deduce that .

Theorem 3.10 (necessary condition for the existence of a concave-convex solution). *If for and , there exists a solution of the NLBVP (2.5) and (2.6) on the interval , then for the following inequalities hold:
*

*Proof. *Since , we have and hence . It follows that, for , close to , the inequality holds. Since , it follows that there exists such that ( is the point where is maximum). The equality implies that the equality (3.14) holds. Taking into account the fact that is a decreasing function on , we deduce (3.15).

Theorem 3.11. *If for , there exists a solution of the NLBVP (2.5) and (2.6), on the interval and , then is concave convex on and there exists , such that is a globally convex solution of the NLBVP (2.5) and (2.6) on . Moreover for the inequality (3.15) holds.*

*Proof. *Since , we have . Inequality implies that . Hence, there exists such that . This equality implies that (3.14) holds. Taking into account the fact that is a decreasing function on *, *we deduce (3.15).

The existence of follows from the fact that for , close to and .

Theorem 3.12. * If for , , there exists a solution of the NLBVP (2.5) and (2.6) on the interval and , then is a convex concave solution of the NLBVP (2.5) and (2.6) on and there exists , , such that is a globally-concave solution of the NLBVP (2.5), (2.6) on . Moreover for the inequality (3.13) holds.*

*Proof. *Since , we have . Inequality implies that . This equality implies that (3.14) holds. Taking into account the fact that is a decreasing function on , we deduce that (3.13) holds.

The existence of follows from the fact that for , close to and *. *

Theorem 3.13 (sufficient condition for the existence of a globally convex solution). *Let be a real number. If and satisfies
**
then there exists in the closed interval such that the solution of the initial value problem
**
is a globally convex solution of the NLBVP (2.5) and (2.6) on .*

*Proof. *Consider the solution of the initial value problem (3.17). Denote by the maximal interval on which the function exists and by the function defined on . for the function the following equality holds:
Since
there exists such that for any the following inequalities hold:

Let now be defined by

It is clear that and for any (3.20) hold. Moreover, exists and satisfies and . Hence is finite, it is strictly positive and for every the inequalities hold: We will show now that and .

In order to show that we assume the contrary, that is, . Under this hypothesis we have for some . Hence . This last inequality is impossible, according to the definition (3.21) of . Therefore, , defined by (3.21), satisfies .

In order to show that , we remark that from the definition (3.21) of it follows that in at least one of the following three equalities hold: Since for any , it follows that the equality is impossible. Hence, we deduce that at at least one of the following two equalities hold: .

We show now that the equality is impossible. For that we assume the contrary, that is, . Under this hypothesis, from (3.17) we have what is impossible.

In this way we obtain that the equality holds.

Taking now we find that the solution of the initial value problem (3.17) on the interval is convex, and it is the meridian curve of an appropriate meniscus.

Corollary 3.14. *If and
**
then there exists such that the solution of the initial value problem (3.17) is a globally convex solution of the NLBVP (2.5) and (2.6) on .*

Corollary 3.15. *If , and the inequalities hold
**
then there exists such that the solution of the initial value problem (3.17) is a globally convex solution of the NLBVP (2.5), (2.6) on .*

The existence of and the inequality follows from Theorem 3.13. The inequality follows from the Corollary 3.4 of Theorem 3.3.

Theorem 3.16 (sufficient condition for the existence of globally concave solution). * Let be a real number. If and satisfies
**
then there exists in the closed interval such that the solution of the initial value problem
**
is a globally concave solution of the NLBVP (2.5) and (2.6) on .*

*Proof. *Consider the solution of IVP (3.29). Denote by the maximal interval on which the function exists and by the function defined on . For the function the following equality holds:
Since
there exists such that for any the following inequalities hold:
Let now be defined by
It is clear that and for any inequalities (3.32) hold. Moreover, exists and satisfies . Hence is finite, it is strictly positive and for every the inequalities hold:
We will show now that and .

In order to show the inequality we assume the contrary, that is, that . Under this hypothesis we have
for some . Since , , and , it follows that . Hence , what is impossible, according to the definition of . Therefore, , defined by (3.33), satisfies .

In order to show that , we remark that from the definition (3.33) of it follows that in at least one of the following three equalities holds:
Since for any , it follows that the equality is impossible. Hence, we deduce that at at least one of the following two equalities holds: .

We show now that the equality is impossible. For that we assume the contrary, that is, . Under this hypothesis, from (3.29) we have , what is impossible.

In this way we obtain that the equality holds.

Taking now we find that the solution of the initial value problem (3.29) on the interval is concave, and it is the meridian curve of an appropriate meniscus.

Corollary 3.17. *If , and the inequalities hold:
**
then there exists such that the solution of the IVP (3.29) is a globally concave solution of the NBVP (2.5), (2.6) on .**The existence of and the inequality follows from Theorem 3.16. The inequality follows from Corollary 3.7. and Theorem 3.6.*

#### 4. Stability of the Solution

*Definition 4.1. *A solution of the NLBVP (2.5), (2.6) stable if is a minimum for the energy functional (2.9).

Theorem 4.2. *If for the NLBVP (2.5), (2.6) has a globally convex solution on and the ratio satisfies:
**
then the solution is stable.*

*Proof. *Since (2.5) is the Euler equation for the energy functional (2.9), it is sufficient to prove that the Legendre and Jacobi conditions are satisfied in this case.

It is easy to see that in this case the Legendre condition reduces to the inequality:
and it is satisfied.

By computation we find that the Jacobi equation in this case is
For the coefficients of (4.3) the following inequality holds:
Hence,
is a Sturm type upper bound [19] for (4.3).

An arbitrary solution of (4.5) is given by
where and are real constants and is given by

Inequality (4.1) implies that the half period of satisfies the inequality . Hence a nonzero solution given by (4.6) has at most one zero on the interval . Since any nonzero solution of (4.5) vanishes at most once on the interval , the solution of (4.3) which satisfies has only one zero on . Hence the Jacobi condition is satisfied in this case.

Theorem 4.3. *If for the NLBVP (2.5) and (2.6) has a globally convex solution on , and the ratio satisfies
**
then the meniscus is unstable.*

*Proof. *We remark now that for the Jacobi equation the following inequalities hold:
Hence
is a Sturm type lower bound [19] of the Jacobi equation (4.3).

An arbitrary solution of (4.10) is given by
where is given by
Inequality (4.8) implies that given by (4.11) vanishes twice on . Therefore the Jacobi condition is not satisfied.

Theorem 4.4. *If for the NLBVP (2.5), (2.6) has a globally-concave solution on and the ratio satisfies:
**
then the solution is stable.*

*Proof. *In order to verify the Jacobi condition in this case, we remark that for the coefficients of the Jacobi equation (4.3) the following inequalities hold:
Hence
is a Sturm type upper bound [19] of the Jacobi equation (4.3).

An arbitrary solution of (4.15) is given by
where and are real constants and is given by
Inequality (4.13) implies that the half period of satisfies the inequality . It follows that a nonzero solution given by (4.16) has at most one zero on the interval . Since any nonzero solution of (4.15) vanishes at most once on the interval , the solution of the Jacobi equation (4.3), which satisfies , has only one zero on . Hence the Jacobi condition is satisfied.

Theorem 4.5. * If for the NLBVP (2.5) and (2.6) has a globally concave solution on and the ratio satisfies
**
then the solution is unstable.*

*Proof. *Remark that in this case for the coefficients of the Jacobi equation (4.3) the following inequalities hold:
From this step the proof is similar to the proof of Theorem 4.3.

#### 5. Numerical Illustration

Numerical computations were performed for NdYAG microfiber growth by pulling down method using the following numerical data. ; ; ; ; ; ; ; .

The objective was to find the ranges for the values of (determined in Section 3) and to find for which meniscus, having prior given shape and size, exists.

(a) Inequality (3.2) establishes the range where has to be when and a globally convex solution of the NLBVP (2.5) and (2.6) on , exists. Computation shows that for the considered numerical data ( this range is .

To identify the value for which a globally convex solution of the NLBVP (2.5) and (2.6) exists on , the IVP (5.1) has to be integrated numerically for and different values of in the above range. The dependence has to be found, where is defined by . The value of for which is equal to is found solving the equation . If the function does not exists (i.e., the growth angle is not reached) or the equation has no solution, then there is no for which a globally convex solution of the NLBVP (2.5) and (2.6) on exists. For the considered numerical data the graphic of the function is represented in Figure 2(a). This figure shows that for Pa the equality holds; that is, meniscus with prior given shape and size exists. The meridian curve at this meniscus is represented in Figure 2(b).

**(a)**

**(b)**

(b) Inequality (3.9) establishes the range where has to be when and a globally-concave solution of the NLBVP (2.5) and (2.6) on exists. Computation shows, that for the considered numerical data, , this range is [1640, 2211] Pa. To identify the value of the IVP (5.1) for and for different values of in the above range has to be solved. The dependence has to be found, where is defined by . The value of is found solving the equation . If the function does not exists (i.e., the growth angle is not reached) or the equation has no solution, then there is no for which meniscus, having the prescribed shape and size, exists. For the considered numerical data the graphic of the function is represented in Figure 3(a). This figure shows that for Pa the equality m holds; that is, meniscus with prescribed shape and size exists. The meridian curve of this meniscus is represented in Figure 3(b).

**(a)**

**(b)**

(c) Inequality (3.13) establishes the range where has to be when and the NLBVP (2.5) and (2.6) has a convex-concave solution on . Computation shows, that for the considered numerical data, , this range is [781, 1867] Pa. The value of can be identified integrating numerically the IVP (5.1) for and for different values of in the above range. The dependence has to be found, where is defined by *. *The value of is found solving the equation . If the function does not exist (i.e., the growth angle is not reached) or the equation has no solution, then there is no for which a meniscus, with the prior given shape and size on exists. For the considered numerical data the graphic of the function is represented on Figure 4(a). This figure shows that equation m has no solution. Therefore, a meniscus with the prior given shape and size does not exist in this case. A convex-concave meniscus obtained for Pa is presented on Figure 4(b), but this is inappropriate because m.

**(a)**

**(b)**

(d) Inequality (3.15) establishes the range where has to be when and the NLBVP (2.5) and (2.6) has a concave-convex solution on . Computation shows that for the considered numerical data, , this range is [1493.72, 1539.2] Pa. The value of can be identified integrating numerically the IVP (5.1) for and for different values of in the above range. The dependence has to be found, where is defined by *. *The value of is found solving the equation . If the function does not exist (i.e., the growth angle is not reached), or the equation has no solution, then there is no for which a meniscus having the prior given shape and size exists on . For the considered numerical data computation shows that for in the range [1493.72, 1539.2] Pa the growth angle on the meridian curve (i.e., the solution of IVP (5.1) for is not reached. Therefore, the function does not exist, and there is no concave-convex meniscus with the prescribed shape and size in this case.

(e) Inequality (3.15) establishes as well the range where has to be when and the NLBVP (2.5) and (2.6) has a concave-convex solution on . Computation shows, that for the considered numerical data, , this range is [1493.72, 1539.2] Pa. For in this range the meniscus shape is convex-concave, as it was shown in case (c). Therefore, for the considered data there is no meniscus with concave-convex shape.

(f) Inequality (3.13) establishes as well the range where has to be when and the NLBVP (2.5), (2.6) has a convex-concave solution on . Computation shows, that for the considered numerical data, , this range is: [1539.2, 1867.18] Pa. The value of can be identified integrating numerically the IVP (5.1) for and for different values of in the above range. Following the method described at (c), computation shows that in this case the function does not exist.

(g) Inequality (3.21) defines a range such that when and is in the range, then there exists in the range and a globally convex solution of the NLBVP (2.5) and (2.6) on . For the considered numerical data, ,