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

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 μm for oxides and about 300–900 μm 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 Ge_𝑥Si_1−𝑥 fibers; 𝑥=0.05. 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 𝑝, 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]: 1𝛾⋅𝑅₁+1𝑅₂=𝑃_𝑎−𝑃_𝑏. (2.1) Here 𝛾 is the melt surface tension; 1/𝑅₁,1/𝑅₂ denote the main normal curvatures of the free surface at a point 𝑀 of the free surface; 𝑃_𝑎 is the pressure in front of the free surface; 𝑃_𝑏 is the pressure behind the free surface (Figure 1).

Figure 1 

Schematic diagram of the μ-pulling-down fiber growth process.

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

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

The pressure difference 𝑃_𝑎−𝑃_𝑏 across the free surface is denoted usually by Δ𝑝 and, according to the above explanations, Δ𝑝=𝑝_𝑔−𝑝_𝑚−𝑝_𝑀−𝜌⋅𝑔⋅𝑧+ℎ_𝑚. (2.2) The part 𝑝 given by: 𝑝=−𝑝_𝑔+𝜌⋅𝑔⋅ℎ_𝑚+𝑝_𝑚+𝑝_𝑀. (2.3) of the pressure difference Δ𝑝 is considered to be known. That is because the major part −𝑝_𝑔+𝜌⋅𝑔⋅ℎ_𝑚 of 𝑝 is known, and 𝑝_𝑚+𝑝_𝑀 in general is small with respect to −𝑝_𝑔+𝜌⋅𝑔⋅ℎ_𝑚 and is mainly determined by the thermal conditions. What is important to retain is that 𝑝 does not depend on spatial coordinate, but it can depend on the moment of time 𝑡.

In the above terms the Young-Laplace equation (2.1) can be written as 1𝛾⋅𝑅₁+1𝑅₂=−𝜌⋅𝑔⋅𝑧−𝑝. (2.4) 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 𝑧^′′=−𝜌⋅𝑔⋅𝑧+𝑝𝛾1+𝑧^′23/2−1𝑟⋅1+𝑧^′2⋅𝑧^′for𝑅_𝑐≤𝑟≤𝑅_𝑑=shaperradius. (2.5) If the solution 𝑧(𝑟) of (2.5) is the meridian curve of the free surface of a meniscus on the interval [𝑅_𝑐,𝑅_𝑑], then it verifies the following boundary conditions: 𝑧^′𝑅_𝑐𝜋=−tan2−𝛼_𝑔;𝑧𝑅_𝑐=ℎ_𝑐𝑧>0,_′𝑅_𝑑=−tan𝛼_𝑐;𝑧𝑅_𝑑𝑧(𝑟)𝑅=0,isstrictlydecreasingon_𝑐,𝑅_𝑑. (2.6)(2.7)(2.8)

The first condition in (2.6) expresses that at the three-phase point (𝑅_𝑐,ℎ_𝑐), 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 ℎ_𝑐>0.

The first condition in (2.7) expresses that, at the three-phase point (𝑅_𝑑,0), where the meridian curve touches the outer edge of the shaper, the catching angle is equal to 𝛼_𝑐.

The second condition in (2.7) expresses that at the three-phase point (𝑅_𝑑,0) 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 𝑧(𝑟) 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: 𝐼(𝑧)=^𝑅𝑑_𝑅𝑐𝛾⋅1+𝑧^′21/2−12⋅𝜌⋅𝑔⋅𝑧²𝑧𝑅−𝑝⋅𝑧⋅𝑟⋅𝑑𝑟,_𝑐=ℎ_𝑐>0,𝑧𝑅_𝑑=0. (2.9) The nonlinear boundary value problem (NLBVP), (2.5) and (2.6), is equivalent to the NLBVP: 𝑑𝑧𝑑𝑟=−tan𝛼,𝑑𝛼=𝑑𝑟𝜌⋅𝑔⋅𝑧+𝑝𝛾⋅1−1cos𝛼𝑟𝑧𝑅⋅tan𝛼,_𝑐=ℎ_𝑐,𝛼𝑅_𝑐=𝜋2−𝛼_𝑔,𝑧𝑅_𝑑=0,𝛼𝑅_𝑑=𝛼_𝑐. (2.10)

3. Existence of the Solution

Consider the differential equation: 𝑧^′′=−𝜌⋅𝑔⋅𝑧+𝑝𝛾1+𝑧^′23/2−1𝑟⋅1+𝑧^′2⋅𝑧^′for0<𝑅_𝑐≤𝑟≤𝑅_𝑑𝑅_𝑑>0. (3.1) And 𝛼_𝑐,𝛼_𝑔 such that 𝛼_𝑐,𝛼_𝑔∈(0,𝜋/2).

Definition 3.1. A solution 𝑧=𝑧(𝑟) of (3.1) is a globally convex solution of the NLBVP (2.5) and (2.6) on the interval [𝑅_𝑐,𝑅_𝑑](0<𝑅_𝑐<𝑅_𝑑) if it possesses the following properties: (a) 𝑧^′(𝑅_𝑐)=−tan(𝜋/2−𝛼_𝑔),  (b) 𝑧^′(𝑅_𝑑)=−tan𝛼_𝑐,  (c) 𝑧(𝑅_𝑑)=0 and 𝑧(𝑟) is strictly decreasing on [𝑅_𝑐,𝑅_𝑑],  (d) 𝑧^′′(𝑟)>0forall𝑟∈[𝑅_𝑐,𝑅_𝑑],   A solution 𝑧(𝑟) of (3.1) is a globally-concave solution of the NLBVP (2.5), (2.6) on the interval [𝑅_𝑐,𝑅_𝑑](0<𝑅_𝑐<𝑅_𝑑) if verifies (a) (b) (c) and in addition the following inequality holds: (e) 𝑧^′′(𝑟)<0forall𝑟∈[𝑅_𝑐,𝑅_𝑑].

Remark 3.2. If 𝛼_𝑐<𝜋/2−𝛼_𝑔, then for a globally concave solution 𝑧(𝑟)(𝑧^′′(𝑟)<0) of (3.1), which satisfies (b) and (c), there is no 𝑅_𝑐∈(0,𝑅_𝑑) such that (a) holds. Therefore, if 𝛼_𝑐<𝜋/2−𝛼_𝑔 and we are interested in solutions of (3.1) which satisfy (b) and (c) and are not globally concave.
If 𝜋/2−𝛼_𝑔<𝛼_𝑐, then for a globally convex solution 𝑧(𝑟)(𝑧^′′(𝑟)<0) of (3.1), which satisfies (b) and (c), and there is no 𝑅_𝑐∈(0,𝑅_𝑑) such that a. holds. Therefore, if 𝜋/2−𝛼_𝑔<𝛼_𝑐, 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 𝛼_𝑐<𝜋/2−𝛼_𝑔 and there exists a globally convex solution 𝑧(𝑟) of the NLBVP (2.5) and (2.6), on [𝑅_𝑐,𝑅_𝑑], then the following inequalities hold: 𝛼𝛾⋅^𝑐+𝛼_𝑔−𝜋/2𝑅_𝑑−𝑅_𝑐⋅cos𝛼_𝑐−𝜌⋅𝑔⋅𝑅_𝑑−𝑅_𝑐𝜋⋅tan2−𝛼_𝑔+𝛾𝑅_𝑑⋅sin𝛼_𝑐𝛼≤𝑝≤𝛾⋅^𝑐+𝛼_𝑔−𝜋/2𝑅_𝑑−𝑅_𝑐⋅sin𝛼_𝑔+𝛾𝑅_𝑐⋅cos𝛼_𝑔. (3.2)

Proof. Let 𝑧(𝑟) be a globally convex solution of the problem (2.5) and (2.6) on [𝑅_𝑐,𝑅_𝑑] and 𝛼(𝑟)=−arctan𝑧^′(𝑟). The function 𝛼(𝑟) verifies the following quation: 𝛼^′(𝑟)=𝜌⋅𝑔⋅𝑧(𝑟)+𝑝𝛾⋅1(𝑟)−1cos𝛼𝑟⋅tan𝛼(𝑟) (3.3) and the boundary conditions: 𝛼𝑅_𝑐=𝜋2−𝛼_𝑔,𝛼𝑅_𝑑=𝛼_𝑐. (3.4) Hence, according to the Lagrange mean value theorem, there exists 𝑟^′∈(𝑅_𝑐,𝑅_𝑑) such that the following equality holds: 𝛼𝑝=𝛾⋅^𝑐+𝛼_𝑔−𝜋/2𝑅_𝑑−𝑅_𝑐⋅cos𝛼𝑟^′−𝜌⋅𝑔⋅𝑧𝑟^′+𝛾𝑟^′⋅sin𝛼𝑟^′. (3.5) Since 𝑧^′′(𝑟)>0 on [𝑅_𝑐,𝑅_𝑑], 𝑧^′(𝑟) is strictly increasing, and 𝛼(𝑟)=−arctan𝑧^′(𝑟) is strictly decreasing on [𝑅_𝑐,𝑅_𝑑]. Therefore, the following inequalities hold: 𝛼_𝑐≤𝛼𝑟^′≤𝜋2−𝛼_𝑔,sin𝛼_𝑔≤cos𝛼𝑟^′≤cos𝛼_𝑐,sin𝛼_𝑐≤sin𝛼𝑟^′≤cos𝛼_𝑔,−𝜌⋅𝑔⋅𝑅_𝑑−𝑅_𝑐𝜋⋅tan2−𝛼_𝑔≤−𝜌⋅𝑔⋅𝑧𝑟^′≤−𝜌⋅𝑔⋅𝑅_𝑑−𝑟^′⋅tan𝛼_𝑐. (3.6) Equality (3.5) and inequalities (3.6) imply (3.2).

Corollary 3.4. In terms of the ratio 𝑛=(𝑅_𝑑/𝑅_𝑐)(𝑛>1) the inequalities (3.2) can be written as 𝛼𝛾⋅^𝑐+𝛼_𝑔−𝜋/2𝑅_𝑑⋅𝑛𝑛−1⋅cos𝛼_𝑐−𝜌⋅𝑔⋅𝑅_𝑑⋅(𝑛−1)𝑛𝜋⋅tan2−𝛼_𝑔+𝛾𝑅_𝑑⋅sin𝛼_𝑐𝛼≤𝑝≤𝛾⋅^𝑐+𝛼_𝑔−𝜋/2𝑅_𝑑⋅𝑛𝑛−1⋅sin𝛼_𝑔+𝛾𝑅_𝑑⋅𝑛⋅cos𝛼_𝑔. (3.7)

Corollary 3.5. If 𝑛→+∞, then 𝑅_𝑐→0 and (3.7) become 𝛼𝛾⋅^𝑐+𝛼_𝑔−𝜋/2𝑅_𝑑⋅cos𝛼_𝑐−𝜌⋅𝑔⋅𝑅_𝑑𝜋⋅tan2−𝛼_𝑔+𝛾𝑅_𝑑⋅sin𝛼_𝑐≤𝑝<+∞. (3.8) If 𝑛→1, then 𝑅_𝑐→𝑅_𝑑 and 𝑝→−∞.

Theorem 3.6 (necessary condition for the existence of a globally concave solution). If 𝜋/2−𝛼_𝑔<𝛼_𝑐, and there exists a globally concave solution 𝑧(𝑟)(𝑧^′′(𝑟)<0) of the NLBVP (2.5) and (2.6) on [𝑅_𝑐,𝑅_𝑑], then the following inequalities hold: 𝛼𝛾⋅^𝑐+𝛼_𝑔−𝜋/2𝑅_𝑑−𝑅_𝑐⋅cos𝛼_𝑐−𝜌⋅𝑔⋅𝑅_𝑑−𝑅_𝑐⋅tan𝛼_𝑐+𝛾𝑅_𝑑⋅cos𝛼_𝑔𝛼≤𝑝≤𝛾⋅^𝑐+𝛼_𝑔−𝜋/2𝑅_𝑑−𝑅_𝑐⋅sin𝛼_𝑔+𝛾𝑅_𝑐⋅sin𝛼_𝑐. (3.9)

Proof. Since 𝑧^′′(𝑟)<0 on [𝑅_𝑐,𝑅_𝑑], 𝑧^′(𝑟) is strictly decreasing and 𝛼(𝑟)=−arctan𝑧^′(𝑟) is strictly increasing on [𝑅_𝑐,𝑅_𝑑]. Therefore the following inequalities hold: 𝜋2−𝛼_𝑔≤𝛼(𝑟)≤𝛼_𝑐,cos𝛼_𝑐≤cos𝛼(𝑟)≤sin𝛼_𝑔,cos𝛼_𝑔≤sin𝛼(𝑟)≤sin𝛼_𝑐,−𝜌⋅𝑔⋅𝑅_𝑑−𝑅_𝑐⋅tan𝛼_𝑐≤−𝜌⋅𝑔⋅𝑧(𝑟)≤−𝜌⋅𝑔⋅𝑅_𝑑−𝑟^′𝜋⋅tan2−𝛼_𝑔 (3.10) for every 𝑟∈[𝑅_𝑐,𝑅_𝑑].

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

Corollary 3.7. In terms of the ratio 𝑛=(𝑅_𝑑/𝑅_𝑐)(𝑛>1) the inequalities (3.9) can be written as 𝛼𝛾⋅^𝑐+𝛼_𝑔−𝜋/2𝑅_𝑑⋅𝑛𝑛−1⋅cos𝛼_𝑐−𝜌⋅𝑔⋅𝑅_𝑑⋅(𝑛−1)𝑛⋅tan𝛼_𝑐+𝛾𝑅_𝑑⋅cos𝛼_𝑔𝛼≤𝑝≤𝛾⋅^𝑐+𝛼_𝑔−𝜋/2𝑅_𝑑⋅𝑛𝑛−1⋅sin𝛼_g+𝛾𝑅_𝑑⋅𝑛⋅sin𝛼_𝑐. (3.11)

Corollary 3.8. If 𝑛→+∞, then 𝑅_𝑐→0 and (3.11) become 𝛼𝛾⋅^𝑐+𝛼_𝑔−𝜋/2𝑅_𝑑⋅cos𝛼_𝑐−𝜌⋅𝑔⋅𝑅_𝑑⋅tan𝛼_𝑐+𝛾𝑅_𝑑⋅cos𝛼_𝑔≤𝑝<+∞. (3.12) If n→1, then R_c→R_d and p→+∞.

Theorem 3.9 (necessary condition for the existence of a convex-concave solution). If for 𝛼_𝑐<𝜋/2−𝛼_𝑔 and 𝑝>(𝛾/𝑅_𝑑)⋅sin𝛼_𝑐and there exists a solution 𝑧(𝑟) of the NLBVP (2.5) and (2.6) on the interval [𝑅_𝑐,𝑅_𝑑](0<𝑅_𝑐<𝑅_𝑑) and then for 𝑝 the following inequalities hold: 𝛾𝑅_𝑑⋅sin𝛼_𝑐𝛾<𝑝<𝑅_𝑐⋅cos𝛼_𝑔. (3.13)

Proof. Since 𝑝>(𝛾/𝑅_𝑑)⋅sin𝛼_𝑐, we have (𝑝/𝛾−1/𝑅_𝑑)⋅sin𝛼_𝑐>0 and hence 𝛼^′(𝑅_𝑑)=(1/cos𝛼_𝑐)⋅[(𝑝/𝛾−1/𝑅_𝑑)⋅sin𝛼_𝑐]>0. It follows that, for 𝑟<𝑅_𝑑, 𝑟 close to 𝑅_𝑑, the following inequality: 𝛼(𝑟)<𝛼(𝑅_𝑑)=𝛼_𝑐 holds. Since 𝛼(𝑅_𝑐)=𝜋/2−𝛼_𝑔>𝛼_𝑐, it follows that there exists 𝑟^′∈(𝑅_𝑐,𝑅_𝑑) such that 𝛼^′(𝑟^′)=0(𝑟^′ is the point where 𝛼(𝑟) is minimum). The equality 𝛼^′(𝑟^′)=0 implies that the following equality holds: 𝛾𝑝=𝑟^′⋅sin𝛼𝑟^′−𝜌⋅𝑔⋅𝑧𝑟^′. (3.14) Taking into account the fact that 𝑧(𝑟) is a decreasing function on [𝑅_𝑐,𝑅_𝑑], we deduce that 𝑝<(𝛾/𝑅_𝑐)⋅cos𝛼_𝑔.

Theorem 3.10 (necessary condition for the existence of a concave-convex solution). If for 𝛼_𝑐>𝜋/2−𝛼_𝑔 and 𝑝<(𝛾/𝑅_𝑑)⋅sin𝛼_𝑐, there exists a solution 𝑧(𝑟) of the NLBVP (2.5) and (2.6) on the interval [𝑅_𝑐,𝑅_𝑑](0<𝑅_𝑐<𝑅_𝑑), then for 𝑝 the following inequalities hold: 𝛾𝑅_𝑑⋅cos𝛼_𝑔𝛾<𝑝<𝑅_𝑑⋅sin𝛼_𝑐. (3.15)

Proof. Since 𝑝<(𝛾/𝑅_𝑑)⋅sin𝛼_𝑐, we have (𝑝/𝛾)−1/𝑅_𝑑⋅sin𝛼_𝑐<0 and hence 𝛼^′(𝑅_𝑑)=(1/cos𝛼_𝑐)⋅[(𝑝/𝛾−1/𝑅_𝑑)⋅sin𝛼_𝑐]<0. It follows that, for 𝑟<𝑅_𝑑, 𝑟 close to 𝑅_𝑑, the inequality 𝛼_𝑐=𝛼(𝑅_𝑑)<𝛼(𝑟) holds. Since 𝛼(𝑅_𝑐)=𝜋/2−𝛼_𝑔<𝛼_𝑐, it follows that there exists 𝑟^′∈(𝑅_𝑐,𝑅_𝑑) such that 𝛼^′(𝑟^′)=0 (𝑟^′ is the point where 𝛼(𝑟) is maximum). The equality 𝛼^′(𝑟^′)=0 implies that the equality (3.14) holds. Taking into account the fact that 𝑧(𝑟) is a decreasing function on [𝑅_𝑐,𝑅_𝑑], we deduce (3.15).

Theorem 3.11. If for 𝛼_𝑐<𝜋/2−𝛼_𝑔, 𝑝<(𝛾/𝑅_𝑑)⋅sin𝛼_𝑐 there exists a solution 𝑧(𝑟) of the NLBVP (2.5) and (2.6), on the interval [𝑅_𝑐,𝑅_𝑑](0<𝑅_𝑐<𝑅_𝑑) and 𝑧"(𝑅_𝑐)<0, then 𝑧(𝑟) is concave convex on [𝑅_𝑐,𝑅_𝑑] and there exists 𝑅¹_𝑐, 𝑅_𝑐<𝑅¹_𝑐<𝑅_𝑑 such that 𝑧(𝑟) is a globally convex solution of the NLBVP (2.5) and (2.6) on [𝑅¹_𝑐,𝑅_𝑑]. Moreover for 𝑝 the inequality (3.15) holds.

Proof. Since 𝑝<(𝛾/𝑅_𝑑)⋅sin𝛼_𝑐, we have 𝛼^′(𝑅_𝑑)<0. Inequality 𝑧"(𝑅_𝑐)<0 implies that 𝛼^′(𝑅_𝑐)>0. Hence, there exists 𝑟^′∈(𝑅_𝑐,𝑅_𝑑) such that 𝛼^′(𝑟^′)=0. This equality implies that (3.14) holds. Taking into account the fact that 𝑧(𝑟) is a decreasing function on [𝑅_𝑐,𝑅_𝑑], we deduce (3.15).
The existence of 𝑅¹_𝑐 follows from the fact that 𝜋/2−𝛼_𝑔=𝛼(𝑅_𝑐)<𝛼(𝑟) for 𝑟>𝑅_𝑐, 𝑟 close to 𝑅_𝑐 and 𝛼(𝑅_𝑑)=𝛼_𝑐<𝜋/2−𝛼_𝑔.

Theorem 3.12. If for 𝛼_𝑐>𝜋/2−𝛼_𝑔, 𝑝>(𝛾/𝑅_𝑑)⋅sin𝛼_𝑐, there exists a solution 𝑧(𝑟) of the NLBVP (2.5) and (2.6) on the interval [𝑅_𝑐,𝑅_𝑑](0<𝑅_𝑐<𝑅_𝑑) and 𝑧^′′(𝑅_𝑐)>0, then 𝑧(𝑟) is a convex concave solution of the NLBVP (2.5) and (2.6) on [𝑅_𝑐,𝑅_𝑑] and there exists 𝑅¹_𝑐, 𝑅_𝑐<𝑅¹_𝑐<𝑅_𝑑, such that 𝑧(𝑟) is a globally-concave solution of the NLBVP (2.5), (2.6) on [𝑅¹_𝑐,𝑅_𝑑]. Moreover for 𝑝 the inequality (3.13) holds.

Proof. Since 𝑝>(𝛾/𝑅_𝑑)⋅sin𝛼_𝑐, we have 𝛼^′(𝑅_𝑑)>0. Inequality 𝑧^′′(𝑅_𝑐)>0 implies that 𝛼^′(𝑟^′)=0. This equality implies that (3.14) holds. Taking into account the fact that 𝑧(𝑟) is a decreasing function on [𝑅_𝑐,𝑅_𝑑], we deduce that (3.13) holds.
The existence of 𝑅¹_𝑐 follows from the fact that 𝜋/2−𝛼_𝑔=𝛼(𝑅_𝑐)>𝛼(𝑟) for 𝑟>𝑅_𝑐, 𝑟 close to 𝑅_𝑐 and 𝛼(𝑅_𝑑)=𝛼_𝑐>𝜋/2−𝛼_𝑔.

Theorem 3.13 (sufficient condition for the existence of a globally convex solution). Let 𝑛>1 be a real number. If 𝛼_𝑐<𝜋/2−𝛼_𝑔 and 𝑝 satisfies 𝛼𝑝<𝛾⋅^𝑐+𝛼_𝑔−𝜋/2𝑅_𝑑⋅𝑛𝑛−1⋅cos𝛼_𝑐−𝜌⋅𝑔⋅𝑅_𝑑⋅𝑛−1𝑛𝜋⋅tan2−𝛼_𝑔+𝛾𝑅_𝑑⋅sin𝛼_𝑐, (3.16) then there exists 𝑅_𝑐 in the closed interval [𝑅_𝑑/𝑛,𝑅_𝑑] such that the solution of the initial value problem 𝑧^′′=−𝜌⋅𝑔⋅𝑧+𝑝𝛾1+𝑧^′23/2−1𝑟1+𝑧^′2⋅𝑧^′0<𝑟<𝑅_𝑑,𝑅_𝑑𝑧𝑅>0_𝑑=0,𝑧^′𝑅_𝑑=−tan𝛼_𝑐0<𝛼_𝑐<𝜋2−𝛼_𝑔,𝛼_𝑔∈𝜋0,2 (3.17) is a globally convex solution of the NLBVP (2.5) and (2.6) on [𝑅_𝑐,𝑅_𝑑].

Proof. Consider the solution 𝑧(𝑟) of the initial value problem (3.17). Denote by 𝐼 the maximal interval on which the function 𝑧(𝑟) exists and by 𝛼(𝑟) the function 𝛼(𝑟)=−arctan𝑧^′(𝑟) defined on 𝐼. for the function 𝛼(𝑟) the following equality holds: 𝛼^′(𝑟)=1cos𝛼𝑝+𝜌⋅𝑔⋅𝑧𝛾−sin𝛼𝑟. (3.18) Since 𝑧^′′𝑅_𝑑𝛼=−^′𝑅_𝑑cos²𝛼𝑅_𝑑1=−cos³𝛼𝑅_𝑑𝑝𝛾−sin𝛼𝑅_𝑑𝑅_𝑑𝑧>0,_′𝑅_𝑑=−tan𝛼_𝑐𝑧<0,_′𝑅_𝑑𝜋>−tan2−𝛼_𝑔, (3.19) there exists 𝑟^′∈𝐼,0<𝑟^′<𝑅_𝑑 such that for any 𝑟∈[𝑟^′,𝑅_𝑑] the following inequalities hold: 𝑧^′′(𝑟)>0,𝑧^′(𝑟)≤−tan𝛼_𝑐<0,𝑧^′(𝑟)𝜋>−tan2−𝛼_𝑔. (3.20)

Let now be 𝑟_∗ defined by 𝑟_∗=inf𝑟^′∈𝐼∣0<𝑟^′<𝑅_𝑑suchthatforany𝑟∈𝑟^′,𝑅_𝑑inequalities(3.20)hold. (3.21)

It is clear that 𝑟_∗≥0 and for any 𝑟∈(𝑟_∗,𝑅_𝑑] (3.20) hold. Moreover, 𝑧^′(𝑟_∗+0)=lim_{𝑟→𝑟∗,𝑟>𝑟∗}𝑧^′(𝑟) exists and satisfies 𝑧^′(𝑟_∗+0)<−tan𝛼_𝑐<0 and 𝑧^′(𝑟_∗+0)≥−tan(𝜋/2−𝛼_𝑔). Hence 𝑧(𝑟_∗+0)=lim_{𝑟→𝑟∗,𝑟>𝑟∗}𝑧(𝑟) is finite, it is strictly positive and for every 𝑟∈(𝑟_∗,𝑅_𝑑] the inequalities hold: 0≤𝑧(𝑟)<𝑧𝑟_∗+0;𝑅_𝑑−𝑟_∗⋅tan𝛼_𝑐≤𝑧𝑟_∗+0≤𝑅_𝑑−𝑟_∗𝜋⋅tan2−𝛼_𝑔. (3.22) We will show now that 𝑟_∗≥𝑅_𝑑/𝑛 and 𝑧^′(𝑟_∗+0)=−tan(𝜋/2−𝛼_𝑔).

In order to show that 𝑟_∗≥𝑅_𝑑/𝑛 we assume the contrary, that is, 𝑟_∗<𝑅_𝑑/𝑛. Under this hypothesis we have 𝛼𝑅_𝑑𝑛−𝛼𝑅_𝑑=−𝛼^′𝑟^′⋅𝑅_𝑑−𝑅_𝑑𝑛=1cos𝛼𝑟^′−𝑝𝛾−𝜌⋅𝑔⋅𝑧𝑟^′𝛾+sin𝛼𝑟^′𝑟^′⋅𝑅_𝑑−𝑅_𝑑𝑛>𝑅_𝑑−𝑅_𝑑/𝑛cos𝛼𝑟^′⋅−𝛼_𝑐+𝛼_𝑔−𝜋/2𝑅_𝑑⋅𝑛𝑛−1⋅cos𝛼_𝑐+𝜌⋅𝑔⋅𝑅_𝑑𝛾⋅𝑛−1𝑛𝜋⋅tan2−𝛼_𝑔−1𝑅_𝑑⋅sin𝛼_𝑐−𝜌⋅𝑔⋅𝑧𝑟^′𝛾+sin𝛼𝑟^′𝑟^′>𝜋2−𝛼_𝑐+𝛼_𝑔 (3.23) for some 𝑟^′∈(𝑅_𝑑/𝑛,𝑅_𝑑). Hence 𝛼(𝑅_𝑑/𝑛)>𝜋/2−𝛼_𝑔. This last inequality is impossible, according to the definition (3.21) of 𝑟_∗. Therefore, 𝑟_∗, defined by (3.21), satisfies 𝑟_∗≥𝑅_𝑑/𝑛.

In order to show that 𝑧^′(𝑟_∗+0)=−tan(𝜋/2−𝛼_𝑔), we remark that from the definition (3.21) of 𝑟_∗ it follows that in 𝑟_∗ at least one of the following three equalities hold: 𝑧^′𝑟_∗+0=−tan𝛼_𝑐,𝑧^′′𝑟_∗+0=0,𝑧^′𝑟_∗𝜋+0=−tan2−𝛼_𝑔. (3.24) Since 𝑧^′(𝑟_∗+0)<𝑧^′(𝑟)≤−tan𝛼_𝑐 for any 𝑟∈(𝑟_∗,𝑅_𝑑], it follows that the equality 𝑧^′(𝑟_∗+0)=−tan𝛼_𝑐 is impossible. Hence, we deduce that at 𝑟_∗ at least one of the following two equalities hold: 𝑧^′′(𝑟_∗+0)=0,𝑧^′(𝑟_∗+0)=−tan(𝜋/2−𝛼_𝑔).

We show now that the equality 𝑧^′′(𝑟_∗+0)=0is impossible. For that we assume the contrary, that is, 𝑧^′′(𝑟_∗+0)=0. Under this hypothesis, from (3.17) we have 𝑝=−𝜌⋅𝑔⋅𝑧𝑟_∗+𝛾+0𝑟_∗⋅sin𝛼𝑟_∗+0≥−𝜌⋅𝑔⋅𝑅_𝑑−𝑟_∗𝜋⋅tan2−𝛼_𝑔+𝛾𝑟_∗⋅sin𝛼𝑟_∗>𝛾+0𝑅_𝑑⋅sin𝛼_𝑐−𝜌⋅𝑔⋅𝑅_𝑑⋅𝑛−1𝑛𝜋⋅tan2−𝛼_𝑔>𝛾𝑅_𝑑⋅sin𝛼_𝑐−𝜌⋅𝑔⋅𝑅_𝑑⋅𝑛−1𝑛𝜋⋅tan2−𝛼_𝑔𝛼+𝛾⋅^𝑐+𝛼_𝑔−𝜋/2𝑅_𝑑⋅𝑛𝑛−1⋅cos𝛼_𝑐 (3.25) what is impossible.

In this way we obtain that the equality 𝑧^′(𝑟_∗+0)=−tan(𝜋/2−𝛼_𝑔) holds.

Taking now 𝑅_𝑐=𝑟_∗ we find that the solution of the initial value problem (3.17) on the interval [𝑅_𝑐,𝑅_𝑑] is convex, and it is the meridian curve of an appropriate meniscus.

Corollary 3.14. If 𝛼_𝑐<𝜋/2−𝛼_𝑔 and 𝛼𝑝<𝛾⋅^𝑐+𝛼_𝑔−𝜋/2𝑅_𝑑⋅cos𝛼_𝑐−𝜌⋅𝑔⋅𝑅_𝑑𝜋⋅tan2−𝛼_𝑔, (3.26) then there exists 𝑅_𝑐∈(0,𝑅_𝑑] 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 [𝑅_𝑐,𝑅_𝑑].

Corollary 3.15. If 𝛼_𝑐<𝜋/2−𝛼_𝑔, 𝑛>𝑛^′>1 and the inequalities hold 𝛼𝛾⋅^𝑐+𝛼_𝑔−𝜋/2𝑅_𝑑⋅𝑛^′𝑛^′−1⋅sin𝛼_𝑔+𝛾𝑅_𝑑⋅𝑛^′cos𝛼_𝑔𝛼<𝑝<𝛾⋅^𝑐+𝛼_𝑔−𝜋/2𝑅_𝑑⋅𝑛𝑛−1⋅cos𝛼_𝑐−𝜌⋅𝑔⋅𝑅_𝑑⋅𝑛−1𝑛𝜋⋅tan2−𝛼_𝑔+𝛾𝑅_𝑑⋅sin𝛼_𝑐, (3.27) then there exists 𝑅_𝑐∈[𝑅_𝑑/𝑛,𝑅_𝑑/𝑛^′] such that the solution of the initial value problem (3.17) is a globally convex solution of the NLBVP (2.5), (2.6) on [𝑅_𝑐,𝑅_𝑑].

The existence of 𝑅_𝑐 and the inequality 𝑅_𝑐>𝑅_𝑑/𝑛 follows from Theorem 3.13. The inequality 𝑅_𝑐<𝑅_𝑑/𝑛^′ follows from the Corollary 3.4 of Theorem 3.3.

Theorem 3.16 (sufficient condition for the existence of globally concave solution). Let 𝑛>1 be a real number. If 𝛼_𝑐>𝜋/2−𝛼_𝑔 and 𝑝 satisfies 𝛼𝑝<𝛾⋅^𝑐+𝛼_𝑔−𝜋/2𝑅_𝑑⋅𝑛𝑛−1⋅sin𝛼_𝑔+𝛾𝑅_𝑑⋅𝑛⋅sin𝛼_𝑐, (3.28) then there exists 𝑅_𝑐 in the closed interval [𝑅_𝑑/𝑛,𝑅_𝑑] such that the solution of the initial value problem 𝑧^′′=−𝜌⋅𝑔⋅𝑧+𝑝𝛾1+𝑧^′23/2−1𝑟1+𝑧^′2⋅𝑧^′0<𝑟<𝑅_𝑑,𝑅_𝑑𝑧𝑅>0_𝑑=0,𝑧^′𝑅_𝑑=−tan𝛼_𝑐𝜋0<2−𝛼_𝑔<𝛼_𝑐<𝜋2 (3.29) is a globally concave solution of the NLBVP (2.5) and (2.6) on [𝑅_𝑐,𝑅_𝑑].

Proof. Consider the solution 𝑧(𝑟) of IVP (3.29). Denote by 𝐼 the maximal interval on which the function 𝑧(𝑟) exists and by 𝛼(𝑟) the function 𝛼(𝑟)=−arctan𝑧^′(𝑟) defined on 𝐼. For the function 𝛼(𝑟) the following equality holds: 𝛼^′(𝑟)=1(𝑟)⋅𝑝cos𝛼𝛾+𝜌⋅𝑔⋅𝑧(𝑟)𝛾−sin𝛼𝑟. (3.30) Since 𝑧^′′𝑅_𝑑𝛼=−^′𝑅_𝑑cos²𝛼𝑅_𝑑1=−cos³𝛼𝑅_𝑑𝑝𝛾−sin𝛼𝑅_𝑑𝑅_𝑑𝑧<0,_′𝑅_𝑑=−tan𝛼_𝑐𝑧<0,_′𝑅_𝑑𝜋<−tan2−𝛼_𝑔, (3.31) there exists 𝑟^′∈𝐼,0<𝑟^′<𝑅_𝑑 such that for any 𝑟∈[𝑟^′,𝑅_𝑑] the following inequalities hold: 𝑧^′′(𝑟)<0,𝑧^′(𝑟)≥−tan𝛼_𝑐,𝑧^′(𝑟)𝜋≤−tan2−𝛼_𝑔. (3.32) Let now 𝑟_∗ be defined by 𝑟_∗=inf𝑟^′∈𝐼∣0<𝑟^′<𝑅_𝑑suchthatforany𝑟∈𝑟^′,𝑅_𝑑inequalities(3.32)hold. (3.33) It is clear that 𝑟_∗≥0 and for any 𝑟∈(𝑟_∗,𝑅_𝑑] inequalities (3.32) hold. Moreover, 𝑧^′(𝑟_∗+0)=lim_{𝑟→𝑟∗,𝑟>𝑟∗}𝑧^′(𝑟) exists and satisfies −tan𝛼_𝑐≤𝑧^′(𝑟)≤−tan(𝜋/2−𝛼_𝑔). Hence 𝑧(𝑟_∗+0)=lim_{𝑟→𝑟∗,𝑟>𝑟∗}𝑧(𝑟) is finite, it is strictly positive and for every 𝑟∈(𝑟_∗,𝑅_𝑑] the inequalities hold: 0≤𝑧(𝑟)≤𝑧𝑟_∗+0;𝑅_𝑑−𝑟_∗𝜋⋅tan2−𝛼_𝑔≤𝑧𝑟_∗+0≤𝑅_𝑑−𝑟_∗⋅tan𝛼_𝑐. (3.34) We will show now that 𝑟_∗≥𝑅_𝑑/𝑛 and 𝑧^′(𝑟_∗+0)=−tan(𝜋/2−𝛼_𝑔).
In order to show the inequality 𝑟_∗≥𝑅_𝑑/𝑛 we assume the contrary, that is, that 𝑟_∗<𝑅_𝑑/𝑛. Under this hypothesis we have 𝛼𝑅_𝑑𝑅−𝛼^𝑑𝑛=𝛼^′𝑟^′⋅𝑅_𝑑−𝑅_𝑑𝑛=𝑅_𝑑−𝑅_𝑑/𝑛cos𝛼𝑟^′𝑝𝛾+𝜌⋅𝑔⋅𝑧𝑟^′𝛾−sin𝛼𝑟^′𝑟^′>𝑅_𝑑⋅((𝑛−1)/𝑛)cos𝛼𝑟^′⋅𝛼_𝑐+𝛼_𝑔−𝜋/2𝑅_𝑑⋅𝑛𝑛−1⋅sin𝛼_𝑔+𝑛𝑅_𝑑⋅sin𝛼_c+𝜌⋅𝑔⋅𝑧𝑟^′𝛾−sin𝛼𝑟^′𝑟^′ (3.35) for some 𝑟^′∈(𝑅_𝑑/𝑛,𝑅_𝑑). Since (𝑛/𝑅_𝑑)⋅sin𝛼_𝑐>(𝑛/𝑟^′)⋅sin𝛼(𝑟^′), sin𝛼_𝑔/cos𝛼(𝑟^′)>1, and 𝜌⋅𝑔⋅𝑧(𝑟^′)/𝛾>0, it follows that 𝛼(𝑅_𝑑)−𝛼(𝑅_𝑑/𝑛)>𝛼_𝑐+𝛼_𝑔−𝜋/2. Hence 𝛼(𝑅_𝑑/𝑛)>𝜋/2−𝛼_𝑔, what is impossible, according to the definition of 𝑟_∗. Therefore, 𝑟_∗, defined by (3.33), satisfies 𝑟_∗≥𝑅_𝑑/𝑛.
In order to show that 𝑧^′(𝑟_∗+0)=−tan(𝜋/2−𝛼_𝑔), we remark that from the definition (3.33) of 𝑟_∗ it follows that in 𝑟_∗ at least one of the following three equalities holds: 𝑧^′𝑟_∗+0=−tan𝛼_𝑐,𝑧^′′𝑟_∗+0=0,𝑧^′𝑟_∗𝜋+0=−tan2−𝛼_𝑔. (3.36) Since 𝑧^′(𝑟_∗+0)>𝑧^′(𝑟)>−tan𝛼_𝑐 for any 𝑟∈(𝑟_∗,𝑅_𝑑], it follows that the equality 𝑧^′(𝑟_∗+0)=−tan𝛼_𝑐 is impossible. Hence, we deduce that at 𝑟_∗ at least one of the following two equalities holds: 𝑧^′′(𝑟_∗+0)=0,𝑧^′(𝑟_∗+0)=−tan(𝜋/2−𝛼_𝑔).
We show now that the equality 𝑧^′′(𝑟_∗+0)=0is impossible. For that we assume the contrary, that is, 𝑧^′′(𝑟_∗+0)=0. Under this hypothesis, from (3.29) we have 𝑝=−𝜌⋅𝑔⋅𝑧(𝑟_∗+0)+(𝛾/𝑟_∗)⋅sin𝛼(𝑟_∗+0)<(𝑛⋅𝛾/𝑅_𝑑)⋅sin𝛼_𝑐, what is impossible.
In this way we obtain that the equality 𝑧^′(𝑟_∗+0)=−tan(𝜋/2−𝛼_𝑔) holds.
Taking now 𝑅_𝑐=𝑟_∗ we find that the solution of the initial value problem (3.29) on the interval [𝑅_𝑐,𝑅_𝑑] is concave, and it is the meridian curve of an appropriate meniscus.

Corollary 3.17. If 𝛼_𝑐>𝜋/2−𝛼_𝑔, 𝑛>𝑛^′>1 and the inequalities hold: 𝛼𝛾⋅^𝑐+𝛼_𝑔−𝜋/2𝑅_𝑑⋅𝑛𝑛−1⋅sin𝛼_𝑔+𝛾𝑅_𝑑⋅𝑛⋅sin𝛼_𝑐𝛼<𝑝<𝛾⋅^𝑐+𝛼_𝑔−𝜋/2𝑅_𝑑⋅𝑛^′𝑛^′−1⋅cos𝛼_𝑐−𝜌⋅𝑔⋅𝑅_𝑑⋅𝑛^′−1𝑛^′⋅tan𝛼_𝑐+𝛾𝑅_𝑑⋅sin𝛼_𝑐, (3.37) then there exists 𝑅_𝑐∈[𝑅_𝑑/𝑛,𝑅_𝑑/𝑛^′] such that the solution of the IVP (3.29) is a globally concave solution of the NBVP (2.5), (2.6) on [𝑅_𝑐,𝑅_𝑑].
The existence of 𝑅_𝑐 and the inequality 𝑅_𝑑/𝑛<𝑅_𝑐 follows from Theorem 3.16. The inequality 𝑅_𝑐<𝑅_𝑑/𝑛 follows from Corollary 3.7. and Theorem 3.6.

4. Stability of the Solution

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

Theorem 4.2. If for 𝛼_𝑐<𝜋/2−𝛼_𝑔 the NLBVP (2.5), (2.6) has a globally convex solution 𝑧(𝑟) on [𝑅_𝑐,𝑅_𝑑] and the ratio 𝑛=𝑅_𝑑/𝑅_𝑐>1 satisfies: 𝑛−1𝑛^1/21<𝜋⋅𝑅_𝑑⋅𝛾^1/2⋅sin^3/2𝛼_𝑔𝜌^1/2⋅𝑔^1/2 (4.1) 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: 𝑟⋅𝛾1+𝑧^′23/2>0 (4.2) and it is satisfied.
By computation we find that the Jacobi equation in this case is 𝑑𝑑𝑟𝑟⋅𝛾1+𝑧^′23/2⋅𝑑𝜂𝑑𝑟+𝜌⋅𝑔⋅𝑧⋅𝜂=0. (4.3) For the coefficients of (4.3) the following inequality holds: 𝑟⋅𝛾1+𝑧^′23/2≥𝑅_𝑑𝑛⋅𝛾⋅sin³𝛼_𝑔,𝜌⋅𝑔⋅𝑧≤𝜌⋅𝑔⋅𝑅_𝑑. (4.4) Hence, 𝑑𝑅𝑑𝑟𝑛^𝑑⋅𝛾⋅sin³𝛼_𝑔⋅𝑑𝜂𝑑𝑟+𝜌⋅𝑔⋅𝑅_𝑑⋅𝜂=0 (4.5) is a Sturm type upper bound [19] for (4.3).
An arbitrary solution of (4.5) is given by 𝜂(𝑟)=𝐴⋅sin𝜔⋅𝑟+𝜑, (4.6) where 𝐴 and 𝜑 are real constants and 𝜔² is given by 𝜔²=𝜌⋅𝑔⋅𝑛𝛾⋅sin³𝛼_𝑔. (4.7)
Inequality (4.1) implies that the half period 𝜋/𝜔 of 𝜂(𝑟) satisfies the inequality 𝜋/𝜔>𝑅_𝑑−𝑅_𝑐. Hence a nonzero solution given by (4.6) has at most one zero on the interval [𝑅_𝑐,𝑅_𝑑]. Since any nonzero solution of (4.5) vanishes at most once on the interval [𝑅_𝑐,𝑅_𝑑], the solution of (4.3) which satisfies 𝜂(𝑅_𝑑)=0,𝜂^′(𝑅_𝑑)=1 has only one zero on [𝑅_𝑐,𝑅_𝑑]. Hence the Jacobi condition is satisfied in this case.

Theorem 4.3. If for 𝛼_𝑐<𝜋/2−𝛼_𝑔 the NLBVP (2.5) and (2.6) has a globally convex solution 𝑧(𝑟) on [𝑅_𝑐,𝑅_𝑑], and the ratio 𝑛=𝑅_𝑑/𝑅_𝑐>1 satisfies 𝑛−1𝑛^1/21>2𝜋⋅𝑅_𝑑⋅𝛾^1/2⋅cos^3/2𝛼_𝑐𝜌^1/2⋅𝑔^1/2, (4.8) then the meniscus is unstable.

Proof. We remark now that for the Jacobi equation the following inequalities hold: 𝑟⋅𝛾1+𝑧^′23/2≤𝑅_𝑑⋅𝛾⋅cos³𝛼_𝑐𝑅,𝜌⋅𝑔⋅𝑧≥𝜌⋅𝑔⋅^𝑑𝑛. (4.9) Hence 𝑑𝑅𝑑𝑟^𝑑𝛾⋅cos³𝛼_𝑐⋅𝑑𝜂𝑅𝑑𝑟+𝜌⋅𝑔⋅^𝑑𝑛⋅𝜂=0 (4.10) is a Sturm type lower bound [19] of the Jacobi equation (4.3).
An arbitrary solution of (4.10) is given by 𝜂(𝑟)=𝐴⋅sin𝜔⋅𝑟+𝜑, (4.11) where 𝜔² is given by 𝜔²=𝜌⋅𝑔𝑛⋅𝛾⋅cos³𝛼_𝑐. (4.12) Inequality (4.8) implies that 𝜂(𝑟) given by (4.11) vanishes twice on [𝑅_𝑐,𝑅_𝑑]. Therefore the Jacobi condition is not satisfied.

Theorem 4.4. If for 𝜋/2−𝛼_𝑔<𝛼_𝑐<𝜋/2 the NLBVP (2.5), (2.6) has a globally-concave solution 𝑧(𝑟) on [𝑅_𝑐,𝑅_𝑑] and the ratio 𝑛=𝑅_𝑑/𝑅_𝑐>1 satisfies: 𝑛−1𝑛^1/21<𝜋⋅𝑅_𝑑⋅𝛾^1/2⋅cos^3/2𝛼_𝑐𝜌^1/2⋅𝑔^1/2 (4.13) 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: 𝑟⋅𝛾1+𝑧^′23/2≥𝑅_𝑑𝑛⋅𝛾⋅cos³𝛼_𝑐,𝜌⋅𝑔⋅𝑧≤𝜌⋅𝑔⋅𝑅_𝑑. (4.14) Hence 𝑑𝑅𝑑𝑟𝑛^𝑑⋅𝛾⋅cos³𝛼_𝑐⋅𝑑𝜂𝑑𝑟+𝜌⋅𝑔⋅𝑅_𝑑⋅𝜂=0 (4.15) is a Sturm type upper bound [19] of the Jacobi equation (4.3).
An arbitrary solution of (4.15) is given by 𝜂(𝑟)=𝐴⋅sin𝜔⋅𝑟+𝜑, (4.16) where 𝐴 and 𝜑 are real constants and 𝜔² is given by 𝜔²=𝜌⋅𝑔⋅𝑛𝛾⋅cos³𝛼_𝑐. (4.17) Inequality (4.13) implies that the half period 𝜋/𝜔 of 𝜂(𝑟) satisfies the inequality 𝜋/𝜔>𝑅_𝑑−𝑅_𝑐. It follows that a nonzero solution given by (4.16) has at most one zero on the interval [𝑅_𝑐,𝑅_𝑑]. Since any nonzero solution of (4.15) vanishes at most once on the interval [𝑅_𝑐,𝑅_𝑑], the solution of the Jacobi equation (4.3), which satisfies 𝜂(𝑅_𝑑)=0,𝜂^′(𝑅_𝑑)=1, has only one zero on [𝑅_𝑐,𝑅_𝑑]. Hence the Jacobi condition is satisfied.

Theorem 4.5. If for 𝜋/2−𝛼_𝑔<𝛼_𝑐<𝜋/2 the NLBVP (2.5) and (2.6) has a globally concave solution 𝑧(𝑟) on [𝑅_𝑐,𝑅_𝑑] and the ratio 𝑛=𝑅_𝑑/𝑅_𝑐>1 satisfies 𝑛−1𝑛^3/21>2𝜋⋅𝑅_𝑑⋅𝛾^1/2⋅sin^3/2𝛼_𝑔𝜌^1/2⋅𝑔^1/2, (4.18) then the solution is unstable.

Proof. Remark that in this case for the coefficients of the Jacobi equation (4.3) the following inequalities hold: 𝑟⋅𝛾1+𝑧^′23/2≤𝑅_𝑑⋅𝛾⋅sin³𝛼_𝑔𝑅,𝜌⋅𝑔⋅𝑧≥𝜌⋅𝑔⋅^𝑑𝑛. (4.19) 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. 𝑅_𝑑=5×10⁻⁴[m]; 𝑅_𝑐=4×10⁻⁴[m]; 𝛼¹_𝑐=0.523[rad];  𝛼²_𝑐=1.4[rad]; 𝛼_𝑔=0.2967[rad]; 𝜌=3.6×10³[kg/m³]; 𝛾=7.81×10⁻¹[N/m]; 𝑔=9.81[m/s²].

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

(a) Inequality (3.2) establishes the range where 𝑝 has to be when 𝛼_𝑐<𝜋/2−𝛼_𝑔 and a globally convex solution of the NLBVP (2.5) and (2.6) on [𝑅_𝑐,𝑅_𝑑], exists. Computation shows that for the considered numerical data (𝛼_𝑐=𝛼¹_𝑐) this range is −4306.62Pa≤𝑝≤−219.94Pa.

To identify the 𝑝 value for which a globally convex solution of the NLBVP (2.5) and (2.6) exists on [𝑅_𝑐,𝑅_𝑑], the IVP (5.1) 𝑑𝑧𝑑𝑡=−tan𝛼𝑑𝛼=𝑑𝑡𝜌⋅𝑔⋅𝑧+𝑝𝛾⋅1−1cos𝛼𝑟𝑧𝑅⋅tan𝛼_𝑑=0,𝛼𝑅_𝑑=𝛼_𝑐 (5.1) has to be integrated numerically for 𝛼_𝑐=𝛼¹_𝑐 and different values of 𝑝 in the above range. The dependence 𝑟_𝑐(𝑝) has to be found, where 𝑟_𝑐(𝑝) is defined by 𝛼(𝑟_𝑐(𝑝))=𝜋/2−𝛼_𝑔. The value of 𝑝 for which 𝑟_𝑐(𝑝) is equal to 𝑅_𝑐 is found solving the equation 𝑟_𝑐(𝑝)=𝑅_𝑐. If the function 𝑟_𝑐(𝑝) does not exists (i.e., the growth angle is not reached) or the equation 𝑟_𝑐(𝑝)=𝑅_𝑐 has no solution, then there is no 𝑝 for which a globally convex solution of the NLBVP (2.5) and (2.6) on [𝑅_𝑐,𝑅_𝑑] exists. For the considered numerical data the graphic of the function 𝑟_𝑐(𝑝) is represented in Figure 2(a). This figure shows that for 𝑝=−2300 Pa the equality 𝑟_𝑐(𝑝)=𝑅_𝑐 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)

Figure 2 

(a) The function 𝑟𝑐(𝑝) for 𝛼𝑐=𝛼1𝑐=0.523rad. (b) The convex meridian curve for 𝑝=−2300Pa, suitable to grow microfiber with 𝑟𝑐(𝑝)=𝑅𝑐=4×10−4m.

(b) Inequality (3.9) establishes the range where 𝑝 has to be when 𝛼_𝑐>𝜋/2−𝛼_𝑔 and a globally-concave solution of the NLBVP (2.5) and (2.6) on [𝑅_𝑐,𝑅_𝑑] exists. Computation shows, that for the considered numerical data, 𝛼_𝑐𝛼²_𝑐, this range is [1640, 2211] Pa. To identify the value of 𝑝 the IVP (5.1) for 𝛼_𝑐=𝛼²_𝑐 and for different values of 𝑝 in the above range has to be solved. The dependence 𝑟_𝑐(𝑝) has to be found, where 𝑟_𝑐(𝑝) is defined by 𝛼(𝑟_𝑐(𝑝))=𝜋/2−𝛼_𝑔. The value of 𝑝 is found solving the equation 𝑟_𝑐(𝑝)=𝑅_𝑐. If the function 𝑟_𝑐(𝑝) does not exists (i.e., the growth angle is not reached) or the equation 𝑟_𝑐(𝑝)=𝑅_𝑐 has no solution, then there is no 𝑝 for which meniscus, having the prescribed shape and size, exists. For the considered numerical data the graphic of the function 𝑟_𝑐(𝑝) is represented in Figure 3(a). This figure shows that for 𝑝=1906.5 Pa the equality 𝑟_𝑐(𝑝)=4×10⁻⁴ 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)

Figure 3 

(a) The function 𝑟𝑐(𝑝) for 𝛼𝑐=𝛼2𝑐=1.4 rad. (b) The concave meridian curve for 𝑝=1906.5 Pa, suitable to grow microfiber with 𝑟𝑐(𝑝)=𝑅𝑐=4×10−4 m.

(c) Inequality (3.13) establishes the range where 𝑝 has to be when 𝛼_𝑐<𝜋/2−𝛼_𝑔 and the NLBVP (2.5) and (2.6) has a convex-concave solution on [𝑅_𝑐,𝑅_𝑑]. Computation shows, that for the considered numerical data, 𝛼_c=𝛼¹_𝑐, this range is [781, 1867] Pa. The value of 𝑝 can be identified integrating numerically the IVP (5.1) for 𝛼_𝑐=𝛼¹_𝑐 and for different values of 𝑝 in the above range. The dependence 𝑟_𝑐(𝑝) has to be found, where 𝑟_𝑐(𝑝) is defined by 𝛼(𝑟_𝑐(𝑝))=𝜋/2−𝛼_𝑔. The value of 𝑝 is found solving the equation 𝑟_𝑐(𝑝)=𝑅_𝑐. If the function 𝑟_𝑐(𝑝) does not exist (i.e., the growth angle is not reached) or the equation 𝑟_𝑐(𝑝)=𝑅_𝑐 has no solution, then there is no 𝑝 for which a meniscus, with the prior given shape and size on [𝑅_𝑐,𝑅_𝑑] exists. For the considered numerical data the graphic of the function 𝑟_𝑐(𝑝) is represented on Figure 4(a). This figure shows that equation 𝑟_𝑐(𝑝)=4×10⁻⁴ 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 𝑝=1000 Pa is presented on Figure 4(b), but this is inappropriate because 𝑟_𝑐(𝑝)≠𝑅_𝑐=4×10⁻⁴ m.

(a)

(b)

Figure 4 

(a) The function 𝑟𝑐(𝑝) for 𝛼𝑐=𝛼1𝑐=0.523 rad. (b) The convex-concave meridian curve for 𝑝=1000 Pa, suitable to grow microfiber with 1×10−4 m.

(d) Inequality (3.15) establishes the range where 𝑝 has to be when 𝛼_𝑐>𝜋/2−𝛼_𝑔 and the NLBVP (2.5) and (2.6) has a concave-convex solution on [𝑅_𝑐,𝑅_𝑑]. Computation shows that for the considered numerical data, 𝛼_𝑐=𝛼²_𝑐, this range is [1493.72, 1539.2] Pa. The value of 𝑝 can be identified integrating numerically the IVP (5.1) for 𝛼_𝑐=𝛼²_𝑐 and for different values of 𝑝 in the above range. The dependence 𝑟_𝑐(𝑝) has to be found, where 𝑟_𝑐(𝑝) is defined by 𝛼(𝑟_𝑐(𝑝))=𝜋/2−𝛼_𝑔. The value of 𝑝 is found solving the equation 𝑟_𝑐(𝑝)=𝑅_𝑐. If the function 𝑟_𝑐(𝑝) does not exist (i.e., the growth angle is not reached), or the equation 𝑟_𝑐(𝑝)=𝑅_𝑐 has no solution, then there is no 𝑝 for which a meniscus having the prior given shape and size exists on [𝑅_𝑐,𝑅_𝑑]. For the considered numerical data computation shows that for 𝑝 in the range [1493.72, 1539.2] Pa the growth angle on the meridian curve (i.e., the solution of IVP (5.1) for 𝛼_𝑐=𝛼²_𝑐) is not reached. Therefore, the function 𝑟_𝑐(𝑝) 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 𝑝 has to be when 𝛼_𝑐<𝜋/2−𝛼_𝑔 and the NLBVP (2.5) and (2.6) has a concave-convex solution on [𝑅_𝑐,𝑅_𝑑]. Computation shows, that for the considered numerical data, 𝛼_𝑐=𝛼¹_𝑐, this range is [1493.72, 1539.2] Pa. For 𝑝 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 𝑝 has to be when 𝛼_𝑐>𝜋/2−𝛼_𝑔 and the NLBVP (2.5), (2.6) has a convex-concave solution on [𝑅_𝑐,𝑅_𝑑]. Computation shows, that for the considered numerical data, 𝛼_𝑐=𝛼¹_𝑐, this range is: [1539.2, 1867.18] Pa. The value of 𝑝 can be identified integrating numerically the IVP (5.1) for 𝛼_𝑐=𝛼¹_𝑐 and for different values of 𝑝 in the above range. Following the method described at (c), computation shows that in this case the function 𝑟_𝑐(𝑝) does not exist.

(g) Inequality (3.21) defines a range such that when 𝛼_𝑐<𝜋/2−𝛼_𝑔 and 𝑝 is in the range, then there exists 𝑟_𝑐 in the range [𝑅_𝑑/𝑛,𝑅_𝑑/𝑛^′] and a globally convex solution of the NLBVP (2.5) and (2.6) on [𝑟_𝑐,𝑅_𝑑]. For the considered numerical data, 𝛼_𝑐=𝛼¹_𝑐, 𝑛=1.1>1.01=𝑛^′>1, the range for 𝑝 is [−33110, −10390] Pa. Integrating numerically the IVP (5.1) for 𝛼_𝑐=𝛼¹_𝑐 and for different values of 𝑝 in this range, we will find 𝑟_𝑐 in the range [𝑅_𝑑/1.1,𝑅_𝑑/1.01]=[4.54;4.95]×10⁻⁴ m and a globally convex solution of the NLBVP (2.5) and (2.6) on [𝑟_𝑐,𝑅_𝑑]. For the considered numerical data the dependence 𝑟_𝑐(𝑝) is represented in Figure 5(a) and the globally convex meniscus shape obtained for 𝑝=−17000 Pa is presented in Figure 5(b).

(a)

(b)

Figure 5 

(a) The function 𝑟𝑐(𝑝) for 𝛼𝑐=𝛼1𝑐=0.523 rad, 𝑛=1.1,𝑛′=1.01. (b) The convex meridian curve for 𝑝=−17000 Pa, suitable to grow microfiber with radius 4.8×10−4 m.

(h)   Inequality (3.37) defines a range such that when 𝛼_𝑐>𝜋/2−𝛼_𝑔 and 𝑝 is in the range, then there exists 𝑟_𝑐 in the range [𝑅_𝑑/𝑛,𝑅_𝑑/𝑛^′] and a globally-concave solution of the NLBVP (2.5) and (2.6) on [𝑟_𝑐,𝑅_𝑑]. For the considered numerical data, 𝛼_𝑐=𝛼²_𝑐, 𝑛=1.1>1.01=𝑛^′>1, the range for 𝑝 is [1836, 2326] Pa. Integrating numerically the IVP (5.1) for 𝛼_𝑐=𝛼²_𝑐 and for different values of 𝑝 in this range, we will find 𝑟_𝑐 in the range [𝑅_𝑑/1.1,𝑅_𝑑/1.01]=[4.54;4.95]×10⁻⁴ m and a globally-concave solution of the NLBVP (2.5) and (2.6) on [𝑟_𝑐,𝑅_𝑑]. For the considered numerical data the dependence 𝑟_𝑐(𝑝) is represented in Figure 6(a) with continuous line. It turns out that for the same 𝛼_𝑐=𝛼²_𝑐 and 𝑝 in the mentioned range we find globally-concave and a convex-concave solution of the NLBVP (2.5) and (2.6). For the convex-concave solution 𝑟_𝑐 is less than 𝑅_𝑑/1.1=4.54×10⁻⁴ m. The radius of these fibers is presented in Figure 6(a) with dot line. In Figure 6(b) such a convex-concave meniscus is presented, obtainable for 𝑝=2200 Pa. What happens in fact is that the growth angle is reached at 𝑟¹_𝑐=4.632×10⁻⁴ m and 𝑟²_𝑐=2.2×10⁻⁴ m, respectively. If the thermal field is fixed such that the solidification conditions are satisfied at the level 𝑧=ℎ¹_𝑐=1.68×10⁻⁴ m, then the fiber radius will be 𝑟¹_𝑐=4.632×10⁻⁴ m. When the thermal field is fixed such that the solidification conditions are satisfied at the level 𝑧=ℎ²_𝑐=7×10⁻⁴ m, then the fiber radius will be 𝑟²_𝑐=2.2×10⁻⁴ m.

(a)

(b)

Figure 6 

(a) The function 𝑟𝑐(𝑝) for 𝛼𝑐=𝛼2𝑐=1.4 rad, 𝑛=1.1,𝑛′=1.01; (b) The convex-concave meridian curve for 𝑝=2200 Pa, suitable to grow microfiber with radius 2.2×10−4 m.

6. Conclusion

The above results are necessary or sufficient conditions for the existence and stability of a nonlinear boundary value problem occurring in microfiber crystal growth by pulling-down method. They can be used for the evaluation of the shape and size of the meniscus.

Acknowledgment

This work was supported by a grant of the Romanian National Authority for Scientific Research, CNCS-UEFISCDI, project no. PN-II-ID-PCE-2011-3-0171.