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

Authors: Agneta M. Balint; Stefan Balint
Journal: Abstract and Applied Analysis
 (2012)

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## 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.


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## Body

## 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 GexSi1-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 inμ-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 byμ-PD method, the free surface of the meniscus at each moment of time is described by the Young-Laplace equation [17, 18]:
(2.1)γ·(1R1+1R2)=Pa-Pb.
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; Pa is the pressure in front of the free surface; Pb is the pressure behind the free surface (Figure 1).Figure 1
Schematic diagram of theμ-pulling-down fiber growth process.The pressurePa in front of the free surface is equal to the pressure of the gas introduced in the furnace and thereafter is denoted by pg (Pa=pg).The pressurePb behind the free surface is the sum of the hydrodynamic pressure pm in the meniscus melt (due to the thermal convection), the Marangoni pressure pM due to the Marangoni convection, and the hydrostatic pressure of the melt column behind the free surface equal to ρ·g·(z+hm) (see Figure 1). Here ρ denotes the melt density; g is the gravity acceleration; z is the coordinate of M with respect to the Ozaxis, directed vertically downwards; hm denotes the melt column height between the horizontal crucible melt level and the shaper bottom level (see Figure 1).The pressure differencePa-Pb across the free surface is denoted usually by Δp and, according to the above explanations,
(2.2)Δp=pg-pm-pM-ρ·g·(z+hm).
The part p given by:
(2.3)p=-pg+ρ·g·hm+pm+pM.
of the pressure difference Δp is considered to be known. That is because the major part -pg+ρ·g·hm of p is known, and pm+pM in general is small with respect to -pg+ρ·g·hm 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
(2.4)γ·(1R1+1R2)=-ρ·g·z-p.
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(2.5)z′′=-ρ·g·z+pγ[1+(z′)2]3/2-1r·[1+(z′)2]·z′forRc≤r≤Rd=shaper  radius.
If the solution z(r) of (2.5) is the meridian curve of the free surface of a meniscus on the interval [Rc,Rd], then it verifies the following boundary conditions:
(2.6)z′(Rc)=-tan(π2-αg);z(Rc)=hc>0,(2.7)z′(Rd)=-tanαc;z(Rd)=0,(2.8)z(r)isstrictlydecreasingon[Rc,Rd].The first condition in (2.6) expresses that at the three-phase point (Rc,hc), 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 hc>0.The first condition in (2.7) expresses that, at the three-phase point (Rd,0), where the meridian curve touches the outer edge of the shaper, the catching angle is equal to αc.The second condition in (2.7)  expresses that at the three-phase point (Rd,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 ofz(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:
(2.9)I(z)=∫RcRd{γ·[1+(z′)2]1/2-12·ρ·g·z2-p·z}·r·dr,z(Rc)=hc>0,z(Rd)=0.
The nonlinear boundary value problem (NLBVP), (2.5) and (2.6), is equivalent to the NLBVP:
(2.10)dzdr=-tanα,dαdr=ρ·g·z+pγ·1cosα-1r·tanα,z(Rc)=hc,α(Rc)=π2-αg,z(Rd)=0,α(Rd)=αc.


## 3. Existence of the Solution

Consider the differential equation:(3.1)z′′=-ρ·g·z+pγ[1+(z′)2]3/2-1r·[1+(z′)2]·z′for0<Rc≤r≤RdRd>0.
And αc,αg such that αc,αg∈(0,π/2).Definition 3.1.
A solutionz=z(r) of (3.1) is a globally convex solution of the NLBVP (2.5) and (2.6) on the interval [Rc,Rd](0<Rc<Rd) if it possesses the following properties:(a)z′(Rc)=-tan(π/2-αg),(b)z′(Rd)=-tanαc,(c)z(Rd)=0 and z(r) is strictly decreasing on [Rc,Rd],(d)z′′(r)>0forallr∈[Rc,Rd],A solutionz(r) of (3.1) is a globally-concave solution of the NLBVP (2.5),  (2.6) on the interval [Rc,Rd](0<Rc<Rd) if verifies (a) (b) (c) and in addition the following inequality holds:(e)z′′(r)<0forallr∈[Rc,Rd].Remark 3.2.
Ifαc<π/2-αg, then for a globally concave solution z(r)(z′′(r)<0) of (3.1), which satisfies (b) and (c), there is no Rc∈(0,Rd) such that (a) holds. Therefore, if αc<π/2-αg and we are interested in solutions of (3.1) which satisfy (b) and (c) and are not globally concave.
Ifπ/2-αg<αc, then for a globally convex solution z(r)(z′′(r)<0) of (3.1), which satisfies (b) and (c), and there is no Rc∈(0,Rd)  such that a. holds. Therefore, if π/2-αg<αc, 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).
Ifαc<π/2-αg and there exists a globally convex solution z(r) of the NLBVP (2.5) and (2.6), on [Rc,Rd], then the following inequalities hold:
(3.2)γ·αc+αg-π/2Rd-Rc·cosαc-ρ·g·[Rd-Rc]·tan(π2-αg)+γRd·sinαc≤p≤γ·αc+αg-π/2Rd-Rc·sinαg+γRc·cosαg.Proof.
Letz(r) be a globally convex solution of the problem (2.5) and (2.6) on [Rc,Rd] and α(r)=-arctanz′(r). The function α(r) verifies the following quation:
(3.3)α′(r)=ρ·g·z(r)+pγ·1cosα(r)-1r·tanα(r)
and the boundary conditions:
(3.4)α(Rc)=π2-αg,α(Rd)=αc.
Hence, according to the Lagrange mean value theorem, there exists r′∈(Rc,Rd) such that the following equality holds:
(3.5)p=γ·αc+αg-π/2Rd-Rc·cosα(r′)-ρ·g·z(r′)+γr′·sinα(r′).
Since z′′(r)>0 on [Rc,Rd], z′(r) is strictly increasing, and α(r)=-arctanz′(r) is strictly decreasing on [Rc,Rd]. Therefore, the following inequalities hold:
(3.6)αc≤α(r′)≤π2-αg,sinαg≤cosα(r′)≤cosαc,sinαc≤sinα(r′)≤cosαg,-ρ·g·(Rd-Rc)·tan(π2-αg)≤-ρ·g·z(r′)≤-ρ·g·(Rd-r′)·tanαc.
Equality (3.5) and inequalities (3.6) imply (3.2).Corollary 3.4.
In terms of the ration=(Rd/Rc)(n>1) the inequalities (3.2) can be written as
(3.7)γ·αc+αg-π/2Rd·nn-1·cosαc-ρ·g·Rd·(n-1)n·tan(π2-αg)+γRd·sinαc≤p≤γ·αc+αg-π/2Rd·nn-1·sinαg+γRd·n·cosαg.Corollary 3.5.
Ifn→+∞, then Rc→0 and (3.7) become
(3.8)γ·αc+αg-π/2Rd·cosαc-ρ·g·Rd·tan(π2-αg)+γRd·sinαc≤p<+∞.
If  n→1, then Rc→Rd and p→-∞.Theorem 3.6 (necessary condition for the existence of a globally concave solution).
Ifπ/2-αg<αc, and there exists a globally concave solution z(r)(z′′(r)<0) of the NLBVP (2.5) and (2.6) on [Rc,Rd], then the following inequalities hold:
(3.9)γ·αc+αg-π/2Rd-Rc·cosαc-ρ·g·[Rd-Rc]·tanαc+γRd·cosαg≤p≤γ·αc+αg-π/2Rd-Rc·sinαg+γRc·sinαc.Proof.
Sincez′′(r)<0 on [Rc,Rd], z′(r) is strictly decreasing and α(r)=-arctanz′(r) is strictly increasing on [Rc,Rd]. Therefore the following inequalities hold:
(3.10)π2-αg≤α(r)≤αc,cosαc≤cosα(r)≤sinαg,cosαg≤sinα(r)≤sinαc,-ρ·g·(Rd-Rc)·tanαc≤-ρ·g·z(r)≤-ρ·g·(Rd-r′)·tan(π2-αg)
for every r∈[Rc,Rd].Equality (3.5) and inequalities (3.10) imply inequalities (3.9).Corollary 3.7.
In terms of the ration=(Rd/Rc)(n>1) the inequalities (3.9) can be written as
(3.11)γ·αc+αg-π/2Rd·nn-1·cosαc-ρ·g·Rd·(n-1)n·tanαc+γRd·cosαg≤p≤γ·αc+αg-π/2Rd·nn-1·sinαg+γRd·n·sinαc.Corollary 3.8.
Ifn→+∞, then Rc→0 and (3.11) become
(3.12)γ·αc+αg-π/2Rd·cosαc-ρ·g·Rd·tanαc+γRd·cosαg≤p<+∞.
If n→1, then Rc→Rd and p→+∞.Theorem 3.9 (necessary condition for the existence of a convex-concave solution).
If forαc<π/2-αg and p>(γ/Rd)·sinαcand there exists a solution z(r) of the NLBVP (2.5) and (2.6) on the interval [Rc,Rd](0<Rc<Rd) and then for p the following inequalities hold:
(3.13)γRd·sinαc<p<γRc·cosαg.Proof.
Sincep>(γ/Rd)·sinαc, we have (p/γ-1/Rd)·sinαc>0 and hence α′(Rd)=(1/cosαc)·[(p/γ-1/Rd)·sinαc]>0. It follows that, for r<Rd, r close to Rd, the following inequality: α(r)<α(Rd)=αc holds. Since α(Rc)=π/2-αg>αc, it follows that there exists r′∈(Rc,Rd) 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:
(3.14)p=γr′·sinα(r′)-ρ·g·z(r′).
Taking into account the fact that z(r) is a decreasing function on [Rc,Rd], we deduce that p<(γ/Rc)·cosαg.Theorem 3.10 (necessary condition for the existence of a concave-convex solution).
If forαc>π/2-αg and p<(γ/Rd)·sinαc,  there exists a solution z(r) of the NLBVP (2.5) and (2.6) on the interval [Rc,Rd](0<Rc<Rd), then for p the following inequalities hold:
(3.15)γRd·cosαg<p<γRd·sinαc.Proof.
Sincep<(γ/Rd)·sinαc, we have (p/γ)-1/Rd·sinαc<0 and hence α′(Rd)=(1/cosαc)·[(p/γ-1/Rd)·sinαc]<0. It follows that, for r<Rd, r close to Rd, the inequality αc=α(Rd)<α(r) holds. Since α(Rc)=π/2-αg<αc, it follows that there exists r′∈(Rc,Rd) 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 [Rc,Rd], we deduce (3.15).Theorem 3.11.
If forαc<π/2-αg, p<(γ/Rd)·sinαc there exists a solution z(r) of the NLBVP (2.5) and (2.6), on the interval [Rc,Rd](0<Rc<Rd) and z"(Rc)<0, then z(r) is concave convex on [Rc,Rd] and there exists Rc1, Rc<Rc1<Rd such that z(r) is a globally convex solution of the NLBVP (2.5) and (2.6) on [Rc1,Rd]. Moreover for p the inequality (3.15) holds.Proof.
Sincep<(γ/Rd)·sinαc, we have α′(Rd)<0. Inequality z"(Rc)<0 implies that α′(Rc)>0. Hence, there exists r′∈(Rc,Rd) 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 [Rc,Rd],we deduce (3.15).
The existence ofRc1 follows from the fact that π/2-αg=α(Rc)<α(r) for r>Rc, r close to Rc and α(Rd)=αc<π/2-αg.Theorem 3.12.
If forαc>π/2-αg, p>(γ/Rd)·sinαc, there exists a solution z(r) of the NLBVP (2.5) and (2.6) on the interval [Rc,Rd](0<Rc<Rd) and z′′(Rc)>0, then z(r) is a convex concave solution of the NLBVP (2.5) and (2.6) on [Rc,Rd] and there exists Rc1, Rc<Rc1<Rd, such that z(r) is a globally-concave solution of the NLBVP (2.5), (2.6) on [Rc1,Rd]. Moreover for p the inequality (3.13) holds.Proof.
Sincep>(γ/Rd)·sinαc, we have α′(Rd)>0. Inequality z′′(Rc)>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 [Rc,Rd], we deduce that (3.13) holds.
The existence ofRc1 follows from the fact that π/2-αg=α(Rc)>α(r) for r>Rc, r close to Rc and α(Rd)=αc>π/2-αg.Theorem 3.13 (sufficient condition for the existence of a globally convex solution).
Letn>1 be a real number. If αc<π/2-αg and p satisfies
(3.16)p<γ·αc+αg-π/2Rd·nn-1·cosαc-ρ·g·Rd·n-1n·tan(π2-αg)+γRd·sinαc,
then there exists Rc in the closed interval [Rd/n,Rd] such that the solution of the initial value problem
(3.17)z′′=-ρ·g·z+pγ[1+(z′)2]3/2-1r[1+(z′)2]·z′0<r<Rd,Rd>0z(Rd)=0,z′(Rd)=-tanαc0<αc<π2-αg,αg∈(0,π2)
is a globally convex solution of the NLBVP (2.5) and (2.6) on [Rc,Rd].Proof.
Consider the solutionz(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:
(3.18)α′(r)=1cosα[p+ρ·g·zγ-sinαr].
Since
(3.19)z′′(Rd)=-α′(Rd)cos2α(Rd)=-1cos3α(Rd)[pγ-sinα(Rd)Rd]>0,z′(Rd)=-tanαc<0,z′(Rd)>-tan(π2-αg),
there exists r′∈I,0<r′<Rd such that for any r∈[r′,Rd] the following inequalities hold:
(3.20)z′′(r)>0,z′(r)≤-tanαc<0,z′(r)>-tan(π2-αg).Let now ber* defined by
(3.21)r*=inf{r′∈I∣0<r′<Rdsuchthatforanyr∈[r′,Rd]inequalities(3.20)hold}.It is clear thatr*≥0 and for any r∈(r*,Rd] (3.20) hold. Moreover, z′(r*+0)=limr→r*,r>r*z′(r) exists and satisfies z′(r*+0)<-tanαc<0 and z′(r*+0)≥-tan(π/2-αg). Hence z(r*+0)=limr→r*,r>r*z(r) is finite, it is strictly positive and for every r∈(r*,Rd] the inequalities hold:
(3.22)0≤z(r)<z(r*+0);[Rd-r*]·tanαc≤z(r*+0)≤[Rd-r*]·tan(π2-αg).
We will show now that r*≥Rd/n and z′(r*+0)=-tan(π/2-αg).In order to show thatr*≥Rd/n we assume the contrary, that is, r*<Rd/n. Under this hypothesis we have
(3.23)α(Rdn)-α(Rd)=-α′(r′)·(Rd-Rdn)=1cosα(r′)[-pγ-ρ·g·z(r′)γ+sinα(r′)r′]·(Rd-Rdn)>Rd-Rd/ncosα(r′)·[-αc+αg-π/2Rd·nn-1·cosαc+ρ·g·Rdγ·n-1n·tan(π2-αg)-1Rd·sinαc-ρ·g·z(r′)γ+sinα(r′)r′{αc+αg-π/2Rd}]>π2-(αc+αg)
for some r′∈(Rd/n,Rd). Hence α(Rd/n)>π/2-αg. This last inequality is impossible, according to the definition (3.21) of  r*. Therefore, r*, defined by (3.21), satisfies r*≥Rd/n.In order to show thatz′(r*+0)=-tan(π/2-αg), we remark that from the definition (3.21) of r* it follows that in r* at least one of the following three equalities hold:
(3.24)z′(r*+0)=-tanαc,z′′(r*+0)=0,z′(r*+0)=-tan(π2-αg).
Since z′(r*+0)<z′(r)≤-tanαc for any r∈(r*,Rd], it follows that the equality z′(r*+0)=-tanαc 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-αg).We show now that the equalityz′′(r*+0)=0is impossible. For that we assume the contrary, that is, z′′(r*+0)=0. Under this hypothesis, from (3.17) we have
(3.25)p=-ρ·g·z(r*+0)+γr*·sinα(r*+0)≥-ρ·g·(Rd-r*)·tan(π2-αg)+γr*·sinα(r*+0)>γRd·sinαc-ρ·g·Rd·n-1n·tan(π2-αg)>γRd·sinαc-ρ·g·Rd·n-1n·tan(π2-αg)+γ·αc+αg-π/2Rd·nn-1·cosαc
what is impossible.In this way we obtain that the equalityz′(r*+0)=-tan(π/2-αg) holds.Taking nowRc=r* we find that the solution of the initial value problem (3.17) on the interval [Rc,Rd] is convex, and it is the meridian curve of an appropriate meniscus.Corollary 3.14.
Ifαc<π/2-αg and
(3.26)p<γ·αc+αg-π/2Rd·cosαc-ρ·g·Rd·tan(π2-αg),
then there exists Rc∈(0,Rd] 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 [Rc,Rd].Corollary 3.15.
Ifαc<π/2-αg, n>n′>1 and the inequalities hold
(3.27)γ·αc+αg-π/2Rd·n′n′-1·sinαg+γRd·n′cosαg<p<γ·αc+αg-π/2Rd·nn-1·cosαc-ρ·g·Rd·n-1n·tan(π2-αg)+γRd·sinαc,
then there exists Rc∈[Rd/n,Rd/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 [Rc,Rd].The existence ofRc and the inequality Rc>Rd/n follows from Theorem 3.13. The inequality Rc<Rd/n′  follows from the Corollary 3.4 of Theorem 3.3.Theorem 3.16 (sufficient condition for the existence of globally concave solution).
Letn>1 be a real number. If αc>π/2-αg and p satisfies
(3.28)p<γ·αc+αg-π/2Rd·nn-1·sinαg+γRd·n·sinαc,
then there exists Rc in the closed interval [Rd/n,Rd] such that the solution of the initial value problem
(3.29)z′′=-ρ·g·z+pγ[1+(z′)2]3/2-1r[1+(z′)2]·z′0<r<Rd,Rd>0z(Rd)=0,z′(Rd)=-tanαc0<π2-αg<αc<π2
is a globally concave solution of the NLBVP (2.5) and (2.6) on [Rc,Rd].Proof.
Consider the solutionz(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:
(3.30)α′(r)=1cosα(r)·[pγ+ρ·g·z(r)γ-sinαr].
Since
(3.31)z′′(Rd)=-α′(Rd)cos2α(Rd)=-1cos3α(Rd)[pγ-sinα(Rd)Rd]<0,z′(Rd)=-tanαc<0,z′(Rd)<-tan(π2-αg),
there exists r′∈I,0<r′<Rd such that for any r∈[r′,Rd] the following inequalities hold:
(3.32)z′′(r)<0,z′(r)≥-tanαc,z′(r)≤-tan(π2-αg).
Let now r* be defined by
(3.33)r*=inf{r′∈I∣0<r′<Rdsuchthatforanyr∈[r′,Rd]inequalities(3.32)hold}.
It is clear that r*≥0 and for any r∈(r*,Rd] inequalities (3.32) hold. Moreover, z′(r*+0)=limr→r*,r>r*z′(r) exists and satisfies -tanαc≤z′(r)≤-tan(π/2-αg). Hence z(r*+0)=limr→r*,r>r*z(r) is finite, it is strictly positive and for every r∈(r*,Rd] the inequalities hold:
(3.34)0≤z(r)≤z(r*+0);[Rd-r*]·tan(π2-αg)≤z(r*+0)≤[Rd-r*]·tanαc.
We will show now that r*≥Rd/n and z′(r*+0)=-tan(π/2-αg).
In order to show the inequalityr*≥Rd/n we assume the contrary, that is, that r*<Rd/n. Under this hypothesis we have
(3.35)α(Rd)-α(Rdn)=α′(r′)·(Rd-Rdn)=Rd-Rd/ncosα(r′)[pγ+ρ·g·z(r′)γ-sinα(r′)r′]>Rd·((n-1)/n)cosα(r′)·[αc+αg-π/2Rd·nn-1·sinαg+nRd·sinαc+ρ·g·z(r′)γ-sinα(r′)r′]
for some r′∈(Rd/n,Rd). Since (n/Rd)·sinαc>(n/r′)·sinα(r′), sinαg/cosα(r′)>1, and ρ·g·z(r′)/γ>0, it follows that α(Rd)-α(Rd/n)>αc+αg-π/2. Hence α(Rd/n)>π/2-αg, what is impossible, according to the definition of r*. Therefore, r*, defined by (3.33), satisfies r*≥Rd/n.
In order to show thatz′(r*+0)=-tan(π/2-αg), we remark that from the definition (3.33) of r* it follows that in r* at least one of the following three equalities holds:
(3.36)z′(r*+0)=-tanαc,z′′(r*+0)=0,z′(r*+0)=-tan(π2-αg).
Since z′(r*+0)>z′(r)>-tanαc for any r∈(r*,Rd], it follows that the equality z′(r*+0)=-tanαc 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-αg).
We show now that the equalityz′′(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/Rd)·sinαc, what is impossible.
In this way we obtain that the equalityz′(r*+0)=-tan(π/2-αg) holds.
Taking nowRc=r* we find that the solution of the initial value problem (3.29) on the interval [Rc,Rd] is concave, and it is the meridian curve of an appropriate meniscus.Corollary 3.17.
Ifαc>π/2-αg,  n>n′>1 and the inequalities hold:
(3.37)γ·αc+αg-π/2Rd·nn-1·sinαg+γRd·n·sinαc<p<γ·αc+αg-π/2Rd·n′n′-1·cosαc-ρ·g·Rd·n′-1n′·tanαc+γRd·sinαc,
then there exists Rc∈[Rd/n,Rd/n′] such that the solution of the IVP (3.29) is a globally concave solution of the NBVP (2.5), (2.6) on [Rc,Rd].
The existence ofRc and the inequality Rd/n<Rc  follows from Theorem 3.16. The inequality Rc<Rd/n follows from Corollary 3.7.  and Theorem 3.6.


## 4. Stability of the Solution

Definition 4.1.
A solutionz(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 forαc<π/2-αg the NLBVP (2.5), (2.6) has a globally convex solution z(r) on [Rc,Rd] and the ratio n=Rd/Rc>1 satisfies:
(4.1)n-1n1/2<π·1Rd·γ1/2·sin3/2αgρ1/2·g1/2
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:(4.2)r·γ[1+(z′)2]3/2>0
and it is satisfied.
By computation we find that the Jacobi equation in this case is(4.3)ddr(r·γ[1+(z′)2]3/2·dηdr)+ρ·g·z·η=0.
For the coefficients of (4.3) the following inequality holds:
(4.4)r·γ[1+(z′)2]3/2≥Rdn·γ·sin3αg,ρ·g·z≤ρ·g·Rd.
Hence,
(4.5)ddr(Rdn·γ·sin3αg·dηdr)+ρ·g·Rd·η=0
is a Sturm type upper bound [19] for (4.3).
An arbitrary solution of (4.5) is given by
(4.6)η(r)=A·sin(ω·r+φ),
where A and φ are real constants and ω2 is given by
(4.7)ω2=ρ·g·nγ·sin3αg.
Inequality (4.1) implies that the half period π/ω of η(r) satisfies the inequality π/ω>Rd-Rc. Hence a nonzero solution given by (4.6) has at most one zero on the interval [Rc,Rd]. Since any nonzero solution of (4.5) vanishes at most once on the interval [Rc,Rd], the solution of (4.3) which satisfies η(Rd)=0,η′(Rd)=1 has only one zero on [Rc,Rd]. Hence the Jacobi condition is satisfied in this case.Theorem 4.3.
If forαc<π/2-αg the NLBVP (2.5) and (2.6) has a globally convex solution z(r) on [Rc,Rd], and the ratio n=Rd/Rc>1 satisfies
(4.8)n-1n1/2>2π·1Rd·γ1/2·cos3/2αcρ1/2·g1/2,
then the meniscus is unstable.Proof.
We remark now that for the Jacobi equation the following inequalities hold:(4.9)r·γ[1+(z′)2]3/2≤Rd·γ·cos3αc,ρ·g·z≥ρ·g·Rdn.
Hence
(4.10)ddr[Rdγ·cos3αc·dηdr]+ρ·g·Rdn·η=0
is a Sturm type lower bound [19] of the Jacobi equation (4.3).
An arbitrary solution of (4.10) is given by
(4.11)η(r)=A·sin(ω·r+φ),
where ω2 is given by
(4.12)ω2=ρ·gn·γ·cos3αc.
Inequality (4.8) implies that η(r) given by (4.11) vanishes twice on [Rc,Rd]. Therefore the Jacobi condition is not satisfied.Theorem 4.4.
If forπ/2-αg<αc<π/2 the NLBVP (2.5), (2.6) has a globally-concave solution z(r) on [Rc,Rd] and the ratio n=Rd/Rc>1 satisfies:
(4.13)n-1n1/2<π·1Rd·γ1/2·cos3/2αcρ1/2·g1/2
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:
(4.14)r·γ[1+(z′)2]3/2≥Rdn·γ·cos3αc,ρ·g·z≤ρ·g·Rd.
Hence
(4.15)ddr[Rdn·γ·cos3αc·dηdr]+ρ·g·Rd·η=0
is a Sturm type upper bound [19] of the Jacobi equation (4.3).
An arbitrary solution of (4.15) is given by
(4.16)η(r)=A·sin(ω·r+φ),
where A and φ are real constants and ω2 is given by
(4.17)ω2=ρ·g·nγ·cos3αc.
Inequality (4.13) implies that the half period π/ω of η(r) satisfies the inequality π/ω>Rd-Rc. It follows that a nonzero solution given by (4.16) has at most one zero on the interval [Rc,Rd]. Since any nonzero solution of (4.15) vanishes at most once on the interval [Rc,Rd], the solution of the Jacobi equation (4.3), which satisfies η(Rd)=0,η′(Rd)=1, has only one zero on [Rc,Rd]. Hence the Jacobi condition is satisfied.Theorem 4.5.
If forπ/2-αg<αc<π/2 the NLBVP (2.5) and (2.6) has a globally concave solution z(r) on [Rc,Rd] and the ratio n=Rd/Rc>1 satisfies
(4.18)n-1n3/2>2π·1Rd·γ1/2·sin3/2αgρ1/2·g1/2,
then the solution is unstable.Proof.
Remark that in this case for the coefficients of the Jacobi equation (4.3) the following inequalities hold:
(4.19)r·γ[1+(z′)2]3/2≤Rd·γ·sin3αg,ρ·g·z≥ρ·g·Rdn.
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.Rd=5×10-4[m]; Rc=4×10-4[m]; αc1=0.523[rad];  αc2=1.4[rad]; αg=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 ofp (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 αc<π/2-αg and a globally convex solution of the NLBVP (2.5) and (2.6) on [Rc,Rd], exists. Computation shows that for the considered numerical data (αc=αc1) this range is -4306.62Pa≤p≤-219.94Pa.To identify thep value for which a globally convex solution of the NLBVP (2.5) and (2.6) exists on [Rc,Rd], the IVP (5.1)
(5.1)dzdt=-tanαdαdt=ρ·g·z+pγ·1cosα-1r·tanαz(Rd)=0,α(Rd)=αc
has to be integrated numerically for αc=αc1 and different values of p in the above range. The dependence rc(p) has to be found, where rc(p) is defined by α(rc(p))=π/2-αg. The value of p for which rc(p) is equal to Rc is found solving the equation rc(p)=Rc. If the function rc(p) does not exists (i.e., the growth angle is not reached) or the equation rc(p)=Rc has no solution, then there is no p for which a globally convex solution of the NLBVP (2.5) and (2.6) on [Rc,Rd] exists. For the considered numerical data the graphic of the function rc(p) is represented in Figure 2(a). This figure shows that for p=-2300 Pa the equality rc(p)=Rc holds; that is, meniscus with prior given shape and size exists. The meridian curve at this meniscus is represented in Figure 2(b).(a) The functionrc(p) for αc=αc1=0.523rad. (b) The convex meridian curve for p=-2300Pa, suitable to grow microfiber with rc(p)=Rc=4×10-4m.
(a)
(b)(b) Inequality (3.9) establishes the range where p has to be when αc>π/2-αg and a globally-concave solution of the NLBVP (2.5) and (2.6) on [Rc,Rd] exists. Computation shows, that for the considered numerical data, αcαc2, this range is [1640, 2211] Pa. To identify the value of p the IVP (5.1) for αc=αc2 and for different values of p in the above range has to be solved. The dependence rc(p) has to be found, where rc(p) is defined by α(rc(p))=π/2-αg. The value of p is found solving the equation rc(p)=Rc. If the function rc(p) does not exists (i.e., the growth angle is not reached) or the equation rc(p)=Rc 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 rc(p) is represented in Figure 3(a). This figure shows that for p=1906.5 Pa the equality rc(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) The functionrc(p) for αc=αc2=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.
(a)
(b)(c) Inequality (3.13) establishes the range where p has to be when αc<π/2-αg and the NLBVP (2.5) and (2.6) has a convex-concave solution on [Rc,Rd]. Computation shows, that for the considered numerical data, αc=αc1, this range is [781, 1867] Pa. The value of p can be identified integrating numerically the IVP (5.1) for αc=αc1 and for different values of p in the above range. The dependence rc(p) has to be found, where rc(p) is defined by α(rc(p))=π/2-αg.The value of  p is found solving the equation rc(p)=Rc. If the function rc(p) does not exist (i.e., the growth angle is not reached) or the equation rc(p)=Rc has no solution, then there is no p for which a meniscus, with the prior given shape and size on [Rc,Rd] exists. For the considered numerical data the graphic of the function rc(p) is represented on Figure 4(a). This figure shows that equation rc(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 rc(p)≠Rc=4×10-4 m.(a) The functionrc(p) for αc=αc1=0.523 rad. (b) The convex-concave meridian curve for p=1000 Pa, suitable to grow microfiber with 1×10-4 m.
(a)
(b)(d) Inequality (3.15) establishes the range where p has to be when αc>π/2-αg and the NLBVP (2.5) and (2.6) has a concave-convex solution on [Rc,Rd]. Computation shows that for the considered numerical data, αc=αc2,  this range is [1493.72, 1539.2] Pa. The value of p can be identified integrating numerically the IVP (5.1) for αc=αc2 and for different values of p in the above range. The dependence rc(p) has to be found, where rc(p) is defined by α(rc(p))=π/2-αg.The value of p is found solving the equation rc(p)=Rc. If the function rc(p) does not exist (i.e., the growth angle is not reached), or the equation rc(p)=Rc has no solution, then there is no p for which a meniscus having the prior given shape and size exists on [Rc,Rd]. 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 αc=αc2) is not reached. Therefore, the function rc(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 αc<π/2-αg and the NLBVP (2.5) and (2.6) has a concave-convex solution on [Rc,Rd]. Computation shows, that for the considered numerical data, αc=αc1, 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 αc>π/2-αg and the NLBVP (2.5), (2.6) has a convex-concave solution on [Rc,Rd]. Computation shows, that for the considered numerical data, αc=αc1, this range is: [1539.2, 1867.18] Pa. The value of p can be identified integrating numerically the IVP (5.1) for αc=αc1 and for different values of p in the above range. Following the method described at (c), computation shows that in this case the function rc(p) does not exist.(g) Inequality (3.21) defines a range such that when αc<π/2-αg and p is in the range, then there exists rc in the range [Rd/n,Rd/n′] and a globally convex solution of the NLBVP (2.5) and (2.6) on [rc,Rd]. For the considered numerical data, αc=αc1,  n=1.1>1.01=n′>1, the range for p is [−33110, −10390] Pa. Integrating numerically the IVP (5.1) for αc=αc1 and for different values of p in this range, we will find rc in the range [Rd/1.1,Rd/1.01]=[4.54;4.95]×10-4 m and a globally convex solution of the NLBVP (2.5) and (2.6) on [rc,Rd]. For the considered numerical data the dependence rc(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) The functionrc(p) for αc=αc1=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.
(a)
(b)(h)   Inequality (3.37) defines a range such that when αc>π/2-αg and p is in the range, then there exists rc in the range [Rd/n,Rd/n′] and a globally-concave solution of the NLBVP (2.5) and (2.6) on [rc,Rd]. For the considered numerical data, αc=αc2,  n=1.1>1.01=n′>1, the range for p is [1836, 2326] Pa. Integrating numerically the IVP (5.1) for αc=αc2 and for different values of p in this range, we will find rc in the range [Rd/1.1,Rd/1.01]=[4.54;4.95]×10-4 m and a globally-concave solution of the NLBVP (2.5) and (2.6) on [rc,Rd].For the considered numerical data the dependence rc(p) is represented in Figure 6(a) with continuous line. It turns out that for the same αc=αc2 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 rc is less than Rd/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 rc1=4.632×10-4 m and rc2=2.2×10-4 m, respectively. If the thermal field is fixed such that the solidification conditions are satisfied at the level z=hc1=1.68×10-4 m, then the fiber radius will be rc1=4.632×10-4 m. When the thermal field is fixed such that the solidification conditions are satisfied at the level z=hc2=7×10-4 m, then the fiber radius will be rc2=2.2×10-4 m.(a) The functionrc(p) for αc=αc2=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.
(a)
(b)


## 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.



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*Source: 582746-2012-12-17.xml*