Collapsing of glass tubes: analytic approaches in a hydrodynamic problem with free boundaries

Let \( (r, \varphi , z) \) be a cylindrical coordinate system with z-axis as the rotation axis of the collapsing tube. Axial symmetry will be assumed throughout so that all physical quantities in question may depend only upon r, z, and time t. In particular, the viscous flow \( \mathbf{v} (r,z,t)\) may be given aswhere \({\hat{\mathbf{e}} }_r,\) \({\hat{\mathbf{e}} }_z\) are the radial and axial unit vectors and \({v_r}(r,z,t)\), \({v_z}(r,z,t) \) the radial and axial flow components, respectively. Molten glass will be considered to be incompressible, that is \({\eta }(r,z,t) \cdot {\varvec{\sigma '}}(r,z,t)\) denotes that part of the total stress tensor originated by the viscous material properties (see, e.g., [41], Chap. 2). \({\eta }(r,z,t)\) is the dynamic viscosity depending, in general, upon space and time, and \({\varvec{\sigma '}}\) is determined by the viscous flow only. For axial symmetry, there are five non-vanishing components of the symmetric tensor \({\varvec{\sigma '}}\) in cylindrical coordinates, where we need [41, 42] whereas \(\sigma '_{\varphi \varphi } = 2 v_r/r \) is unimportant in the following. Regarding the space dependence of \(\eta \), the Navier–Stokes equation for incompressible fluids reads (see, e.g., [41, 42])where \(\rho \) is the mass density of the glass and \(p = p(r,z,t)\) is the hydrodynamic pressure related to the viscous flow. The gravity, as well as the centrifugal force in a rotating collapsing tube will be neglected. As mentioned above, the space and time dependence of \(\eta \) is caused by its temperature dependence. The same holds true for \(\rho \), but this is unimportant, regarding approaches introduced later. Below, we will show that the kinematics of the collapsing process, as well as the course of steady-state collapsing profiles is ultimately governed by the temperature dependence of \(\eta \). It is accepted to describe the temperature dependence of the dynamic viscosity of glasses by Arrhenius relations (see, e.g., [3, 39] and references cited therein). For silica, one may adopt, for a wide temperature range from about 1300 K up to 2500 K, an Arrhenius fit¹ according towith \(\eta _\mathrm{emp}(T) \) being the empirical, temperature-dependent dynamic viscosity (in [\(\mathrm {Pa} \cdot \mathrm{s}\)]) and T the temperature (in [ \(\mathrm {K}\) ]). There are other empirical Arrhenius-like viscosity relations for silica and other glasses (see, e.g., [3, 39]), too, where, for silica, the logarithm of the former may differ from that of (2.7) in the ten percent range.

We will analyze steady-state collapsing induced by a heat source moving uniformly in z-direction with the velocity \(v_T\). The mathematics become substantially simplified if we distinguish between the frame, where the flow components are measured, and coordinates on which the flow components depend. Precisely, (i) we measure the viscous flow in the laboratory system (see our remark at the beginning of Sect. 1); (ii) we change from laboratory coordinates (r, z) to new coordinates \((\tilde{r},\tilde{z})\) (such as “traveling wave coordinates”), which comove with the heat source, and then we have \({\tilde{r}}=r\), \({\tilde{z}}=z-v_T t\); and (iii) the viscous flow components measured in the laboratory system will be calculated as the functions of the comoving coordinates. A comoving observer may measure a steady-state axial temperature distribution. Regarding the temperature dependence of the viscosity given, e.g., by (2.7), a variable viscosity may be provided according to \({\eta }(z,t)={\tilde{\eta }}({\tilde{z}})\). Thus, it is suggested to look for solutions \(\mathbf{v}(r,z,t) = {\tilde{\mathbf{v}}}(r,{{\tilde{z}}},t)\), \(p(r,z,t)={{\tilde{p}}}(r,{{\tilde{z}}},t)\). The Navier–Stokes equation (2.6), if rewritten according to (i) to (iii), does not preserve its original form. Nevertheless, we can conclude, following the well-known argument [41, 42], that the l.h.s. is small, on the order of the Reynolds number \(\mathrm{Re}=LU{\bar{\rho }}/{\bar{\eta }}\) compared with the r.h.s., where L , \({\bar{\rho }}\), and \({\bar{\eta }}\) stands for the order of the characteristic length scale (e.g., axial length of the heater), density, and viscosity of our problem, respectively, and U is a characteristic velocity which now stands for \(v_T\), too. For collapsing experiments, we have \(L\approx \mathrm {10^{-2}\;m}\), \(U\approx \mathrm {10^{-5}\,m/s}\) (the order of \(v_T\)), \({\bar{\rho }}\approx \mathrm {10^{3}\;kg/m^3}\), and \({\bar{\eta }}\approx \mathrm {10^{4}\;Pa\; s}\) (the order of the minimum viscosity for maximum temperatures of about 2300 K), resulting in \(\mathrm{Re}\approx 10^{-8}\). Therefore, the Stokes approximation (see, e.g., [41], Chap. 2) for highly viscous fluids can be applied, neglecting the l.h.s. of (2.6) in comoving coordinates. The resulting Stokes equation [41, 42] preserves, in fact, its form carrying out (i) to (iii). These findings are important from physical, as well as mathematical viewpoint and will be discussed below. In the same way, the definitions (2.3) to (2.5) of the components of \({\varvec{\sigma '}}\) remain the same substituting z by \({\tilde{z}}\).

Here we will make the first simplification of notation by denoting quantities in comoving coordinates with the axial coordinate denoted by z (not \(\tilde{z}\)) (in case of doubt, the choice of the coordinate system will be outlined).

For incompressible fluids, the Stokes equations read The steady-state viscous flow \(\mathbf{v} (r,z)\) and pressure p(r, z) must satisfy two groups of boundary conditions. The first one concerns the force balance at the surface of the collapsing glass tube and the second one the behavior of the viscous flow at the axial boundaries of the collapsing region. In the following, we will call them the radial boundary conditions (RBC) and the axial boundary conditions (ABC), respectively. According to the collapsing model introduced above, the RBC embrace the balance of the internal frictional force from the viscous flow and the external force due to the surface tension. The steady-state outer and inner surfaces of the collapsing glass tube according to (ii) may be represented by \(r={R_0}(z)\) and \(r={R_1}(z)\), respectively. The associated tangential and normal unit vectors may be denoted by \({\mathbf{t}_0}(z)\), \({\mathbf{t}_1}(z)\), and \({\mathbf{n}_0}(z)\), \({\mathbf{n}_1}(z)\), respectively, where both the normal vectors may be directed outward with respect to the collapsing glass tube. We have where \(i = 0,1\) may indicate the outer and inner boundaries, respectively. The RBC can be decomposed into two groups of equations corresponding to the balance of the normal and tangential force components, respectively. The external force at the outer and inner surfaces is given by \(-(-1)^{i}\cdot { \mathbf{n}_i}(z) \cdot 2{\tau }{H_i}(z)\), respectively, where \(\tau \) is the surface tension. For silica, \(\tau \) is widely temperature independent for temperatures that are accessible for collapsing [3], but in general this is not true for other (doped) glasses [39]. In what follows, we will deal with constant \(\tau \), for simplification only. We will show in the next section that analysis of the RBC can be straightforwardly extended if \(\tau \) depends upon z via its temperature dependence. \({H_i} (z)\) denotes the mean curvature of the outer and inner surfaces, respectively (see, e.g., [41], Chap. 7). Consequently, the external force enters the first group of the RBC only. For a surface of revolution given by \(r=R(z)\) in cylindrical coordinates, the mean curvature is (see, e.g., [43])The first group of the RBC reads (see, e. g. [41], Chap. 2) \(i = 0,1\). If an additional hollow pressure \(p_h\) is impressed to carry out a joint determination of viscosity and surface tension through two independent measurements (see, e.g., [3]), the term \(p({R_i}(z),z)\) in (2.13) for \(i=1\) has to be substituted by \(p({R_i}(z),z)-p_h\). In the same way, an external torch pressure \(p_T\) from streaming hot gases impinging the outer tube can be taken into account substituting \(p({R_i}(z),z)\) in (2.13) by \(p({R_i}(z),z)-p_T\) for \(i=0\). The second group of RBC reads, observing \({ \mathbf{t}_i}(z)\cdot { \mathbf{n}_i}(z)=0\):The ABC are simply written down according to (i)–(iii). We have the obvious physical condition that the viscous flow must become slowed down to zero if the glass approaches a rigid body outside the heating zone. Then the ABC for collapsing of an infinitely extended tube becomeif \(\lim _{|z| \rightarrow \infty }{\eta } (r,z,t)=\infty \).

Finally, the RBC and ABC have to be found by kinematic equations of the free boundaries, which implies that the movement of the free boundaries is governed, vice versa, by the viscous flow. For steady-state collapsing profiles in comoving coordinates, a simple geometrical consideration results in the two first-order ordinary differential equations:where \(i=0,1\). These kinematic equations close the mathematical problem of steady-state collapsing.

The dimensionless coordinates and geometric and hydrodynamic quantities introduced in the following are specified to collapsing and based on an unconventional velocity benchmark. This treatment is distinguished from that commonly used for drawing (see, e.g., [9, 32]). In this section only, the dimensionless quantities are temporarily denoted with an overbar. The length will be measured in units of the benchmark \(\Lambda \), specified below. Dimensionless cylindrical coordinates \(( {\bar{r}}, {\bar{\varphi }}, {\bar{z}})\) are introduced via \({\bar{r}}=r/\Lambda \), \({\bar{z}}=z/\Lambda \), \({\bar{\varphi }}=\varphi \), and the steady-state dimensionless outer and inner profile functions in moving, dimensionless cylindrical coordinates are denoted by \({\bar{r}}= {{\bar{ R}}_0}({\bar{z}})\) and \({\bar{r}}= {{\bar{ R}}_1}({\bar{z}})\), respectively. The corresponding tangential and normal unit vectors and the dimensionless curvature \({{\bar{ H}}}({\bar{z}})\) are found by applying (2.10), (2.11), and (2.12) in dimensionless coordinates, respectively, where \({H}(z)=(1/\Lambda )\cdot {{\bar{ H}}}({\bar{z}})\). The axial coordinate z will be specified such that the measured temperature distribution T(z), and the viscosity \(\eta (z)\) calculated from T(z) assumes the maximum \(T_{\mathrm{max}}\) and minimum \(\eta _{\mathrm{min}}\), respectively, at \(z=0\). We introduce dimensionless viscosity \({\bar{\eta }}\), viscous flow \({\bar{\mathbf{v}}}\), pressure \({\bar{p}}\), and tensor \(\bar{{\varvec{\sigma }}}'\), all depending upon the dimensionless coordinates, by \({\bar{\eta }}({\bar{z}})=\eta (\Lambda {\bar{z}})/\eta _{\mathrm{min}}\), \({\bar{\mathbf{v}}}({\bar{r}},{\bar{z}})=({\eta _{\mathrm{min}}}/\tau )\cdot \mathbf{v}(\Lambda \bar{r},\Lambda {\bar{z}})\), \({\bar{ p}}({\bar{r}},{\bar{z}})=(\Lambda /\tau )\cdot p(\Lambda {\bar{r}},\Lambda {\bar{z}}) \), and \(\bar{{\varvec{\sigma '}}} ({{\bar{r}},}{\bar{z}})= (\Lambda \cdot \eta _{\mathrm{min}}/\tau ){\varvec{\sigma }}' (\Lambda {\bar{r}},\Lambda {\bar{z}})\). The dimensionless velocity \({\bar{u}}\) of the heat source is \({{\bar{u}}}=(\eta _{\mathrm{min}}/\tau )\cdot v_T\) , and the dimensionless time is \({\bar{t}}=t\cdot \tau /(\Lambda \cdot \eta _{\mathrm{min}})\). The velocity benchmark \(\tau /\eta _{\mathrm{min}}\) is the capillary velocity at the maximum temperature.

The hydrodynamic equations (2.8), (2.9), the RBC (2.13), (2.14), and the kinematic equations (2.16), if rewritten in dimensionless coordinates and quantities, are invariant against the transformation to dimensionless coordinates and quantities defined above, with the formal substitution \(\tau =1\). This invariance means that all physical quantities entering the collapsing problem become scaled by the three basic parameters \(\Lambda \), \(\eta _{\mathrm{min}}\), and \(\tau \). In particular, any quantities x, t, v, and p with dimensions of length, time, velocity, and pressure related to the corresponding dimensionless ones by \(x=\Lambda {\bar{x}}\), \(t=({{\eta _{\mathrm{min}}}\Lambda /\tau })\cdot {\bar{t}}\) , \(v=(\tau /\eta _{\mathrm{min}})\cdot {\bar{v}}\), and \(p=(\tau /\Lambda )\cdot {\bar{p}}\). Any quantities \({\bar{x}}\), \({\bar{t}}\),\({\bar{v}}\), and \({\bar{p}}\) thought to result from the dimensionless analysis of steady-state collapsing will be determined by the dimensionless tube geometry at the start, the course \({\bar{\eta }}({\bar{z}})\) of the dimensionless viscosity, and the dimensionless velocity \({\bar{u}}\) of the heat source only.

If the temperature dependence of the surface tension cannot be neglected, we have to take into account a steady-state z-dependent surface tension \({\breve{\tau }}(z)\), such as the z-dependent viscosity. The dimensional parameter \(\tau \) may be defined, at the beginning, as the surface tension at the maximum temperature \(T=T_{\mathrm{max}}\), that is \(\tau ={\breve{\tau }}(0)\). We introduce, like the dimensionless viscosity \({\bar{\eta }}\), the dimensionless surface tension \({\bar{\tau }}\) as \({\bar{\tau }}({\bar{z}})={\breve{\tau }}(\Lambda z)/\tau \). Then \({\bar{\tau }}({\bar{z}})\) appears as a factor on the r.h.s. of the non-dimensionalized RBC (2.13).

In the following, we will make the second simplification of notation that, starting from (2.8), (2.9) together with (2.13) to (2.16), we may set \(\tau =1\) and denote all dimensionless quantities by omitting the overbar.

It is worth discussing some basic aspects to succeed in the analytic modeling of collapsing. Physically speaking, the Stokes flow does not exhibit any intrinsic dynamics and responds to any time-dependent change of the boundary conditions (surface profiles) without delay. This is the prerequisite that the kinematics of surface profiles become decoupled from the hydrodynamics. In fact, assuming a solution manifold \(\mathbf{v} = \mathbf{v}(r,z,\{{R_0},{R_1\}})\), \(p = p(r,z,\{{R_0},{R_1\}})\) of the Stokes equations (2.8), (2.9) and the RBC and ABC would be known, including the functional dependence upon any arbitrary boundary courses \({R_0}(z)\), \({R_1}(z)\). We would arrive, after substitution of \(\mathbf{v} = \mathbf{v}(r,z,\{{R_0},{R_1\}})\) into (2.16), at autonomous kinematic equations, which would cover the entity of steady-state collapsing profiles. Thus, the governing step to succeed in analytic approaches of collapsing is to establish appropriate approaches of hydrodynamic quantities for appropriately prescribed boundary profiles.

This aim requires further advanced ideas for an obvious modeling because the viscous flow will be determined, in general, to depend in a non-local way upon the entire course of \(\eta (z)\), \({R_0}(z)\) and \({R_1}(z)\), whereas this knowledge would be required for arbitrary courses of \({R_0}(z)\), and \({R_1}(z)\). One way to succeed is to look for local approaches where the viscous flow \(\mathbf{v}(r,z)\) at the axial position z becomes determined by \(\eta (z)\), \({R_0}(z)\), \({R_1}(z)\), and local derivatives at the same z-position only. If existing, such a local approach implies the elimination of hydrodynamics in favor of kinematics, which leads to a strong mathematical simplification. Namely, the substitution of \(\mathbf{v}(r,z)\) into the steady-state kinematic equations (2.16) would result in a closed system of ordinary differential equations to determine the steady-state profiles \({R_0}(z)\), \({R_1}(z)\) for a moving heat source.

The leading idea to arrive at such a local description in the theory of collapsing, like in the drawing theory [9], is the assumption of two different governing length scales \({\tilde{L}}\) and \({\tilde{h}}\), which stand for characteristic axial and radial lengths (e.g., length of the heating zone and maximum tube diameter), respectively, where \({{\tilde{h}}}/{{\tilde{L}}}\ll 1\). The 1D theory of collapsing characterized the above changes for \({{\tilde{h}}}/{{\tilde{L}}}\rightarrow 0\) straightforwardly into a highly symmetric limiting case, which can be exactly solved. This is the base to establish the more complex asymptotic multiscale analysis (AMSA) of Sect. 3, to arrive at a local 2D theory of collapsing. In particular, the 1D theory of collapsing is derived in leading AMSA order, meanwhile the 2D description requires approaches beyond the leading order. This AMSA treatment is unconventional insofar as the flow field is now calculated depending on local derivatives of \({R_0}(z)\) and \({R_1}(z)\), too, so that the kinematic equations (2.16) become implicit differential equations. This circumstance causes further restrictions beyond the basic AMSA preconditions, as shown in Sect. 3.5 in detail. Finally, it should be mentioned that, in principle, the AMSA treatment could be straightforwardly extended to drawing by providing for an additional leading-order axial flow with non-zero axial gradient across the heating zone only, together with the additional radial flow induced through the condition of incompressibility, and modified ABC and RBC. As always concluded (see, e.g., [9]), the leading-order AMSA shows that the drawing phenomenon is explainable without consideration of the surface tension. We will not study more details, which are out of our topic.

As emphasized in Sect. 1, we want to demonstrate an analysis of collapsing beyond the leading order. Therefore, we will perform an unconventional analysis of the Stokes equations (2.8), (2.9) for incompressible fluids with a space-dependent viscosity. Our aim, in particular, is (i) to eliminate (2.8) strictly to satisfy the incompressibility condition within each perturbation order, if AMSA is applied, and (ii) to derive the starting equations allowing a convenient way to deal with the space-dependent viscosity, namely \(\eta = {\eta }(z)\). To eliminate (2.8), we introduce the vector field \(\mathbf{A}(r,z)\) via where \({\hat{\mathbf{e}} }_{\varphi }\) is the azimuthal unit vector. Thus, \(A_{\varphi } (r,z)\) appears as a streaming function for rotational symmetric problems with an incompressible flow. Physically speaking, \(\mathbf{A}(r,z)\) is the vector potential belonging to the incompressible viscous flow, as shown in Appendix 1.

Instead of (2.9), we change to coupled equations to calculate \({A_{\varphi } }(r,z)\) together with a pressure-related function \(Q''(r,z)\). The resulting equations derived in Appendix 1 are whereand \(p_0\) is an arbitrary constant.

Equation (2.22) is simply understood. It relates the viscous flow through \(A_{\varphi }(r,z)\) to the driving force through \(Q''(r,z)\) with \(1/\eta \) being the kinetic coefficient. For illustration, we may choose \(\eta (z)=\mathrm const\) and consider the simple solution \(Q''(r,z)=Cr\), \(A_{\varphi }(r,z)=(C/4\eta )({r^3}/2-{R^2}r)\) of (2.21), (2.22) where C is an arbitrary constant. We find \(\partial p/\partial r=0\), \(\partial p/\partial z= 2C\) from (5.6), and \({v_r}(r,z)=0\), \({v_z}(r,z)= (C/2\eta )({r^2}-{R^2})\) from (2.19), (2.20). This is the well-known laminar axial flow within a tube of radius R in \(+z\)-direction for \(\partial p/\partial z<0\) (see, e.g., [41, 42]), which satisfies the no-slip condition \({v_z}(r,z)=0\) at \(r=R\).

The principles of the asymptotic multiscale analysis (AMSA) as used in this work are outlined, e.g., in [44], Chap. 10. Our method is similar to Adomian’s decomposition method [45] applied to linear partial differential equations with variable coefficients. First, we will extend AMSA to perform a joint analysis of the Stokes equations and boundary conditions, to construct the local approach of Sect. 2.2 for steady-state \({R_0}(z)\) and \({R_1}(z)\) in comoving coordinates. The kinematics will be discussed later. \(R_{0\mathrm{max}}\), \(R_{0\mathrm{min}}\) and \(R_{1\mathrm{max}}\), \(R_{1\mathrm{min}}\) may be the outer and inner tube radii before and after collapsing, respectively. We consider an infinitely extended axial region so that \({R_{0\mathrm{max}}}=\lim _{z \rightarrow \infty }{R_0}(z)\), \({R_{0\mathrm{min}}}=\lim _{z \rightarrow -\infty }{R_0}(z)\), and so on. The dimensionless units of Sect. 2.2 will be specified choosing \(\Lambda =R_{0\mathrm{max}}\). Occasionally, the outer and inner tube radii will also be denoted by \(R_{0 }^{(0)}\) and \(R_{1 }^{(0)}\), respectively.

The AMSA given here exploits the two governing length scales \({\tilde{L}}\) and \({\tilde{h}}\), as discussed in Sect. 2.2 (for further specification, see Sect. 3.5.2). The AMSA will be based on the exact solution of hydrodynamic equations and boundary conditions for the limiting case \({{\tilde{h}}}/{{\tilde{L}}} \rightarrow 0\). This implies the knowledge of the dependence upon parameters entering this problem. AMSA aims to construct solutions where relevant parameters become “slowly” z-dependent on the length scale \({\tilde{L}}\), with the condition to turn into the known solution for \({{\tilde{h}}}/{{\tilde{L}}} \rightarrow 0\). A trivial approach for sufficiently small \( {{\tilde{h}}}/{{\tilde{L}}}\) would leave this known solution, but taking the parameters in question to become slowly z-dependent. Even, this is the choice of the zeroth-order AMSA. Subsequent corrections are found through an “entangled” perturbation analysis in powers of \({{\tilde{h}}}/{{\tilde{L}}}\), which shows a characteristic interplay between perturbing effects from differential equations and boundary conditions (see Sect. 3.3 and Appendix 2).

The 1D theory [2] corresponds to the starting approach of AMSA for \({\tilde{L}}\rightarrow \infty \), \({{\tilde{h}}}/{{\tilde{L}}} \rightarrow 0\). Then the ABC have to be replaced by symmetry conditions of invariance against translations and inversions. This implies \(\eta (z)= \eta =1\), \(v_{r}(r,z)=v_{r}\), \(v_{z}(r,z)=0\), \(p(r,z)=p(r)\), \(R_{0}(z)=R_{0 }^{(0)}=\mathrm const \), and \(R_{1}(z)=R_{1 }^{(0)}=\mathrm const \) (where \(\eta \) will be explicitly outlined for reasons given below). There is one solution set of (2.21) to (2.24) satisfying these symmetry conditions, namely \(Q''(r,z)=0\), \(A_{\varphi }(r,z)={c_A} \cdot z/r \), and \(p(r,z)={c_P} =\mathrm const \), with two arbitrary coefficients \(c_A\), \(c_P\). For symmetry reasons, the normal and tangential vectors at the boundaries are in radial and axial directions, respectively. Thus, the balance of the tangential (axial) force components (2.14) is satisfied because of \({\sigma '_{rz}}=0\). Then \(c_A\), \(c_P\) are determined from the balance of the normal (radial) forces (2.13) of the RBC only. \(v_r\) and \( {\sigma '_{rr}}\) are calculated using (2.19) and (2.3): so that the radial force balance on both the boundaries \(r=R_{0 }^{(0)}\), \(r=R_{1 }^{(0)}\) becomeswith the lower sign to be taken for \(r= R_{1 }^{(0)}\). Then we find

The relevant parameters of the 1D theory are \(\eta \), \(R_{0 }^{(0)}\), and \(R_{1 }^{(0)}\), to be considered to vary in axial direction on the length scale \({{\tilde{L}}}\gg {{\tilde{h}}}\). To underline this slow axial dependence, we introduce the formal and non-specified perturbation parameter \(\varepsilon \), considered to be small, on the order of \({{\tilde{h}}}/{{\tilde{L}}}\ll 1\), and the ”slow” axial variable \(Z=\varepsilon z\), writing \(\eta =\eta (Z)\), \({R_0}={R_0}(Z)\), \({R_1}={R_1}(Z)\). Then \(\mathrm{d}\eta (z)/ \mathrm{d}z=\varepsilon \mathrm{d}\eta (Z)/ \mathrm{d}Z\), \(\mathrm{d}{R_0}(z)/ \mathrm{d}z = \varepsilon \mathrm{d}{R_0}(Z)/ \mathrm{d}Z\), and \(\mathrm{d}{R_1}(z)/ \mathrm{d}z = \varepsilon \mathrm{d}{R_1}(Z)/ \mathrm{d}Z\). The terms \(\mathrm{d}\eta (Z)/ \mathrm{d}Z\) and so on with \(\varepsilon \) as a prefactor are now O(1). The formal perturbation parameter \(\varepsilon \) is required to systematize the perturbation analysis. In the following, Z and \(\varepsilon \) will be outlined as arguments of functions we are looking for, writing \({\overline{\overline{Q}}}''(r,z,Z,\varepsilon )\), \({{\overline{\overline{A}}}_{\varphi }}(r,z,Z,\varepsilon )\), and \({\overline{\overline{p}}}(r,z,Z,\varepsilon )\) instead of \(Q''(r,z)\) , \({A_{\varphi }(r,z)}\), and p(r, z), and so on. Then in the basic equations (2.21), (2.22), and (5.8), (2.23), the substitutionmust be carried out. Upon completion of the perturbation analysis, we may set \(\varepsilon =1\) and verify that the terms in \(\varepsilon ^n\) become quantities of the order \(({{\tilde{h}}}/{{\tilde{L}}})^n\) due to their concrete parameter dependence.

To solve the differential equations, which have been brought in AMSA form in this way, Z is formally treated as a free parameter, and all Z-derivatives may be collected on the r.h.s., so that the order in \(\varepsilon \) of the r.h.s. is enhanced compared with the l.h.s. Looking for solutions \({\overline{\overline{Q}}}''(r,z,Z,\varepsilon )\), \({{\overline{\overline{A}}}_{\varphi }}(r,z,Z,\varepsilon )\) as perturbation series in \(\varepsilon \), the idea is that the error due to the incomplete treatment of the axial dependence in lower perturbation orders becomes corrected in subsequent perturbation orders.

In the following, we present a strict AMSA version, assuming that any axial dependence is a slow one, suppressing the rapid axial variable z. Then we arrive at a rapidly proceeding perturbation analysis of the differential equations and boundary conditions in powers of \(\varepsilon ^2\). (2.21), (2.22) now read in AMSA version Equations (3.7), (3.8) are supplied by the AMSA version of (2.23) to determine the pressure \({\overline{\overline{p}}}(r,Z,\varepsilon )\) from known \({\overline{\overline{Q}}}''\) , \({\overline{\overline{A}}}_{\varphi }\) according toWe will solve (3.7), (3.8) by series expansions according to After substitution of (3.10), (3.11) into (3.7), (3.8), the comparison in powers of \(\varepsilon \) results in a chain of coupled ordinary differential equations with respect to r only, to determine the higher-order terms of (3.10), (3.11) from the lowest order ones. Further hydrodynamic quantities in AMSA version can be calculated as power series in \(\varepsilon \) from (3.9) to (3.11), applying (2.19), (2.20), and (2.3) to (2.5), with observing (3.6). In particular, the series expansions for the pressure, and the rr- and rz-component of the stress tensor (divided by \(\eta (Z)\)) read (in (3.14), the term \(\propto 1/\varepsilon \) vanishes according to (5.13), (5.16) given below).

In AMSA, the ABC for an infinitely extended axial region become satisfied in an elementary way. We provide throughout that \(1/\eta (Z)\) is represented as exponential \(1/\eta (Z)=\mathrm{exp} (-f(Z))\), where \(f(Z)\ge 0\), with maximum \(1/\eta (0)=1\) at \(Z=0\). f(Z) may be an at least second-order polynomial \(f(Z)={f_2}{Z^2}+{f_3}{Z^3}+\cdots \), \({f_2}>0\), with \(f(Z)\rightarrow \infty \) for \(Z\rightarrow \pm \infty \). If the flow derived from the lowest order term in (3.11) satisfies the ABC for \(Z\rightarrow \pm \infty \), this holds true in each perturbation order. These mathematical findings are simply verified from the perturbation analysis of (3.7), (3.8), noting that the exponential in question, if entering into the lowest order approach of \({{\overline{\overline{A}}}_{\varphi }}\), becomes reproduced in each perturbation order and in all quantities derived from \({{\overline{\overline{A}}}_{\varphi }}\). Our prerequisite regarding the viscosity course is physically motivated through the temperature dependence of the glass viscosity of Arrhenius type, like (2.7), and the characteristic axial temperature course across the collapsing region, where the temperature maximum is surrounded by steeply descending edges. Thus, AMSA naturally describes the suppression of the viscous flow beyond the heating zone for the preconditions outlined above.

The analysis of the Z-dependent RBC is a much more complex task. First, we will make the agreement that in this and subsequent sections, R stands for \({R_0}(Z)\) and \({R_1}(Z)\), respectively. Our starting assumption is that any solution pair \({\overline{\overline{Q}}}''(r,Z,\varepsilon )\), \({{\overline{\overline{A}}}_{\varphi }}(r,Z,\varepsilon )\) of (3.7), (3.8) satisfying the RBC, as well as the hydrodynamic functions derived from the former can be represented like the series expansions (3.10), (3.11), and (3.12) to (3.14), respectively. The l.h.s. of (2.13) and (2.14), divided by \(\eta (Z)\), may be abbreviated as with \({\overline{\overline{\mathbf{n}}}}\) and \({\overline{\overline{\mathbf{t}}}}\) being the axial courses of the normal and tangential vectors at the boundaries, respectively. Then the series expansions of (3.15), (3.16) are found through expanding (2.10), (2.11) in powers of \(\varepsilon \), with observing (3.6). In the same way, the mean curvature at the boundaries given by (2.12) will be written as \(H(z)={\overline{\overline{H}}}(R,Z,\varepsilon )\). We arrive at expansions in even and odd powers of \(\varepsilon \) only, according to The conditions (2.13) and (2.14) of the normal and tangential force balance, respectively, at the boundaries can now be rewritten according to with the lower sign in (3.20) to be taken for \(R={R_1}(Z)\). After substitution of (3.17) to (3.19) into (3.20), (3.21), we see that (i) the RBC become satisfied through the normal force balance only in leading order \(\varepsilon ^0\), and (ii) in addition, through the tangential force balance only in \(\varepsilon ^1\), beyond the leading order. Continuing, contributions to the normal and tangential force balance will appear in higher even and odd powers of \(\varepsilon \) only, respectively.

The AMSA treatment of the RBC implies, as already emphasized in Sects. 3.1 and 3.2, that any force balance condition in \(\varepsilon ^n\) will be satisfied with respect to the rapid (radial) variable only, meanwhile its Z-dependence is taken into account through Z-dependent coefficients. The resulting error will be corrected in two steps: (i) to arrive at the solutions of (3.7), (3.8), this Z-dependence implies corrections in subsequent orders, as described below (3.10), (3.11); and (ii) these corrections on their part will induce additional higher-order contributions to \({\overline{\overline{F}}}_\mathrm{norm}(R,Z,\varepsilon )\) and \({\overline{\overline{F}}}_\mathrm{tan}(R,Z,\varepsilon )\), to be taken into account in the corresponding higher-order force balance conditions. Details are given in Appendix 2.

As a result, the outlined quantities (3.10) to (3.16) as the solutions of (3.7), (3.8) and RBC can be represented, due to the successive settlement of the RBC in \(\varepsilon ^0\), \(\varepsilon ^1\),..., as series, e.g.,where \({\overline{\overline{A}}}_{\varphi 0}=O(1/\varepsilon )\), \({\overline{\overline{A}}}_{\varphi 1}=O(\varepsilon )\). These series could be rearranged to proceed strictly in powers of \(\varepsilon \), which will not be done here.

We will summarize the zeroth- and first-order results for the viscous flow and refer, for details, to Appendix 3. From general AMSA, the radial flow \({\overline{\overline{v}}}_r(r,Z,\varepsilon )\) and axial flow \({\overline{\overline{v}}}_z(r,Z,\varepsilon )\) result as series in even and odd powers of \(\varepsilon \), respectively, observing that in the lowest order \({\overline{\overline{A}}}_{\varphi 0}\propto (1/\varepsilon ) \cdot (1/r)\). For simplification, we will write in the following From the zeroth- and first-order AMSA, we have As argued in Sect. 1, the leading AMSA order for collapsing is governed by the radial flow component \({\overline{\overline{v}}}_{r0}(r,Z)\) only; meanwhile, the axial flow one \({\overline{\overline{v}}}_{z1}(r,Z)\) appears as the first-order effect beyond the leading order. Restricting to the first-order AMSA, \({\overline{\overline{v}}}_{z1}(r,Z)\) is determined, according to (3.26), as a linear function of \(\mathrm{d}(1/\eta (Z))/\mathrm{d}Z\), \(\mathrm{d}R_0(Z)/\mathrm{d}Z\), and \(\mathrm{d}R_1(Z)/\mathrm{d}Z\). The courses of the latter functions even characterize the slow axial parameter dependence discussed in Sects. 3.1 and 3.2.

We subdivide \({\overline{\overline{v}}}_{z1}(r,Z)\) into a kinematic part \(^{(\mathrm kin)}{\overline{\overline{v}}}_{z1}^{(0)}\cdot \mathrm{d}R_0(Z)/\mathrm{d}Z+^{(\mathrm kin)}{\overline{\overline{v}}}_{z1}^{(1)}\cdot \mathrm{d}R_1(Z)/\mathrm{d}Z\), and a non-kinematic one \(^{(\eta )}{\overline{\overline{v}}}_{z1}\cdot \mathrm{d}(1/\eta (Z))/\mathrm{d}Z\). In the lowest order, the former quantifies that part of the axial flow component generated through the axial change of the boundary profiles of the collapsing tube, meanwhile the latter describes the axial flow contribution due to the axial change of the viscosity only. As shown in Appendix 3, both the contributions arise from the tangential force balance at the boundaries, taken into account within the first-order AMSA. We writeThe radial flow as well as both the parts of the axial flow component depend upon governing parameters entering the collapsing problem in a quite distinct way. For a rough estimation, we introduce the variable wall thickness \(W(Z)=R_0(Z)-R_1(Z)\) of the collapsing tube, approaching \(R_0(Z)\approx 1\), \(W(Z)\approx w\), \(r\approx 1\), \(1/\eta (Z)\approx 1\), \(|\mathrm{d}(1/\eta (Z))/\mathrm{d}Z|\approx \beta \). The dummy w and \(\beta \) may estimate the wall thickness and the axial distance where the reciprocal viscosity significantly changes, respectively. We find

Free boundaries and a variable viscosity characterize collapsing as an outstanding hydrodynamic problem beyond common analytic tools of mathematical physics. Analytic approaches only facilitate the detailed physical understanding and knowledge of basic parameter dependencies of this complex phenomenon. If collapsing is applied to measure the temperature-dependent glass viscosity [3, 4], correct analytic approaches of geometry parameters can significantly reduce the effort of data evaluation, compared with finite element (FEM) simulations. On the other hand, FEM simulations can verify the consistency of analytic approaches, and vice versa, correct analytic approaches allow for a rigorous test of FEM algorithms. The latter aspect is important insofar as the same FEM algorithms will be extended, of course, to parameter areas where analytical approaches fail.

In the following, the dimensionless results from above will be rewritten in physical (dimensional) quantities. We will focus on the “shrinking ratio” (outer radius after collapsing \(R_{0\mathrm{min}}\) divided by the outer radius \(R_{0\mathrm{max}}\) before collapsing), as it depends upon several parameters entering the collapsing problem. The reciprocal viscosity in comoving coordinates may be given as Gaussian (3.44). Comparative FEM calculations have been carried out with the \(COMSOL^{(R)}\) Multiphysics 4.2a program package in laboratory coordinates and physical units, to simulate collapsing as an advancing time-dependent process, like in laboratory. A rigid wall cut-off of the viscosity course at \(\eta (z)=10^{12} \, \mathrm{Pa \; s}\), corresponding to \(10^7 \,\eta _{\mathrm{min}}\), has been introduced, and the successive shape deformation across the collapsing cone has been taken into account through the ALE moving mesh module in COMSOL. \(R_{0\mathrm{min}}/R_{0\mathrm{max}}\) has been determined with an accuracy of \(10^{-4}\).

In Fig. 1a, b, we show the fitted results of \(R_{0\mathrm{min}}/R_{0\mathrm{max}}\) from FEM. The parameters are \(R_{1\mathrm{max}}/R_{0\mathrm{max}}=0.7\) and 0.9, \(0.20\le \alpha \le 1.00\), where \(\alpha =R_{0\mathrm{max}}/z_e\).

Numerical data of \(1-R_{0\mathrm{min}}/R_{0\mathrm{max}}\) from FEM, from the analytic 1D and 2D theories, and the estimated absolute error of the AMSA data are outlined in Tables 1–8. The 1D data have been calculated from the zeroth-order AMSA formulae, combining (3.43), (3.39), and (3.46). The 2D data are based, in addition, on the AMSA formulae up to the second order (5.29) to (5.31), (5.35), and (3.52). The fifth column of Tables 1–8 outlines the estimations of absolute uncertainties of AMSA data via extrapolation to non-performed higher-order perturbation steps. They summarize non-kinematic contributions, based on the second and third remainders of (3.53) (with \(\beta =\alpha \)), and the kinematic ones from the remainders in (3.49) and (3.52). Tables 1, 2, 3 and 4 are arranged to demonstrate the limit of AMSA with respect to the non-kinematic perturbation for increasing \(\alpha \), and Tables 5, 6, 7 and 8 with respect to the kinematic perturbation for decreasing u. In particular, the data of both Tables 6 and 8 nearly approach the geometric collapsing limit \(R_{1\mathrm{min}}\rightarrow 0\) that would be reached for \(1-R_{0\mathrm{min}}/R_{0\mathrm{max}}\rightarrow 0.286\) if \(R_{1\mathrm{max}}/R_{0\mathrm{max}}=0.7\), and \(1-R_{0\mathrm{min}}/R_{0\mathrm{max}}\rightarrow 0.564\) if \(R_{1\mathrm{max}}/R_{0\mathrm{max}}=0.9\) .

We refer to the striking fact that, the error estimation of the AMSA data notwithstanding, the numerical differences of \(1-R_{0\mathrm{min}}/R_{0\mathrm{max}}\) are the marginal ones, comparing the 1D and 2D results. In addition, these findings also hold true in comparison with the outlined FEM data. At least for the Gaussian (3.44), the analytic result for \(1-R_{0\mathrm{min}}/R_{0\mathrm{max}}\) derived from the 1D theory through combining (3.43), (3.39), and (3.46) represents an excellent fit of the FEM data for the full parameter area outlined in Tables 1–8.

Comparing the 1D and 2D theories, the minor differences of \(1-R_{0\mathrm{min}}/R_{0\mathrm{max}}\) may be partially explained through peculiarities of the low-order AMSA perturbation analysis discussed in Sect. 3.5.3, namely the suppression of the non-kinematic perturbation. Regarding the estimated error outlined in column 5 of Tables 1–8, the predicted accuracy of AMSA widely varies, namely with respect to \(\alpha \), from being very precise beyond FEM for \(\alpha \le 0.2\) to extremely erroneous for \(\alpha >0.6\). The latter region is characterized by a strong non-kinematic perturbation, so that AMSA is no longer justified from its fundamentals.