Hydrodynamics of Collapsing Glass Tubes and Measuring of Glass Viscosities: Analytic Results beyond Asymptotic Approaches for Rapidly Varying Viscosities

Round 1

Reviewer 1 Report

The author has aimed to investigate the Navier-Stokes equation with a variable viscosity applied for molten glasses and considering the set of boundary conditions together with the system of equations. The axially-dependent reciprocal viscosity was investigated for infinitely extended tubes with constant radii.

The notations and derivation of equations are clear. The main goal is to give a method for the back-calculation of the viscosity from experimental input data by giving appropriate error estimation.

The article is very well written, not an easy read. I looked through the 64 pages, it would take more than a week to check the calculations and read the links. Overall, it is a well-structured review paper that I recommend for publication without change.

Author Response

Thank you for your comments.

Reviewer 2 Report

Re: Hydrodynamics of collapsing glass tubes and measuring of glass viscosities: Analytic results beyond asymptotic approaches for rapidly varying viscosities

The author presents an interesting analytical treatment of the axial-symmetric boundary value problem of the Stokes equation for incompressible liquids with rapidly varying viscosity. It is a good study that will interest the fluid dynamics community and spark some interesting discussions. The reviewer has some comments to enhance the clarity of the manuscript for the author to consider in revising the submission:

• This statement was made at the end of the introduction to explain the contribution: “At present, there would appear to be no standard analytic method to solve boundary value problems of the Stokes equation with a variable viscosity beyond the preconditions of the asymptotic analysis, as a role model to treat our problem in question. Thus, our mathematical approach is novel, and without exploiting abstract functional-analytic methods”. This needs to be backed up with a solid review of the previous attempts at this solution or similar and their drawbacks and/or advantages. Currently, the case made is not convincing enough.
• As you’ve already mentioned steady state, the time argument on the LHS of eq. 2.6 should be removed. Or restate the preceding statement to capture the general case with time dependence, which you simplified later in the paragraph. This is necessary purely for completeness.
• Required from your solution are plots of the viscosity, temperature T(x, z), flow velocities V(x, z), and stress τzz(x, z) fields calculated from model
• The real test of the solution will a comparison with experimental data. Since your solution was obtained for the case of a collapsing glass tube, then simple experiments can be conducted for this purpose if appropriate secondary data is not available. 
• Your main assumptions were not stated. 
• Will the solution apply to non-newtonian fluids, and which type if yes? 
• A table of nomenclature defining all variables will be helpful due to the large number of symbols throughout the manuscript.
• It will also be useful to state that this will not be applicable to the turbulent case
• 2nd line of Last paragraph in Page 5: The statement is not complete: "...(see [11] and refs. cited therein)" - which other refs?

Author Response

see attachment

Author Response File: Author Response.pdf

Round 2

Reviewer 2 Report

The reviewer is satisfied with the changes made to the manuscript to enhance its clarity and is happy for it to be accepted for publication. It is indeed an elegant piece of work and the author is thanked for their hard work and submission. Best of luck!