Stokes Equation in a Semi-Infinite Region: Generalization of the Lamb Solution and Applications to Marangoni Flows

Round 1

Reviewer 1 Report

The manuscript “Stokes equation in a semi-infinite region: generalization of Lamb solution and applications to Marangoni flows” by G. Kolesky and T. Bickel is devoted to the extension of the Lamb solution of the Stokes equation to a semi-infinite geometry. The obtained results are applied to consideration of Marangoni flows induced by a local heat source at the liquid-air interface. The presented results could be of particular interest to the community of “Fluids” readers. However, the manuscript is overloaded with detailed descriptions of equation transformations and sometimes looks like a set of mathematical exercises. At the same time, the abundance of these mathematical “trees” around the reader complicates the overall perception of the physical “forest” associated with the problem being solved. In addition, the initial statement of the particular problems in some cases raises certain questions. In my opinion, additional consideration of some points suggested below will improve the manuscript quality and make it acceptable for publication in “Fluids”.

1. The presented statement and consideration of the heat transfer problem (expressions 11 – 13) seems questionable. In the absence of heat exchange with the upper half-space (Fig. 1) and constant power of the heat source (Q), the temperature field in the lower half-space cannot be stationary. This is contrary to the principle of energy conservation, and the non-stationary heat equation should be applied instead of the stationary one (Eq. 11). Analogies with an electrostatic field are inappropriate here. Besides, the heat transfer problem should be stated more correctly. Eq. (11) in its present form is inconsistent. If Q is the power of a point-like source ([J/s]) and k is the thermal capacity ([J/K]), then the dimension of the right-hand side of Eq. 11 is defined as [K/s]. At the same time, the dimension of the left-hand side is defined as [K/m^2]. Please, check the correctness and rewrite this part.

2. The presented solutions exhibit a non-physical behavior in the vicinity of the origin (the inverse power law divergence). Accordingly, the minimum distance from the origin must be determined, for which the obtained solutions retain their physical meaning. Intuitively, we can conclude that this minimum distance depends on the medium parameters and the temperature difference.

3. The quality of the presented work could be better, if the reported findings will be illustrated some examples based on the numerical estimates for some real systems under real conditions. This could be useful for further practical applications of this work.

Author Response

Please see the attachment.

Author Response File: Author Response.pdf

Reviewer 2 Report

In this paper, a generalization of Lamb solution for the Stokes flow in a semi-infinite domain is derived. The results presents several features that are rather unconventional and have wide application. The expression of the velocity components of the results can be used directly for any problem with hemispherical geometry. Clearly, the goal of the paper is desirable, and the results obtained may be reasonable. In my view, the paper is acceptable for publication. 

Author Response

We thank the referee for his careful reading of the manuscript. We are delighted that he finds our paper acceptable for publication.

Round 2

Reviewer 1 Report

Revisions and corrections provided by the authors improved the quality of the work and made it more acceptable for publication. Nevertheless, regarding the solution of the heat transfer problem, it is necessary to note the following. There is no doubt that the solution to the problem at large times converges to a stationary case due to the exponential term exp(-r^2/4at) approaching to 1 (a is the thermal diffusivity). There is no need to explain these trivial things to the reviewer, and even more so to talk about his wrongness. I meant that the initial formulation of the problem was given by the authors in an incorrect form and requires the presence of the time derivative of temperature in the equation (11). Additionally, the authors should have noted that the solution is considered for the case of large time scales or small distances from the source, when the condition r^2/4at<<1 is satisfied and the temperature dependence on time can be neglected. Such a rigorous and consistent formulation of the problem of heat transfer will be more acceptable in the methodological sense. I strongly recommend that authors make these minor but methodically important changes.

Author Response

We thank the referee for his attentive proofreading of our manuscript. We agree with the referre that the discussion regarding the heat equation might be improved. We have thus modified the text in section 3.1 starting at line 72. In the new version, we now first give the time-dependent heat equation, and subsequently discuss the stationary limit r<<(Dt)^1/2. We have also modified the numbering of the references [14] to [17], that did not appear in the right order in the previous version.

We hope that, with these minor changes, our article is now suitable for publication in Fluids. With our best regards,

Thomas Bickel