Bursting Sand Balloons

Round 1

Reviewer 1 Report

Comments and Suggestions for Authors

The manuscript "Bursting sand balloons" is very well summarized by its title. It contains an experimental study of the burst of a balloon full of Ottawa sand; and a theoretical description of a decompression wave. To my knowledge, this is the first attempt to study this problem, which makes it relevant and exciting. 

 

However, some aspects should be improved or clarified. I will mention them by going through the manuscript sections. 

 

Section 2. 

It is unclear if the balloon included sand when pressure measurements (as a function of the radius) were performed. If not, how can we be confident that the pressure will be the same when including sand?

 

Section 3. 

Around Fig. 2, there is this phrase: "in Figs. 2(b)-2(c) it can be observed how a decompression front penetrates into the sand mass provoking a fast radial expansion of the sand mass." From the images, I can see the radial expansion of sand. However, I am not convinced that any decompression front could be visible from the same pictures. Could you give more evidence about the decompression front? 

 

In Fig 3, one can see that the growth in the displacement takes longer for a larger radius. Is this referred to as "time delays"? If not, could you explain what "time delays" means? 

As the concept comes again later, it is better to clarify it. 

Do you have any physical idea about the origin of these time delays? It seems an interesting ingredient that deserves further discussion. For instance, it increases with R_0, similarly as the displacement slope decreases.  

 

Section 4

Please discuss further the use of the Heaviside function. What is its physical explanation in the context of this model?

 

Section 5

I call your attention to the fact that, given the values presented in Tables 1 and 2, the ratio p/K_e is not truly << 1. It takes values between 0.35 and 0.5, which is < 1 but not much less than 1. Does it compromise the approximation?

 

From equation 7, I understand that r(t) is meaningful when it is larger than R_0 because, in that case, there is an expansion. Why do you use r = 0.03 m and r = 0.04 m? Also, contrary to Figure 4, which compares with measurements in Figure 3, I don't see the contribution of Figure 5 (what knowledge or insights it adds to the ideas presented in the manuscript).

 

Section 6. 

Most of the conclusions are well supported by the presented material. However, there are two very relevant exceptions (at the beginning and the end of the section):

1) The phrase "In this work we have experimentally shown that the problem of the sand balloon bursting differs greatly from water balloon bursting" appears here without much previous discussion (apart from a brief mention of crack propagation, which is not the main topic of this study). In that regard, I cannot see how this conclusion emerges from the manuscript. The previous sections do not support this conclusion well.

(2) The main conclusion (also mentioned in the abstract), "In conclusion, all theoretical and experimental results appear to indicate that both the elastic wave and the granular front behaves in a similar manner, but in opposite directions." is not well justified either. The "opposition" is mentioned only in the abstract and conclusion but not adequately discussed in the main text. It would help if you made it appear clearly in the main text. 

Comments on the Quality of English Language

I suggest a revision of the English, with a native speaker if possible. It generally reads well, but some phrases seemed unnatural to me in English. 

Also, I found a few typos. For instance a "cuases" instead of "causes"

Author Response

Response to Reviewer 1

Fluids-2787222

We wish to thank the referee for his/her thoughtful reading of the paper and the comments and criticism in his/her report.

Here we reply point-by-point each one of the questions:

The manuscript "Bursting sand balloons" is very well summarized by its title. It contains an experimental study of the burst of a balloon full of Ottawa sand; and a theoretical description of a decompression wave. To my knowledge, this is the first attempt to study this problem, which makes it relevant and exciting. 

 

However, some aspects should be improved or clarified. I will mention them by going through the manuscript sections. 

 

Section 2. 

It is unclear if the balloon included sand when pressure measurements (as a function of the radius) were performed. If not, how can we be confident that the pressure will be the same when including sand?

 

Answer. – To clarify this point, we added the last paragraph of Section 2:

 

In our experiments, when we fill the balloons, the air volume that allowed to reach

the radius R0 is exchanged by the sand volume, we assume that the respective inflation

pressure p(R = R0) now acts on the granular mass. In this sense the plot in Fig. 1(b) will

be useful.

 

To reinforce our confident on the pressure measurements, we found that the pressure decreases as the ballon radii increases because such radii fall in the first decreasing branch of plot in Fig. 1b.

 

Section 3. 

Around Fig. 2, there is this phrase: "in Figs. 2(b)-2(c) it can be observed how a decompression front penetrates into the sand mass provoking a fast radial expansion of the sand mass." From the images, I can see the radial expansion of sand. However, I am not convinced that any decompression front could be visible from the same pictures. Could you give more evidence about the decompression front? 

 

Answer. – We agree with this criticism. Alluding to new Fig. 3, we have changed the figure caption of this figure to clarify the events in the snapshots:

 

Figure 3. Example of a series of snapshots at the very early time instants of the bursting balloon of mass M = 1.00 kg: (a) t = 0 s, (b) t = 3.75 × 10⁻³ s (c) t = 1.00 × 10⁻² s and (d) t = 2.18 × 10⁻² s. In (a) the balloon is laid to hang and an accelerated peeling of the rubber film occurs. From (b) to (d) the decompression occurs, simultaneously, as a dense granular front (measured in plot of Fig 4) and as a dilute front of not interacting single grains (measured in plot of Fig.5).

 

 a new Fig. 5 and two paragraphs after such a figure were also included the evolution of the front:

 

The radial expansion observed in snapshots of Fig. 3 has interesting behaviors like those occurring in the explosive dispersal process [24]: in the early time of the sudden decompression, an expansion develops close to the free surface and the dynamics simultaneously obeys a dense granular expansion quantified by measurements in Fig. 4, and a dilute gas-solid mixture (η very low). This latest case is shown in the inset of Fig. 5, in which the particles located at the left hand side of the lower region of the granular mass shown in Fig. 3(b), appear as discrete and far from each other; the probability of a particle-particle collision is very low and hence the interactions between the particles can be neglected.

 

In Fig. 5 we found that in the dilute regime some particles (particles 1 and 2 in

the inset) acquire rapidly, as will be seen in Section 5, a similar motion as that of the

decompression wave, but in an opposite direction as the wave itself, whereas large

particles respond slowly, as in Fig. 4, due to their large inertia. It is very possible that

the more external and smaller particles of our sand samples could be part of the dilute

regime, both of which reach an average velocity vg = 1.43 ± 0.07 m/s. Afterwards, we

will return to this matter in order to understand other aspects of the decompression.

 

In Fig 3, one can see that the growth in the displacement takes longer for a larger radius. Is this referred to as "time delays"? If not, could you explain what "time delays" means? 

As the concept comes again later, it is better to clarify it. 

Do you have any physical idea about the origin of these time delays? It seems an interesting ingredient that deserves further discussion. For instance, it increases with R_0, similarly as the displacement slope decreases.  

 

Answer.- The delay times are related to the decompression waves which do not penetrates immediately into the sand masses, because they behave as an elastic medium, we added the following three paragraphs:

 

In the paragraph following the Table 1 we added the next text:

In the plot of Fig. 4, the flat part, for each radius R₀ refers that the average displacement (given by the sloped straight lines) starts after a given time, i.e., there is a delay time for each one balloon. We observe that the delay time is longest for a larger mass, later on (Section 5) we shall discuss that this behavior can be directly associated to the decompression waves that go inward the granular masses, which also present temporal delays.

For the new Figs. 6 and 7 we added as a second and third paragraph of Section 5:

It is possible to explain graphically the behavior of the elastic waves given by Eq. (7). In Fig. 6 the behavior of the decompression waves u(r, t), for r =constant, is given: in Fig. 6(a) for r = 0.03 m and in Fig. 6(b) for r = 0.04 m, this latest value is closer to any of the initial radius of the balloons compressing the sand, which are indicated in the inset of each plot. The flat parts in plots of Fig. 6 indicate that the decompression wave arrives to the radial positions r = 0.03 m or r = 0.04 m at different times, larger than t = (R0 − r)/c, this relationship is obtained from Eq. (7) when the argument β = ct + (r − R0) is equal to zero in order to obtain u = 0. For instance, if we consider the balloon of radius R0 = 5.30 × 10⁻² m, we found that the wave arrives at r = 0.03 m after t = 8. 77 × 10⁻³ s, as is observed in the green dashed line in Fig. 6(a), while if r = 0.04 m the wave reaches such a radial position after t = 4. 96 × 10⁻³ s, as is seen in the green dashed line in Fig. 6(b); for smaller balloons these times are shortest.

Similarly, the plots of u(r, t), for t =constant, are given in Fig. 7. The flat parts occur again if β = 0 and in this case, we used the formula r = R₀ − ct which gives the limit for which u = 0. Consequently, for the time t = 0.019 s and R₀ = 5.30 × 10⁻² m the previous formula produces that r = 3 × 10−3 m, it means that between r = 0 m and r = 3 × 10⁻³ m the wave is flat and u = 0 m, as can be seen Fig. 7(a). In Fig. 7(b), when t = 0.020 s, u = 0 m, between r = 0 m and r = 0.6×10⁻³ m. Finally, we observe that the other waves have not plane parts.

 

Section 4

Please discuss further the use of the Heaviside function. What is its physical explanation in the context of this model?

Answer.- Done. We included its formal definition and the physical meaning was explained in the initial manuscript. Now, the paragraph after Eq. (3) is:

 

where H is the Heaviside step function, defined as H(t) = 0 if t < 0 and H(t) = 1 if t ≥ 0.

These are a symmetry condition at the center of the balloon and the condition that the

pressure of the rubber film is instantly released at t = 0.

 

Section 5

I call your attention to the fact that, given the values presented in Tables 1 and 2, the ratio p/K_e is not truly << 1. It takes values between 0.35 and 0.5, which is < 1 but not much less than 1. Does it compromise the approximation?

 

Answer.- In agreement with Landau and Lifshitz (Ref. [14]) in hydrostatic compression of a body -p/K_e=u_ll, where u_ll is the change in the volume and it must be small, consequently if  p/K_e<1 the approximation is not compromised. We did this change in the paragraph following Table 2, in Section 5.

 

From equation 7, I understand that r(t) is meaningful when it is larger than R_0 because, in that case, there is an expansion. Why do you use r = 0.03 m and r = 0.04 m? Also, contrary to Figure 4, which compares with measurements in Figure 3, I don't see the contribution of Figure 5 (what knowledge or insights it adds to the ideas presented in the manuscript).

Answer.- In our treatment u(r,t) is a wave that penetrates into the masses, i.e. r<R₀, and the displacement (defined by Eq. (1) is for r>R₀ More details were given in the answers of Section 3.

 

Section 6. 

Most of the conclusions are well supported by the presented material. However, there are two very relevant exceptions (at the beginning and the end of the section):

1) The phrase "In this work we have experimentally shown that the problem of the sand balloon bursting differs greatly from water balloon bursting" appears here without much previous discussion (apart from a brief mention of crack propagation, which is not the main topic of this study). In that regard, I cannot see how this conclusion emerges from the manuscript. The previous sections do not support this conclusion well.

(2) The main conclusion (also mentioned in the abstract), "In conclusion, all theoretical and experimental results appear to indicate that both the elastic wave and the granular front behaves in a similar manner, but in opposite directions." is not well justified either. The "opposition" is mentioned only in the abstract and conclusion but not adequately discussed in the main text. It would help if you made it appear clearly in the main text. 

 

Answer.- We have discussed along the main text that the granular front occurs in response to the decompression wave. In the Conclusions we summarized these facts. Now, Section 6 reads:

 

In this work we have experimentally and theoretically shown that the problem of the sand balloon bursting is physically interesting and complex. Through the current study, our main purpose was to show that the sudden release of the initially confined granular mass behaves as a genuine elastic body, producing a linear decompression wave that travels into the granular material, at the earliest times of the pressure release, and that in response the sand generates a complex expansion front. To have a complete and consistent theoretical treatment we proposed a new model to compute the effective compression modulus, which was used in plots of Figs. 6 and 7, giving waves 12% slower

than those computed with the Walton effective compression modulus [15]. Similarly, by assuming spherical balloons, a theoretical model based on the hydrostatic compression approach allowed us to analytically find radial decompression waves, which generate radial granular fronts with physical characteristics very similar as those of the elastic wave. Due to the granular mass is opaque we could not observe the decompression waves and only through the resulting granular fronts we have detected the effects of the waves. For instance, for three b alloons of different sizes, it was found that the granular fronts expand in a similar manner as the respective decompression wave: the fronts move at a constant velocity (rate of deformation in Table 1) after delay times very similar to those of the waves. Through Fig. 5 we found that the front is composed simultaneously of a dense and a dilute front. It is possible to compute at the free surface the theoretical velocity of the granular front induced by the decompression wave, this velocity is vf = pc/Ke. Incidentally, in order of magnitude, such a velocity appears to be more adequate for the most external single grains radially expelled at velocity vg. The velocity of the dense front was characterized experimentally by the rate deformation and it is slower than vf . Due to it we have proposed, through a heuristic hypothesis, that simply the velocity of the dense front could be equal to the factor ηvf because it is of the same order of magnitude as the rate deformation. Clearly, such a hypothesis merits more theoretical and experimental studies.

 

Comments on the Quality of English Language

I suggest a revision of the English, with a native speaker if possible. It generally reads well, but some phrases seemed unnatural to me in English. 

Answer.- We will use the English editing services of FLUIDS.

Also, I found a few typos. For instance a "cuases" instead of "causes"

Answer.- Done

 

Author Response File: Author Response.pdf

Reviewer 2 Report

Comments and Suggestions for Authors

The manuscript presents evidence that the initial dynamics for an expanding granular system behaves, however shortly, as an elastic medium.

The manuscript is well written, but a few typos can be detected such as

a)      Pg 2, 3^rd paragraph: instead of “of the body but no change its shape”, it should be “of the body but do not change its shape”;

b)     Pg 5, first line of 4.1 should be “causes”;

There could be more. The authors should recheck the writing.

More importantly, the work is composed of an experimental and a theoretical part. The experiment consists by flame bursting sand filled balloons, under an initial cohesion pressure. The theoretical part consists of modelling the pressure and density waves inside the granular material, assuming an elastic model. I think the experiment by itself is interesting enough, but I have a few questions I would like the authors to answer before accepting the work.

c)      The authors do not comment on the effect that the induced flows, by the flame gas, has on the expanding grains. Is it completely negligible?

d)     I miss the information about the grain’s coefficient of restitution (CR), in section 2. It is important to make sure that viscoelastic friction does not induce important wave energy loss terms, since that is related to the dynamic CR.

e)      The temperature information is missing in figure 1.

f)      An elastic model for grains only works when the network of contacts, and the force arches thus formed, are intact. Many experiments have shown that the duration of such arches is very small. It is necessary to estimate the typical arch-network survival time in the model, otherwise the elastic equations should not apply. Certainly, the out-going external mass of grains do not fit the elastic model. My question is, for how long does the internal cluster can still be considered a connected network of grains? Is that time compatible with the experimental times (dozens of milliseconds)?

I will gladly consider accepting the manuscript after the authors answer my queries above.

Comments for author File: Comments.pdf

Comments on the Quality of English Language

The manuscript is well written, but a few typos can be detected such as

a)      Pg 2, 3^rd paragraph: instead of “of the body but no change its shape”, it should be “of the body but do not change its shape”;

b)     Pg 5, first line of 4.1 should be “causes”;

There could be more. The authors should recheck the writing.

Author Response

Response to Reviewer 2

Fluids-2787222

We wish to thank the referee for his/her thoughtful reading of the paper and the comments and criticism in his/her report.

Here we reply point-by-point each one of the questions:

The manuscript is well written, but a few typos can be detected such as

1. a) Pg 2, 3rd paragraph: instead of “of the body but no change its shape”, it should be “of the body but do not change its shape”;
2. b) Pg 5, first line of 4.1 should be “causes”;

Answer.- Done. The amendments were made in the manuscript.

 

There could be more. The authors should recheck the writing.

 

More importantly, the work is composed of an experimental and a theoretical part. The experiment consists by flame bursting sand filled balloons, under an initial cohesion pressure. The theoretical part consists of modelling the pressure and density waves inside the granular material, assuming an elastic model. I think the experiment by itself is interesting enough, but I have a few questions I would like the authors to answer before accepting the work.

1. c) The authors do not comment on the effect that the induced flows, by the flame gas, has on the expanding grains. Is it completely negligible?

Answer.- In Section 3 we added more text in the first paragraph as follows:

Once the sand is in the balloon, it was laid to hang, see Fig. 2, and then, it was burster by exposing it to an intense flame produced by a plumber blow torch (butane/propane gas) whose hottest point is approximately 1372.2 K, the actual temperature a heated component can attain is much lower than this and depends on the burner, and the thermal properties of the component and its surroundings. After any burst, we immediately collected manually the sand and no sensitive temperature change was noticed. In the same line we highlight that if the air warm occurs, its density decreases, and so the ratio ρ_g/ρ_air must be larger than ρ_g/ρ_air = 2240 which was computed at T_room = 298.2 K. A large value of this quotient indicates that the dynamics of the grains dominates on that of the air [22].

 

Reference [22] was added: [22] Bagnold, R.A. The physics of blown sand and desert dunes, Chapman & Hall, London, 1941.

 

2. d) I miss the information about the grain’s coefficient of restitution (CR), in section 2. It is important to make sure that viscoelastic friction does not induce important wave energy loss terms, since that is related to the dynamic CR.

Answer.- In the first paragraph of Section 2 we included the value of the coefficient of restitution of rounded quartz grains as:

coefficient of restitution COR≈1 for velocities lower than 1.6 m/s [17], …

Reference [17] was added: [17] Ge, J.; Monroe, C.A. The effect of coefficient of restitution in modeling of sand granular flow for core making: Part I Free-Fall Experiment and theory, Intl. J. Metalcasting 2019, 13, 753- 767.

1. e) The temperature information is missing in figure 1.

Answer.- The figure caption of Fig. 1, now is:

FIG. 1: (a) Plot of the cumulative percentage under- and over-size. The average diameter of the grains is taken to be the median (vertical straight line) which is the value separating the higher half from the lower half of the sizes distribution. Thus, the average diameter of the grains of Ottawa sand is 0.0093 m. (b) Plot of the inflation pressure p, as a function of the average stretch, R, of the near spherical rubber balloon, measurements were performed at room temperature Troom = 298.2 K. Notice the strong non linear behaviour of the pressure as a function of R.

 

1. f) An elastic model for grains only works when the network of contacts, and the force arches thus formed, are intact. Many experiments have shown that the duration of such arches is very small. It is necessary to estimate the typical arch-network survival time in the model, otherwise the elastic equations should not apply. Certainly, the out-going external mass of grains do not fit the elastic model. My question is, for how long does the internal cluster can be still considered a connected network of grains? Is that time compatible with the experimental times (dozens of milliseconds)?

Answer.- In the last three paragraphs previous to Section 4, we discussed a little on the chains forces or network of contacts, in the context of our study. These paragraphs say:

A last issue concerning with the inner structure of the granular mass is related to the possible existence of force chains, which are typical in granular masses subjected to strong compression stresses [25–30]. Chain forces in two-dimensional granular materials have been found experimentally through photoelasticity [25–27], force sensors [28] and by using DEM and Lattice Boltzmann simulations [25,27].

Transparent materials with a non-crystalline molecular structure are optically isotropic when unstressed, i.e., the polarization of the incident light is not altered by the material. However, such materials become optically anisotropic (birefringent) when put under stress. The polarization of the incident light is changed in the stressed material in a way similar to the behavior of birefringent crystals. After unloading, the material becomes optically isotropic again.

Measurements show that spatially extended and strong force chains (much larger than the mean force) occur when the applied stresses are large (but they are exponentially rare [29]) and conversely, short and weak force chains shall be produced for small stresses [25,27]. Moreover, the force-force spatial distribution function and contact point radial distribution function indicate that spatial correlations between the contact forces and positions of the contacts extend out only to approximately three particle diameters. This shows that force correlations dissipate quickly in the bulk and that the force transmission network propagates locally but becomes diffuse rapidly [30]. Taking in to account all these previous facts, that the typical inflation pressures involved in experiments are relatively small (Table 1) and that the decompression wave only penetrates small distances respect to free surface (see Section 5) before the gravity action plays an important role, it is possible that effect of the chains force on the decompression wave could be marginal.

Moreover, based on Eq. (2) we have estimated the time for which the decompression (lost of contact of grains) occurs for each granular mass of size R₀, so we added after Eq. (2), the next paragraph:

An order of magnitude estimation of the time for which the decompression occurs in a fast manner (in absence of gravity) is obtained through Eq. (2). If the inertial force dominates on the hydrostatic compression, when the granular mass has a size r = R₀, then, ρu/t² > K_eu/R₀², it gives that t<(ρR₀² /K_e)^1/2. Therefore, our theoretical analysis will be valid for times that fulfills such a condition.

After, in the second paragraph of Subsection 5.1 we used this inequality to estimate the times of decompression:

Previous to show the referred plots, we must remember the result given in Section 4 which states that an order of magnitude estimation of the time for which the decompression occurs obeys the inequality t<(ρR₀² /K_e)^1/2. For the balloon of radius R₀ = 4.15 × 10⁻² m results that t < 1.50 × 10⁻² s, similarly if the balloon has radius R₀ = 4.45×10⁻² m then t < 1. 60×10⁻² s and for the balloon of radius R₀ = 5.30×10⁻² m we find that t < 2.00 × 10⁻² s. Times longer than the previously computed implies that the granular masses stop to be a connected network of grains.

Author Response File: Author Response.pdf

Reviewer 3 Report

Comments and Suggestions for Authors

The paper reports interesting results and hence may be accepted for publication. Still I have a couple of comments: 

1)In Fig.1 a there is no explanation for the black and red curves. I guess that the black curve refers to the differential size distribution and red curve -- to the integral distribution. Please provide the respective explanations. 

2) Do you have any qualitative explanation for the shape of the cure in Fig. 1b?  Please provide. 

3)  An additional figure, illustrating the preparation of the experimental setup (schematic) in the beginning of the section "Materials" would be very helpful 

4)In the caption to Fig. 3 there are two times "of the", please correct. 

5)Please discuss (provide the qualitative explanation) the flat (zero) parts on the Figs. 3, 4, 5, 

Comments on the Quality of English Language

The English is adequate 

Author Response

Response to Reviewer 3

Fluids-2787222

We wish to thank the referee for his/her thoughtful reading of the paper and the comments and criticism in his/her report.

Here we reply point-by-point each one of the questions:

Comments and Suggestions for Authors

The paper reports interesting results and hence may be accepted for publication. Still, I have a couple of comments:

 

1) In Fig.1 a there is no explanation for the black and red curves. I guess that the black curve refers to the differential size distribution and red curve -- to the integral distribution. Please provide the respective explanations.

Answer. – Done.

In the first paragraph of Section 2 Materials, when we give the average diameter we added a note, Ref. [16], which presents more detail on the cumulative graph:

[16] Once the size distribution has been measured, it is relatively simple to develop a suitable graph. An alternative, and often more useful approach is to present the data as a cumulative graph in which particle size is plotted along the horizontal axis and the ordinate represents cumulative percentage undersize or oversize (indicated in Figure 1a). The principal advantage of this latter type of graph is that values not determined experimentally are reliably predicted.

 

Additionally, in Fig. 1a we highlight that the red and black curves refers to the cumulative distribution of particles undersize (red curve) and oversize (black curve).

 

2) Do you have any qualitative explanation for the shape of the curve in Fig. 1b?  Please provide.

Answer.- To clarify the curve behavior in Fig. 1b, we added a new third paragraph an Section 2 Materials:

The physics of inflation of rubber balloons, evinces interesting facts. The pressure-deformation curve in spherical balloons quickly reaches a maximum in pressure (because in this stage the pressure depends on the inverse of the balloon radius which initially is very small), this part of the curve is the first increasing branch. Upon further inflation, the pressure decreases because the balloon radius increases (such a region is the first decreasing branch), after pressure increases rapidly again until the bursting point. The of the molecular chain structure [21].   

3)  An additional figure, illustrating the preparation of the experimental setup (schematic) in the beginning of the section "Materials" would be very helpful

Answer.- Done. We added the new figure 2, of the experimental setup, in Section 3, Experiments, because in this place is more specific.

4) In the caption to Fig. 3 there are two times "of the", please correct.

Answer. -Done

5) Please discuss (provide the qualitative explanation) the flat (zero) parts on the Figs. 3, 4, 5,

Answer. -  In the paragraph after Table 1, we added the next text:

In the plot of Fig. 4, the flat part, for each radius R₀ refers that the average displacement (given by the sloped straight lines) starts after a given time, i.e., there is a delay time for each one balloon. We observe that the delay time is longest for a larger mass, later on (Section 5) we shall discuss that this behavior can be directly associated to the decompression waves that go inward the granular masses, which also present temporal delays.

For the new Figs. 6 and 7 we added as a second and third paragraph of Section 5:

It is possible to explain graphically the behavior of the elastic waves given by Eq. (7). In Fig. 6 the behavior of the decompression waves u(r, t), for r =constant, is given: in Fig. 6(a) for r = 0.03 m and in Fig. 6(b) for r = 0.04 m, this latest value is closer to any of the initial radius of the balloons compressing the sand, which are indicated in the inset of each plot. The flat parts in plots of Fig. 6 indicate that the decompression wave arrives to the radial positions r = 0.03 m or r = 0.04 m at different times, larger than t = (R0 − r)/c, this relationship is obtained from Eq. (7) when the argument β = ct + (r − R0) is equal to zero in order to obtain u = 0. For instance, if we consider the balloon of radius R0 = 5.30 × 10⁻² m, we found that the wave arrives at r = 0.03 m after t = 8. 77 × 10⁻³ s, as is observed in the green dashed line in Fig. 6(a), while if r = 0.04 m the wave reaches such a radial position after t = 4. 96 × 10⁻³ s, as is seen in the green dashed line in Fig. 6(b); for smaller balloons these times are shortest.

Similarly, the plots of u(r, t), for t =constant, are given in Fig. 7. The flat parts occur again if β = 0 and in this case, we used the formula r = R₀ − ct which gives the limit for which u = 0. Consequently, for the time t = 0.019 s and R₀ = 5.30 × 10⁻² m the previous formula produces that r = 3 × 10−3 m, it means that between r = 0 m and r = 3 × 10⁻³ m the wave is flat and u = 0 m, as can be seen Fig. 7(a). In Fig. 7(b), when t = 0.020 s, u = 0 m, between r = 0 m and r = 0.6×10⁻³ m. Finally, we observe that the other waves have not plane parts.

Author Response File: Author Response.pdf

Round 2

Reviewer 2 Report

Comments and Suggestions for Authors

I believe the manuscript has improved and might be accepted.

However, a small correction should be made: page 4, first paragraph: change “if the air warm occurs” to “if the air warming occurs”

Author Response

Response to Reviewer 2

Fluids-2787222

We wish to thank the referee for his/her thoughtful reading of the paper and the comments and criticism in his/her report.

 

Comments and suggestions for the authors:

I believe the manuscript has improved and might be accepted.

However, a small correction should be made: page 4, first paragraph: change “if the air warm occurs” to “if the air warming occurs”.

 

Answer.- Done. We amended such a part of the first paragraph on page 4, as can be seen in the manuscript with bold typeface.

 

Author Response File: Author Response.pdf