# The Problem of Filling a Spherical Cavity in an Aqueous Solution of Polymers

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## Abstract

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## 1. Introduction

_{2}CHCONH

_{2}-) to water affects the moment of occurrence of cavitation, i.e., a decrease in the number of cavitation is observed [14]. It turned out that the effect of polymers on single bubble dynamics is very small [15,16,17]. In the experimental work [18], the behavior of a bubble near a solid wall is studied and it is concluded that polymer additives do not significantly affect the bubble dynamics. A study on the dynamics of a spherical gas bubble in an incompressible power-law non-Newtonian fluid [19] showed that for some certain indices, there is no energy concentration at all. An experimental study [20] has shown that polymeric additives in water affect the cavitation phenomena and reduce the critical cavitation number. For instance, polymeric additives suppress the erosion of materials during cavitation developing in a flow [21]. The formation and collapse of a vapor bubble in the aqueous solution of PAM is studied experimentally in [22]. The experiments did not reveal a noticeable deceleration of bubble collapse. Our goal is to consider this problem for the case of a relaxing fluid.

## 2. Mathematical Models in the Dynamics of Polymer Solution

^{2}. The relaxation time of an aqueous solution of PAM with a concentration of ${10}^{-2}$ percent is of the order of ${10}^{-4}$ s. In the case of the relaxation viscosity coefficient, the authors of the model did not provide its characteristic values, though one can assume that they are sufficiently small. In [26], we discussed the possibility of the experimental determination of this parameter.

## 3. Problem Formulation and Similarity Criteria

^{3}, $\nu =0.01$ cm

^{2}/s, $\sigma =72.8$ dyn/cm, then $\beta =22.89$. When calculating the parameter $\tau $, considering the order of the relaxation time of an aqueous solution of PAM with a concentration of approximately ${10}^{-2}$%, specified in [23], ${10}^{-4}$ s is taken, which gives $\tau ={10}^{3}$. As for the parameter $\gamma ,$ it is more difficult to indicate its characteristic values since there are no direct experiments from which the value of the quantity $\kappa $ is found. However, even at $\kappa ={10}^{-4}$ cm

^{2}, the values of this parameter will be very large, $\gamma ={10}^{5}$. The last of the four similarity criteria is the Reynolds number. This parameter is at our disposal. However, there are physical limitations from above on the value of $\mathrm{Re}$, which are related to the fact that regarding large bubble diameters, it is difficult to ensure the sphericity of its shape. Assuming ${r}_{0}=0.2$ cm, we find $\mathrm{Re}=6372$, and for ${r}_{0}=0.01$ cm, we get $\mathrm{Re}=318.63$.

## 4. Hereditary Model

^{2}/s, $\rho =1.26$ g/cm

^{3}, which corresponds to glycerol at a temperature of 293 K and $\kappa ={10}^{-4}$ cm

^{2}, we find $\gamma =3.67$. By choosing the relaxation time $\theta ={10}^{-3}$ s, we get $\tau =54.34$. For such values of the chosen parameters and $\beta =0.0048$, $\mathrm{Re}=4$, we ensure the finiteness of the bubble boundary velocity at the moment of focusing. Figure 2 demonstrates the result of numerical solution of problem (22), (23) at $\mathrm{Re}=4,\beta =0.0048,\gamma =3.67,$ and $\tau =54.43$.

## 5. Pavlovskii Model

## 6. Pressure

## 7. Discussion

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**Cavity radius versus time for $\beta =22.89,\tau ={10}^{3},\text{}\gamma ={10}^{5},$ and various values of the Reynolds number $\mathrm{Re}$.

**Figure 2.**Cavity radius versus time for $\mathrm{Re}=4,\beta =0.0048,\gamma =3.67$, and $\tau =54.43$.

**Figure 3.**Cavity radius versus time for $\mathrm{Re}=5,\beta =0.0048,\gamma =5,$ and various values of the parameter $\tau $.

**Figure 4.**Cavity radius versus time for $\mathrm{Re}=318.63,\text{}\tau ={10}^{3},\text{}\gamma ={10}^{5},\text{}$ and various values of the parameter $\beta $.

**Figure 5.**Cavity radius versus time at $\mathrm{Re}=500,\text{}\tau ={10}^{3},\text{}\gamma ={10}^{5},$ and various values of the parameter $\beta $.

**Figure 6.**Cavity radius versus time at $\mathrm{Re}=10,\text{}\tau =10,\text{}\gamma =100,$ and $\beta =0.$.

**Figure 7.**Cavity radius versus time for $\beta =22.89,\text{}\gamma ={10}^{5},$ and various values of the Reynolds number Re.

**Figure 8.**Cavity radius versus time for $\mathrm{Re}=318.63,\text{}\beta =22.89,$ and various values of the parameter $\gamma $.

**Figure 9.**Cavity radius versus time for $\mathrm{Re}=318.63,\text{}\gamma ={10}^{5}$, and various values of the parameter $\beta $.

**Figure 10.**Cavity radius versus time for $\mathrm{Re}=500,\text{}\gamma ={10}^{5},$ and various values of the parameter $\beta $.

**Figure 11.**Pressure distribution in the fluid surrounding the bubble for $\mathrm{Re}=318.63,\beta =22.89,\gamma ={10}^{5},$ and at the time $t=1247$.

**Figure 12.**Pressure distribution in the fluid surrounding the bubble for $\mathrm{Re}=318.63,\beta =22.89,\gamma ={10}^{3}$, and at the time $t=293$.

**Figure 13.**Pressure distribution in the fluid surrounding the bubble for $\mathrm{Re}=318.63,\beta =22.89,\gamma =100,$ and at the time $t=273$.

**Figure 14.**Pressure distribution in the fluid surrounding the bubble for $\mathrm{Re}=318.63,\beta =22.89,\gamma =0$, and at the time $t=271$.

**Figure 15.**Normal stress in the fluid surrounding the bubble for $\mathrm{Re}=318.63,\beta =22.89$, and various $t$ and $\gamma $.

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**MDPI and ACS Style**

Frolovskaya, O.A.; Pukhnachev, V.V.
The Problem of Filling a Spherical Cavity in an Aqueous Solution of Polymers. *Polymers* **2022**, *14*, 4259.
https://doi.org/10.3390/polym14204259

**AMA Style**

Frolovskaya OA, Pukhnachev VV.
The Problem of Filling a Spherical Cavity in an Aqueous Solution of Polymers. *Polymers*. 2022; 14(20):4259.
https://doi.org/10.3390/polym14204259

**Chicago/Turabian Style**

Frolovskaya, Oxana A., and Vladislav V. Pukhnachev.
2022. "The Problem of Filling a Spherical Cavity in an Aqueous Solution of Polymers" *Polymers* 14, no. 20: 4259.
https://doi.org/10.3390/polym14204259