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Bubble Systems pp 99–132Cite as

Thermally Controlled Bubble Growth

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Part of the book series: Mathematical Engineering ((MATHENGIN))

Abstract

The mathematical formulation of the heat input governed vapor bubble growth in a bulk of uniformly heated liquid is presented. Using the theory of dimensions, the structure of the solution was analyzed qualitatively. A historical survey of theoretical works devoted to the considered problem is presented. The set of degenerate solutions of the problem is obtained and studied systematically. The complete analytical solution of the problem is obtained for the first time. Formulas for the calculation of the bubble growth rate in the whole domain of possible variations in regime parameters are presented. The conclusion is made that the influence of permeability of the interface has a significant effect on the bubble growth rate. It is shown that the Plesset–Zwick formula, which is commonly accepted in computational practice, is not applicable at both small and large Jakob numbers and its good agreement with the experiment is determined to a large extent by a combination of the imperfectness of the theoretical analysis and the experimental error. The conclusion is made that, for many liquids, the ultimately achievable value of the dimensionless superheating parameter (Stefan number) can exceed unity. In this case, the regularities in the bubble growth acquire some features unexplored to date.

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Notes

  1. 1.

    The left branch of the function for the case of condensation \(({\text{S}} < 0)\) is shown on Fig. 1.9.

  2. 2.

    The mathematical part of this investigation was presented in the earlier paper Plesset and Zwick (1952), as well as in the report Zwick (1954).

  3. 3.

    Their arguments were based on the analysis presented in the classical book by Sedov (1993), the first Russian edition of this book dates back to 1944.

  4. 4.

    The difference between the ‘planar’ and ‘spherical’ problems will be manifested only after expanding the parameter С.

  5. 5.

    Calculations by ‘the mean enthalpy’ heat capacity of a liquid, \(\left\langle {c_{pl} } \right\rangle = {{\left( {h_{ * } - h_{s} } \right)} \mathord{\left/ {\vphantom {{\left( {h_{ * } - h_{s} } \right)} {\left( {T_{ * } - T_{s} } \right)}}} \right. \kern-0pt} {\left( {T_{ * } - T_{s} } \right)}}\), gives the value of the Stefan number smaller by approximately 10 %.

  6. 6.

    In his original paper, Labuntsov (1974) proposed the fourth asymptotical bubble growth rate assuming the governing role of molecular-kinetic effects relating to evaporation of liquid from superheated interfacial boundary. This approach is valid in the case of highly intensive evaporation processes, when the velocity of motion of resultant vapour is commensurable with the sound velocity.

  7. 7.

    We recall that formula (4.105) is, strictly speaking, an asymptotical law of bubble dynamical growth, which does not apply at early growth stages (see Sect. 2.4).

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    Alexander A. Avdeev

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Avdeev, A.A. (2016). Thermally Controlled Bubble Growth. In: Bubble Systems. Mathematical Engineering. Springer, Cham. https://doi.org/10.1007/978-3-319-29288-5_4

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  • DOI: https://doi.org/10.1007/978-3-319-29288-5_4

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