Distorted gas bubbles at large Reynolds number

The distortion of a gas bubble rising steadily in an inviscid incompressible liquid of infinite extent under the action of surface tension forces is investigated theoretically using an appropriate extension of the tensor virial theorem. A convenient parameter for distinguishing the bubble shape is the Weber numberW. The virial method leads to an expression relatingWand the axis ratio χ, of the transverse and longitudinal axes of the bubble. To first order inW, this relation agrees with the linear theory established by Moore (1959). Also, comparison of the results with his (1965) approximate theory reveals similar features and excellent agreement up to χ = 2. In particular, it confirms his prediction of the existence of a maximum Weber number. Although the present work does not consider the stability of these bubbles, it is interesting to note that the maximum value of 3.271 attained byWdiffers only by about 2.8% from the critical Weber number obtained by Hartunian & Sears (1957) for the onset of instability.An approximate method for the study of slightly distorted spheroidal gas bubbles is also formulated and the resulting boundary-value problem solved numerically. The theory is then extended to include gravity. The joint effect of surface tension as well as gravitational forces has not been included in earlier theories. The shapes of the bubbles are traced and compared with the unperturbed spheroids. Comparisons for the velocity of bubble rise are made between the present predictions and some experimental results. In particular the results are compared with recent experimental data for the motion of gas bubbles in liquid metals.

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On the instability of small gas bubbles moving uniformly in various liquids

Journal of Fluid Mechanics ◽

10.1017/s0022112057000464 ◽

1957 ◽

Vol 3 (1) ◽

pp. 27-47 ◽

Cited By ~ 105

Author(s):

R. A. Hartunian ◽

W. R. Sears

Keyword(s):

Surface Tension ◽

Weber Number ◽

Gas Bubbles ◽

Hydrodynamic Pressure ◽

Terminal Velocity ◽

Approximate Methods ◽

Stability Calculation ◽

Fluid Medium ◽

Critical Weber Number ◽

The Stability

The instability of small gas bubbles moving uniformly in various liquids is investigated experimentally and theoretically.The experiments consist of the measurement of the size and terminal velocity of bubbles at the threshold of instability in various liquids, together with the physical properties of the liquids. The results of the experiments indicate the existence of a universal stability curve. The nature of this curve strongly suggests that there are two separate criteria for predicting the onset of instability, namely, a critical Reynolds number (202) and a critical Weber number (1.26). The former criterion appears to be valid for bubbles moving uniformly in liquids containing impurities and in the somewhat more viscous liquids, whereas the latter criterion is for bubbles moving in pure, relatively inviscid liquids.The theoretical analysis is directed towards an investigation of the possibility of the interaction of surface tension and hydrodynamic pressure leading to unstable motions of the bubble, i.e. the existence of a critical Weber number. Accordingly, the theoretical model assumes the form of a general perturbation in the shape of a deformable sphere moving with uniform velocity in an inviscid, incompressible fluid medium of infinite extent. The calculations lead to divergent solutions above a certain Weber number, indicating, at least qualitatively, that the interaction of surface tension and hydrodynamic pressure can result in instabilities of the bubble motion.A subsequent investigation of the time-independent equations, however, shows the presence of large deformations in shape of the bubble prior to the onset of unstable motion, which is not compatible with the approximation of perturbing an essentially spherical bubble. This deformation and its possible effects on the stability calculation are therefore determined by approximate methods. From this it is concluded that the deformation of the bubble serves to introduce quantitative, but not qualitative, changes in the stability calculation.

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Analysis of Surface Tension and Viscosity of Liquid Metals

Metallurgical and Materials Transactions B ◽

10.1007/s11663-021-02157-y ◽

2021 ◽

Author(s):

Yuri N. Starodubtsev ◽

Vladimir S. Tsepelev

Keyword(s):

Surface Tension ◽

Liquid Metals

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Gas diffusion from ascending gas bubbles

Journal of Fluid Mechanics ◽

10.1017/s002211206900139x ◽

1969 ◽

Vol 35 (4) ◽

pp. 711-719 ◽

Cited By ~ 11

Author(s):

Paul H. Leblond

Keyword(s):

Surface Tension ◽

Gas Exchange ◽

Gas Bubbles ◽

Gas Diffusion ◽

Gas Pressure ◽

Pressure Balance ◽

Spherical Bubble ◽

Gas Leakage ◽

Quantitative Results ◽

Ascent Velocity

General qualitative rules are derived for the behaviour of the volume of an ascending spherical bubble and of the gas pressure within it. Three modes of behaviour are discerned, corresponding to as many possible orderings of the relative influences of ascent velocity, gas leakage and surface tension on the volume and the pressure balance. These general results are nearly independent of the particular forms of the ascent velocity and gas exchange functions. Quantitative results are presented for the Stokes law régime.

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Cavity Detachment from a Wedge with Rounded Edges and the Surface Tension Effect

Journal of Marine Science and Engineering ◽

10.3390/jmse9111253 ◽

2021 ◽

Vol 9 (11) ◽

pp. 1253

Author(s):

Yuriy N. Savchenko ◽

Georgiy Y. Savchenko ◽

Yuriy A. Semenov

Keyword(s):

Surface Tension ◽

Weber Number ◽

Solution Process ◽

Cavity Flow ◽

Surface Tension Effect ◽

Force Coefficient ◽

Hodograph Method ◽

Wide Range ◽

Critical Weber Number ◽

Cavity Boundary

Cavity flow around a wedge with rounded edges was studied, taking into account the surface tension effect and the Brillouin–Villat criterion of cavity detachment. The liquid compressibility and viscosity were ignored. An analytical solution was obtained in parametric form by applying the integral hodograph method. This method gives the possibility of deriving analytical expressions for complex velocity and for potential, both defined in a parameter plane. An expression for the curvature of the cavity boundary was obtained analytically. By using the dynamic boundary condition on the cavity boundary, an integral equation in the velocity modulus was derived. The particular case of zero surface tension is a special case of the solution. The surface tension effect was computed over a wide range of the Weber number for various degrees of cavitation development. Numerical results are presented for the flow configuration, the drag force coefficient, and the position of cavity detachment. It was found that for each radius of the edges, there exists a critical Weber number, below which the iterative solution process fails to converge, so a steady flow solution cannot be computed. This critical Weber number increases as the radius of the edge decreases. As the edge radius tends to zero, the critical Weber number tends to infinity, or a steady cavity flow cannot be computed at any finite Weber number in the case of sharp wedge edges. This shows some limitations of the model based on the Brillouin–Villat criterion of cavity detachment.

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