On the shape of a two-dimensional bubble in uniform motion

Shankar, PN (1992) On the shape of a two-dimensional bubble in uniform motion. Journal of Fluid Mechanics, 244. pp. 187-200.

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Consider a two-dimensional bubble moving with speed U through an unbounded, inviscid fluid. Let all lengths be normalized by T/ rhoU quot;SUP 2quot; where T is the surface tension. Then the shape of the bubble depends on a single parameter GAMMA=2 DELTAp/ rhoU quot;SUP 2quot; -1, where deltap=p quot;SUB bquot; -p quot;SUB infinityquot; is the difference between the bubble pressure and the ambient pressure. We obtain solutions for the bubble shape over the whole range of GAMMA-values that are physically relevant. The formulation involves a mapping from an auxiliary circle plane zeta where the flow field is known. The problem then reduces to solving an infinite set of nonlinear algebraic equations for the coefficients in the mapping function. To a first approximation, when GAMMA right arrow infinity, the bubble takes an elliptical shape of aspect ratio (1+ 2/3 GAMMA quot;SUP -1quot; )/(1-2/3 GAMMA quot;SUP -1quot; ) flattened in the flow direction. The solution correct to order GAMMA quot;SUP -5quot; is then obtained which is fairly accurate for GAMMA as low as 2. When GAMMA=0 the exact, nonlinear solution for the bubble shape is given by x= 1/3(1/3cos phi-1/27cos3 phi), y= 1/3(5/3sin phi+1/27sin3 phi). We can then obtain a perturbation solution for GAMMA right arrow0 correct to order GAMMA quot;SUP 6quot; . This solution, useful in the range 0.75 GAMMA-0.4537, even gives reasonable descriptions of non-convex bubble shapes for GAMMA0 down to the pinch-off limit GAMMA * when the bubble ceases to be simply connected. It is remarkable that a simple analytical representation correct to order GAMMA quot;SUP 2quot; analytically yields a value for GAMMA * of -0.4548, i.e. within 0.3% of the correct value; naturally, the higher-order approximations are even more accurate. While the present results eliminate the need for direct numerical computations over most of the range of GAMMA, such results, too, are presented. Finally, the dependence of the bubble geometrical parameters, Weber number and added mass on GAMMA is determined. (A)

Item Type: Article
Uncontrolled Keywords: Bubbles; Modelling-Mathematical; Two-Phase Flow(Gas/Liquid;
Subjects: AERONAUTICS > Aerodynamics
Depositing User: Users 3 not found.
Date Deposited: 10 Mar 2005
Last Modified: 24 May 2010 04:08
URI: http://nal-ir.nal.res.in/id/eprint/377

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