Rise of an Ellipsoidal Bubble at High Reynolds Numbers

Rise of an Ellipsoidal Bubble at High Reynolds Numbers 

Let us consider the motion of a gas bubble at high Reynolds numbers. For small We, the bubble shape is nearly spherical. The Weber numbers of the order of 1 constitute an intermediate range of We, very important in practice, when the bubble, though essentially deformed, conserves its symmetry with respect to the midsection. For such We, the bubble shape is well approximated by an ellipsoid with semiaxes a and b = xa oblate in the flow direction the semiaxis b is directed across the flow, and x 1  

The requirement that the boundary condition for the normal stress be satisfied at the front and rear critical points, as well as along the boundary of the midsection of the bubble, leads to the following relationship between the Weber number We and the ratio x of the major semiaxis to the minor semiaxis of the ellipsoid [291]  

Numerical estimates in [291] show that the maximal deviation of the true curvature from the corresponding value for the approximating ellipsoid does not exceed 5% for We 1 (x 1.5) and 10% for We 1.4 (x 2). 

For usual liquids like water, we have Mo 10 10, and one must take account of the bubble deformation starting from Re 102. (For oil, Mo 10-2, and the bubble deformation is essential even for low Reynolds numbers.) 

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