Suppose that an elongated bubble is rising at its terminal velocity U in an open, liquid-filled tube, as in Fig. P7.14. The bubble is shaped as a cylinder of radius a and length L with hemispherical caps, such that L ≫ a. The tube radius is b. Between the bubble and tube wall is a liquid film of thickness h = b – a. Assume that h⁄L is small enough that the flow is fully developed in almost the entire film, and that the liquid pressure elsewhere is nearly static. For simplicity, assume also that h/a ≪ 1, in which case the boundaries of the film can be approximated as planar. The objective is to predict U for a given set of linear dimensions.

(a) It is helpful to use Cartesian coordinates fixed on the tube wall, as shown. Solve for v_x(y) in the liquid film in terms of

Explain why the liquid velocity at y = h will not equal the mean gas velocity U. (Sketch the circulating flow pattern within the bubble.)

(b) By considering the rate at which the bubble displaces liquid, evaluate the mean velocity in the film and find 

(c) Use an overall force balance on the bubble to relate U to the fluid properties and the dimensions.

Transcribed Image Text:

L b a Liquid Gas h X Figure P7.14 Long bubble rising in a liquid-filled tube. The relative lengths have been distorted for clarity in labeling; actually, h<< a <