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The flow fields in and around a droplet moving axially within a tube

Published online by Cambridge University Press:  29 March 2006

G. Hetsroni ,
S. Haber  and
E. Wacholder
G. Hetsroni
Affiliation:
Department of Nuclear Science
S. Haber
Affiliation:
Department of Aeronautical Engineering
E. Wacholder
Affiliation:
Department of Nuclear Science Technion, Israel Institute of Technology, Haifa
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Abstract

A solution is presented for the flow field in and around a single spherical droplet or bubble moving axially at an arbitrary radial location, within a long circular tube. In the tube there is viscous fluid flowing with a constant Poiseuillian velocity distribution far from the droplet.

The settling velocity of the droplet or bubble is \begin{eqnarray*} U = \frac{2(\rho_i-\rho_e)ga^2}{9\mu_e}\frac{1+\alpha}{\frac{2}{3}+\alpha}\left[1-\frac{2+3\alpha}{3(1+\alpha)}\left(\frac{a}{R_0}\right)f\left(\frac{b}{R_0}\right)\right]+U_0\left[1-\left(\frac{b}{R_0}\right)^2\right.\\ \left. - \frac{2\alpha}{2+3\alpha}\left(\frac{a}{R_0}\right)^2\right] + O\left(\frac{a}{R_0}\right)^3. \end{eqnarray*} This is a general equation and it contains as special cases the familiar solutions of Stokes, Hadamard-Rybczynski, Brenner & Happel, Greenstein & Happel and Haberman & Sayre.

The function describing the deviation of the interface from sphericity is solved and an iterative procedure is suggested for obtaining higher order solutions.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 41 , Issue 4 , 15 May 1970 , pp. 689 - 705
Copyright
© 1970 Cambridge University Press

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References

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