In the classical works of Prandtl and Taylor devoted to the analysis of the problem of the rise of a Taylor bubble in a round tube, a solution of the Laplace equation is used, which contains divergent infinite series. The present paper outlines a method for the correct analysis of the mentioned problem. Using the method of superposition of “elementary flows,” a solution was obtained for flow of an ideal fluid over a body of revolution in a pipe. Satisfying the free surface condition in the vicinity of the stagnation point and using the limiting transition with respect to the main parameter lead to the relation for the rise velocity of a Taylor bubble expressed in terms of the Froude number. In order to validate the method of superposition, it was applied to the problem of the rise of a plane Taylor bubble in a flat gap, which also has an exact analytical solution obtained with the help of the complex variable theory.
Received 12 October 2012Accepted 22 April 2013Published online 13 May 2013
The author is grateful to Professor Bernhard Weigand (Head of Institute of Aerospace Thermodynamics, University Stuttgart), Dr. Igor V. Shevchuk (MBtech Group GmbH & Co. KGaA), and Dipl.-Ing. Hassan Gomaa (Research Associate at the Institute of Aerospace Thermodynamics, University Stuttgart) for their very useful comments and numerous discussions.
Article outline: I. INTRODUCTION II. SUPERPOSITION OF “ELEMENTARY FLOWS” III. FLOW IN THE VICINITY OF THE STAGNATION POINT IV. ASYMPTOTICAL CASES OF THE SOLUTIONS V. PLANE TAYLOR BUBBLE VI. CORRECT APPROXIMATE SOLUTION VII. DISCUSSION OF THE RESULTS A. Axisymmetric problem B. Plane problem VIII. CONCLUSIONS
7.G. B. Wallis, One-Dimensional Two-Phase Flow (McGraw-Hill, New York, 1969).
8.G. K. Batchelor, An Introduction to Fluid Dynamics (Cambridge University Press, Cambridge, 2000).
9.A. D. Polyanin, Handbook of Linear Partial Differential Equations for Engineers and Scientists (Chapman and Hall, Boca Raton, 2002).
10.M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions: With Formulas, Graphs, and Mathematical Tables (National Bureau of Standards, Washington, 1964).
11.B. Weigand, Analytical Methods for Heat Transfer and Fluid Flow Problems (Springer, Berlin, 2004).
12.I. V. Shevchuk, Convective Heat and Mass Transfer in Rotating Disk Systems (Springer, Berlin, 2009).
13.I. V. Shevchuk, “Unsteady conjugate laminar heat transfer of a rotating non-uniformly heated disk: Application to the transient experimental technique,” Int. J. Heat Mass Transfer49(19–20), 3530–3537 (2006).