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Similarity solutions for stratified rotating-disk flow

Published online by Cambridge University Press:  21 April 2006

J. D. Goddard ,
J. B. Melville  and
K. Zhang
J. D. Goddard
Affiliation:
Department of Chemical Engineering, University of Southern California, Los Angeles, CA 90089–1211, USA
J. B. Melville
Affiliation:
Department of Chemical Engineering, University of Southern California, Los Angeles, CA 90089–1211, USA
K. Zhang
Affiliation:
Department of Chemical Engineering, University of Southern California, Los Angeles, CA 90089–1211, USA
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Abstract

This work treats the vertically stratified system of two homogeneous fluid layers confined between horizontal infinite rotating disks which rotate steadily about a common vertical axis. Allowance is made for uniform injection of fluid at either disk. With appropriate restrictions on disk rotational speeds and injection rates a flat interface is possible, and the problem admits similarity solutions to the Navier-Stokes equations of the Kármán-Bödewadt-Batchelor variety. This type of flow allows for a uniformly accessible surface of interphase mass and heat transfer at the two-fluid interface, and, with that as the primary motivation, the present work provides exploratory numerical solutions of the above equations for both corotation and counter-rotation combined with injection.

A linearized theory is given for the case of nearly rigid rotation, with explicit analytical results for the large-Reynolds-number boundary-layer limit. Also, we offer a theoretical discussion of the inviscid limit for arbitrary rotation and injection rates. Based on the type of Euler-cell solutions identified in previous work, we derive the remarkably simple formula \[ \rho_1\omega_1^2\cot^2\frac{\omega_1d_1}{V_1} = \rho_2\omega^2_2\cot^2\frac{\omega_2d_2}{V_2} \] connecting densities ρ, depths d, rotation speeds ω and injection velocities V.

Sample calculations and comparisons are given for property ratios typical of water-kerosene layers. In this case, the linearized theory works exceedingly well for corotation with small injectional Rossby numbers V/ωd. The simple inviscid theory cited above shows excellent agreement with the numerical computations for Reynolds numbers greater than 500 and for Rossby numbers > 1/π, corresponding to strong blowing in the inviscid regime. The larger-wavelength inviscid cell structure appears to provide the kind of stagnation-flow pattern essential to the application envisaged.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 182 , September 1987 , pp. 427 - 446
Copyright
© 1987 Cambridge University Press

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