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The Penetration of a Finger into a Viscous Fluid in a Channel and Tube

Reinelt, D. A. and Saffman, P. G. (1985) The Penetration of a Finger into a Viscous Fluid in a Channel and Tube. SIAM Journal on Scientific and Statistical Computing, 6 (3). pp. 542-561. ISSN 0196-5204.

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The steady-state shape of a finger penetrating into a region filled with a viscous fluid is examined. The two-dimensional and axisymmetric problems are solved using Stokes equations for low Reynolds number flow. To solve the equations, an assumption for the shape of the finger is made and the normal-stress boundary condition is dropped. The remaining equations are solved numerically by covering the domain with a composite mesh composed of a curvilinear grid which follows the curved interface, and a rectilinear grid parallel to the straight boundaries. The shape of the finger is then altered to satisfy the normal-stress boundary condition by using a nonlinear least squares iteration method. The results are compared with the singular perturbation solution of Bretherton (J. Fluid Mech., 10 (1961), pp. 166–188). When the axisymmetric finger moves through a tube, a fraction $m$ of the viscous fluid is left behind on the walls of the tube. The fraction $m$ was measured experimentally by Taylor (J. Fluid Mech., 10 (1961), pp. 161–165) as a function of the dimensionless parameter µU/T. The numerical results are compared with the experimental results of Taylor.

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Additional Information:©1985 Society for Industrial and Applied Mathematics. Received by the editors September 13, 1983, and in revised form January 13, 1984. This work was supported by the U.S. Department of Energy, Office of Basic Energy Sciences, and the Office of Naval Research. We wish to thank Prof. H.O. Kreiss for suggesting the composite mesh method and B. Kreiss for help with the initial implementation.
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Department of EnergyUNSPECIFIED
Office of Naval ResearchUNSPECIFIED
Subject Keywords:viscous fluid, fingering, singular perturbation, composite mesh
Issue or Number:3
Record Number:CaltechAUTHORS:REIsiamjssc85
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Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:12019
Deposited By: Archive Administrator
Deposited On:20 Oct 2008 16:57
Last Modified:03 Oct 2019 00:24

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