Summary
A general solution of the creeping flow equations suitable for a flow that is bounded by a nondeforming planar interface is presented. New compact representations for the velocity and pressure fields are given in terms of two scalar functions which describe arbitrary Stokes flow. A general reflection theorem is derived for a fluid-fluid interface problem containing Lorentz reflection formula as a particular case. The theorem allows a better interpretation of the image system for various singularities in the presence of a planar interface. The general solution is further used to describe the first-order approximation of the deformed interface by performing normal stress balance. It is found that the normal stress imbalance and the interface displacement are independent of the viscosity ratio of two fluids (!) and only depend on the location of initial singularity.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Lorentz, H. A.: A general theorem concerning the motion of a viscous fluid and few consequences derived from it. Versl. Konigl. Akad. Wetensch. Amst.5, 168–175 (1896). See also J. Engng. Math.30, 19–24 (1996).
Oseen, C. W.: Hydrodynamik, pp. 97–107. Leipzig: Akad. Verlagsgesellschaft 1927.
Chwang, A. T., Wu, T. Y.: Hydromechanics of low Reynolds-number flow. J. Fluid Mech.67, 787–815 (1975).
Jones, R. B., Felderhof, B. U., Deutch, J. M.: Diffusion of polymers along a fluid-fluid interface. Macromolecules8, 680–684 (1975).
Blake, J. R.: A note on the image system for a Stokeslet in a no-slip boundary. Proc. Camb. Phil. Soc.70, 303–310 (1971).
Blake, J. R., Chwang, A. T.: Fundamental singularities of viscous flow. Part I: The image systems in the vicinity of a stationary no-slip boundary. J. Engng. Math.8, 23–29 (1974).
Aderogba, K., Blake, J. R.: Action of force near the planar surface between two semi-infinite immiscible liquids at very low Reynolds numbers. Bull. Austral. Math. Soc.18, 345–356 (1978).
Aderogba, K.: On Stokeslets in a two-fluid space. J. Engng. Math.10, 143–151 (1976).
Hasimoto, H., Sano, O.: Stokeslets and eddies in creeping flow. Ann. Rev. Fluid Mech.12, 335–363 (1980).
Palaniappan, D., Nigam, S. D., Amaranath, T., Usha, R.: Lamb's solution of Stokes's equations: a sphere theorem. Q. J. Mech. Appl. Math.45, 47–56 (1992).
Happel, J., Brenner, H.: Low Reynolds number hydrodynamics, p. 87. The Hague: Matinus Nijhoff 1983.
Lee, S. H., Chadwick, R. S., Leal, L. G.: Motion of a sphere in the presence of a plane interface. Part 1. An approximate solution by the generalization of the method of Lorentz. J. Fluid Mech.93, 705–726 (1979).
Pozrikidis, C.: Boundary integral and singularity methods for linearised viscous flow, p. 212. Cambridge: Cambridge University Press 1992.
Feng, J., Ganatos, P., Weinbaum, S.: The general motion of a disk in a Brinkman Medium. Phys. Fluids10, 2137–2146 (1998).
Landau, L. D., Lifshitz, E. M.: Fluid mechanics. Oxford: Pergamon Press 1959.
Lamb, H.: Hydrodynamics, 6th ed., p. 594. New York: Dover 1945.
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Palaniappan, D. General slow viscous flows in a two-fluid system. Acta Mechanica 139, 1–13 (2000). https://doi.org/10.1007/BF01170178
Received:
Revised:
Issue Date:
DOI: https://doi.org/10.1007/BF01170178