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Added mass of a pair of disks at small separation

Published online by Cambridge University Press:  01 December 2016

C. ATKINSON  and
J. D. SHERWOOD
C. ATKINSON
Affiliation:
Department of Mathematics, Imperial College, Queen's Gate, South Kensington, London SW7 2NB, UK email: c.atkinson@imperial.ac.uk
J. D. SHERWOOD
Affiliation:
Department of Applied Mathematics and Theoretical Physics, University of Cambridge, Wilberforce Road, Cambridge CB3 0WA, UK email: jds60@cam.ac.uk
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Abstract

Inviscid irrotational flow around a pair of coaxial disks is considered in the limit in which the distance 2h between the disks is small compared to their radius a. The disks have zero thickness and accelerate away from one another along their common axis. The added mass M of each accelerating disk is increased by the presence of the other disk. Analytic predictions are obtained when h/a ≪ 1, with M ~ πa/(8h)-ln(h/a)/2+0.77875+. . . The term O(a/h) can be obtained by means of an inviscid analysis of approximately unidirectional flow within the gap between the disks, but the correction terms have not been reported previously. The irrotational flow problem satisfies Neumann boundary conditions on the surface of the disks, but is otherwise analogous to the Dirichlet problem of the capacitance of a pair of charged disks, which has been the subject of much study and controversy.

Keywords

Laplace EquationAdded MassDisks
Type
Papers
Information
European Journal of Applied Mathematics , Volume 28 , Issue 4 , August 2017 , pp. 687 - 706
Copyright
Copyright © Cambridge University Press 2016 

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References

[1] Maxwell, J. C. (1866) The Bakerian lecture: On the viscosity or internal friction of air and other gases. Phil. Trans R. Soc. 156, 249–268.Google Scholar
[2] Kirchhoff, G. R. (1877) Zur Theorie des Condensators. Monatsberichte der Königlich Preussischen Akademie der Wissenschaften zu Berlin. 144–162.Google Scholar
[3] Cooke, J. C. (1958) The coaxial circular disc problem. Z. Angew. Math. und Mech. 38, 349–356.Google Scholar
[4] Hutson, V. (1963) The circular plate condenser at small separations. Proc. Camb. Philos. Soc. 59, 211–224.Google Scholar
[5] Sneddon, I. N. (1966) Mixed Boundary Value Problems in Potential Theory, North-Holland, Amsterdam.Google Scholar
[6] Shaw, S. J. N. (1970) Circular-disc viscometer and related electrostatic problems. Phys. Fluids 13, 1935–1947.Google Scholar
[7] Atkinson, C. & Leppington, F. G. (1983) The asymptotic solution of some integral equations. IMA J. Appl. Math. 31, 169–182.Google Scholar
[8] Lawrence, C. J., Kuang, Y. & Weinbaum, S. (1985) The inertial draining of a thin fluid layer between parallel plates with a constant normal force. Part 2. Boundary layer and exact numerical solutions. J. Fluid Mech. 156, 479–494.Google Scholar
[9] Moss, E. A., Krassnokutski, A., Skews, B. W. & Paton, R. T. (2011) Highly transient squeeze-film flows. J. Fluid Mech. 671, 384–398.Google Scholar
[10] Brosse, N. & Ern, P. (2011) Paths of stable configurations resulting from the interaction of two disks falling in tandem. J. Fluids Struct. 27, 817–823.Google Scholar
[11] Brosse, N. & Ern, P. (2014) Interaction of two axisymmetric bodies falling in tandem at moderate Reynolds numbers. J. Fluid Mech. 757, 208–230.Google Scholar
[12] Sherwood, J. D. (2011) Added mass of a pair of discs. Phys. Fluids 23, 103601.Google Scholar
[13] Tranter, C. J. (1956) Integral Transforms in Mathematical Physics, Methuen, London.Google Scholar
[14] Tranter, C. J. (1968) Bessel Functions with Some Physical Applications, The English Universities Press, London.Google Scholar
[15] Chen, C. Y. & Atkinson, C. (1997) The axisymmetric problem of a disc in a bimaterial sandwich — (the history dependent) scalar potential case. Int. J. Engng Sci. 35, 229–251.CrossRefGoogle Scholar
[16] Martin, P. A. & Farina, L. (1997) Radiation of water waves by a heaving submerged horizontal disk. J. Fluid Mech. 337, 365–379.Google Scholar
[17] Farina, L. (2010) Water wave radiation by a heaving submerged horizontal disk very near the free surface. Phys. Fluids 22, 057102.Google Scholar
[18] Love, E. R. (1949) The electrostatic field of two equal circular co-axial conducting disks. Q. J. Mech. Appl. Math. 2, 428–451.Google Scholar
[19] Batchelor, G. K. (1973) An Introduction to Fluid Dynamics, Cambridge University Press, Cambridge.Google Scholar
[20] Collins, W. D. (1959) On the solution of some axisymmetric boundary value problems by means of integral equations, II: further problems for a circular disc and a spherical cap. Mathematika 6, 120–133.Google Scholar
[21] MATLAB Release 2014a, The MathWorks, Inc., Natick, Massachusetts, United States.Google Scholar