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Axisymmetric waves in compressible Newtonian liquids contained in rigid tubes: steady-periodic mode shapes and dispersion by the method of eigenvalleys

Published online by Cambridge University Press:  29 March 2006

H. A. Scarton  and
W. T. Rouleau
H. A. Scarton
Affiliation:
Mechanics Division, Rensselaer Polytechnic Institute, Troy, N.Y.
W. T. Rouleau
Affiliation:
Department of Mechanical Engineering, Carnegie-Mellon University
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Abstract

In this paper the first thirty-two axisymmetric modes for steady-periodic waves in viscous compressible liquids contained in rigid, impermeable, circular tubes are calculated. These results end long speculation over the effects of viscosity on guided acoustic waves. Sixteen of the modes belong to a family of rotation-dominated modes whose existence was previously unknown. The thirty-two modes were computed for a wide range of frequencies, viscosities and wave-lengths.

The modes were found through the use of the method of eigenvalleys, which also led to the discovery of backward-propagating waves, an exact analytical expression for the zeroth rotational mode eigenvalue, definitive boundaries between low and intermediate frequencies and between intermediate and high frequencies, and a new type of boundary layer, called a dilatational boundary layer.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 58 , Issue 3 , 8 May 1973 , pp. 595 - 621
Copyright
© 1973 Cambridge University Press

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References

Brown, F. T. 1962 Transient response of fluid lines. Trans. A.S.M.E., J. Basic Enyng, D 84, 547–553.Google Scholar
Burns, S. H. 1966 Finite-amplitude travelling waves with boundary dissipation. Harvard
Burns, S. H. 1967 Finite-amplitude distortion in air a t high acoustic pressures. J. Acoust. Soc. Am. 41, 1157–1168.Google Scholar
Cohen, H. & Tu, Y. 1962 Viscosity and boundary effects in the dynamic behavior of hydraulic systems. Trans. A.S.M.E., J. Basic Engng, D 84, 593–601.Google Scholar
Elco, R. A. & Hughes, W. F. 1962 Acoustic waveguide mode interference and damping with viscous fluids. 4th Int. Congr. Acoustics. Copenhagen, pp. 21–28.
Fitz-Gerald, J. M. 1969 Blood flow in narrow capillaries. Ph.D. dissertation, Imperial College, London.
Fitz-Gerald, J. M. 1972 Plasma motions in narrow capillary flow. J. Fluid Mech. 51, 463–476.Google Scholar
Gerlach, C. R. & Parker, J. D. 1967 Wave propagation in viscous fluid lines including higher mode effects. Trans. A.S.M.E, J. Basic Engng, D 89, 782–788.Google Scholar
Gurtin, M. E. 1972 The linear theory of elasticity. Handbuch der Physik (ed. C. Truesdell), Via/2, pp. 232–242. Springer.
Iberall, A. S. 1950 Attenuation of oscillatory pressures in instrument lines. Nat. Bur. Stds. J. Res. 45, 85–108.Google Scholar
Kirchhoff, G. 1868 Über den Einfluss der Wärmeleitung in einen Gase auf die Schall-bewegung. Ann. Phys. Chem. 134, 177–193.Google Scholar
Lighthill, M. J. 1956 Viscosity effects in sound waves of finite amplitude. In Surveys in Mechanics (ed. G. K. Batchelor & R. M. Davies), pp. 251–350. Cambridge University Press.
Lighthill, M. J. 1965 Group velocity. J. Inst. Maths. Applics. 1, 1–28.Google Scholar
Nehari, Z. 1965 Introduction to Complex Analysis. Allyn & Bacon.
Olson, R. M. 1962 Engineering Fluid Mechanics. International Textbook Co.
Onoe, M., Mcniven, H. D. & Mindlin, R. D. 1962 Dispersion of axially symmetric waves in elastic rods. Trans. A.S.M.E., J. Appl. Mech. E84, 729–734.Google Scholar
Rayleigh, Lord 1896 The Theory of Sound. Dover.
Redwood, M. 1960 Mechanical Waveguides. Pergamon.
Robinson, E. L. & Scarton, H. A. 1972 Contor: a Fortran subroutine to plot smooth contours of a single-valued arbitrary three-dimensional surface. J. Comp. Phys. 10, 242–251.Google Scholar
Rosenhead, L. 1954 The second coefficient of viscosity: a brief review of fundamentals. Proc. Roy. Soc. A 226, 1–6.Google Scholar
Rubinow, S. I. & Keller, J. B. 1971 Wave propagation in a fluid filled tube. J. Acoust. Soc. Am. 50, 198–223.Google Scholar
Scarton, H. A. 1970 Waves and stability in viscous and inviscid compressible liquids contained in rigid and elastic tubes by the method of eigenvalleys. Ph.D. dissertation, Carnegie-Mellon University, Pittsburgh. (Available from University Microfilms, Ann Arbor, Michigan 48106: order no. 70-18, 029E.)
Scarton, H. A. 1971 Double precision Fortran subroutines to compute both ordinary and modified Bessel functions of the first kind and of integer order with arbitrary complex argument: Jn(x+jy) and In(x+jy). J. Comp. Phys. 8, 295–299.Google Scholar
Scarton, H. A. 1973 The method of eigenvalleys. J. Comp. Phys. 11, 1–143.Google Scholar
Sohlichting, H. 1960 Boundary Layer Theory. McGraw-Hill.
Serceev, S. I. 1966 Fluid oscillations in pipes at moderate Reynolds numbers. In Fluid Dynamics, vol. 1, pp. 121–122. Faraday Press.
Shields, F. D., Lee, K. P. & Wiley, W. J. 1965 Numerical solution for sound velocity and absorption in cylindrical tubes. J. Acoust. Soc. Am. 37, 724–729.Google Scholar
Sternberg, E. 1960 On the integration of the equations of motion in the classical theory of elasticity, Arch. Rat. Mech. Anal. 6, 34–50.Google Scholar
Tijdeman, H. 1969 Remarks on the frequency response of pneumatic lines. Trans. A.S.M.E., J. Basic Engng, D 91, 325–327.Google Scholar