Ritz solution of the Helmholtz equation in a stadium-shaped domain with zero normal derivatives – applications to fluid sloshing and thermo-convective stability

• Autorzy: Wang C. Y.
• Języki publikacji: EN

Abstrakty

EN

The eigenvalues and eigenfunctions of the Helmholtz equation with Neumann conditions are obtained for the stadium-shaped domain. The variational Ritz method is found to be accurate and efficient in determining these eigenvalues and eigenfunctions. The eigenfunctions show the evolution and switching of mode shapes from a long rectangular strip to a circle. These new results are applied to the sloshing of a liquid in a tank, and to the onset of thermo-convective stability in a confined porous layer.

Słowa kluczowe

EN

Helmholtz equation; stadium-shape; fluid sloshing; porous stability

• Wydawca: Instytut Podstawowych Problemów Techniki PAN
• Czasopismo: Archives of Mechanics
• Rocznik: 2017
• Tom: Vol. 69, nr 6
• Strony: 471--483
• Opis fizyczny: Bibliogr. 25 poz., rys.

Twórcy

autor

Wang C. Y.

• Departments of Mathematics and Mechanical Engineering Michigan State University East Lansing, MI 48824, U.S.A.

Bibliografia

• 1. J.W.S. Rayleigh, The Theory of Sound, 2nd ed., Dover, New York, 1945.
• 2. C.Y. Wang, C.M. Wang, Structural Vibration, CRC Press, Boca Raton, Florida, 2014.
• 3. R.F. Harrington, Time-Harmonic Electromagnetic Fields, Wiley, New York, 2001.
• 4. P. Lagasse, J. van Bladel, Square and rectangular waveguides with rounded corners, IEEE Transactions, 20, 5, 331–337, 1972.
• 5. Z. Shen, X. Lu, Modal analysis of a rectangular waveguide with rounded sides, Microwave and Optical Technology Letters, 33, 365–368, 2001.
• 6. N. Don, A. Kirilenko, A. Poyedinchuk, V. Tkachenko, Calculation of cutoff wave numbers in the waveguides of rectangular cross section with rounded corners, International Conference on Antenna Theory and Technique, Sevastopol, Ukraine, pp. 779–781, 2003.
• 7. J.A. Ruiz-Cruz, J.M. Rebollar, Eigenmodes of waveguides using a boundary contour mode-matching method with an FFT scheme, International Journal of RF and Microwave Computer Aided Engineering, 15, 286–295, 2005.
• 8. K. Rektorys, Variational Methods in Mathematics, Science and Engineering, Reidel, Dordrecht, 1980.
• 9. P.M. Morse, H. Feshbach, Methods of Theoretical Physics, McGraw-Hill, New York, 1953.
• 10. R. Weinstock, Calculus of Variations, McGraw-Hill, New York, 1952.
• 11. H. Lamb, Hydrodynamics, Dover, New York, 1945.
• 12. J.W.S. Rayleigh, On waves, Philosophical Magazine (5) 1: 257–259, 1876 (in Scientific Papers, vol. 1).
• 13. H. Jeffreys, The free oscillations of water in an elliptic lake, Proceedings of the London Mathematical Society, s2-23, 1, 455–476, 1925.
• 14. J. Proudman, On the tides in a flat semicircular sea of uniform depth, Monthly Notices of the Royal Astronomical Society (Geophysical Supplements), 2, s1, 32–44, 1928.
• 15. D.A. Nield, A. Bejan, Convection in Porous Media, 3rd ed., Springer, New York, 2006.
• 16. J.L. Beck, Convection in a box of porous material saturated with fluid, Physics of Fluids, 15, 1377–1383, 1972.
• 17. A. Zebib, Onset of natural convection in a cylinder of water saturated porous media, Physics of Fluids, 21, 699–700, 1978.
• 18. H.H. Bau, K.E. Torrance, Onset of convection in a permeable medium between vertical coaxial cylinders, Physics of Fluids, 24, 382–385, 1981.
• 19. C.Y. Wang, Onset of natural convection in a sector-shaped box containing a fluidsaturated porous medium, Physics of Fluids, 9, 3570–3571, 1997.
• 20. V.A. Kazhan, On the onset of free convection in vertical channels with cross sections in the form of circular or annular sectors, Technical Physics, 43, 8, 917–920, 1998.
• 21. R.A. Wooding, The stability of a viscous liquid in a vertical tube containing porous material, Proceedings of the Royal Society of London, A252, 120–134, 1959.
• 22. A. Barletta, L. Storesletten, Adiabatic eigenflows in a vertical porous channel, Journal of Fluid Mechanics, 749, 778–793, 2014.
• 23. K.B. Haugen, P.A. Tyvand, Onset of thermal convection in a vertical porous cylinder with conducting wall, Physics of Fluids, 15, 9, 2661–2667, 2003.
• 24. C.W. Horton, F.T. Rogers, Convection currents in a porous medium, Journal of Applied Physics, 16, 367–370, 1945.
• 25. E.R. Lapwood, Convection of a fluid in a porous medium, Proceedings of the Cambridge Philosophical Society, 44, 508–521, 1948.