Advertisement

International Applied Mechanics

, Volume 40, Issue 10, pp 1092–1128 | Cite as

Variational methods of solving dynamic problems for fluid-containing bodies

  • I. A. Lukovskii
Article
  • 44 Downloads

Abstract

A variational approach to solving linear and nonlinear problems for a body with cavities partially filled with a perfect incompressible fluid is enunciated. The approach applies a nonclassical variational principle to describe the spatial motion of a finite fluid with a free surface and the classical variational principle, which is widely used in rigid body dynamics. These principles are used to formulate variational problems that are the basis of direct methods of solving nonlinear and linear dynamic problems for body-fluid systems. The approach allows us to derive an infinite system of nonlinear ordinary differential equations describing the joint motion of the rigid body and fluid and to develop an algorithm for determining the hydrodynamic coefficients. Linearized differential equations of motion of the mechanical system are presented and approximate methods are given to solve linear boundary-value problems and to determine the hydrodynamic coefficients.

Keywords

body with cavities body-fluid system variational approach free surface nonlinear sloshing natural frequencies 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

REFERENCES

  1. 1.
    M. Ya. Barnyak and I. A. Lukovskii, “Determining the natural frequencies and modes of a perfect fluid in a vessel of high relative depth,” Prikl. Mekh., 10, No. 5, 109–115 (1974).Google Scholar
  2. 2.
    M. Ya. Barnyak and I. A. Lukovskii, “The method of orthogonal projections in the problem on natural sloshing of a fluid in a vessel,” Dynamics and Stability of Multidimensional Systems [in Russian], Inst. Mat. AN USSR, Kiev (1974), pp. 77–85.Google Scholar
  3. 3.
    M. Ya. Barnyak and I. A. Lukovskii, “A modification of the variational method for solving the problem on natural sloshing of a fluid in a vessel,” Prikl. Mekh., 13, No. 7, 83–89 (1977).Google Scholar
  4. 4.
    M. Ya. Barnyak, I. A. Lukovskii, and G. A. Shvets, “Numerical implementation of the variational method for solving the problem on natural sloshing of a fluid in a vessel,” Dynamics and Stability of Controllable Systems [in Russian], Inst. Mat. AN USSR, Kiev (1977), pp. 62–73.Google Scholar
  5. 5.
    M. V. Bekker and I. A. Druzhinin, “Natural sloshing of a fluid in a spherical cavity. Anumerical-analytical alternative of the variational method,” Izv. RAN, Zh. Vych. Mat. Tekhn. Fiz., 40, No. 4, 633–637 (2000).MathSciNetGoogle Scholar
  6. 6.
    I. B. Bogoryad, I. A. Druzhinin, G. Z. Druzhinina, and É. E. Libin, Introduction to Dynamics of Vessels with Fluid [in Russian], Izd. Tomskogo Univ., Tomsk (1977).Google Scholar
  7. 7.
    A. N. Guz, V. D. Kubenko, and A. É. Babaev, “Dynamics of shell systems interacting with a liquid,” Int. Appl. Mech., 38, No. 3, 260–301 (2002).Google Scholar
  8. 8.
    L. V. Dokuchaev and I. A. Lukovskii, “A method for determining the hydrodynamic characteristics of a moving vessel with partitions,” Izv. AN SRSR, Mekh. Zhidk. Gaza, No. 4, 205–213 (1968).Google Scholar
  9. 9.
    N. E. Zhukovskii, Motion of a Body with Cavities Filled with a Homogeneous Dropping Fluid [in Russian], Vol. 2 of the 5-volume Collected Works [in Russian], Gosnauchtekhizdat, Moscow-Leningrad (1931).Google Scholar
  10. 10.
    K. S. Kolesnikov, Liquid-Fuel Rocket as a Control Object [in Russian], Mashinostroenie, Moscow (1969).Google Scholar
  11. 11.
    A. N. Komarenko, I. A. Lukovskii, and S. F. Feshchenko, “Eigenvalue problem with a parameter in the boundary conditions,” Ukr. Mat. Zh., 17, No. 6, 22–30 (1965).MATHGoogle Scholar
  12. 12.
    A. N. Komarenko and I. A. Lukovskii, “Stability of nonlinear sloshing of a fluid in a vessel moving harmonically,” Prikl. Mekh., 10, No. 10, 97–102 (1974).Google Scholar
  13. 13.
    V. D. Kubenko and V. V. Dzyuba, “Axisymmetric interaction problem for a sphere pulsating inside an elastic cylindrical shell filled with and immersed into a liquid,” Int. Appl. Mech., 38, No. 10, 1210–1219 (2002).Google Scholar
  14. 14.
    V. D. Kubenko, P. S. Koval’chuk, L. G. Boyarshina et al., Nonlinear Dynamics of Axisymmetric Bodies with Fluid [in Russian], Naukova Dumka, Kiev (1992).Google Scholar
  15. 15.
    P. S. Koval’chuk and L. A. Kruk, “Chaotic modes of forced nonlinear vibrations of fluid-filled cylindrical shells,” Int. Appl. Mech., 39, No. 12, 1452–1457 (2003).Google Scholar
  16. 16.
    H. L. Lamb, Hydrodynamics, 6th ed., Cambridge Univ. Press (1932).Google Scholar
  17. 17.
    O. S. Limarchenko, “Variational formulation of the problem on the motion of a tank with a fluid,” Dokl. AN USSR, Ser. A, No. 10, 904–908 (1978).Google Scholar
  18. 18.
    O. S. Limarchenko, “Analyzing the efficiency of discrete models in solving problems for a tank with a fluid under impulsive excitation,” Mat. Fiz. Nonlin. Mekh., 4, 44–48 (1985).Google Scholar
  19. 19.
    O. S. Limarchenko, “A direct method for solving the problem on the spatial motion of a body-fluid system,” Prikl. Mekh., 19, No. 8, 77–84 (1983).Google Scholar
  20. 20.
    I. O. Lukovskii, “Low-amplitude wave motions of a homogeneous incompressible fluid in vessels having the form of bodies of revolution,” Dop. AN URSR, No. 8, 1013–1017 (1961).Google Scholar
  21. 21.
    I. O. Lukovskii, “Equations of disturbed motion of a body with a cavity that has the form of a body of revolution and is partially filled with a fluid,” Dop. AN URSR, No. 6, 749–753 (1962).Google Scholar
  22. 22.
    I. A. Lukovskii, “The coefficients of the equations of disturbed motion of a body with axisymmetric cavities partially filled with a fluid,” Proc. Conf. Young Scientists of the Institute of Mathematics AS USSR [in Russian], Kiev (1963), pp. 106–110.Google Scholar
  23. 23.
    I. A. Lukovskii, “An approximate method for determining the hydrodynamic coefficients of the equations of disturbed motion of a body with cavities partially filled with a fluid,” Hydroaeromechanics [in Russian], Issue 1, Izd. KhGU, Kharkov (1965), pp. 62–72.Google Scholar
  24. 24.
    I. A. Lukovskii, “Applying the eigenfunction expansion method to solve boundary-value problems in the theory of disturbed motion of a fluid-containing body,” Proc. 1st Republ. Math. Conf. Young Researchers [in Russian], Issue 1, Inst. Mat. AN USSR, Kiev (1965), pp. 462–469.Google Scholar
  25. 25.
    I. A. Lukovskii, “Calculating the characteristics of motion of a fluid in a cavity in the form of an ellipsoid of revolution,” Prikl. Mekh., 1, No. 7, 101–106 (1965).Google Scholar
  26. 26.
    I. A. Lukovskii, “Determining the hydrodynamic characteristics of disturbed motion of a body with cavities separated by radial partitions and partially filled with a fluid,” Hydroaeromechanics, Issue 1, Izd. KhGU, Kharkov (1965),pp. 53–61.Google Scholar
  27. 27.
    I. A. Lukovskii, “Studying the motion of a body with a nonlinearly sloshing fluid,” Prikl. Mekh., 3, No. 6, 119–127 (1967).Google Scholar
  28. 28.
    I. A. Lukovskii, “Solving problems for a sloshing fluid in vessels of complex geometry,” Mat. Fiz., 3, 274–283 (1967).Google Scholar
  29. 29.
    I. O. Lukovskii, “Solving the nonlinear problem on the natural sloshing of the fluid in vessels of arbitrary geometry,” Dop. AN URSR, Ser. A, No. 3, 207–210 (1969).Google Scholar
  30. 30.
    I. A. Lukovskii, “Studying the nonlinear sloshing of the fluid in a vessel in the form of a body of revolution,” Mat. Fiz., 9, 57–72 (1971).MathSciNetGoogle Scholar
  31. 31.
    I. A. Lukovskii, Studying the Nonlinear Sloshing of Fluid in Moving Conic Vessels [in Russian], Mat. Fiz., 10, 70–79 (1971).Google Scholar
  32. 32.
    I. A. Lukovskii, Nonlinear Sloshing of Fluid in Vessels of Complex Geometry [in Russian], Naukova Dumka, Kiev (1975).Google Scholar
  33. 33.
    I. A. Lukovskii, “Methods of approximate solution of nonlinear dynamic boundary-value problems for a finite fluid with free boundary,” Mathematization of Knowledge and Scientific and Technological Advance [in Russian], Naukova Dumka, Kiev (1975), pp. 129–148.Google Scholar
  34. 34.
    I. A. Lukovskii, “The variational method in nonlinear dynamic problems for a finite fluid with free surface,” Vibration of Structures with Fluid [in Russian], TsNTI “Volna,” Moscow (1976), pp. 260–265.Google Scholar
  35. 35.
    I. A. Lukovskii and A. S. Korneyeva, “Spatial nonlinear motions of the fluid in a right circular cylinder,” Dynamics and Stability of Controllable Systems [in Russian], Inst. Mat. AN USSR, Kiev (1977), pp. 93–103.Google Scholar
  36. 36.
    I. A. Lukovskii, Variational Method in Nonlinear Problems of the Theory of Motion of a Body with a Cavity Partially Filled with a Fluid [in Russian], Preprint 78.22, Inst. Mat. AN USSR, Kiev (1978).Google Scholar
  37. 37.
    I. A. Lukovskii and A. M. Pil’kevich, “Studying the nonlinear sloshing of fluid in coaxial circular cylinders by the variational method,” Boundary-Value Problems of Mathematical Physics [in Russian], Inst. Mat. AN USSR, Kiev (1978), pp. 79–91.Google Scholar
  38. 38.
    I. A. Lukovskii, “Variational formulation of nonlinear dynamic boundary-value problems for a finite fluid executing prescribed three-dimensional motion,” Prikl. Mekh., 16, No. 2, 102–108 (1980).Google Scholar
  39. 39.
    I. A. Lukovskii, “Applying Ostrogradskii’s variational principle to solve nonlinear dynamic problems for a body with cavities containing a fluid,” Dynamics and Stability of Mechanical Systems [in Russian], Inst. Mat. AN USSR, Kiev (1980), pp. 3–15.Google Scholar
  40. 40.
    I. A. Lukovskii, “Determining interaction forces in the nonlinear dynamic problem for a body with a cavity containing a fluid,” Nonlinear Boundary-Value Problems [in Russian], Inst. Mat. AN USSR, Kiev (1980), pp. 181–190.Google Scholar
  41. 41.
    I. A. Lukovskii, “Approximate method for solving nonlinear dynamic problems for a fluid in a moving vessel,” Prikl. Mekh., 17, No. 2, 89–96 (1981).Google Scholar
  42. 42.
    I. A. Lukovskii, “Determining the Zhukovskii potentials for the nonlinear wave motions of the fluid in a vessel executing angular displacements,” Proc. 4th Seminar on Dynamics of Rigid and Elastic Bodies Interacting with Fluid [in Russian], Izd. Tomsk. Gos. Univ., Tomsk (1981), pp. 128–139.Google Scholar
  43. 43.
    I. A. Lukovskii, “Determining the natural frequencies and modes of the fluid in a vessel using Bateman’s variational principle,” Analytical Methods for Dynamic Study of Complex Systems [in Russian], Inst. Mat. AN USSR, Kiev (1982), pp. 3–11.Google Scholar
  44. 44.
    I. A. Lukovskii, M. Ya. Barnyak, and A. N. Komarenko, Approximate Methods for Solving Dynamic Problems for a Finite Fluid [in Russian], Naukova Dumka, Kiev (1984).Google Scholar
  45. 45.
    I. A. Lukovskii and A. M. Pil’kevich, Nonlinear Equations of Spatial Motions of a Body with a Cylindrical Cavity Containing a Fluid [in Russian], Preprint 84.18, Inst. Mat. AN USSR, Kiev (1984).Google Scholar
  46. 46.
    I. A. Lukovskii and A. M. Pil’kevich, Tables of Hydrodynamic Coefficients for the Nonlinear Equations of Spatial Motions of a Body with a Cylindrical Cavity Containing a Fluid [in Russian], Preprint 84.39, Inst. Mat. AN USSR, Kiev (1984).Google Scholar
  47. 47.
    I. A. Lukovskii and A. M. Pil’kevich, “Determining the apparent moments of inertia of a fluid by the variational and perturbation methods,” Proc. 5th Seminar on Dynamics of Rigid and Elastic Bodies Interacting with a Fluid [in Russian], Izd. Tomsk. Gos. Univ., Tomsk (1984), pp. 102–112.Google Scholar
  48. 48.
    I. A. Lukovskii and A. N. Bilyk, “Forced nonlinear sloshing of a fluid in moving axisymmetric conic vessels,” Numerical-Analytical Methods for Dynamic and Stability Studies of Multidimensional Systems [in Russian], Inst. Mat. AN USSR, Kiev (1985), pp. 12–26.Google Scholar
  49. 49.
    I. A. Lukovskii and A. M. Pil’kevich, “Sloshing of a fluid in a rectangular tank,” Direct Methods in Dynamic and Stability Problems for Multidimensional Systems [in Russian], Inst. Mat. AN USSR (1986), pp. 13–18.Google Scholar
  50. 50.
    I. A. Lukovskii, “Applying the variational principle to derive the nonlinear equations of disturbed motion of a body-fluid system,” Spacecraft Dynamics and Space Exploration [in Russian], Mashinostroenie, Moscow (1986), pp. 182–194.Google Scholar
  51. 51.
    I. A. Lukovskii and G. A. Shvets, “A rapidly converging variational algorithm for the problem on natural sloshing of a fluid in a vessel,” Spacecraft Dynamics and Space Exploration [in Russian], Mashinostroenie, Moscow (1986), pp. 137–142.Google Scholar
  52. 52.
    I. A. Lukovskii, V. A. Trotsenko, and V. I. Usyukin, Interaction of Thin-Walled Elastic Elements with a Fluid in Moving Containers [in Russian], Naukova Dumka, Kiev (1989).Google Scholar
  53. 53.
    I. A. Lukovskii and A. M. Pil’kevich, “Sloshing of a fluid in a cylindrical container with convex bottom,” Applied Problems of Dynamics and Stability of Multidimensional Systems [in Russian], Inst. Mat. AN USSR, Kiev (1987),pp. 15–21.Google Scholar
  54. 54.
    I. A. Lukovskii, Introduction to Nonlinear Dynamics of a Body with Cavities Containing a Fluid [in Russian], Naukova Dumka, Kiev (1990).Google Scholar
  55. 55.
    I. A. Lukovskii, “Constructing finite-dimensional models in dynamics of a finite fluid based on variational principles,” Modeling Dynamic Interaction in Body-Fluid Systems [in Russian], Inst. Mat. AN USSR, Kiev (1990), pp. 5–15.Google Scholar
  56. 56.
    I. O. Lukovskii, “Variational principle in the nonlinear theory of motion of floating bodies with containers partially filled with a fluid,” Dop. NAN Ukrainy, No. 8, 63–66 (1993).Google Scholar
  57. 57.
    I. A. Lukovskii, O. S. Limarchenko, and A. N. Timokha, “Nonlinear models in applied dynamic problems for bodies containing a fluid with free surface,” Prikl. Mekh., 28, No. 11, 75–83 (1992).MathSciNetGoogle Scholar
  58. 58.
    I. O. Lukovskii, Boundary-Value Problems in the Hydrodynamic Theory of Ships with Compartments Containing Liquid Cargo [in Russian], Preprint 94.16,Inst. Mat. NAN Ukrainy, Moscow (1994).Google Scholar
  59. 59.
    I. O. Lukovskii and M. Ya. Barnyak, “A modified variational method for solving the problem on natural sloshing of a fluid in an oblique cylinder,” Dop. NAN Ukrainy, No. 5, 62–66 (1997).Google Scholar
  60. 60.
    I. A. Lukovskii, “Determining the forces of interaction in dynamic problems for a floating body containing a fluid,” Prikl. Mekh., 33, No. 9, 74–81 (1997).MathSciNetGoogle Scholar
  61. 61.
    I. O. Lukovskii, “The theory of nonlinear sloshing of a finite, weakly viscous, incompressible fluid,” Dop. NAN Ukrainy, No. 10, 80–84 (1997).Google Scholar
  62. 62.
    I. A. Lukovskii, “Nonlinear equations of wave motions of a fluid in moving tanks in the form of a circular cone,” Nonlinear Boundary-Value Problems of Mathematical Physics and Their Application [in Russian], Inst. Mat. NAN Ukrainy, Kiev (1998), pp. 139–143.Google Scholar
  63. 63.
    I. A. Lukovskii and G. F. Zolotenko, “Numerical simulation of the sloshing of a fluid in a moving closed rectangular vessel,” GidroMekh., 72, 72–87 (1998).Google Scholar
  64. 64.
    I. O. Lukovskii, “Solving spectral problems in the linear theory of sloshing of a fluid in finite tanks,” Dop. NAN Ukrainy, No. 5, 53–58 (2002).Google Scholar
  65. 65.
    I. O. Lukovskii and D. V. Ovchinnikov, “Nonlinear mathematical model of the fifth order in the problem on the sloshing of a fluid in a cylindrical vessel,” Dynamic and Stability Problems for Multidimensional Systems [in Russian], Inst. Mat. NAN Ukrainy, Kiev (2003), pp. 119–160.Google Scholar
  66. 66.
    I. O. Lukovskii and O. V. Solodun, “Nonlinear model of motion of a fluid in cylindrical vessels with compartments,” Dop. NAN Ukrainy, No. 5, 50–55 (2001).Google Scholar
  67. 67.
    I. O. Lukovskii and O. V. Solodun, “Studying the forced nonlinear sloshing of a fluid in circular cylindrical vessels using a seven-mode model of the third order,” Dynamic and Stability Problems for Multidimensional Systems [in Russian], Inst. Mat. NAN Ukrainy, Kiev (2003), pp. 161–179.Google Scholar
  68. 68.
    I. A. Lukovskii and A. N. Timokha, “Nonlinear theory of sloshing of the fluid in moving containers: classical and nonclassical problems,” Problems of Analytical Mechanics and Its Applications [in Russian], Inst. Mat. NAN Ukrainy, Kiev (1999), pp. 169–201.Google Scholar
  69. 69.
    I. A. Lukovskii, G. F. Zolotenko, and A. M. Pil’kevich, “Comparative analysis of two variational models in the nonlinear theory of relative motion of fluid,” Prikl. GidroMekh., 5, No. 4, 12–43 (2003).MathSciNetGoogle Scholar
  70. 70.
    A. I. Lur’e, Analytical Mechanics [in Russian], Fizmatgiz, Moscow (1961).Google Scholar
  71. 71.
    G. N. Mikishev and B. I. Rabinovich, Dynamics of a Body with Cavities Partially Filled with a Fluid [in Russian], Mashinostroenie, Moscow (1968).Google Scholar
  72. 72.
    G. N. Mikishev, Experimental Methods in Spacecraft Dynamics [in Russian], Mashinostroenie, Moscow (1978).Google Scholar
  73. 73.
    N. N. Moiseev, “Theory of nonlinear sloshing of a finite fluid,” Prikl. Mat. Mekh., 22, 612–621 (1958).MathSciNetGoogle Scholar
  74. 74.
    N. N. Moiseev and V. V. Rumyantsev, Dynamics of Bodies with Cavities Containing a Fluid [in Russian], Nauka, Moscow (1965).Google Scholar
  75. 75.
    N. N. Moiseev and A. A. Petrov, Numerical Methods for Calculating the Natural Frequencies of a Finite Fluid[inRussian], VTs AN SSSR, Moscow (1966).Google Scholar
  76. 76.
    G. S. Narimanov, “Motion of a vessel partially filled with a fluid executing low-amplitude motions,” Prikl. Mat. Mekh., 21, No. 4, 513–524 (1957).MathSciNetGoogle Scholar
  77. 77.
    G. S. Narimanov, L. V. Dokuchaev, and I. A. Lukovskii, Nonlinear Dynamics of Fluid-Carrying Aircraft [in Russian], Mashinostroenie, Moscow (1977).Google Scholar
  78. 78.
    M. V. Ostrogradskii, Complete Works [in Russian], Vol. 1, Izd. AN USSR, Kiev (1959).Google Scholar
  79. 79.
    B. I. Rabinovich, Introduction to Launch Vehicle Dynamics [in Russian], Mashinostroenie, Moscow (1975).Google Scholar
  80. 80.
    O. V. Solodun, “Studying forced sloshing in circular cylindrical vessels with a diametral partition,” Nonlinear Oscillations, 5, No. 1, 90–106 (2002).MATHMathSciNetGoogle Scholar
  81. 81.
    J. J. Stocker, Water Waves, Interscience, New York (1957).Google Scholar
  82. 82.
    E. M. Stazhkov, “A rapidly converging method based on the optimal choice of the origin of coordinates in the liquid sloshing problem,” Proc. 3rd Seminar on Dynamics of Elastic and Rigid Bodies Interacting with Fluid [in Russian], Izd. Tomskogo Univ., Tomsk (1978), pp. 125–132.Google Scholar
  83. 83.
    V. A. Trotsenko, “Solving dynamic boundary-value problems for a fluid in horizontal cylindrical containers with partitions,” Nonlinear Oscillations, 6, No. 3, 401–427 (2003).MATHMathSciNetGoogle Scholar
  84. 84.
    S. F. Feshchenko, I. A. Lukovskii, B. I. Rabinovich, and L. V. Dokuchaev, Method for Determining the Apparent Masses of a Fluid in Moving Containers [in Russian], Naukova Dumka, Kiev (1969).Google Scholar
  85. 85.
    H. N. Abramson, The Dynamic Behavior of Liquids in Moving Containers, NASA SP-106, Washington (1966).Google Scholar
  86. 86.
    H. N. Abramson, W. H. Chu, and D. D. Kana, “Some studies of nonlinear lateral sloshing in rigid containers,” J. Appl. Mech., 33, No. 4, 66–74 (1966).Google Scholar
  87. 87.
    H. Bateman, Partial Differential Equations in Physics, Dover, New York (1944).Google Scholar
  88. 88.
    H. F. Bauer, “Nonlinear mechanical model for the description of propellant sloshing,” AJAA J., 4, No. 9, 1662–1668 (1966).Google Scholar
  89. 89.
    F. T. Dodge, D. D. Kana, and H. N. Abramson, “Liquid surface oscillation in longitudinally excited rigid cylindrical containers,” AIAA J., 3, No. 4, 685–695 (1965).CrossRefMATHGoogle Scholar
  90. 90.
    O. M. Faltinsen, O. F. Rognebakke, I. A. Lukovsky, and A. N. Timokha, “Multidimensional modal analysis of nonlinear sloshing in a rectangular tank with finite water depth,” J. Fluid Mech., 407, 201–234 (2000).MATHADSMathSciNetGoogle Scholar
  91. 91.
    O. M. Faltinsen, O. F. Rognebakke, and A. N. Timokha, “Resonant three-dimensional nonlinear sloshing in a square base basin,” J. Fluid Mech., 487, 1–47 (2003).MATHADSMathSciNetGoogle Scholar
  92. 92.
    I. P. Gavrilyuk, I. A. Lukovsky, and A. N. Timokha, “A multimodal approach to nonlinear sloshing in a circular cylindrical tank,” Hybrid Methods in Engineering, 2, No. 4, 463–483 (2000).Google Scholar
  93. 93.
    V. D. Kubenko, P. S. Koval’chuk, and L. A. Kruk, “On multimode nonlinear vibrations of filled cylindrical shells,” Int. Appl. Mech., 39, No. 1, 85–92 (2003).Google Scholar
  94. 94.
    P. S. Koval’chuk, N. P. Podchasov, and V. V. Kholopova, “Analysis of nonlinear bulging of liquid-filled cylindrical shells under local dynamic loading,” Int. Appl. Mech., 39, No. 3, 312–317 (2003).Google Scholar
  95. 95.
    P. S. Koval’chuk and V. G. Filin, “Circumferential traveling waves in filled cylindrical shells,” Int. Appl. Mech., 39, No. 2, 192–196 (2003).Google Scholar
  96. 96.
    J. C. Luce, “A variational principle for a fluid with a free surface,” J. Fluid Mech., 27, 395–397 (1976).ADSGoogle Scholar
  97. 97.
    P. McIver, “Sloshing frequencies for cylindrical and spherical containers filled to an arbitrary depth,” J. Fluid Mech., 201, 243–257 (1989).MATHADSMathSciNetGoogle Scholar
  98. 98.
    J. W. Miles, “Nonlinear surface waves in closed basins,” J. Fluid Mech., 75, Pt. 3, 419–448 (1976).MATHADSMathSciNetGoogle Scholar
  99. 99.
    J. W. Miles, “Internally resonant surface waves in a circular cylinder,” J. Fluid Mech., 149, 1–14 (1984).MATHADSMathSciNetGoogle Scholar

Copyright information

© Springer Science+Business Media, Inc. 2004

Authors and Affiliations

  • I. A. Lukovskii
    • 1
  1. 1.Institute of MathematicsNational Academy of Sciences of UkraineKiev

Personalised recommendations