Brooke Benjamin's original theories of fluid mechanical phenomena changed our basic understanding of cavitation bubbles, surface and internal waves, gravity currents, instabilities of shear flow over flexible surfaces, and swirling flows. For some types of finite-amplitude wave phenomena, he generated integral constraints and derived new partial differential equations; by establishing their general properties he showed how they have wide application. He developed a complementary approach based on functional analysis that was quite new to fluid mechanics. He demonstrated methods for deriving, without detailed calculation, the essential features of nonlinear and indeterminate flow problems that are otherwise intractable.


Article metrics loading...

Loading full text...

Full text loading...


Literature Cited

  1. Belcher SE, Hunt JCR. 1998. Turbulent flow over hills and waves. Annu. Rev. Fluid Mech. 30:507–38 [Google Scholar]
  2. Benjamin TB. 1955. Cavitation in liquids PhD thesis. Univ. Cambridge, UK:
  3. Benjamin TB. 1957. Wave formation in laminar flow down an inclined plane. J. Fluid Mech. 2:554–74 [Google Scholar]
  4. Benjamin TB. 1958. Pressure waves from collapsing cavities. Proc. 2nd Symp. Naval Hydrodyn. Nat. Acad. Sci. 515:207–29 [Google Scholar]
  5. Benjamin TB. 1959. Shearing flow over a wavy boundary. J. Fluid Mech. 6:161–205 [Google Scholar]
  6. Benjamin TB. 1960. Effects of a flexible boundary on hydrodynamic stability. J. Fluid Mech. 9:513–32 [Google Scholar]
  7. Benjamin TB. 1961. The development of three-dimensional disturbances in an unstable film of liquid flowing down an inclined plane. J. Fluid Mech. 10:401–19 [Google Scholar]
  8. Benjamin TB. 1962a. The solitary wave on a stream with an arbitrary distribution of vorticity. J. Fluid Mech. 12:97–116 [Google Scholar]
  9. Benjamin TB. 1962b. Standing waves in a stream of water. In Encyclopaedic Dictionary of Physics London: Pergamon [Google Scholar]
  10. Benjamin TB. 1962c. Theory of vortex breakdown phenomenon. J. Fluid Mech. 14:593–629 [Google Scholar]
  11. Benjamin TB. 1963. The threefold classification of unstable disturbances in flexible surfaces bounding inviscid flows. J. Fluid Mech. 16:436–50 [Google Scholar]
  12. Benjamin TB. 1964a. Effects of surface contamination on wave formation in falling liquid films. Arch. Mech. Stosow. (Proc. Polish Acad. Sci. 3) 16:615–26 [Google Scholar]
  13. Benjamin TB. 1964b. Fluid flow with flexible boundaries. In Proc. Int. Congr. Appl. Mech., 11th ed. H Görtler pp. 109–28 Berlin: Springer-Verlag [Google Scholar]
  14. Benjamin TB. 1965. Significance of the vortex breakdown phenomenon. Trans. ASME J. Basic. Eng. 87:518–22 [Google Scholar]
  15. Benjamin TB. 1966. Internal waves of finite amplitude and permanent form. J. Fluid Mech. 25:241–70 [Google Scholar]
  16. Benjamin TB. 1967a. Some developments in the theory of vortex breakdown. J. Fluid Mech. 28:65–84 [Google Scholar]
  17. Benjamin TB. 1967b. Instability of periodic wave trains in nonlinear dispersive systems. Proc. R. Soc. London Ser. A 299:59–75 [Google Scholar]
  18. Benjamin TB. 1967c. Internal waves of permanent form in fluids of great depth. J. Fluid Mech. 29559–92 [Google Scholar]
  19. Benjamin TB. 1968. Gravity currents and related phenomena. J. Fluid Mech. 31:209–48 [Google Scholar]
  20. Benjamin TB. 1970. Upstream influence. J. Fluid Mech. 40:49–79 [Google Scholar]
  21. Benjamin TB. 1971. A unified theory of conjugate flows. Philos. Trans. R. Soc. London Ser. A 269:587–643 [Google Scholar]
  22. Benjamin TB. 1972. Conjugate-flow theory for heterogeneous compressible fluids with applications to non-uniform suspensions of gas bubbles in liquids. J. Fluid Mech. 54:545–63 [Google Scholar]
  23. Benjamin TB. 1974. Lectures on nonlinear wave motion. Lect. Appl. Math. Am. Math. Soc. 15:3–47 [Google Scholar]
  24. Benjamin TB. 1975. The alliance of practical and analytical insights into the nonlinear problems of fluid mechanics. In Appl. Methods Funct. Anal. Problems Mech., Proc. Jt. Symp. Marseille, Fr. pp. 8–29 Berlin: Springer-Verlag [Google Scholar]
  25. Benjamin TB. 1976. Applications of Leray-Schauder degree theory to problems of hydrodynamic stability. Cambridge Philos. Soc. Math. Proc. 79:373–92 [Google Scholar]
  26. Benjamin TB. 1978a. Bifurcation phenomena in steady flows of a viscous fluid. Part I. Theory Proc. R. Soc. London Ser. A 359:1–26 [Google Scholar]
  27. Benjamin TB. 1978b. Bifurcation phenomena in steady flows of a viscous fluid. Part II. Experiments. Proc. R. Soc. London Ser. A 35927–43 [Google Scholar]
  28. Benjamin TB. 1979. Computing methods in applied sciences and engineering. Lect. Notes Phys. 91:62–71 [Google Scholar]
  29. Benjamin TB. 1984. Impulse, flow force and variational principles. IMA J. Appl. Math. 32:3–68 [Google Scholar]
  30. Benjamin TB. 1986a. On the Boussinesq model for two-dimensional wave motions in heterogeneous fluids. J. Fluid Mech. 165:445–74 [Google Scholar]
  31. Benjamin TB. 1986b. Note on added mass and drift. J. Fluid Mech. 169:251–56 [Google Scholar]
  32. Benjamin TB. 1987. Hamiltonian theory for motions of bubbles in an infinite liquid. J. Fluid Mech. 181:349–79 [Google Scholar]
  33. Benjamin TB. 1992. A new kind of solitary wave. J. Fluid Mech. 245:401–11 [Google Scholar]
  34. Benjamin TB. 1995. Verification of the Benjamin-Lighthill conjecture about steady water waves. J. Fluid Mech. 295:337–56 [Google Scholar]
  35. Benjamin TB, Boczar-Karakiewicz B, Pritchard WG. 1987. Reflection of water waves in a channel with a corrugated bed. J. Fluid Mech. 185:249–74 [Google Scholar]
  36. Benjamin TB, Bona JL. 1972. The stability of solitary waves. Proc. R. Soc. London Ser. A 328:153–83 [Google Scholar]
  37. Benjamin TB, Bona JL, Bose DK. 1990. Solitary-wave solutions of nonlinear problems. Philos. Trans. R. Soc. Ser. A 331:195–244 [Google Scholar]
  38. Benjamin TB, Bona JL, Mahony JJ. 1972. Model equation for long waves in nonlinear dispersive systems. Philos. Trans. R. Soc. Ser. A 272:47–78 [Google Scholar]
  39. Benjamin TB, Bowman S. 1987. Discontinuous solutions of one-dimensional Hamiltonian systems. Proc. R. Soc. London Ser. A 413:263–95 [Google Scholar]
  40. Benjamin TB, Bridges TJ. 1997a. Reappraisal of the Kelvin-Helmholtz problem. Part 1. Hamiltonian structure. J. Fluid Mech. 333301–25 [Google Scholar]
  41. Benjamin TB, Bridges TJ. 1997b. Reappraisal of the Kelvin-Helmholtz problem. Part 2. Interaction of the Kelvin-Helmholtz, superharmonic and Benjamin-Feir instabilities. J. Fluid Mech. 333:327–73 [Google Scholar]
  42. Benjamin TB, Ellis AJ. 1990. Self-propulsion of asymmetrically vibrating bubbles. J. Fluid Mech. 212:65–80 [Google Scholar]
  43. Benjamin TB, Ellis AT. 1966. The collapse of cavitation bubbles and the pressures thereby produced against solid boundaries. Philos. Trans. R. Soc. Ser. A 260:221–40 [Google Scholar]
  44. Benjamin TB, Feir JE. 1967. The disintegration of wave trains on deep water. Part I. Theory J. Fluid Mech. 27:417–30 [Google Scholar]
  45. Benjamin TB, Lighthill MJ. 1954. On cnoidal waves and bores. Proc. R. Soc. London Ser. A 224:448–60 [Google Scholar]
  46. Benjamin TB, Mahony JJ. 1971. On an invariant property of water waves. J. Fluid Mech. 49:385–89 [Google Scholar]
  47. Benjamin TB, Mullin T. 1980. Transition to oscillatory motion in the Taylor experiment. Nature 280:567–69 [Google Scholar]
  48. Benjamin TB, Mullin T. 1982. Notes on the multiplicity of steady flows in the Taylor experiment. J. Fluid Mech. 121:219–30 [Google Scholar]
  49. Benjamin TB, Mullin T. 1988. Buckling instabilities in layers of viscous liquid subjected to shearing. J. Fluid Mech. 195:523–40 [Google Scholar]
  50. Benjamin TB, Olver PJ. 1982. Hamiltonian structure, symmetries and conservation laws for water waves. J. Fluid Mech. 125:137–85 [Google Scholar]
  51. Benjamin TB, Pathak SK. 1987. Cellular flows of a viscous liquid that partly fills a horizontal rotating cylinder. J. Fluid Mech. 183:399–420 [Google Scholar]
  52. Benjamin TB, Ursell F. 1954. The stability of the plane free surface of a liquid in vertical periodic motion. Proc. R. Soc. London Ser. A 225:505–15 [Google Scholar]
  53. Benney DJ. 1962. Non-linear gravity wave interactions. J. Fluid Mech. 14:577–84 [Google Scholar]
  54. Blake JR, Gibson DC. 1987. Cavitation bubbles near boundaries. Annu. Rev. Fluid Mech. 19:99–123 [Google Scholar]
  55. Boussinesq J. 1871. Théorie de l'intumescence liquide appele onde solitaire oude translation se propageant dans un canal rectangulaire. C.R. Acad. Sci. Paris 73755 [Google Scholar]
  56. Brenner MP, Lohse D, Dupont T. 1995. Bubble shape oscillations and the onset of sono luminescence. Phys. Rev. Lett. 75:954–57 [Google Scholar]
  57. Cliffe KA. 1988. Numerical calculations of the primary-flow exchange process in the Taylor problem. J. Fluid Mech. 197:57–79 [Google Scholar]
  58. Coles D. 1965. Transition in circular Couette flow. J. Fluid Mech. 21:385–425 [Google Scholar]
  59. Craik A. 2005. The origins of water wave theory. Annu. Rev. Fluid Mech. 36:1–28 [Google Scholar]
  60. Darwin CG. 1953. Note on hydrodynamics. Proc. Cambridge Philos. Soc. 49:342–54 [Google Scholar]
  61. Davey A, DiPrima RC, Stuart JT. 1968. On the instability of Taylor vortices. J. Fluid Mech. 29:691–704 [Google Scholar]
  62. Davies RM, Taylor GI. 1950. The mechanics of large bubbles rising through extended liquids and through liquids in tubes. Proc. R. Soc. London Ser. A 200:375–90 [Google Scholar]
  63. Davis RE, Acrivos A. 1967. Solitary internal waves in deep water. J. Fluid Mech. 29:691–704 [Google Scholar]
  64. Dickens C. 1865. Our Mutual Friend London: Chapman & Hall
  65. Drazin PG. 1970. Kelvin-Helmholtz instability of finite amplitude. J. Fluid Mech. 65:781–97 [Google Scholar]
  66. Dubreil-Jacotin ML. 1937. Sur les théorems d'existence relatifs aux ondes permantes periodiques à deux dimensions dans les liquides hétérogenes. J. Math Pure Appl. 19:43 [Google Scholar]
  67. Eames I, Belcher SE, Hunt JCR. 1994. Drift, partial drift and Darwin's proposition. J. Fluid Mech. 275:201–23 [Google Scholar]
  68. Faraday M. 1831. On a peculiar class of acoustical figures; and on certain forms assumed by groups of particles upon vibrating elastic surfaces. Phil. Trans. R. Soc. London 121:299–340 [Google Scholar]
  69. Feir JE. 1967. The disintegration of wave trains on deep water PhD thesis. Univ. Cambridge:
  70. Gaster M. 1968. The development of three-dimensional wave packets in a boundary layer. J. Fluid Mech. 32:173–84 [Google Scholar]
  71. Hasselmann K. 1962. On the non-linear energy transfer in a gravity wave spectrum. Part 1. General theory J. Fluid Mech. 12:481–500 [Google Scholar]
  72. Hunt JCR. 2003. Thomas Brooke Benjamin. Biogr. Mem. Fellows R. Soc. 49:39–68 [Google Scholar]
  73. Hunt JCR, Orr A, Rottman JW, Capon R. 2004. Coriolis effects in mesoscale flows with sharp changes in surface conditions. Q. J. R. Meteorol. Soc. 130:2703–31 [Google Scholar]
  74. Jeffreys H. 1925. On the formation of waves by wind. Proc. R. Soc. London Ser. A 107:189 [Google Scholar]
  75. Keady G, Norbury J. 1975. Water waves and conjucate streams. J. Fluid Mech. 70:663–71 [Google Scholar]
  76. Keulegan GH. 1957. An experimental study of the motion of saline water from locks into fresh water channels. US Nat. Bur. Stand. Rep.5168
  77. Kirchoff GR. 1869. Über die bewegung eines rotationskörpen in ein flüssigkeit. Crelle 71:237 [Google Scholar]
  78. Klemp JB, Rotunno R, Skamarock WC. 1997. On the propogation of internal bores. J. Fluid Mech. 331:81–106 [Google Scholar]
  79. Kornfield M, Suvorov L. 1944. On the destructive action of cavitation. J. Appl. Phys. 15:495–506 [Google Scholar]
  80. Korteweg DJ, deVries F. 1895. On the change of form of long waves advancing in a rectangular canal, and on a new type of long stationary waves. Philos. Mag. 39:422–43 [Google Scholar]
  81. Kramer MO. 1960. Boundary layer stabilisation by distributed damping. J. Am. Soc. Nav. Eng. 72:25 [Google Scholar]
  82. Kulikovskii AA. 1962. The structure of shock waves. Prikl. Mat. Mekh. 26:631–41 [Google Scholar]
  83. Landahl M. 1962. On the stability of a laminar incompressible boundary layer over a flexible surface. J. Fluid Mech. 13:609–32 [Google Scholar]
  84. Leibovich S. 1983. The form and dynamics of Langmuir circulations. Annu. Rev. Fluid Mech. 15:391–427 [Google Scholar]
  85. Levi-Civita T. 1925. Détermination rigoureuse des ondes permanents d'ampleur fini. Math. Ann. 93264 [Google Scholar]
  86. Lighthill MJ. 1978. Waves in Fluids Cambridge, UK: Cambridge Univ. Press
  87. Lin CC. 1955. Hydrodynamic Stability Cambridge, UK: Cambridge Univ. Press
  88. Long RR. 1953. Some aspects of the flow of stratified fluids. 1: A theoretical investigation. Tellus 5:42–58 [Google Scholar]
  89. Longuet-Higgins MS, Phillips OM. 1962. Phase velocity effects in tertiary wave interactions. J. Fluid Mech. 12:333–36 [Google Scholar]
  90. Miles JW. 1957. On the generation of surface waves by shear flows. J. Fluid Mech. 3185–204 Erratum. 1959. J. Fluid Mech. 6:583–98
  91. Milies JW. 1980. Solitary waves. Annu. Rev. Fluid Mech. 12:11–43 [Google Scholar]
  92. Miles JW, Howard LN. 1964. Note on heterogeneous shear flow. J. Fluid. Mech. 20:331–36 [Google Scholar]
  93. Mougin G, Magnaudet J. 2002. Path instability of a rising bubble. Phys. Rev. Lett. 88:014502 [Google Scholar]
  94. Mullin T. 1982. Mutations of steady cellular flows in the Taylor experiment. J. Fluid Mech. 121:207–18 [Google Scholar]
  95. Ono H. 1975. Algebraic solitary waves in stratified fluids. J. Phys. Soc. Jpn. 39:1082–91 [Google Scholar]
  96. Peregrine DH. 1966. Calculations of the development of an undular bore. J. Fluid Mech. 25:321–30 [Google Scholar]
  97. Phillips OM. 1960. On the dynamics of unsteady gravity waves of finite amplitude. Part 1. The elementary interactions. J. Fluid Mech. 9:193–217 [Google Scholar]
  98. Plesset MS, Mitchell TL. 1956. On the stability of the spherical shape of vapour cavity in a liquid. Q. Appl. Math. 13:419–30 [Google Scholar]
  99. Rayleigh Lord. 1876. On waves. Philos. Mag. (5 1:257 [Google Scholar]
  100. Rayleigh Lord. 1883. On maintained vibrations. Philos. Mag. 15:229–35 [Google Scholar]
  101. Rayleigh Lord. 1910. Aerial plane waves of finite amplitude. Proc. R. Soc. London Ser. A 84:247–84 [Google Scholar]
  102. Rayleigh Lord. 1914. On the theory of long waves and bores. Proc. R. Soc. London Ser. A 90:324 [Google Scholar]
  103. Rayleigh Lord. 1917. On the pressure developed in a liquid during the collapse of a spherical void. Philos. Mag. 34:(6)94 [Google Scholar]
  104. Russell JS. 1865. Report on waves. In Report 14th Meet. Br. Assoc. Adv. Sci., pp. 311–90 London: John Murray [Google Scholar]
  105. Saffman PG. 1967. The self propulsion of a deformable body in a perfect fluid. J. Fluid Mech. 28:385–89 [Google Scholar]
  106. Simpson JE. 1997. Gravity Currents: In the Environment and the Laboratory Cambridge, UK:: Cambridge Univ. Press
  107. Stokes GG. 1845. On the theories of internal friction of fluids in motion. Trans. Cambridge Philos. Soc. 8:287–305 [Google Scholar]
  108. Tanaka M, Dold JW, Lewy M, Peregrine DH. 1987. Instability and breaking of a solitary wave. J. Fluid Mech. 185:235–48 [Google Scholar]
  109. Taylor GI. 1910. The conditions necessary for discontinuous motion in gases. Proc. R. Soc. London Ser. A 84:371–77 [Google Scholar]
  110. Taylor GI. 1931. Effects of variation in density on the stability of superposed streams of fluid. Proc. R. Soc. London Ser. A 132:499–523 [Google Scholar]
  111. Taylor GI. 1968. Instability of jet, threads and sheets of viscous fluid. Proc. 12th Int. Congr. Appl. Mech., Stanford pp. 382–88 [Google Scholar]
  112. Thomson W, Tait PG. 1879. Principles of Mechanics and Dynamics Cambridge, UK: Cambridge Univ. Press
  113. Tobak M, Peake DJ. 1982. Topology of three dimensional separated flow. Annu. Rev. Fluid Mech. 14:61–85 [Google Scholar]
  114. von Karman T. 1940. The engineer grapples with non-linear problems. Bull. Am. Meteorol. Soc. 46615 [Google Scholar]
  115. Whitham GB. 1962. Mass, momentum and energy flux in water waves. J. Fluid Mech. 12:135–47 [Google Scholar]
  116. Whitman GB. 1966. Non-linear dispersion of water waves. J. Fluid Mech. 27:399–412 [Google Scholar]
  117. Whitman GB. 1967. Variational methods and applications to water waves. Proc. R. Soc. London Ser. A 299:6–25 [Google Scholar]
  118. Wood IR, Simpson JE. 1984. Jumps in layered miscible fluids. J. Fluid Mech. 140:329–42 [Google Scholar]
  119. Yih CS. 1965. Dynamics of Non-Homogeneous Fluids New York: Macmillan
  120. Zakharov VE. 1968. Stability of periodic waves of finite amplitude on the surface of a deep fluid. J. Appl. Mech. Tech. Phys. 2190 [Google Scholar]
  121. Zukoski EE. 1966. Influence of viscosity, surface tension, and inclination angle on motion of long bubbles in closed tubes. J. Fluid Mech. 25:821–37 [Google Scholar]
  • Article Type: Review Article
This is a required field
Please enter a valid email address
Approval was a Success
Invalid data
An Error Occurred
Approval was partially successful, following selected items could not be processed due to error