Abstract:
Recently, a new approach to certain flow problems, called the theory of fluid sheets, has been developed by Green and Naghdi [1974,1987]. This theory models threedimensional unsteady flow and is based on a special two dimensional surface known as a directed (or Cosserat) surface. The theory is still three-dimensional in character, but the independent variables are only two space variables and time. The application of the theory to water waves was first limited to shallow-water problems, but was later extended to treat waves in water of infinite depth. An alternative derivation of the theory using a variational approach according to Kantorovich is given here. This approach is similar to the work of Shields [1986] and Shields and Webster [1988], but it is not restricted to shallow-water. Two problems are treated: solitary waves in shallow water with surface tension, and the characteristics of deep water gravity waves (without surface tension). The characteristics of solitary waves with surface tension are studied using fluid sheet theory and other nonlinear shallow-water wave theories, such as the KortewegdeVries equation and the so-called "Super Korteweg-deVries" equation. The exact asymptotic behavior of these waves far from the crest is determined using a procedure similar to that used by Stokes for solitary gravity waves without surface tension. Five separate regions are identified from an analysis of these results. It is found that the higher level Green-Naghdi theories capture this asymptotic behavior significantly better than the competing theories both near the critical value of Weber number equal to 3, and for small and large values of surface tension. Computations of solitary waves are made for various combinations of Froude number and Weber number. It is found that there exist waves of depression for F < 1 near the critical value [tau] = 1/3. The critical value is the boundary between solitary waves of depression, whose wave profile lies wholly beneath the still water level ([tau] > 1/3), and solitary waves, part of whose profile lies above the still water level ([tau] < 1/3). The behavior of solitary waves as surface tension is decreased is examined. This result is used to clarify the unexpected behavior reported by Vanden-Broeck & Hunter [1983], in which solitary waves of depression were not obtained as surface tension decreased and solitary waves of elevation did not result in the limit of long capillary-gravity waves. In order to assess the validity of the direct theory in deep water, two-dimensional, steady, periodic waves were also studied. These results are compared with the accurate solution by Cokelet [1977] and Schwartz [1974] and the Stokes fifth order wave theory. It is found that the direct theory gives excellent agreement with the accurate solution for wave shape and celerity, and to a lesser extent to its internal kinematics. The first level direct theory is found comparable to the Stokes fifth order theory for steep waves.