Exact time-dependent flows for non-shallow water equations
The authors discuss two classes of exact solutions of nonlinear, non- shallow water equations. Invoking several transformation of variables of the original systems, the first class is shown to be governed by a third order nonlinear ordinary differential equation which is found to be singular at both ends of the interval of interest. This equation is solved by a numerical technique and found to describe flows between two intersecting planes. The second class of solution is expressed in terms of incomplete beta functions and delineates flows under a free surface which is a function of time only. The solutions are then compared with those obtained earlier by \textit{N. C. Freeman} [J. Fluid Mech., 56, 257- 263 (1972; Zbl 0256.76009)], \textit{P. L. Sachdev} [J. Fluid Mech., 96, 797-802 (1980; Zbl 0418.76013)] and \textit{M. S. Longuet-Higgins} [J. Fluid Mech., 55, 529-543 (1972; Zbl 0257.76013)]. Several special solutions are examined. Numerical results are also presented.
scientific article; zbMATH DE number 3898429