The characteristic function method and exact solutions of nonlinear sheared flows with free surface under gravity

Method of characteristic function has been applied for solving a system of first order nonlinear sheared flows with a free surface. The application of a one-parameter group of infinitesimal transformations reduces the number of independent variables by one, and consequently, by applying this method twice, the system of partial differential equations, in three independent variables, with the boundary conditions reduces to a system of ordinary differential equations with the appropriate corresponding conditions. The obtained differential equations are solved analytically and the forms of the free surface, the horizontal component and vertical component of the velocity are obtained in closed form for different cases. The results are illustrated graphically for different parameters.