Direct functional separation of variables and new exact solutions to axisymmetric unsteady boundary-layer equations

• Equations of unsteady axisymmetric boundary layer are studied.
• New solutions to the boundary layer equations are obtained.
• All solutions involve two to four arbitrary functions.
• Direct method of functional separation of variables is described.
• Many solutions are expressed in terms of elementary functions.

The paper deals with equations describing the unsteady axisymmetric laminar boundary layer on an extensive body of revolution as well as axisymmetric jet flows. Such equations are shown to reduce to a single nonlinear third-order PDE with variable coefficients wtz+wzwxz−wxwzz=νzwzzz+F(t,x),wtz+wzwxz−wxwzz=νzwzzz+F(t,x),where w is a modified stream function. We describe a number of new generalized and functional separable solutions to this equation, which depend on two to four arbitrary functions of a single argument (a few solutions depend on an arbitrary function of two arguments). We use three methods to construct the exact solutions: (i) direct method for symmetry reductions, (ii) direct method of functional separation of variables (a special form of solutions with six undetermined functions is preset and particular solutions to an auxiliary ODE are used), and (iii) method of generalized separation of variables. Most of the solutions obtained are expressed in terms of elementary functions, provided that the arbitrary functions are also elementary. Such solutions, having relatively simple form and presenting significant arbitrariness, can be especially useful for testing numerical and approximate analytical methods for nonlinear hydrodynamic-type PDEs and solving certain model problems. The direct method of functional separation of variables used in this paper can also be effective for constructing exact solutions to other nonlinear PDEs.