Unsteady axisymmetric boundary-layer equations: Transformations, properties, exact solutions, order reduction and solution method

• Equations of unsteady axisymmetric boundary layer are studied.
• New solutions to the boundary layer equations are obtained.
• All solutions involve two to five arbitrary functions.
• Many solutions are obtained by a new method.
• The method can be effective for other non-linear PDEs.

The paper deals with equations describing the unsteady axisymmetric boundary layer on a body of revolution. The shape of the body is assumed to be arbitrary. The axisymmetric boundary-layer equation for the stream function is shown to reduce to a plane boundary-layer equation with a streamwise-coordinate-dependent viscosity of the formwtz+wzwxz−wxwzz=νr2(x)wzzz+F(t,x).wtz+wzwxz−wxwzz=νr2(x)wzzz+F(t,x).We describe a number of new generalized and functional separable solutions to this non-linear equation, which depend on two to five arbitrary functions. The solutions are obtained with a new method (direct method of functional separation of variables) based on using particular solutions to an auxiliary ODE. Many of the solutions are expressed in terms of elementary functions, provided that the arbitrary functions are also elementary. Two theorems are stated that enable one to generalize exact solutions of unsteady axisymmetric boundary-layer equations by including additional arbitrary functions. Furthermore, we specify a von Mises-type transformation that reduces the unsteady axisymmetric boundary-layer equation to a non-linear second-order PDE. We also present several new exact solutions to the plane boundary-layer equation and solve a boundary layer problem for a non-uniformly heated flat plate in a unidirectional fluid flow with temperature dependent viscosity.The method proposed in this paper can also be effective for constructing exact solutions to many other non-linear PDEs.