Existence results for nano boundary layer flows with nonlinear Navier boundary condition

The standard no slip boundary condition of classical fluid mechanics is no longer valid at the micro- and nano-scale and should be replaced by a boundary condition that allows some degree of tangential slip. In the present work, the classical laminar boundary layer equation of the flow away from the origin past a wedge with the no-slip boundary condition replaced by a nonlinear Navier boundary condition is revisited. This boundary condition includes an arbitrary index parameter, denoted by n>0, which appears in the coefficients of the differential equation to be solved. It is proved corresponding to the value n=13, there are exactly three situations for the problem: (i) there is no solution; (ii) there exist two solutions; (iii) there exist four solutions. Furthermore, the exact analytical solution of the problem is given in terms of parabolic cylinder functions for further physical interpretations.