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Nano boundary layer equation with nonlinear Navier boundary condition. (English) Zbl 1207.76050
Summary: At the micro and nano scale the standard no slip boundary condition of classical fluid mechanics does not apply and must be replaced by a boundary condition that allows some degree of tangential slip. In this study the classical laminar boundary layer equations are studied using Lie symmetries with the no-slip boundary condition replaced by a nonlinear Navier boundary condition. This boundary condition contains an arbitrary index parameter, denoted by n>0, which appears in the coefficients of the ordinary differential equation to be solved. The case of a boundary layer formed in a convergent channel with a sink, which corresponds to n=1/2, is solved analytically. Another analytical but non-unique solution is found corresponding to the value n=1/3, while other values of n for n>1/2 correspond to the boundary layer formed in the flow past a wedge and are solved numerically. It is found that for fixed slip length the velocity components are reduced in magnitude as n increases, while for fixed n the velocity components are increased in magnitude as the slip length is increased.
MSC:
76D05Navier-Stokes equations (fluid dynamics)
35A30Geometric theory for PDE, characteristics, transformations
35Q35PDEs in connection with fluid mechanics
76A02Foundations of fluid mechanics
76M60Symmetry analysis, Lie group and algebra methods (fluid mechanics)