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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:
76D05 Navier-Stokes equations for incompressible viscous fluids
35A30 Geometric theory, characteristics, transformations in context of PDEs
35Q35 PDEs in connection with fluid mechanics
76A02 Foundations of fluid mechanics
76M60 Symmetry analysis, Lie group and Lie algebra methods applied to problems in fluid mechanics
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