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Classical Boundary Layer Problems

In: Algebraic Approaches to Partial Differential Equations

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  • Xiaoping Xu

    (Academy of Mathematics and System Science, Institute of Mathematics)

Abstract

In 1904 Prandtl observed that, in the flow of a slightly viscous fluid past a body, the frictional effects are confined to a thin layer of fluid adjacent to the surface of the body. Classical unsteady boundary layer equations are fundamental nonlinear partial differential equations in the boundary layer theory of fluid dynamics. In this chapter, we introduce various schemes with multiple parameter functions to solve these equations and obtain many families of new explicit exact solutions with multiple parameter functions. Symmetry transformations are used to simplify our arguments. The moving-frame technique is applied in the three-dimensional case in order to capture the rotational properties of the fluid. In particular, we obtain a family of solutions singular on any moving plane, which may be used to study abrupt high-speed rotating flows. Many other solutions are analytic and related to trigonometric and hyperbolic functions, which reflect various wave characteristics of the fluid. Our solutions may also help engineers to develop more effective algorithms to find physical numerical solutions to practical models.

Suggested Citation

  • Xiaoping Xu, 2013. "Classical Boundary Layer Problems," Springer Books, in: Algebraic Approaches to Partial Differential Equations, edition 127, chapter 0, pages 317-383, Springer.
    • Handle: RePEc:spr:sprchp:978-3-642-36874-5_10
    DOI: 10.1007/978-3-642-36874-5_10
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