# Boundary layer

A region of large values of the gradient of a function, in particular, in hydrodynamics it is a region of the flow of a viscous liquid (gas) the transversal thickness of which is small in comparison with its longitudinal dimensions and which is produced at the surface of a solid or at the boundary between two liquid flows with different velocities, temperatures or chemical compositions. A boundary layer is characterized by a sharp change in the velocity in the transversal direction (a shear layer) or a sharp change in the temperature (a thermal, or temperature, boundary layer), or else in the concentrations of the individual chemical components (a diffusion, or concentration, boundary layer). The concept of a boundary layer and the term itself were introduced by L. Prandtl (1904) in connection with the solution of a boundary value problem for non-linear partial differential equations in the hydrodynamics of viscous liquids. The needs of aviation have led to the development of a boundary-layer theory in aerohydrodynamics. In the mid-20th century, the mathematical boundary-layer theory developed, as well as applications in the theory of heat transfer, diffusion, processes in semi-conductors, etc.

#### Comments

Although the concept of a boundary layer has its origin in fluid dynamics, the recent mathematical development of this concept, as well as its applications, are in the field of singular perturbations. For a partial review one may consult [a1].

#### References

[a1] | W. Eckhaus, "Boundary layers in linear elliptic singular perturbation problems" SIAM Review , 14 (1972) pp. 225–270 MR0600325 Zbl 0234.35009 |

**How to Cite This Entry:**

Boundary layer.

*Encyclopedia of Mathematics.*URL: http://encyclopediaofmath.org/index.php?title=Boundary_layer&oldid=28156