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Figure 14. Sketch of the viscous boundary layer on a flat plate.

where is the Reynolds number (and we have used subscripts throughout to denote
partial derivatives). We will consider the problem of steady flow with
the boundary conditions for uniform flow at infinity are

which imply that the pressure p constant away from the plate (and we will not
analyse the nature of the flow near x = 0); the plate will extend to infinity
The presence of the small parameter multiplying the highest derivatives, is the
hallmark of a boundary-layer problem. In particular, the (inviscid) problem can satisfy
v = 0 on y = 0, but not as (in x > 0), so we expect a boundary-layer
scaling in y; see figure 14.
Outside the boundary layer, the solution is written

and so equations (5.109) give

subject to the boundary conditions (5.110a,b,d), for zero-subscripted variables; this
has the solution

(It is clear that this solution has additional problems for on y = 0, where the
stagnation point exists at the leading edge of the plate.) Note that this solution can be
expressed in terms of the stream function: (where, in general,

The region of the boundary layer is described by the scaled variable where
as and x is unscaled. (We would need to scale x near excep-
tional points such as the leading edge, a point of separation and the trailing edge of a