Page 149 - Bird R.B. Transport phenomena

§4.4 Flow near Solid Surfaces by Boundary-Layer Theory 133

Fig. 4.3-5. Potential flow into a rectangular channel
with separation, as calculated by H. von Helmholtz,
Phil. Mag., 36, 337-345 (1868). The streamlines for
^ = ±TT separate from the inner surface of the channel.
The velocity along this separated streamline is con-
У = 0 stant. Between the separated streamline and the wall
is an empty region.
Х Р = -7 Г

X = -

of flow. Potential-flow analyses are not useful in the separated region. They can, how-
ever, be used upstream of this region if the location of the separation streamline is known.
Methods of making such calculations have been highly developed. Sometimes the posi-
tion of the separation streamline can be estimated successfully from potential-flow the-
ory. This is true for flow into a channel, and, in fact, Fig. 4.3-5 was obtained in this way. 9
For other systems, such as the flow around the cylinder, the separation point and separa-
tion streamline must be located by experiment. Even when the position of the separation
streamline is not known, potential flow solutions may be valuable. For example, the flow
field of Ex. 4.3-1 has been found useful for estimating aerosol impaction coefficients on
cylinders. 10 This success is a result of the fact that most of the particle impacts occur near
the forward stagnation point, where the flow is not affected very much by the position of
the separation streamline. Valuable semiquantitative conclusions concerning heat- and
mass-transfer behavior can also be made on the basis of potential flow calculations ig-
noring the separation phenomenon.
The techniques described in this section all assume that the velocity vector can be
written as the gradient of a scalar function that satisfies Laplace's equation. The equation
of motion plays a much less prominent role than for the viscous flows discussed previ-
ously, and its primary use is for the determination of the pressure distribution once the
velocity profiles are found.

§4.4 FLOW NEAR SOLID SURFACES
BY BOUNDARY-LAYER THEORY
The potential flow examples discussed in the previous section showed how to predict
the flow field by means of a stream function and a velocity potential. The solutions for
the velocity distribution thus obtained do not satisfy the usual "no-slip" boundary con-
dition at the wall. Consequently, the potential flow solutions are of no value in describ-
ing the transport phenomena in the immediate neighborhood of the wall. Specifically,
the viscous drag force cannot be obtained, and it is also not possible to get reliable de-
scriptions of interphase heat- and mass-transfer at solid surfaces.
To describe the behavior near the wall, we use boundary-layer theory. For the descrip-
tion of a viscous flow, we obtain an approximate solution for the velocity components in
a very thin boundary layer near the wall, taking the viscosity into account. Then we
"match" this solution to the potential flow solution that describes the flow outside the

H. von Helmholtz, Phil Mag. (4), 36, 337-345 (1868). Herman Ludwig Ferdinand von Helmholtz
9
(1821-1894) studied medicine and became an army doctor; he then served as professor of medicine and
later as professor of physics in Berlin.
10
W. E. Ranz, Principles oflnertial Impaction, Bulletin #66, Department of Engineering Research,
Pennsylvania State University, University Park, Pa. (1956).