Flow Modeling - Environmental Modeling - page 215

The mean velocity is given by:

v mean ¼ D z 2
12

D p
D x

and the flux per unit width by:

q ¼ D z 3
12

D p
D x

The equation states a linear relationship between fluid flux q and the
pressure gradient

D p= D x . So far, one-dimensional flow between two plates
in cartesian coordinates has been studied as most simple situation. one-
dimensional flow within a pipe can be treated similarly using a two-dimen-
sional coordinate system. Instead of the z -coordinate, the radial coordinate r
has to be considered, and the differential equation for v x

in cylinder

coordinates takes the form:

@r v x

1
r

@
@r

r @ þ D p

D x ¼

0

In analogy to the procedure demonstrated just before one obtains the
solution:

r 2
R 2

v x ¼ v max 1

where R denotes the radius of the pipe. The paraboloid shape is depicted
in Fig. 11.4 .

R 2
4

D p
D x

v max ¼

v

Fig. 11.4 Laminar flow of
paraboloid shape in a pipe
with circular cross-section

( continued )