Mechanics of Fluids

must be zero so as to meet the requirement of no slip. Thus u = 0atthe
lower plane where y = 0, and substituting these values in eqn 6.18 gives
B = 0. Further, u = 0 at the upper plane where y = c and the substitution
of these values gives
�
δp∗ �
c
A =
δx 2
Inserting the values of A and B in eqn 6.18 and rearranging gives the value
of u at any distance y from the lower plane:
u = 1
�
δp∗ �
(y
2µ δx
2 − cy) (6.19)
As shown in Fig. 6.7, the velocity profile is in the form of a parabola with
its vertex (corresponding to the maximum velocity) mid-way between the
planes as is to be expected from symmetry. Putting y = c/2 in eqn 6.19
gives the maximum velocity as −(c2 /8µ)(δp∗ /δx), the minus sign indicating,
not that the fluid flows backwards, but that (δp∗ /δx) is negative, that is,
p∗ decreases in the direction of flow.
The total volume flow rate between the two planes may readily be calculated.
Consider an elemental strip, of thickness δy and fixed breadth b,
perpendicular to the page. The breadth b is assumed very large so that end
effects associated with it are negligible. In other words, there is a region,
at each end of the strip, in which the velocity u may be somewhat different
from the velocity in the centre; if b is sufficiently large this region is small
compared with the total breadth, and the velocity u may then fairly be taken
as the average velocity across the breadth. With this proviso, the discharge
through the strip is ub δy and the total discharge is:
� c � c �
1 δp∗ �
Q = ub dy = b
(y
0
0 2µ δx
2 − cy)dy
= b
�
δp∗ �
2µ δx
�
y3 �c cy2
−
3 2
0
=− bc3
12µ
�
∂p∗ �
∂x
Steady laminar flow between parallel planes 201
(6.20)
Dividing this result by the area of the cross-section bc gives the mean velocity
as −c 2 (δp ∗ /δx)/12µ which is two-thirds of the maximum value.
Fig. 6.7