Mechanics of Fluids

Thus for any value of y the velocity of the fluid may now be expressed by
�
δp∗ �
y2 ��
y δp∗ � �
c µV
u =
− −
δx 2µ µ δx 2 c
�
δp∗ �
1
=
δx 2µ (y2 − cy) + Vy
c
For a fixed breadth b perpendicular to the page, and with provisos as before
concerning end effects, the total volume flow rate Q may be calculated by
the same method as before.
� �
c
�δp∗ � �
1 y
Q = ub dy = b
0
δx 2µ
3
�
cy2
− +
3 2
Vy2
�c 2c
0
� �
δp∗ � �
c3 Vc
= b −
+ (6.23)
δx 12µ 2
It is important to remind ourselves here that V must be measured in the
same direction as the flow. If the movement of the boundary is in fact opposite
to the direction of flow, then, in eqn 6.23, V is negative. The velocity
profile (Fig. 6.10) is modified by the movement of one of the boundary surfaces
and the maximum velocity no longer occurs in the centre of the section.
Indeed, if V is sufficiently large the greatest velocity may occur at the moving
surface as in Fig. 6.11.
Steady laminar flow between moving planes 205
Equation 6.23 shows that flow may occur even without a difference of Couette flow
piezometric pressure provided that one boundary is moving. In these circumstances
δp∗ /δx is zero and Q = 1
2bVc. Such flow, caused only by the
movement of a boundary, is known as Couette flow (after M. F. A. Couette
(1858–1943)). Couette flow, however, is not necessarily laminar.
Fig. 6.10
Fig. 6.11