Page 390 - Introduction to Continuum Mechanics

374 Plane-Poiseuille Flow

We shall now prove that in the presence of gravity and independent of the inclination of
the channel, the Poiseuille flow always has the velocity profile given by Eq. (6.12,3),
Let kbe a unit vector pointing upward in the vertical direction, then the body force is:

and the components of the body force in the jq^2 and #3 directions are:

Let rbe the position vector of a fluid particle and let y be its vertical coordinate. Then

and

Now, using Eq. (vii) we can write the body force components Eq. (v) as follows:

Thus, the Navier-Stokes equations can be written

These equations are the same as Eqs. (6.12.1) except that the pressure/? is replaced by
+
P Pgy- From these equations, one clearly will obtain the same parabolic velocity profile, except
that the driving force in this case is the gradient ofp+pgy in the flow direction , instead of
simply the gradient of/7. We note that [p l(pg) + y] has been defined in Example 6.7.2 as the
piezometric head. We can also say that the driving force is the gradient of the piezometric head
and the piezometric head is a constant along any direction perpendicular to the flow.

6.13 Hagen-Poiseuille Flow

The so-called Hagen-Poiseuille flow is a steady unidirectional axisymmetric flow in a
circular cylinder. Thus, we look for the velocity field in cylindrical coordinates in the following
form