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Calculating Viscous Flow: Velocity Profiles in Rivers and
Pipes
Michael Fowler, UVa
6/12/06
Introduction
In this lecture, we’ll derive the velocity distribution for two examples of laminar flow.
First
we’ll consider a wide river, by which we mean wide compared with its depth (which we take to
be uniform) and we ignore the more complicated flow pattern near the banks. Our second
example is smooth flow down a circular pipe.
For the wide river, the water flow can be thought
of as being in horizontal “sheets”, so all the water at the same depth is moving at the same
velocity.
As mentioned in the last lecture, the flow can be pictured as like a pile of printer paper
left on a sloping desk: it all slides down, assume the bottom sheet stays stuck to the desk, each
other sheet moves downhill a little faster than the sheet immediately beneath it.
For flow down a
circular pipe, the laminar “sheets” are hollow tubes centered on the line down the middle of the
pipe. The fastest flowing fluid is right at that central line.
For both river and tube flow, the drag
force between adjacent small elements of neighboring sheets is given by force per unit area
( )
dv z
F
A
dz
η
=
where now the
z
-direction means perpendicular to the small element of sheet.
A Flowing River: Finding the Velocity Profile
For a river flowing steadily down a gentle incline under gravity, we’ll assume all the streamlines
point in the same direction, the river is wide and of uniform depth, and the depth is much smaller
than the width.
This means almost all the flow is well away from the edges (the river banks), so
we’ll ignore the slowing down there, and just analyze the flow rate
per meter of river width
,
taking it to be uniform across the river.
The simplest basic question is: given the slope of the land and the depth of the river, what is the
total flow rate?
To answer, we need to find the speed of flow
v
(
z
) as a function of depth (we know the water in
contact with the river bed isn’t flowing at all), and then add the flow contributions from the
different depths (this will be an integral) to find the total flow.
The function
v
(
z
) is
called the
“velocity profile”.
We’ll prove it looks something like this:

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