No Flow in Liquid Container Despite Pressure Difference
Date: Winter 2011-2012
Why is it that a liquid in a container cannot flow from bottom to top even though there is a pressure difference?
The pressure difference for a liquid in a container is due to gravity. The pressure supports the weight of the liquid above it. The pressure at the top supports the atmosphere, countering the atmospheric pressure. The pressure at the bottom of the container supports the atmosphere plus the weight of the liquid within the container. The pressure difference pushes up, but gravity pulls down. The two effects cancel each other, so the liquid just sits still in the container.
Dr. Ken Mellendorf
Illinois Central College
Remember that gravity is pulling the fluid down. This is much larger than the pressure difference.
The stream of water returning from bottom to top,
always has its own pressure difference built-in,
equal and opposite to the pressure difference in the container,
because it is filled with same density of water and has the same height as the column of water in the container.
You can draw your own picture of it:
a square container, 7/8 full of water,
with a recirculating pipe
that comes out of one side 1/8 of the side-length from the bottom,
goes sideways a short distance,
turns 90degrees and goes up to 1/8 from the top,
turns another 90 degrees and goes sideways back to the container,
and then re-joins the container.
The pipe has zero pressure at the top, and [density x height x gravity] pressure at the bottom.
Exactly the same is true in the short wide container.
There is no net force to overcome the inertia or the viscosity of the water.
If the water in the pipe was to flow, would it flow up or down?
The case for each is equal. The net result is no flow.
In college we would have done an integral of the pressure gradient around the whole loop of water,
and shown that the integral is zero.
There are also many such loops within the water in the container,
but they too have zero loop integral.
Heating the water on one side more than the other changes that
to small-but-non-zero loop integral.
Then you get circulating convection flow.
Blowing bubbles in the side-pipe greatly changes the average density,
and a strong flow results.
The reason has to do with energy. When things move spontaneously, it is
because they are moving from a high energy state to a lower-energy
state. In order for spontaneous mixing of a liquid to occur, there
must be some transition from a higher-energy state to a lower-energy
state. In this case, you are talking about water only. If there is no
temperature or chemical gradient in the water, then the system before
and after inversion would have the same energy (you can get
spontaneous mixing if there is hot water below cold water, for
example). With no transfer of energy, there is no driving force to cause
If you think in terms of individual droplets of water, it takes energy
to move a droplet of water up, but it takes an equal amount of energy
to move a different droplet. In other words, the pressure pushing on
the water that might move a droplet up also acts equally to prevent a
higher droplet from moving down. The net effect is zero energy transfer,
and therefore, no spontaneous mixing.
Hope this helps,
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Update: June 2012