11-15-99
Sections 10.10 - 10.13
Real-life fluids
Real-life fluids, like air, water, oil, blood, shampoo, or anything like that, often don't perfectly obey the fairly straight-forward Bernoulli's principle, and in some cases Bernoulli's principle doesn't really come close to describing the behavior of real-life fluids when they're flowing in real-life situations. Even static fluids exhibit unusual behavior, particularly associated with surface tension. We should get away from our ideal world (at least for one day!) and get into some more realistic situations.
Viscosity
The viscosity of a fluid is basically a measure of how sticky it is. Water has a fairly low viscosity; things like shampoo or syrup have higher viscosities. Viscosity also depends on temperature : engine oil, for instance, is much less viscous at high temperatures than it is in a cold engine in the middle of winter.
For fluids flowing through pipes, the viscosity produces a resistive force. This resistance can basically be thought of as a frictional force acting between parts of the fluid that are traveling at different speeds. The fluid very close to the pipe walls, for instance, travels more slowly than the fluid in the very center of the pipe.
Poiseuille's equation
The equation that governs fluid flowing through a pipe or tube is known as Poiseuille's equation. It accounts for the fluids viscosity, although it really is valid only for streamline (non-turbulent) flow. Blood flowing through blood vessels in the human body isn't exactly streamline, but applying Poiseuille's equation in that situation is a reasonable first approximation, and leads to some interesting implications.
For blood, the coefficient of viscosity is about 4 x 10-3 Pa s.
The most important thing to notice about the volume rate of flow is how strongly it depends on the radius of the tube. The flow rate is proportional to r4, so a relatively small change in radius can produce a significant change in flow. Decreasing the radius by a factor of two, for instance, reduces the flow rate by a factor of 16! A This is why it's so important to worry about cholesterol levels, or to worry about other things that can clog the arteries in our bodies - even a minor change in the size of the blood vessels can have a significant impact on the rate at which blood is pumped around our bodies, as well as on how much work our hearts have to do to move that blood around.
Surface tension
You've probably noticed the interesting behavior that can take place at the surfaces of liquids. According to Archimedes' principle, for instance, a steel needle should sink in water. A needle placed carefully on water, however, can be supported by the surface tension - the liquid responds in a way similar to a stretched membrane. Try it at home - see what you can get to float on water.
One way to think of surface tension is in terms of energy. The larger the surface, the more energy there is. To minimize energy, most fluids assume the shape with the smallest surface area This is why small drops of water are round, for instance - a sphere is the shape with the minimum suface area for a given volume. Soap bubbles also tend to form themselves into shapes with minimal surface area.
It takes work to increase the surface area of a liquid. The surface tension can be defined in terms of this work W, as follows:
If you have a thin film of fluid, and try to stretch it, the film
resists. The surface tension can also be defined as the force F per
unit length L tending to pull the surface back :
Water is often used for cleaning, but the surface tension makes it hard
for water to penetrate into small crevices or openings, such as are
found in clothes. Soap is added to water to reduce the surface tension,
so clothes (or whatever else) get much cleaner.
