Physics 125

# Physics 125, Fall 1995 - Lecture 21: Fluids: Bouyancy and Surface Tension

The term fluids applies to both liquids and gases. the liquid state is less tightly bound than the solid state of the a substance. Liquids generally take the shape of their container, but are subject to the forces due to surface tension. Most liquids can be treated as incompressible, that is the density depends only slightly on applied pressure. And liquids have viscosity, a kind of friction that produces drag in a moving fluid or on an object moving through a liquid.

A gas is a collection of essentially unbound molecules. A gas is compressible, that is the density can be readily changed by applying pressure. Gases can be contained but do not have appreciable surface tension.

## Pressure

The pressure in a fluid is the normal force the fluid applies on any surface such as the container walls. Consider a flat bottomed glass. When the glass is empty, or more accurately filled with air, the force the bottom of the glass exerts on the air is equal in magnitude to the force that the entire column of air exerts on the bottom of the glass: . To demonstrate this, place a piece of newspaper on top of a meter stick or rod on a table. Try to rapidly lift the paper by way of the rod. You will meet resistance because, in essence, you must lift a column of air many kilometers high. In fact, the air squishes around and the actual result of your experience will depend on how fast you try to lift the rod, but the meaning of pressure should be obvious.

The mass of the column of air above the glass is not uniform in density. In fact it is most dense at the surface of the earth. As the altitude increases, the density of the air decreases exponentially. At an altitdude of 10 km, about the height of Mount Everest the pressure of air is about 28% (210 mm of Hg) and at 20 km the air pressure is 5.5% (42 mm Hg) of that at sea level. The air pressure at sea level is about 10 Pascals (Pa). The unit of pressure is Pascals (Pa) where 1 Pa = 1 N/m.

Another way to be convinced of the pressure of the air is to use a manometer. This device is a U-tube with an incompressible liquid in it, often mercury (Hg). If both ends of the U-tube are open to the atmosphere, the level of Hg is equal on the two sides. We can evacuate one side of the U-tube so that there is no air pressure. The level of Hg is higher on the evacuated side by about 0.76 m (760 mm or about 30 inches). This column of Hg is supported by the force of the air on the side open to the atmosphere. The force is given by the density and volume of Hg:

where

is the cross sectional area of the tube. The pressure of the atmosphere is

Now consider an addditional mass of water sitting on the bottom of the glass. That is, fill the glass. The force exerted on the glass is increased by an amount equal to

. The pressure is the normal force per unit area:

The expression for the extra pressure due to the water is meaningful if water is an incompressible fluid, that is when

is a constant. The air is not incompressible. The deeper the water, the greater the pressure. This is common experience. At 3 meters, about 10 feet, below the surface of a swimming pool, the pressure exceeds atmospheric pressure by

Pa. You can definitely feel this in your ears before equalizing pressure by blowing with your nose plugged.

The term ``gauge pressure'' corresponds to the pressure above normal atmospheric pressure, that is

. Thus gauge pressure of the atmosphere at sea level is ZERO, and gauge pressure at a depth of 3 meters is 30 kPa.

Try this: put a drinking straw into a cup full of your favorite drink. Seal the top of the straw with your finger, and lift the straw out of the cup. Note that the level of liquid in the straw remains. Why? This is how a pipette works.

## Pascal's Principle

A remarkable aspect of fluids is that the pressure is uniformly distributed throughout the fluid. Thus the force exerted on the walls of a fluid filled container is related to the pressure by

where

is the force exerted normal or perpendicular to the surface of area

. This, combinded with their relative incompressibility, makes fluids amazingly useful in hydraulics. Consider a device made up of a tube with very small cross section at one end and very large cross section at the other end. An excess pressure

at the small end requires a normal force

At the large end, this can produce a force

Thus the ratio of cross sections

provides a mechanical advantage that is often utilized, for example in an automobile lift.

## Buoyancy

When an object is placed in a contianer filled with fluid, the fluid exerts a force on the object due to the pressure. Consider a flat bottomed boat that floats on water. The water exerts an upward force proportional to the are of the boat's bottom and to the pressure at the bottom of the boat, . Therefore the upward force is proportional to the product of the area and the depth or draught of the boat, that is the upward force is proportional to the volume of the water displaced by the boat. The force is equal to , that is it is equal to the ``weight'' of the fluid displaced.

Such buoyant forces exist for all fluids, that is for liquids and gases. A helium filled balloon is buoyed upward because the total weight of the filled balloon is less that the weight of the air displaced. A boat floats because the weight of the water displaced is equal to the weight of the boat.

Sunken objects, that is objects that weight more than the fluid displaced, are subject to buoyant forces. Consider this: when an anchor is thrown out of a boat, does the level of water in the lake go up or down?

## Surface Tension

Liquids are characterized by cohesive forces due to interatomic forces. In the bulk of the liquid these forces lead to incompressibility and viscosity. At the interface between a liquid and some other material (solid, liquid or gas) another kind of force exists that is a manifestation of the two dimensional nature of the surface. Surface tension is a force per unit length. A soap bubble forms on a hoop because of the forces between the liquid soap and the material of the hoop.

Positive surface tension indicates that the force between a surface and substrate is attractive. In some cases, a fluid is repelled from a substrate and the surface tension is negative. Capillary action results from positive surface tension. The meniscus formed between water and a glass, or between mercury and a glass is due to surface tension.