Fluids and Pressure: Density, Buoyancy and Flow

A kilogram of feathers and a kilogram of lead weigh the same, but the lead takes up far less room. Density measures how tightly mass is packed:

ρ = m ÷ V

The symbol ρ is the Greek letter rho. Fresh water has a density of about 1,000 kg/m³, which is the same as 1.00 g/cm³: one liter has a mass of one kilogram. Seawater is a little denser, about 1,025 kg/m³, because of the dissolved salt. Air at room temperature is only about 1.2 kg/m³. Ice, at about 917 kg/m³, is less dense than liquid water, which is unusual and is why lakes freeze from the top down.

Density belongs to a material, not to a particular object. Cut a gold bar in half and each piece still has a density of about 19,300 kg/m³. That's why density helps identify substances: Archimedes is said to have used the idea to test whether a king's crown was pure gold.

Press a thumbtack into a board and the sharp end goes in while the flat end doesn't hurt your thumb. The force is the same at both ends; the area is very different. Pressure is force spread over area:

P = F ÷ A

Pressure is measured in pascals (Pa), named after Blaise Pascal. One pascal is tiny, about the pressure of a sheet of paper lying flat on a table, so you'll often see kilopascals (kPa). Sharp knives, needles and stiletto heels concentrate a force on a small area to make a large pressure. Snowshoes, tractor tires and wide building foundations spread a force out to make the pressure small.

The air around us presses on everything too. Atmospheric pressure at sea level is about 101 kPa, which is like a 10-tonne mass resting on every square meter. We don't feel crushed because the fluids inside our bodies push back out just as hard. Climb a mountain and there is less air above you, so the pressure falls. That's why a sealed bag of chips puffs up on a mountain drive or a flight, where the cabin is kept at lower pressure than at sea level.

The deeper you go in a fluid, the more fluid is stacked above you, and the more it weighs. The extra pressure from a column of fluid of depth h is

P = ρgh

where g = 9.8 N/kg. In water, every 10 m of depth adds about 98 kPa, close to one more atmosphere. A diver at 10 m feels about twice the pressure at the surface. At the bottom of the Mariana Trench, nearly 11 km down, the pressure is more than 1,000 times sea-level air pressure.

Fluid pressure behaves in three ways that can seem odd. It pushes in every direction at once, sideways and upward as well as down. It depends only on depth and density, not on how wide the container is, so a narrow tube of water 10 m tall presses on its base just as hard as a lake 10 m deep, which is why dams are built thickest at the bottom. And it pushes at right angles to every surface it touches.

The total pressure under water is the air pressure on the surface plus ρgh. Tire gauges and blood pressure cuffs read gauge pressure, the amount above atmospheric pressure.

Lift a friend in a swimming pool and they seem to weigh much less. The water is helping. An object in a fluid has its bottom deeper than its top, so the fluid pushes up on the bottom harder than it pushes down on the top. The difference is an upward buoyant force. Archimedes' principle says how big it is:

F_b = weight of fluid displaced = ρ_fluidVg

Here V is the volume of fluid the object pushes aside, which is the part of the object that's under the surface. Every object in a fluid feels this force, including ones that sink. A rock seems lighter under water; it just isn't light enough to float.

Whether something floats depends on comparing densities. If an object's average density is less than the fluid's, it floats, sinking only until it has displaced its own weight of fluid. If it's denser, it sinks. A steel ship floats because its hull holds a lot of air, so its average density is less than seawater's. Ice, at 917 kg/m³, floats in seawater with about 90% of its volume under the surface, which is why most of an iceberg is out of sight. Submarines and fish adjust how much they float by taking in or letting out water or gas.

Squeeze a sealed tube of toothpaste anywhere and paste comes out the other end. In 1653 Blaise Pascal described the rule behind this. Pascal's principle: a change in pressure applied to a trapped fluid is passed on undiminished to every part of the fluid and to the walls of its container.

That makes liquids into force multipliers. In a hydraulic lift, a small force F₁ on a small piston of area A₁ creates a pressure F₁ ÷ A₁. The same pressure acts on a large piston of area A₂, so

F₂ = F₁ × A₂ ÷ A₁

If the large piston has 50 times the area, the force out is 50 times bigger. Energy is still conserved: the small piston has to move 50 times farther than the large one rises. Car brakes, dentist's chairs, excavators and aircraft controls all use hydraulics. Liquids work well because they barely compress; air in brake lines makes the pedal feel soft, because the air squashes instead of passing the pressure on.

Put your thumb over the end of a garden hose and the water shoots out faster. Liquids barely compress, so the same volume has to pass every point in a pipe each second. That gives the continuity equation:

A₁v₁ = A₂v₂

Where the pipe narrows, the fluid speeds up. In 1738 Daniel Bernoulli showed what happens to the pressure. To speed up, the fluid needs a push from behind, so the pressure in the slow, wide part must be higher than in the fast, narrow part. Bernoulli's principle: along a smooth flow, where the speed is higher, the pressure is lower.

You can see it with a sheet of paper: hold one edge under your lip, let it droop and blow across the top. The paper rises. Venturi meters use a narrow section of pipe to measure flow speed, and perfume atomizers and old carburetors use the low pressure in a fast air stream to draw up liquid. Bernoulli's principle is also part of how wings make lift, together with the way a wing turns air downward. One popular explanation, that air over the top must rejoin air under the bottom at the same moment, is wrong: the air over the top actually arrives first.

Archimedes worked out the rule for floating more than two thousand years ago, but most of what we know about fluid pressure came in the 1600s and 1700s. Torricelli showed that air has weight, Pascal showed that pressure falls with height and explained hydraulics, and Bernoulli and Leonhard Euler wrote the first mathematics of moving fluids.