Fluids in four verbs

Density tells you how much mass is packed into each cubic metre.

Seen itOil floats on water because it is less dense.

TrapEqual masses do not give the simple average. The equal-mass answer is always the smaller one. Relative density has no unit.

Pressure is force shared over an area. The same push on a smaller area gives a bigger pressure.

Seen itA sharp knife cuts, but a blunt one doesn't. A camel's wide feet don't sink into sand.

TrapPressure is a scalar. At a point it is the same in every direction. The force a fluid at rest exerts on a surface is always normal (perpendicular) to that surface.

The deeper you go, the more liquid sits on top of you, so the pressure rises by ρgh. Only the vertical depth matters.

Seen itDams are built thicker at the bottom.

TrapThe shape of the vessel and the amount of liquid do not matter. In a connected liquid at rest, all points at the same level have the same pressure.

We live at the bottom of an ocean of air. A pressure gauge reads only the pressure above atmospheric.

Seen itA drinking straw works because the atmosphere pushes the drink up the straw.

TrapGas laws and bubble problems need absolute pressure. When atmosphere pushes on both sides (dam walls, lids), use gauge pressure.

Press on a trapped liquid and the extra pressure reaches every part of it undiminished. A small force on a small piston can lift a big load.

Seen itHydraulic brakes, car-service lifts and JCB arms.

TrapIf diameters are given, square their ratio to get the force ratio. A hydraulic lift multiplies force, not work: the small piston moves much farther.

A liquid pushes up with a force equal to the weight of the liquid the body pushes aside.

Seen itA steel ship floats, and only the tip of an iceberg shows above the sea.

TrapUse the submerged volume, not the whole volume. Once a body is fully under the surface, the buoyant force doesn't change with depth.

Liquid can't pile up inside a pipe. Where the pipe narrows, the liquid has to speed up.

Seen itPut your thumb over a hose and the water shoots out faster.

TrapHalving the diameter cuts the area to ¼, so the speed becomes 4 times larger, not 2 times.

Energy per unit volume stays constant along the flow, so wherever a fluid moves faster, its pressure is lower.

Seen itStorms blow roofs off. Two boats moving side by side are pulled towards each other.

TrapBernoulli applies only to ideal flow: incompressible, non-viscous, steady, and along a streamline. It is the law of energy conservation.

Water comes out of a hole as fast as a stone would be moving after falling from the water surface down to the hole.

Seen itOn a leaking tank, water from the lower holes spurts out faster.

TrapMeasure h down from the free surface, not up from the bottom. The efflux speed doesn't depend on the liquid's density.

Narrow the pipe and measure how much the pressure drops. The size of the drop tells you how fast the fluid flows.

Seen itPerfume sprayers, carburettors and Bunsen burners all use this.

TrapThe throat has the highest speed and the lowest pressure. A venturimeter needs continuity and Bernoulli together.

Air moves faster over the top of a wing, so the pressure on top is lower and the wing is pushed up.

Seen itAeroplane wings, and a spinning cricket ball that swerves (the Magnus effect).

TrapA spinning ball moves towards the side where its spin adds to the airflow. The air is faster there, so the pressure is lower. Reverse the spin and it curves the other way.

In an accelerating tank, the water surface tilts until it is perpendicular to the “effective gravity”.

Seen itTea sloshes to the back of the cup when a bus starts.

TrapThe surface rises at the back, on the side opposite the acceleration. In a lift, the fraction of a floating block under water does not change, because its weight and the buoyancy scale together.

Viscosity is friction between liquid layers as they slide over each other. Honey has a lot of it; water has a little.

Seen itHoney pours slowly, and engine oil keeps metal parts from rubbing.

TrapWhen heated, liquids become less viscous but gases become more viscous. Convert poise to Pa·s (÷ 10) before you substitute.

Flow through a thin tube depends on the fourth power of its radius.

Seen itWhen an artery narrows a little, blood flow drops a lot.

TrapBecause of the fourth power, halving r divides Q by 16. The speed at the axis is twice the mean speed.

A falling ball speeds up until drag plus buoyancy balance its weight. After that it falls at a constant speed, the terminal velocity.

Seen itRaindrops land gently instead of hitting like bullets.

TrapUse (ρ − σ), never ρ alone. If σ > ρ, as for an air bubble, v_t points upward.

One number tells you whether a flow moves in calm layers or chaotic swirls. It compares inertia with viscosity.

Seen itSmoke from an incense stick rises smoothly at first, then breaks into swirls.

TrapHigher viscosity gives a lower Re, so the flow stays streamline up to a higher speed. Re is a pure number with no dimensions.

A liquid surface behaves like a stretched elastic skin. T is the pull on each metre of its edge.

Seen itA pond skater walks on water, a steel needle can float, and raindrops are round.

TrapA soap film has two surfaces, so F = 2TL. T decreases as temperature rises. Soap lowers T; dissolved salt (NaCl) raises it.

Making new surface costs energy: T joules for every square metre of new surface.

Seen itBlowing a big soap bubble takes effort.

TrapWhen drops merge, the area shrinks and energy is released, which warms the drop. Splitting a drop needs work. Count both surfaces of a film.

The contact angle shows whether a liquid “likes” a solid. Below 90° it spreads out and wets the solid; above 90° it beads up.

Seen itWater beads up on a lotus leaf or a raincoat but spreads out on clean glass.

TrapDetergents lower θ, so the liquid wets better. Waterproofing agents make θ obtuse.

A curved surface squeezes whatever is inside it. The smaller the radius, the larger the extra pressure inside.

Seen itA balloon is hardest to blow at the very start, when it is smallest.

TrapIf two soap bubbles are connected, air flows from the smaller one into the larger one. A soap bubble has two surfaces; an air bubble inside a liquid has only one.

In a thin tube, the surface skin pulls water upward. The narrower the tube, the higher the water climbs.

Seen itOil climbs a lamp wick, blotting paper soaks up ink, and water rises through soil and plants.

TrapIf the tube is shorter than h, water does not overflow; the meniscus just flattens. In a tilted tube the vertical height stays h, and the length along the tube is h/cos α.