Mechanical Properties of Fluids: common doubts, answered
The questions students ask most often about Mechanical Properties of Fluids, each with a short answer. For the full chapter, read the Mechanical Properties of Fluids notes.
Fluids and pressure
Read this section in the notes →Is pressure a vector or a scalar?
Pressure is a scalar. Although it is defined from a force, the force it produces always acts perpendicular to whatever surface it presses on, whichever way that surface faces. Pressure at a point has no direction of its own; the direction comes from the surface, not from the pressure.
Why does a sharp knife cut more easily than a blunt one?
Because pressure is force divided by area, P = F/A. A sharp edge has a tiny contact area, so the same push produces a much larger pressure on the material being cut. A blunt edge spreads the force over a wider area, lowering the pressure. The same idea explains why snowshoes and wide tyres help avoid sinking into soft ground.
Pascal's law and pressure with depth
Read this section in the notes →Does the shape of a container affect the pressure at its bottom?
No. Pressure at a depth h in a liquid depends only on the depth, the liquid's density and g, through P = P_a + ρgh. A wide tank and a narrow tube filled to the same height have the same pressure at the bottom. This surprising result is sometimes called the hydrostatic paradox.
What does Pascal's law state?
Pascal's law says that a pressure change applied anywhere in a confined fluid reaches every other point of the fluid, and the container walls, in full. In addition, all points at the same horizontal level in a fluid at rest share one pressure. Hydraulic lifts and brakes rely on this to transmit a push from a small piston to a large one.
Atmospheric and gauge pressure
Read this section in the notes →What is the difference between gauge pressure and absolute pressure?
Absolute pressure is the total pressure at a point, including the atmosphere. Gauge pressure is the excess above atmospheric pressure, so absolute pressure = atmospheric pressure + gauge pressure. Tyre gauges and blood pressure readings show gauge pressure. At depth h in water, the gauge pressure is ρgh.
Why is mercury used in barometers instead of water?
Because mercury is about 13.6 times denser than water, so the column needed to balance atmospheric pressure is only about 76 cm tall. A water barometer would need a column over 10 m high, which is impractical. Mercury also does not wet glass, so it does not cling to the tube walls.
Hydraulic machines
Read this section in the notes →Does a hydraulic lift give more work than you put in?
No. It multiplies force, not energy. The small piston must move through a much larger distance, since A₁L₁ = A₂L₂, so the work done on the small piston equals the work done by the large piston, ignoring losses. Force is gained at the cost of distance.
How is force multiplied in a hydraulic lift?
The same pressure acts on both pistons, so F₂ = F₁A₂/A₁. The force ratio equals the area ratio, which is the square of the diameter ratio. A large piston with ten times the diameter of the small one gives a hundredfold increase in force, not tenfold.
Streamline flow and continuity
Read this section in the notes →Why does water flow faster when you squeeze a hose pipe?
Because of the equation of continuity, A₁v₁ = A₂v₂. For an incompressible liquid, the same volume must pass every cross-section each second, so narrowing the opening makes the water speed up. The product of area and speed stays constant along the flow.
What is the difference between streamline and turbulent flow?
In streamline flow, each fluid particle follows a smooth path and particles passing the same point all have the same velocity there, so streamlines never cross. In turbulent flow the motion is irregular, with eddies and swirls. Flow becomes turbulent at high speeds, and simple results such as Bernoulli's equation then no longer apply.
Bernoulli's principle
Read this section in the notes →Is pressure higher or lower where a pipe narrows?
Lower. In the narrow part the fluid moves faster, and by Bernoulli's principle a higher speed comes with a lower pressure at the same height. Many students expect the squeeze to raise pressure, but the fluid has to be pushed faster from a higher-pressure region into a lower-pressure one.
When does Bernoulli's equation apply?
It applies to steady, streamline flow of an incompressible fluid with negligible viscosity, along a single streamline. It is an energy statement: P + ½ρv² + ρgh stays constant. It fails for turbulent flow and when viscous losses are large, because then mechanical energy is lost to heat.
What is Torricelli's law?
It gives the speed of liquid flowing out of a small hole in an open tank: v = √(2gh), where h is the depth of the hole below the free surface. This is the same speed a body gains falling freely through height h. It follows from Bernoulli's equation when the tank is wide, so the surface falls very slowly.
Dynamic lift
Read this section in the notes →How does an aeroplane wing produce lift?
The wing is shaped and tilted so that air flows faster over its top surface than under it. By Bernoulli's principle, faster air means lower pressure, so the pressure below the wing exceeds that above. This pressure difference acting on the wing area produces an upward force, called dynamic lift.
Why does a spinning ball swing in the air?
A spinning ball drags air around with it, so on one side the air moves faster relative to the ball and on the other side slower. The pressure is lower on the faster side, so the ball is pushed sideways towards it. This effect, called the Magnus effect, makes a spinning ball curve away from a straight path.
Viscosity and Stokes' law
Read this section in the notes →What happens to viscosity when a liquid or gas is heated?
The viscosity of a liquid falls as it is heated, while the viscosity of a gas rises. In liquids, heating weakens the forces between molecules, so layers slide more easily. In gases, viscosity comes from molecules carrying momentum between layers, and faster molecules at higher temperature do this more.
What is terminal velocity?
It is the constant speed a body reaches when falling through a viscous fluid, once the viscous drag and buoyancy together balance its weight. Since drag grows with speed, the body stops accelerating at that point. For a small sphere, v_t = 2a²(ρ − σ)g/(9η), so larger and denser spheres fall faster.
Surface energy and surface tension
Read this section in the notes →Why are raindrops and soap bubbles spherical?
Surface tension makes a liquid surface behave like a stretched skin that tries to shrink to the smallest area possible. For a given volume, a sphere has the least surface area, so small drops and bubbles become spherical when gravity is too weak to flatten them. Larger drops are flattened because their weight matters more.
Angle of contact, drops and bubbles
Read this section in the notes →Why is the excess pressure in a soap bubble 4S/r and not 2S/r?
Because a soap bubble has two surfaces, an inner and an outer one, each with surface tension. A liquid drop or an air cavity inside a liquid has only one surface, giving excess pressure 2S/r. The bubble's double film doubles this to 4S/r. Smaller bubbles therefore have greater excess pressure inside.
Capillary rise
Read this section in the notes →Why does water rise in a capillary tube but mercury falls?
Because water wets glass, making an acute angle of contact, so the surface curves upward and surface tension pulls the column up. Mercury does not wet glass; its angle of contact is obtuse, cos θ is negative, and the level is pushed down. The height follows h = 2S cos θ/(ρga), where a is the tube's radius.
Why is capillary rise greater in a narrower tube?
Because the rise h = 2S cos θ/(ρga) is inversely proportional to the tube's radius a. The upward pull of surface tension depends on the circumference, while the weight lifted depends on the cross-sectional area, which grows faster with radius. So a thinner tube supports a taller column. Use the radius, not the diameter.
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