Mechanical Properties of Fluids: NEET notes

1. Fluids and pressure

NCERT §9.1, §9.2

• Liquids and gases can flow, so both are called fluids. Flowing is the basic property that separates them from solids.
• A fluid has no shape of its own. A solid or a liquid keeps a fixed volume (at atmospheric pressure), while a gas spreads to fill whatever container holds it.
• Solids and liquids have far lower compressibility than gases: a change of outside pressure alters their volume only slightly.
• Fluids resist shear stress very weakly: a tiny shear stress changes their shape. Their shearing stress is roughly a million times smaller than that of solids.
• The same force does more damage when it is spread over a smaller area: a needle pierces skin, the back of a spoon does not. A wide plank on a performer's chest protects him from an elephant's foot.
• A fluid at rest pushes on any surface in contact with it at right angles to that surface. A parallel component would, by Newton's third law, make the fluid flow along the surface, which a fluid at rest cannot do.
• Average pressure P_av = F/A, with F the normal force on area A. Taking the area to zero gives the pressure at a point: P = lim (ΔF/ΔA) as ΔA → 0.
• Pressure is a scalar: only the normal component of force appears in its definition. Dimensions [ML⁻¹T⁻²]; SI unit N m⁻², named the pascal (Pa) after Blaise Pascal.
• 1 atm is the pressure of the atmosphere at sea level: 1 atm = 1.013 × 10⁵ Pa.
• Density ρ = m/V, dimensions [ML⁻³], unit kg m⁻³, a positive scalar. A liquid's density hardly changes with pressure; a gas's density changes a great deal.
• Water at 4 °C (277 K) has density 1.0 × 10³ kg m⁻³. Relative density = density ÷ density of water at 4 °C, a pure number. Aluminium: relative density 2.7, so ρ = 2.7 × 10³ kg m⁻³.
• Two thighbones, each 10 cm² in cross-section, holding up 40 kg of upper body (g = 10 m s⁻²): A = 20 × 10⁻⁴ m², F = 400 N, so the average pressure is 2 × 10⁵ N m⁻².

2. Pascal's law and pressure with depth

NCERT §9.2.1, §9.2.2

• Pascal's observation: in a fluid at rest, any two points at one common height share one common pressure.
• Proof with a tiny right-angled prism of fluid: the normal forces balance (F_b sin θ = F_c, F_b cos θ = F_a) and the face areas obey the same relations (A_b sin θ = A_c, A_b cos θ = A_a), so P_a = P_b = P_c.
• So pressure at a point acts equally in every direction; like other stresses it has no direction. The force on any surface inside a resting fluid is normal to that surface, whatever its orientation.
• A horizontal bar of fluid in equilibrium must have equal pressures at its two ends. Unequal pressures in a horizontal plane would drive a flow, so without flow the pressure is uniform across any horizontal plane.
• A vertical cylinder of fluid, base area A and height h: (P₂ − P₁)A = mg and m = ρhA, so P₂ − P₁ = ρgh.
• With point 1 at an open surface, P₁ = P_a (atmospheric) and P = P_a + ρgh. The excess P − P_a = ρgh at depth h is the gauge pressure.
• Neither the area nor the shape of the container appears in P = P_a + ρgh: only the height of the fluid column matters.
• Hydrostatic paradox: vessels A, B and C of different shapes, joined at the bottom by a horizontal pipe, fill to the same level although they hold different amounts of water, because the pressure at the bottom is the same under each.
• Swimmer 10 m below a lake surface (ρ = 1000 kg m⁻³, g = 10 m s⁻²): P = 1.01 × 10⁵ + 1000 × 10 × 10 = 2.01 × 10⁵ Pa ≈ 2 atm, double the surface value. At 1 km depth the rise is about 100 atm, which submarines must be built to withstand.

3. Atmospheric and gauge pressure

NCERT §9.2.3

• Atmospheric pressure at a point is the weight of the air standing above it, per unit area: a column of 1 m² section running up to where the air ends. At sea level it is 1.013 × 10⁵ Pa (1 atm).
• Torricelli's mercury barometer: fill a long glass tube, sealed at its lower end, with mercury and turn it upside down into a dish of mercury. The space above the column holds only mercury vapour at negligible pressure, so P_A = 0.
• Point B inside the column and point C on the open trough surface are at the same level, so both are at P_a. Hence P_a = ρgh, with ρ the density of mercury and h the column height.
• At sea level the column stands about 76 cm high, which is 1 atm. Pressure is often quoted in cm or mm of mercury; 1 mm of Hg is called a torr (after Torricelli), and 1 torr = 133 Pa.
• Doctors and physiologists quote pressures in torr or mm of Hg. Weather science uses the bar and millibar: 1 bar = 10⁵ Pa.
• Open-tube manometer: a U-tube of liquid, one end open to air and the other joined to the system being tested. The level difference h gives the gauge pressure P − P_a = ρgh. Oil (low density) suits small pressure differences, mercury (high density) large ones.
• Pressure is equal at the same level in both arms of a U-tube holding one fluid at rest. Liquid density barely changes with pressure and temperature, so liquids are treated as incompressible; gas density changes a lot.
• If air kept its sea-level density 1.29 kg m⁻³ all the way up: ρgh = 1.01 × 10⁵ Pa with g = 9.8 m s⁻² gives h = 7989 m ≈ 8 km. In reality both air density and g fall with height and the atmosphere thins out over more than 100 km.
• Sea-level pressure is not always 760 mm of Hg: a fall of 10 mm or more in the barometer points to an approaching storm.
• 1000 m down in the sea (ρ = 1.03 × 10³ kg m⁻³, g = 10 m s⁻²): absolute pressure 104.01 × 10⁵ Pa ≈ 104 atm; gauge pressure 103 × 10⁵ Pa ≈ 103 atm. A submarine kept at 1 atm inside feels only the gauge pressure on a 20 cm × 20 cm window (0.04 m²): F = 4.12 × 10⁵ N.
• Tyre-pressure gauges and the blood-pressure gauge (sphygmomanometer) read gauge pressure.

4. Hydraulic machines

NCERT §9.2.4

• Push the piston of a horizontal cylinder fitted with three vertical tubes, and the liquid climbs to the same new height in every tube: the extra pressure reaches every part of the liquid.
• Second form of Pascal's law: a pressure change applied to any part of an enclosed fluid is passed on, without loss and equally in all directions, to every point of the fluid and the vessel walls.
• Hydraulic lift: a small piston of area A₁ pushed with force F₁ sets up pressure P = F₁/A₁. The same P acts on a large piston of area A₂, giving an upward force F₂ = PA₂ = F₁A₂/A₁.
• The force is multiplied by A₂/A₁, the mechanical advantage of the device. Changing F₁ raises or lowers the platform carrying a car or truck.
• Liquid is incompressible, so the volume swept in by the small piston equals the volume swept out by the large one: A₁L₁ = A₂L₂. The large piston moves a shorter distance, in the ratio A₁/A₂.
• Two water-filled syringes joined by a rubber tube, piston diameters 1.0 cm and 3.0 cm: 10 N on the small one gives 10 × (3/1)² = 90 N on the large one. Pushing the small piston in 6.0 cm moves the large one out 6.0/9 ≈ 0.67 cm. Atmospheric pressure acts on both and cancels.
• Car lift with pistons of radius 5.0 cm and 15 cm, car mass 1350 kg, g = 9.8 m s⁻²: F₁ = 1350 × 9.8 × (5/15)² = 1470 N ≈ 1.5 × 10³ N. The air pressure needed, F₁/(π × 0.05²) ≈ 1.9 × 10⁵ Pa, is almost twice atmospheric pressure.
• Hydraulic brakes: a light push on the pedal moves the master piston; brake oil carries the pressure to larger pistons that press the brake shoes against the lining. The same pressure reaches the cylinders at all four wheels, so the braking effort is equal on every wheel.

5. Streamline flow and continuity

NCERT §9.3

• Fluid dynamics studies fluids in motion. A tap opened slowly gives a smooth stream; opened fully, the stream loses its smoothness.
• Steady flow: at any fixed point the velocity of each fluid particle passing it stays the same in time. Velocities may differ from point to point, but every particle reaching a given point behaves like the one before it.
• A streamline is the path of a fluid particle in steady flow: a curve whose tangent at every point gives the direction of the fluid velocity there. The map of streamlines does not change with time.
• Two streamlines never cross: at a crossing a particle would have two possible velocities and the flow could not be steady.
• A tube of flow is bounded by the same streamlines, so the mass crossing sections P, R and Q in time Δt is the same: ρ_P A_P v_P Δt = ρ_R A_R v_R Δt = ρ_Q A_Q v_Q Δt.
• For an incompressible fluid the densities are equal and A_P v_P = A_R v_R = A_Q v_Q, or Av = constant. This is the equation of continuity, a statement of conservation of mass.
• Av is the volume flux (flow rate). Where the tube narrows and streamlines crowd together the speed rises; where it widens the speed falls. The fluid speeds up passing from a wide section R to a narrow section Q.
• Steady flow needs low speeds. Above a critical speed the flow becomes turbulent, like the foamy whirls of white-water rapids where a fast stream meets rocks.
• Laminar flow: velocities at different points may differ in size but point in parallel directions. A jet of air hitting a flat plate held across it gives turbulent flow.

6. Bernoulli's principle

NCERT §9.4, §9.4.1

• In a pipe of varying cross-section and height, continuity forces the fluid's speed to change. The acceleration needs a net force from the surrounding fluid, so the pressure must differ from region to region.
• Daniel Bernoulli (1738) related the pressure difference between two points to the change in speed (kinetic energy) and the change in height (potential energy), using conservation of energy.
• Work done on a slug of fluid of volume ΔV: P₁ΔV at the inlet end minus P₂ΔV at the outlet end, (P₁ − P₂)ΔV. It supplies ΔK = ½ρΔV(v₂² − v₁²) and ΔU = ρgΔV(h₂ − h₁).
• Dividing by ΔV: P₁ + ½ρv₁² + ρgh₁ = P₂ + ½ρv₂² + ρgh₂, or P + ½ρv² + ρgh = constant along a streamline.
• In words: along a streamline, pressure + kinetic energy per unit volume (½ρv²) + potential energy per unit volume (ρgh) stays constant.
• Assumptions: no energy lost to internal friction (zero viscosity), an incompressible fluid (no elastic energy), and steady flow. It fails for turbulent or non-steady flow, where speed and pressure fluctuate. It still works well for low-viscosity, incompressible fluids.
• Fluid at rest (v = 0 everywhere): Bernoulli's equation reduces to P₁ − P₂ = ρg(h₂ − h₁), the same as the result for pressure with depth.
• Horizontal pipe: where the fluid moves faster, its pressure is lower. A narrow throat therefore has lower pressure than the wide part feeding it.
• Torricelli's law (efflux means outflow): tank of liquid, small side hole at height y₁, surface at y₂ with pressure P above it. If the tank area is much larger than the hole, the surface is nearly at rest, and v₁ = √[2gh + 2(P − P_a)/ρ] with h = y₂ − y₁.
• When P ≫ P_a and 2gh can be ignored, the container pressure alone sets the outflow speed, as in rocket propulsion.
• Tank open to the air (P = P_a): v₁ = √(2gh), the same speed as a body falling freely through h.

7. Dynamic lift

NCERT §9.4.2

• Dynamic lift is the force on a body such as an aeroplane wing, a hydrofoil or a spinning ball because it moves through a fluid.
• In cricket, tennis, baseball and golf a spinning ball swerves away from its parabolic path; Bernoulli's principle explains this in part.
• Ball without spin: the streamlines are symmetric, so air moves equally fast above and below at matching points. No pressure difference, no vertical force from the air.
• Ball with spin: the spinning ball drags air round with it (more if its surface is rough). Relative to the ball the air above moves faster and the air below slower; streamlines crowd above and spread below.
• The slower air below is at higher pressure, giving a net upward force. This lift caused by spin is the Magnus effect.
• An aerofoil is a solid shaped to give upward lift when it moves horizontally through air; aircraft wing sections look roughly like one. Its shape and tilt crowd the streamlines above it, so air flows faster on top than underneath and the lift balances the plane's weight.
• Loaded aircraft of mass 3.3 × 10⁵ kg, wing area 500 m², level flight at 960 km/h, air density 1.2 kg m⁻³: ΔP = (3.3 × 10⁵ × 9.8)/500 = 6.5 × 10³ N m⁻².
• Ignoring the small height difference, ΔP = ½ρ(v₂² − v₁²), so (v₂ − v₁)/v_av = ΔP/(ρv_av²). With v_av = 960 km/h = 267 m s⁻¹ this is ≈ 0.08: air above the wing need move only about 8% faster than below.

8. Viscosity and Stokes' law

NCERT §9.5, §9.5.1

• Real fluids resist motion through a kind of internal friction between layers moving relative to one another. This is viscosity.
• Oil between two glass plates, lower plate fixed, upper plate moved at constant velocity v: honey in place of oil needs a bigger force for the same v, so honey is more viscous.
• The fluid touching a surface moves with that surface: the top layer at v, the bottom layer at rest, with speed rising uniformly between. Each layer is pulled forward by the one above and back by the one below; this is laminar flow, like pages of a book sliding when the cover is pushed.
• In a pipe the layer along the axis moves fastest and the speed falls to zero at the walls; the speed is constant over any cylindrical surface coaxial with the tube.
• A shape ABCD in the fluid becomes AEFD after a time Δt, a shear strain Δx/l that keeps growing. Unlike a solid, the stress depends on the strain rate Δx/(lΔt) = v/l, not on the strain itself.
• Coefficient of viscosity η = shearing stress ÷ strain rate = (F/A)/(v/l) = Fl/(vA). SI unit poiseuille (Pl), also N s m⁻² or Pa s; dimensions [ML⁻¹T⁻¹].
• Viscosities (mPl), thin liquids low and thick ones high; blood's viscosity relative to water stays constant from 0 °C to 37 °C: water 1.0 at 20 °C and 0.3 at 100 °C; blood 2.7 at 37 °C; machine oil 113 at 16 °C and 34 at 38 °C; glycerine 830 at 20 °C; honey 200; air 0.017 at 0 °C and 0.019 at 40 °C.
• The viscosity of a liquid falls as temperature rises (its molecules become more mobile); the viscosity of a gas rises (random motion increases).
• Block of area 0.10 m² on a 0.30 mm liquid film, pulled by a 0.010 kg hanging mass over an ideal pulley, slides at a steady 0.085 m s⁻¹: F = 9.8 × 10⁻² N, shear stress 0.98 N m⁻², strain rate 0.085/(0.30 × 10⁻³) s⁻¹, η = 3.46 × 10⁻³ Pa s.
• Stokes' law: a sphere of radius a moving at speed v through a fluid of viscosity η feels a drag F = 6πηav, opposite to its motion and proportional to its speed. Raindrops and swinging pendulum bobs meet this drag.
• A falling sphere speeds up until viscous drag plus buoyancy equals its weight; then it falls at a constant terminal velocity: 6πηa v_t = (4π/3)a³(ρ − σ)g, so v_t = 2a²(ρ − σ)g/(9η), with ρ the sphere's density and σ the fluid's. v_t grows as the square of the radius and falls inversely with the viscosity.
• Copper ball, radius 2.0 mm, falling through oil at 20 °C with terminal velocity 6.5 cm s⁻¹ (ρ = 8.9 × 10³, σ = 1.5 × 10³ kg m⁻³, g = 9.8 m s⁻²): η = 2a²(ρ − σ)g/(9v_t) = 9.9 × 10⁻¹ kg m⁻¹ s⁻¹.

9. Surface energy and surface tension

NCERT §9.6, §9.6.1, §9.6.2

• Oil and water do not mix; water wets people but not ducks; mercury does not wet glass while water clings to it; oil climbs a cotton wick; sap reaches the top of a tree; a paint brush forms a fine tip only when lifted out of water. All of these come from free liquid surfaces.
• A liquid has a definite volume but no definite shape, so in a container it forms a free surface. That surface carries extra energy; the effect is surface tension. Gases have no free surface, so surface tension belongs to liquids.
• A molecule deep inside a liquid is attracted by all its neighbours, giving it a negative potential energy. The large heat needed to evaporate a liquid shows how deep this is: for water about 40 kJ/mol.
• A molecule at the surface has neighbours only on the liquid side, so its negative potential energy is roughly half that of an interior molecule. Surface molecules therefore have extra energy, and a liquid takes the least surface area its surroundings allow.
• Bringing a molecule to the surface costs roughly half the energy needed to remove it from the liquid entirely, about half the heat of evaporation. The surface is not razor-sharp: density falls to zero over a few molecular sizes.
• A horizontal film on a frame with a sliding bar of length l: moving the bar a distance d adds area 2dl, because the film has two surfaces. The work Fd becomes surface energy: S(2dl) = Fd, so S = F/2l.
• Surface tension S is the surface energy per unit area of the interface, and equally the force per unit length acting in the plane of the interface. It is the extra energy of interface molecules compared with interior ones.
• Across any line drawn on a liquid surface, equal and opposite forces S per unit length act perpendicular to the line and in the plane of the surface. At a true edge of the interface only the inward force S per unit length remains.
• The surface energy belongs to the interface between two materials and depends on both: mutual attraction lowers it, repulsion raises it.
• Surface tension (N/m) with heat of vaporisation (kJ/mol): helium 0.000239 at −270 °C (0.115); oxygen 0.0132 at −183 °C (7.1); ethanol 0.0227 at 20 °C (40.6); water 0.0727 at 20 °C (44.16); mercury 0.4355 at 20 °C (63.2).
• Surface tension usually falls as temperature rises, as viscosity does for liquids.
• Measuring it: a vertical glass plate hangs from one arm of a balance with its lower edge just above the liquid. The raised liquid touches and pulls the plate down; the extra weight W needed to just free it gives S_la = W/2l = mg/2l, with l the length of the plate edge.

10. Angle of contact, drops and bubbles

NCERT §9.6.3, §9.6.4

• A liquid sticks to a solid if the solid-liquid surface energy is less than the sum of the solid-air and liquid-air surface energies.
• Angle of contact θ: where liquid meets solid, draw the tangent to the liquid surface; θ is the angle from the solid surface to that tangent, measured through the liquid. It depends on the particular liquid-solid pair.
• θ decides whether a liquid spreads or forms droplets: water beads up on a lotus leaf but spreads over a clean plastic plate.
• Balancing the three interfacial tensions at the line of contact: S_la cos θ + S_sl = S_sa.
• θ is obtuse when S_sl > S_la (water on a leaf, water on wax or oil, mercury on any surface): the liquid's molecules attract one another strongly and the solid's weakly, so the liquid does not wet the solid.
• θ is acute when S_sl < S_la (water on glass or plastic, kerosene on almost anything): strong liquid-solid attraction lowers S_sl, cos θ grows and the liquid spreads.
• Soaps, detergents and dyeing substances are wetting agents that make θ small so they penetrate well. Waterproofing agents make θ large between water and fibres.
• For a given volume, the sphere has the least surface area, so free drops and bubbles are spherical when gravity and air resistance can be ignored.
• Pressure inside a drop exceeds that outside. Growing a drop of radius r by Δr costs surface energy 8πrΔr S_la and gains work (P_i − P_o)4πr²Δr, so P_i − P_o = 2S_la/r.
• Across any curved liquid-gas surface the concave side is at the higher pressure. An air bubble (cavity) inside a liquid has one surface: excess pressure 2S/r.
• A soap bubble in air has two surfaces, inner and outer: P_i − P_o = 4S_la/r. That is why blowing a soap bubble needs a little extra pressure, but not too much.
• Capillary of diameter 2.00 mm dipped 8.00 cm into water (S = 7.30 × 10⁻² N m⁻¹, g = 9.80 m s⁻², 1 atm = 1.01 × 10⁵ Pa), hemispherical bubble of radius 1.00 mm at its end: P_o = 1.01 × 10⁵ + 0.08 × 1000 × 9.8 = 1.01784 × 10⁵ Pa; excess pressure 2S/r = 146 Pa; P_i = 1.02 × 10⁵ Pa to three significant figures.

11. Capillary rise

NCERT §9.6.5

• Water climbs a narrow tube against gravity; capilla is Latin for hair, and the thinner the tube the higher the climb.
• Water meets glass at an acute angle, so the water surface in a capillary of radius a is concave. The surface is part of a sphere of radius r = a sec θ, giving a pressure difference 2S/r = (2S/a) cos θ across it.
• The water just below the meniscus is therefore below atmospheric pressure. Points A (inside the tube, level with the outer surface) and B (on the outer surface) must be at equal pressure, which lifts a column of height h.
• hρg = 2S cos θ / a, so h = 2S cos θ/(ρga). The rise comes from surface tension and is larger for a narrower tube, typically a few cm for fine capillaries.
• Water in a tube of radius a = 0.05 cm with S = 0.073 N m⁻¹ (θ taken as 0): h = 2 × 0.073/(10³ × 9.8 × 5 × 10⁻⁴) = 2.98 × 10⁻² m = 2.98 cm.
• If the meniscus is convex, as for mercury in glass, cos θ is negative and the liquid stands lower inside the capillary than outside: capillary depression.

Must-know facts

1. Pressure P = F/A (normal force); scalar; unit Pa = N m⁻²; dimensions [ML⁻¹T⁻²].
2. 1 atm = 1.013 × 10⁵ Pa (1.01 × 10⁵ in the summary); 1 bar = 10⁵ Pa; 1 torr = 1 mm of Hg = 133 Pa.
3. Density of water at 4 °C = 1.0 × 10³ kg m⁻³; relative density is a pure number.
4. Pascal's law: equal pressure at equal heights in a fluid at rest; applied pressure is transmitted undiminished to every point.
5. P = P_a + ρgh; gauge pressure P − P_a = ρgh; container shape does not matter (hydrostatic paradox).
6. Barometer: P_a = ρgh, about 76 cm of mercury at sea level.
7. Hydraulic lift: F₂ = F₁A₂/A₁, mechanical advantage A₂/A₁; distances A₁L₁ = A₂L₂.
8. Syringes 1.0 cm and 3.0 cm diameter: 10 N becomes 90 N; 6.0 cm in gives 0.67 cm out.
9. Streamlines never cross in steady flow; above a critical speed flow turns turbulent.
10. Continuity: Av = constant for an incompressible fluid (conservation of mass).
11. Bernoulli: P + ½ρv² + ρgh = constant along a streamline (energy conservation, ideal fluid, steady flow).
12. Torricelli: v = √(2gh) from an open tank, the free-fall speed.
13. Magnus effect: a spinning ball gets lift because air moves faster on one side than the other.
14. Aircraft example: ΔP = 6.5 × 10³ N m⁻², upper-surface air only about 8% faster.
15. η = (F/A)/(v/l); unit Pa s = N s m⁻² = poiseuille; dimensions [ML⁻¹T⁻¹].
16. Liquid viscosity falls with temperature; gas viscosity rises.
17. Stokes: F = 6πηav; terminal velocity v_t = 2a²(ρ − σ)g/(9η) ∝ a².
18. Surface tension S = F/l per surface = surface energy per unit area; unit N m⁻¹ = J m⁻²; water 0.0727 N m⁻¹ at 20 °C.
19. Surface tension usually falls with temperature.
20. Angle of contact: acute for water-glass (wets), obtuse for mercury-glass (does not wet); S_la cos θ + S_sl = S_sa.
21. Excess pressure: drop or air bubble in a liquid 2S/r; soap bubble in air 4S/r.
22. Capillary rise h = 2S cos θ/(ρga); 2.98 cm for water in a 0.05 cm-radius tube; mercury is depressed.

Common traps

Formulas

Key terms

Fluid

A substance that can flow: a liquid or a gas.

Pressure

Normal force per unit area; a scalar measured in pascals.

Relative density

Density of a substance divided by the density of water at 4 °C.

Gauge pressure

Pressure above atmospheric pressure, P − P_a.

Hydrostatic paradox

Differently shaped connected vessels fill to the same level because only depth sets the pressure.

Torr

Pressure of 1 mm of mercury, 133 Pa.

Steady flow

Flow in which the velocity at each fixed point does not change with time.

Streamline

The path of a fluid particle in steady flow; its tangent gives the local velocity.

Turbulent flow

Unsteady, irregular flow that sets in above the critical speed.

Equation of continuity

Av = constant: conservation of mass for an incompressible fluid.

Magnus effect

Sideways or upward force on a spinning ball moving through a fluid.

Viscosity

Internal friction between layers of a fluid in relative motion.

Terminal velocity

Constant speed of a body falling through a fluid when drag and buoyancy balance its weight.

Surface tension

Force per unit length, or energy per unit area, of a liquid's interface.

Angle of contact

Angle inside the liquid between the liquid surface and the solid at their line of contact.

Capillary rise

The climb of a wetting liquid in a narrow tube, driven by surface tension.