Chapter 4: Hydrodynamics

• What a fluid is; the difference between liquids and gases.
• Density and relative density; their applications (car battery, blood, urine).
• Laminar flow and turbulent flow; streamlines; steady flow.
• Volume flow rate, mass flow rate and the continuity equation A₁v₁ = A₂v₂.
• A glimpse of Bernoulli's principle (enrichment).
• Viscosity, the coefficient of viscosity (η = Fd/Av), and applications of viscosity.

A fluid is any material that has the ability to flow and has no shape of its own. Fluids include liquids and gases.

Because liquids are practically incompressible, we treat liquid flow as incompressible laminar flow: the density of the moving liquid does not change.

Density is a basic property of every material: it is the mass of a unit volume of the substance.

ρ = m / Vol

• SI unit: kg/m³ (also g/cm³; 1 g/cm³ = 1000 kg/m³).
• Dimensional formula: ML⁻³.

Why does density differ from one material to another?

1. The atoms or molecules have different masses.
2. The spaces between the atoms or molecules (inter-atomic or intermolecular spaces) differ. (In gases the spaces are huge, so gas densities are small.)

Denser materials sink; objects of lower density float on liquids of higher density.

Typical values: water 1000 kg/m³, sea water ≈ 1030, oil ≈ 800–900, mercury 13 600, aluminium 2700, iron ≈ 7800, gold 19 320, air at room temperature ≈ 1.2 kg/m³.

Relative density of a material = (density of the material at a given temperature) ÷ (density of water at 4 °C).

relative density = ρ_material / ρ_water(4 °C)

Since the density of water at 4 °C is 1000 kg/m³ (1 g/cm³),

relative density = mass of a certain volume of the material ÷ mass of the same volume of water at 4 °C.

• It is a ratio of two quantities of the same kind → it has no unit (and no dimensions).
• Its numerical value equals the density in g/cm³.

Example 1. A liquid of volume 250 cm³ has a mass of 200 g. ρ = 200/250 = 0.8 g/cm³ = 800 kg/m³; relative density = 0.8.

Example 2. Find the mass of 3 litres of mercury (ρ = 13 600 kg/m³). Vol = 3 L = 3 × 10⁻³ m³; m = ρVol = 13 600 × 3 × 10⁻³ = 40.8 kg.

Example 3. An iron block has dimensions 10 cm × 5 cm × 4 cm and mass 1.56 kg. Vol = 200 cm³ = 2 × 10⁻⁴ m³; ρ = 1.56/(2 × 10⁻⁴) = 7800 kg/m³; relative density = 7.8.

Example 4. The relative density of a certain oil is 0.9. Its density is 0.9 × 1000 = 900 kg/m³, and 2 m³ of it has mass 1800 kg.

Measuring density is an important analytical technique in science and in practical life.

1. Charge state of a car battery. The electrolyte is dilute sulphuric acid. When the battery discharges, sulphuric acid is used up in reacting with the lead plates to form lead sulphate, so the density of the electrolyte decreases. When it is recharged, sulphate ions return to the solution from the plates, so the density increases. Measuring the density of the electrolyte (with a hydrometer) therefore tells us how charged the battery is.
2. Medicine.
   • Blood density normally lies between 1040 and 1060 kg/m³. If it is higher, the concentration of blood cells is higher; if it is lower, the concentration of blood cells is lower, which may indicate anaemia.
   • Urine density is normally about 1020 kg/m³. Some diseases increase the amount of salts excreted, so the density of the urine increases.
3. Gold jewellery (critical thinking). Pure (24-carat) gold has a density of about 19.32 g/cm³; 18-carat gold about 15.45 g/cm³ and 14-carat gold about 13.1 g/cm³. Alloys differ in density because gold is mixed with lighter metals (such as copper or silver), which lowers the average density. By measuring the density of a piece of jewellery (mass ÷ volume) we can estimate its carat value.

Liquid flow is classified into laminar and turbulent flow.

• In laminar flow the liquid moves so that adjacent layers slide smoothly and gently over one another.
• Each small portion of liquid follows a continuous path called a streamline.
• A real or imaginary flow tube can be represented by a set of streamlines.

Properties of streamlines

• They never intersect (a point cannot have two velocity directions at the same time).
• The tangent to a streamline at any point gives the direction of the instantaneous velocity of the liquid particles at that point.
• The number of streamlines crossing perpendicularly a unit area at a point is the streamline density; it indicates the flow speed. Streamlines are close together where the speed is high and far apart where it is low.

• When the speed of the liquid exceeds a certain limit, the flow is no longer smooth: small circular eddies (vortices) appear.
• The same happens with gases (smoke rising from a cigarette or incense stick starts as steady streamlines and then becomes turbulent with eddies).
• Turbulence also results from violent change of flow or from the rapid motion of an object through the liquid, or when a gas diffuses from a narrow space to a larger one or from high to low pressure.

Turbulent flow is accompanied by noise: a doctor listens with a stethoscope to detect turbulent blood flow in an artery, which indicates illness.

• Flow is steady if the velocity of the liquid at each point on its path does not change with time (different points may have different velocities).
• Laminar flow may be considered steady flow.

Conditions for steady flow in a pipe

1. The liquid completely fills the pipe.
2. The amount of liquid entering one end in a given time equals the amount leaving the other end in the same time (the flow rate is constant along the tube because the liquid is incompressible).
3. The velocity of the liquid at every point does not change with time.

Consider steady flow of liquid of density ρ in a tube, and take a cross-section of area A at which the speed is v.

• In a time Δt the liquid at that section advances a distance Δx = vΔt, so the volume crossing the section is ΔVol = AΔx = A v Δt.

• Volume flow rate: the volume of liquid flowing through the section per unit time:

  Q_v = A v (unit m³/s)

• Mass flow rate: the mass of liquid passing per unit time:

  Q_m = ρ Q_v = ρ A v (unit kg/s)

Useful consequences:

• Volume flowing in time t: Vol = Q_v t = A v t = π r² v t (for a circular pipe)
• Mass flowing in time t: M = Q_m t = ρ A v t = π r² v ρ t

• Take two cross-sections of the tube, area A₁ with speed v₁ and area A₂ with speed v₂.
• Because the liquid is incompressible and cannot accumulate or disappear in the tube, the mass passing each section per second is the same (conservation of mass):

ρ A₁ v₁ = ρ A₂ v₂ → A₁ v₁ = A₂ v₂ (the continuity equation)

or v₁/v₂ = A₂/A₁.

Derivation (by volumes). In a time Δt the liquid at section 1 moves Δx₁ = v₁Δt, so the volume entering is A₁v₁Δt. The same volume must leave through section 2 (incompressible liquid): A₂v₂Δt. Therefore A₁v₁Δt = A₂v₂Δt, i.e. A₁v₁ = A₂v₂.

Meaning. The speed of the liquid at any point is inversely proportional to the cross-sectional area of the tube at that point:

• Wide section (large A) → slow flow, streamlines far apart.
• Narrow section (small A) → fast flow, streamlines close together.

For circular pipes A = πr² = πd²/4, so v₁/v₂ = (r₂/r₁)² = (d₂/d₁)². Halving the diameter makes the speed four times larger.

Both Q_v (m³/s) and Q_m (kg/s) are constant along the tube.

• Putting a finger over the end of a garden hose makes the water spray faster and farther: the exit area is smaller, so the speed is greater.
• A river flows fast where it is narrow and slowly where it is wide and deep.
• Blood flows fast in the aorta and slowly in the capillaries, because the total cross-section of all the capillaries is far larger than that of the aorta. The slow flow allows time for exchange of oxygen, carbon dioxide and nutrients with the tissues.

Example 1 (pipe). Water flows at 1.5 m/s in a pipe of diameter 4 cm that narrows to a diameter of 2 cm. (a) Find the speed in the narrow part. (b) Find the volume and mass of water passing per minute. (a) A₁v₁ = A₂v₂: (4)² × 1.5 = (2)² × v₂ → v₂ = 16 × 1.5 / 4 = 6 m/s. (b) A₁ = π(0.02)² = 1.257 × 10⁻³ m²; Q_v = A₁v₁ = 1.257 × 10⁻³ × 1.5 = 1.885 × 10⁻³ m³/s. Volume per minute = 1.885 × 10⁻³ × 60 = 0.113 m³. Mass per minute = 1000 × 0.113 = 113 kg. (Q_m = 1.885 kg/s.)

Example 2 (hose and nozzle). Water leaves a hose of inner diameter 1.8 cm at 2 m/s through a nozzle of diameter 0.6 cm. Speed at the nozzle: (1.8/0.6)² × 2 = 9 × 2 = 18 m/s.

Example 3 (blood). The aorta has radius 0.5 cm and blood flows in it at 0.20 m/s. It branches finally into 5000 capillaries (ignoring intermediate vessels) of radius 0.05 cm each. A₁ = π(0.5)² = 0.25π cm²; A₂ = 5000 × π(0.05)² = 12.5π cm². v₂ = A₁v₁/A₂ = 0.25 × 0.20/12.5 = 0.004 m/s. The blood slows down 50 times.

Example 4 (time to fill). A tap delivers water through a pipe of cross-section 1.5 cm² at a speed of 4 m/s. Q_v = 1.5 × 10⁻⁴ × 4 = 6 × 10⁻⁴ m³/s = 0.6 L/s. A 60 L tank fills in 60/0.6 = 100 s.

Example 5 (area from flow rate). Water flows at a steady 0.003 m³/s in a pipe in which its speed is 6 m/s. A = Q_v/v = 0.003/6 = 5 × 10⁻⁴ m² = 5 cm².

Bernoulli's principle

• relates the pressure of a fluid to its speed: where the speed of a fluid increases, its pressure decreases. Application: the wing of an aeroplane has a curved upper surface and a flat lower surface, so air flows faster over the top than below it.
• The pressure above the wing (P₁) is therefore lower than below (P₂); the pressure difference gives an upward lift force that supports the aircraft.

1. Funnels. Pour equal volumes of alcohol and glycerin into two identical funnels. Alcohol flows out faster — alcohol flows more easily than glycerin.
2. Stirring. Stir water and honey with a glass rod. The rod moves more easily in water; after the rod is removed, the motion in water continues for longer, while honey stops almost at once.
3. Falling ball. Drop identical steel balls into two tall cylinders, one with water and one with glycerin. The ball takes longer to reach the bottom in glycerin.

Conclusion.

• Liquids such as water and alcohol have low viscosity: they flow easily and resist the motion of objects through them only slightly.
• Liquids such as honey and glycerin have high viscosity: they flow with difficulty and strongly resist the motion of objects through them.

• Imagine a liquid between two flat plates: the lower plate is fixed, the upper plate moves with velocity v.
• The layer of liquid touching the fixed plate is stationary, and the layer touching the moving plate moves with the same velocity v.
• The layers between move with velocities increasing steadily from 0 to v; each layer moves slower than the one above it.
• This difference in speed is due to: a) Attraction (adhesion) between the molecules of the solid plate and the molecules of the liquid layer next to it, which holds that layer and makes it move with the plate.
• b) Frictional-like forces between neighbouring layers of liquid, which oppose their sliding over one another, producing a difference in velocity between each layer and the next.

Viscosity is the property of a fluid by which resistance (friction) arises between the layers of the fluid, which hinders their sliding over one another.

To keep the upper plate moving at constant velocity v, a force F must be applied equal to the viscous force. Experiments show that F is

• directly proportional to the velocity v of the moving plate,

• directly proportional to the area A of the moving plate (in contact with the liquid), and

• inversely proportional to the distance d between the plates.

  F ∝ A v / d → F = η A v / d

The constant of proportionality η (eta) is the coefficient of viscosity:

η = F d / (A v)

Definition. The coefficient of viscosity of a liquid is the tangential force acting on a unit area that produces a velocity difference of one unit between two layers of the liquid separated by a unit distance (perpendicular to the flow).

• Unit: N·s/m² (= Pa·s) = kg/(m·s).
• Dimensional formula: [F][d]/([A][v]) = MLT⁻² · L / (L² · LT⁻¹) = ML⁻¹T⁻¹.
• η depends on the type of liquid and on its temperature (it falls as a liquid is heated; engine oil flows more easily when hot).

Example 1 — finding η. A plate of area 0.04 m² moves at 0.1 m/s over a layer of oil 2 mm thick when a force of 0.5 N is applied. η = Fd/(Av) = 0.5 × 0.002 / (0.04 × 0.1) = 0.001/0.004 = 0.25 N·s/m².

Example 2 — finding the force. A liquid of η = 0.8 N·s/m² lies between two plates 4 mm apart; the upper plate has area 0.02 m² and moves at 0.05 m/s. F = ηAv/d = 0.8 × 0.02 × 0.05 / 0.004 = 0.2 N.

Example 3 — effect of doubling. If the distance d between the plates is doubled while the same v and area are used, the force is halved (F ∝ 1/d). If the velocity is doubled, the force is doubled.

Example 4 — comparing liquids. To move the same plate at the same velocity needs a force of 0.3 N in oil A and 1.2 N in oil B. Then η_B/η_A = 4: oil B is four times as viscous.

1. Lubrication. Machine parts must be oiled from time to time. Lubrication a) reduces the heat produced by friction, and b) protects the parts from wear. The lubricant must have high viscosity: it sticks to the moving parts and does not run off quickly. Water (low viscosity) would flow away from the parts at once.
2. Moving vehicles. When a car reaches its maximum speed, most of the engine's work is used to overcome air resistance (due to the viscosity of air) and tyre friction. At low or moderate speeds air resistance is proportional to the speed; above a certain speed it becomes proportional to the square of the speed, so fuel consumption rises rapidly. A skilled driver therefore limits the speed to save fuel.
3. Medicine — the blood sedimentation rate. A ball falling in a liquid is acted on by three forces: its weight (down), the buoyant force (up) and the viscous drag (up). The ball reaches a constant final (terminal) velocity that increases with the radius of the ball. In the same way, red blood cells in a blood sample settle at a rate that depends on their size:
   • In rheumatic fever the red cells stick together to form larger clumps (larger radius), so the sedimentation rate is higher than normal.
   • In anaemia the red cells break down and become smaller, so the rate is lower than normal.

• Choosing engine oil: the viscosity of oil must suit the engine's working temperature and clearances; oil that is too thin cannot form a protective film, wears the parts, and leaks; too thick oil resists the motion, wastes energy and flows poorly when the engine is cold.
• Drip irrigation / pipe networks: designers use the continuity equation to choose pipe diameters so that the desired flow rate and pressure reach every outlet.

• Fluids flow: liquids and gases; ρ = m/Vol (kg/m³); relative density = ρ/ρ_water (no unit).
• Laminar flow: smooth layers, streamlines that never cross; turbulent flow: eddies at high speed.
• Steady flow: tube full, same amount in and out, velocity at each point constant.
• Q_v = Av (m³/s); Q_m = ρAv (kg/s); Vol = Avt = πr²vt; M = ρAvt.
• Continuity equation: A₁v₁ = A₂v₂ (v inversely proportional to A).
• Bernoulli (enrichment): faster flow → lower pressure (lift on a wing).
• Viscosity: resistance between layers; η = Fd/(Av), unit N·s/m² = kg/(m·s), dimensions ML⁻¹T⁻¹; high-viscosity oils for lubrication; used in the blood sedimentation test.