Fluids
From the pressure on a diver to the lift on a wing — how substances that flow push, float, and speed up
- The two quantities that matter for fluids — density \(\rho = m/V\) and pressure \(p = F/A\) — and why pressure is a scalar.
- How hydrostatic pressure grows with depth: \(p = p_{0} + \rho g h\), and the difference between absolute and gauge pressure.
- Pascal's principle and the hydraulic lever (\(F_{o} = F_{i}\,A_{o}/A_{i}\)) that magnifies force without magnifying work.
- Archimedes' principle — a buoyant force equal to the weight of displaced fluid (\(F_{b} = m_{f}g\)) — and the rule for floating.
- Ideal-fluid flow: the equation of continuity (\(Av = \text{const}\)) and Bernoulli's equation (\(p + \tfrac{1}{2}\rho v^{2} + \rho g y = \text{const}\)).
What Is Physics?
The physics of fluids underpins hydraulic engineering, which turns up almost everywhere: the cooling system of a nuclear reactor, the blood flow in an aging patient's arteries, the irrigation of farmland, a diver's safety, the wing flaps that let a jet land — even the hydraulics that raise and lower huge Broadway and Las Vegas sets. Before any of that, though, we need to answer a deceptively simple question: what is a fluid?
What Is a Fluid?
A fluid is a substance that can flow and so conforms to the boundaries of any container. The defining feature, in the language of the previous chapter, is that a fluid cannot sustain a shearing stress — a force tangent to its surface. It can push perpendicular to a surface, but it cannot resist being sheared, so it simply flows. Even slow-moving pitch counts as a fluid; it conforms eventually. We lump liquids and gases together because neither has the rigid long-range atomic order of a crystalline solid like ice — liquid water and steam are, at the molecular level, more alike than either is to ice.
Density and Pressure
With a rigid body we track mass and force. With a fluid — an extended substance whose properties vary from point to point — it is far more useful to track density and pressure. Density is mass per unit volume, treated as smooth on scales much larger than atoms:
| Material or object | Density | Material or object | Density |
|---|---|---|---|
| Best laboratory vacuum | 10⁻¹⁷ | Iron | 7.9 × 10³ |
| Air (20 °C, 1 atm) | 1.21 | Mercury | 13.6 × 10³ |
| Ice | 0.917 × 10³ | Earth (average) | 5.5 × 10³ |
| Water (20 °C, 1 atm) | 0.998 × 10³ | Sun (average) | 1.4 × 10³ |
| Seawater (20 °C, 1 atm) | 1.024 × 10³ | Sun (core) | 1.6 × 10⁵ |
| Whole blood | 1.060 × 10³ | Neutron star (core) | 10¹⁸ |
Pressure is the magnitude of the normal force per unit area that a fluid exerts on a surface:
| Location | Pressure | Location | Pressure |
|---|---|---|---|
| Center of the Sun | 2 × 10¹⁶ | Automobile tire (gauge) | 2 × 10⁵ |
| Center of Earth | 4 × 10¹¹ | Atmosphere at sea level | 1.0 × 10⁵ |
| Deepest ocean trench | 1.1 × 10⁸ | Normal blood systolic (gauge) | 1.6 × 10⁴ |
| Spike heels on a dance floor | 10⁶ | Best laboratory vacuum | 10⁻¹² |
Fluids at Rest
Every diver knows pressure rises with depth; every mountaineer knows it falls with altitude. To find the rule, isolate an imaginary cylinder of fluid in static equilibrium and balance the three vertical forces on it — the downward push on its top, the upward push on its bottom, and its own weight. The pressures at the upper and lower faces (at heights \(y_{1}\) and \(y_{2}\), measured positive upward) must differ by exactly the weight of fluid between them:
The total (absolute) pressure at depth \(h\) is the atmospheric pressure \(p_{0}\) pressing on the surface plus the column term \(\rho g h\). The gauge pressure is the excess over atmospheric, \(\rho g h\) — what a tire gauge or blood-pressure cuff reads. It is positive when \(p > p_{0}\) (a tire) and negative when \(p < p_{0}\) (sucking on a straw).
Measuring Pressure
Two classic instruments follow directly from \(p = p_{0} + \rho g h\). A mercury barometer is a tube of mercury inverted over a dish; the near-vacuum above the column means atmospheric pressure alone supports the column height \(h\):
The barometer's reading in millimetres of mercury equals the pressure in torr only at the standard \(g = 9.80665\,\mathrm{m/s^{2}}\) and 0 °C; elsewhere small corrections apply, because the column height depends on both \(g\) and mercury's temperature-dependent density.
Pascal's Principle
Squeeze one end of a toothpaste tube and paste comes out the other end — that is Pascal's principle, stated in 1652: a change in the pressure applied to an enclosed incompressible fluid is transmitted undiminished to every part of the fluid and to the walls of its container. Adding lead shot to a piston raises the pressure by \(\Delta p_{\text{ext}}\) everywhere at once, independent of depth.
A small input force on a small piston produces a large output force on a large piston — the same pressure acting over a bigger area. But the output piston moves a proportionally smaller distance (\(d_{o} = d_{i}\,A_{i}/A_{o}\)), so the work is unchanged: \(W = F_{o}d_{o} = F_{i}d_{i}\). A hydraulic jack lifts a car you could never lift directly — at the cost of many small pumps of the handle.
Archimedes' Principle
Picture a thin sack of water held submerged in a pool: it neither rises nor sinks, so the surrounding water must push up on it with a force equal to the weight of the water inside. That upward push — present because pressure is greater at the bottom of the sack than the top — is the buoyant force. Replace the sack's contents with a stone or a block of wood of the same shape and the surrounding water cannot tell the difference; the buoyant force is unchanged.
A fully or partially submerged body feels an upward buoyant force equal to the weight of the fluid it displaces. If \(F_{b}\) exceeds the body's weight it rises (wood); if less, it sinks (stone).
A body floats when the buoyant force grows to match its weight, \(F_{b} = F_{g}\) — so a floating body displaces its own weight of fluid. When a buoyant force acts, a scale reads less than the true weight; this apparent weight is
Ideal Fluids in Motion
Real flow is fearsomely complex, so we model an ideal fluid with four simplifying assumptions: flow is steady (velocity at a fixed point does not change in time, i.e. laminar not turbulent), incompressible (constant density), nonviscous (no internal friction draining kinetic energy to heat), and irrotational (a speck of dust carried along does not spin about its own center).
The Equation of Continuity
Pinch a garden hose and the water speeds up — the flow speed depends on the cross-sectional area. For a steady, incompressible flow, the same volume that enters a tube segment per second must leave it per second. Equating the volumes gives the equation of continuity:
Bernoulli's Equation
Apply conservation of energy to a tube of flowing ideal fluid — accounting for the work done by pressure pushing fluid in and out, plus the change in gravitational and kinetic energy — and you arrive at Bernoulli's equation, the fluid-flow restatement of energy conservation:
Three competing terms: static pressure, kinetic-energy density \(\tfrac{1}{2}\rho v^{2}\), and gravitational term \(\rho g y\). Set \(v = 0\) and it reduces to the hydrostatic law; set \(y\) constant and it predicts the headline result below.
Putting It to Work
Problem. (a) What does the air in a living room of dimensions 3.5 m × 4.2 m × 2.4 m weigh at 1.0 atm? (b) What downward force does the atmosphere exert on the top of your head (area ≈ 0.040 m²)?
Solution. Use \(\rho = m/V\) with the density of air (1.21 kg/m³), and \(F = pA\) with \(p = 1.0\,\mathrm{atm} = 1.01 \times 10^{5}\,\mathrm{Pa}\).
The 4 kN on your head is the weight of the entire air column above it, balanced by the air pressure pushing up from below — which is why you never feel it.
Problem. A diver fills his lungs at depth \(L\) and (against instructions) holds his breath while surfacing. At the surface the difference between his lung pressure and the outside pressure is 9.3 kPa. From what depth did he start?
Solution. The trapped lung air keeps the pressure it had at depth, while the outside drops to \(p_{0}\). The leftover difference is the gauge pressure \(\rho g L\).
Less than a metre — yet that ~9% of atmospheric pressure is enough to rupture lung tissue and force air into the blood. Exhaling steadily on the way up keeps the pressures equalized and removes the danger.
Problem. A block of density \(\rho = 800\,\mathrm{kg/m^{3}}\) and height \(H = 6.0\,\mathrm{cm}\) floats face-down in a fluid of density \(\rho_{f} = 1200\,\mathrm{kg/m^{3}}\). (a) To what depth \(h\) is it submerged? (b) If pushed fully under and released, what is its acceleration?
Solution. Floating means buoyancy equals weight: \(\rho_{f}LWh\,g = \rho LWH\,g\). Fully submerged, the buoyant force grows (full height displaces fluid), and Newton's second law gives the upward acceleration.
Two-thirds of the block sits below the surface (the density ratio), and when forced under it springs upward at half a \(g\).
Problem. Water leaves a tap and narrows as it falls. The cross-section shrinks from \(A_{0} = 1.2\,\mathrm{cm^{2}}\) to \(A = 0.35\,\mathrm{cm^{2}}\) over a drop of \(h = 45\,\mathrm{mm}\). Find the volume flow rate.
Solution. Continuity (\(A_{0}v_{0} = Av\)) plus free-fall kinematics (\(v^{2} = v_{0}^{2} + 2gh\)) give \(v_{0}\); then \(R_{V} = A_{0}v_{0}\).
The stream necks because gravity speeds it up, and continuity then demands a smaller area to keep the volume flow rate constant.
Problem. Ethanol (\(\rho = 791\,\mathrm{kg/m^{3}}\)) flows through a horizontal pipe that tapers from \(A_{1} = 1.20 \times 10^{-3}\,\mathrm{m^{2}}\) to \(A_{2} = A_{1}/2\). The pressure difference between the wide and narrow sections is 4120 Pa. Find the volume flow rate.
Solution. Combine continuity (\(v = R_{V}/A\), with \(A_{2} = A_{1}/2\)) and Bernoulli on the horizontal pipe. The narrow section is faster, so its pressure is lower — meaning \(p_{1} - p_{2} = +4120\,\mathrm{Pa}\).
The "faster means lower pressure" rule settled the sign: had we guessed the other way, the square root would have gone imaginary.
Chapter Summary
\(\rho = m/V\) and \(p = F/A\). Pressure is a scalar; gauge pressure is the difference from atmospheric.
\(p = p_{0} + \rho g h\) — pressure depends on depth, not container shape. Same level ⟹ same pressure.
Barometer: \(p_{0} = \rho g h\). Open-tube manometer: \(p_{g} = \rho g h\).
An applied pressure transmits undiminished throughout the fluid. Hydraulic lever: \(F_{o} = F_{i}A_{o}/A_{i}\), same work.
Buoyant force \(F_{b} = m_{f}g\) (weight of displaced fluid). Floating: \(F_{b} = F_{g}\); apparent weight = weight − \(F_{b}\).
\(R_{V} = Av = \text{const}\) and \(R_{m} = \rho A v = \text{const}\) — narrow the channel, speed rises.
\(p + \tfrac{1}{2}\rho v^{2} + \rho g y = \text{const}\) — energy conservation for ideal flow; faster flow means lower pressure.
Steady, incompressible, nonviscous, irrotational. Streamlines never cross; velocity is tangent to them.
Problems
Statics problems lean on \(p = p_{0} + \rho g h\) and "same level, same pressure"; buoyancy problems on \(F_{b} = m_{f}g\) and the floating condition; flow problems on continuity and Bernoulli together. Useful values: \(1\,\mathrm{atm} = 1.01 \times 10^{5}\,\mathrm{Pa}\), \(\rho_{\text{water}} \approx 1000\,\mathrm{kg/m^{3}}\), \(g = 9.8\,\mathrm{m/s^{2}}\).
- Find the pressure increase in the fluid in a syringe when a nurse pushes its circular piston (radius 1.1 cm) with a force of 42 N.
- An office window measures 3.4 m by 2.1 m. During a storm the outside pressure drops to 0.96 atm while inside stays at 1.0 atm. What net outward force pushes on the window?
- At a depth of 10.9 km in the Marianas Trench, what hydrostatic pressure (in atmospheres) must a vessel withstand? Take seawater density as 1024 kg/m³.
- Calculate the hydrostatic difference in blood pressure between the brain and the foot of a person 1.83 m tall. Blood density is \(1.06 \times 10^{3}\,\mathrm{kg/m^{3}}\).
- A hydraulic press has a small piston of diameter 3.80 cm and a large one of diameter 53.0 cm. What force on the small piston balances a 20.0 kN force on the large one?
- An iron anchor (density 7870 kg/m³) appears 200 N lighter in water than in air. Find (a) its volume and (b) its weight in air.
- A block of wood floats in fresh water with two-thirds of its volume submerged, and in oil with 0.90 of its volume submerged. Find the density of (a) the wood and (b) the oil.
- What fraction of an iceberg (density 917 kg/m³) is visible when it floats in (a) seawater (1024 kg/m³) and (b) fresh water (1000 kg/m³)?
- A garden hose of internal diameter 1.9 cm feeds a sprinkler with 24 holes, each 0.13 cm in diameter. If the water moves at 0.91 m/s in the hose, how fast does it leave the holes?
- Water enters a basement at 0.90 m/s and 170 kPa through a 2.5 cm pipe, which then tapers to 1.2 cm and rises 7.6 m to the second floor. Find the (a) speed and (b) pressure upstairs.
- A large tank is filled with water to depth \(D = 0.30\,\mathrm{m}\). A hole of area 6.5 cm² in the bottom lets water drain. At what rate (m³/s) does water flow out?
- A cylindrical tank holds water to height \(H = 40\,\mathrm{cm}\); a hole at depth \(h = 10\,\mathrm{cm}\) lets water out. (a) How far from the base does the stream strike the floor? (b) At what depth should a hole give the same range?