Fluid Mechanics · Unit 1 · Lesson 3 of 13
Who Figured Out How Fluids Behave?
Harappan cities had working drains 4,500 years ago, but the full equations of fluid flow arrived only in the 1800s. Why did it take so long?
1. Real-World Problem
Every city needs water in and waste water out. People solved that by trial and error for thousands of years. Explaining why their canals and pipes worked needed mathematics that did not exist yet, and then experiments to test that mathematics.
2. Predict
Make your guess before you try the experiment. You will see the answer after the next step.
Predict
3. Experiment
Scroll through the timeline from the Indus Valley to modern computers. Use the Practice and Theory chips to see who built things and who wrote equations.
Open the Torricelli card and drag the water height h. Compare the ideal jet speed \(\sqrt{2gh}\) with the speed of a real jet from a sharp-edged hole.
4. Understand
The Main Milestones
| When | Who or where | What they found or built |
|---|---|---|
| 1960s onwards | Computers | Flow equations solved numerically: computational fluid dynamics (CFD), now used alongside wind tunnels and lab tests |
| 1941 | Andrey Kolmogorov | A theory of how energy cascades in turbulence |
| 1904 | Ludwig Prandtl | Boundary layer: the thin layer near a surface where viscosity matters. It joined theory and experiment |
| 1883 | Osborne Reynolds | Dye experiment showing smooth (laminar) and chaotic (turbulent) flow; the Reynolds number |
| 1839 to 1841 | Gotthilf Hagen, Jean Poiseuille | Laws of smooth flow in narrow pipes |
| 1822 and 1845 | Claude-Louis Navier, George Stokes | Navier-Stokes equations: motion of real fluids, including viscosity |
| 1757 | Leonhard Euler | Equations of motion for an ideal (frictionless) fluid |
| 1738 | Daniel Bernoulli | Hydrodynamica: link between pressure and speed in a flowing fluid |
| 1687 | Isaac Newton | Laws of motion, and the idea that fluid friction grows with how fast layers slide |
| 1640s to 1650s | Blaise Pascal | Pressure in a fluid is passed on equally; the pascal (Pa) is named after him |
| 1643 to 1644 | Evangelista Torricelli | Mercury barometer, and the speed of water leaving a hole in a tank |
| About 1500 | Leonardo da Vinci | Detailed sketches of eddies and waves; noticed rivers flow faster where they are narrow |
| 312 BC onwards | Roman engineers | Aqueducts carrying water many kilometres by gentle slopes |
| About 250 BC | Archimedes | Principle of buoyancy; the screw pump |
| About 2500 BC | Indus Valley (Mohenjo-daro, Harappa) | Covered drains and the Great Bath, built from experience |
The pattern: for thousands of years people built first and explained later. From the 1700s, mathematicians wrote elegant equations for ideal fluids that often disagreed with real pipes and canals. Prandtl's boundary layer in 1904 brought theory and experiment together. From the 1960s, computers made it possible to solve the flow equations numerically for real shapes. That greatly widened what engineers can analyse, but CFD results are still checked against experiments before they are trusted.
5. Derive
Torricelli's Law
Step 1. Take a small mass m of water at the surface of a large tank, a height h above a hole. As it falls to the hole it loses potential energy mgh.
Step 2. If no energy is lost to friction, all of it becomes kinetic energy at the hole: ½mv² = mgh.
Step 3. The mass cancels:
v = \sqrt{2 g h}v is the ideal jet speed (m/s), h the height of the water surface above the hole (m). The same result comes from Bernoulli between the tank surface and the jet, both at atmospheric pressure: the water at the hole is pushed out by the pressure ρgh of the water above it.
6. Calculate
Water stands 1.25 m above a small hole. Ideal jet speed: v = √(2 × 9.81 × 1.25) = √24.5 = 4.95 m/s. A real sharp-edged hole gives about 0.975 of this: about 4.83 m/s.
7. Apply
- The unit of pressure, the pascal, honours Pascal.
- The Reynolds number is still one of the first things engineers check when designing pipe flow.
- Cars, aircraft and turbines are designed with a combination of CFD and physical tests (wind tunnels, test rigs); each kind of result is used to check the other.
- Torricelli's speed is used to estimate how fast a tank drains, and appears in the Lesson 6 and Lesson 11 simulations.
8. Validate
On the Torricelli card set h = 1.25 m: ideal 4.95 m/s, measured about 4.8 m/s. The actual jet velocity is slightly below the ideal prediction because of energy losses, represented by the coefficient of velocity C_v (about 0.97 to 0.98). Jet contraction is a separate effect: it reduces the jet's flow area, and so the discharge, through the coefficient of contraction C_c (about 0.62 for a sharp-edged hole). Together they give the discharge coefficient C_d = C_c × C_v ≈ 0.6, the value used in the Lesson 11 tank simulation. This is the lesson of the whole timeline in one example: theory gives the form of the answer, experiment supplies the correction.
Easy To Mix Up
- Bernoulli's equation is from Daniel Bernoulli, not his father Johann, though both worked on fluids.
- The Navier-Stokes equations had two authors, 23 years apart.
- CFD does not replace experiments. Every CFD model has assumptions, and they are tested against measurements.
9. Quiz
Four quick questions: a concept, a calculation and an interpretation. Each answer comes with an explanation. Score 3 or more to mark this lesson completed.
Check Your Understanding
Four questions: a concept, a calculation and an interpretation. Every answer comes with an explanation. Score 3 of 4 to mark the lesson completed.
Pour water into a glass and it takes the glass's shape. Put a stone in and it does not. What is happening inside each one?
Go to Lesson 4: Why Does Water Take The Shape Of Its Glass? →Read More
- Cengel and Cimbala, "Introduction and Basic Concepts" (brief history of fluid mechanics); Rouse and Ince, History of Hydraulics; Bansal, Fluid Mechanics and Hydraulic Machines, "Orifices and Mouthpieces" (C_v, C_c, C_d).
- NPTEL, Introduction to Fluid Mechanics (Prof. Suman Chakraborty, IIT Kharagpur): Week 1.