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Every JEE aspirant meets Bernoulli's equation in Class 11, usually as a formula to plug numbers into. The person behind it is more interesting than that. Daniel Bernoulli was a medical doctor from a family of feuding mathematicians, and he wrote the founding book on moving fluids while working in Russia. This post covers his life, what is actually inside Hydrodynamica, one worked problem in its spirit, and what his way of working can teach you.
Who was Daniel Bernoulli?
His MacTutor biography gives his dates as 8 February 1700, in Groningen, to 17 March 1782, in Basel. He was the son of the mathematician Johann Bernoulli and nephew of Jacob Bernoulli.
His father wanted him to be a merchant, then insisted on medicine, saying there was no money in mathematics. Daniel did qualify in medicine, and his doctoral thesis was on the mechanics of breathing. His interest in blood flow and blood pressure pulled him towards fluids.
In 1725 he took up the chair of mathematics at St Petersburg, and from 1727 he worked there alongside Leonhard Euler. He later returned to Basel, where he taught botany and physiology before holding the chair of physics from 1750 to 1776. MacTutor records that he won the Grand Prize of the Paris Academy ten times, mostly for astronomy and ship design.
What is actually in Hydrodynamica?
The title page of the 1738 first edition reads Hydrodynamica, sive de viribus et motibus fluidorum commentarii: Hydrodynamics, or commentaries on the forces and motions of fluids. It was printed in Strasbourg by Johann Reinhold Dulsecker and describes itself as an academic work put together while the author was in St Petersburg. The word "hydrodynamics" itself comes from this title.
Its index lists thirteen sections. Among them:
Section 3, on the speeds of fluids flowing out of a vessel of any shape through an opening. MacTutor calls this the first correct analysis of water flowing from a hole, based on the conservation of energy he had learned from his father.
Section 7, which shows how useful the principle of conservation of "living forces" (vis viva, an early form of kinetic energy) is, even when some of it is lost.
Section 10, on elastic fluids, especially air. Here Bernoulli treated a gas as particles in motion, an early form of the kinetic theory of gases you study today.
Section 12, a "new statics of moving fluids, which can be called hydraulico-statics": the pressure a moving fluid exerts on the walls of its pipe.
Section 13, on the reaction of jets and its use for driving ships.
MacTutor also records an unhappy footnote. His father published a book called Hydraulica and dated it 1732, although it probably appeared around 1739, to make it look as if Daniel had borrowed from him.
How do you use Bernoulli's idea in a pipe problem?
The NCERT Class 11 chapter on fluids gives the modern form: along a streamline in steady, non-viscous, incompressible flow, P + ½ρv² + ρgh stays constant. It also warns that the equation does not hold for non-steady or turbulent flow. Bernoulli's principle and its applications are listed in the JEE (Main) 2026 syllabus.
Here is a worked example on Section 12's theme, pressure in moving water.
Water flows through a horizontal pipe that narrows from 4.0 cm to 2.0 cm in diameter. In the wide part the speed is 0.50 m/s. Two thin vertical tubes are fitted, one on each part. How different are the water levels in them?
Continuity first. A₁v₁ = A₂v₂. Area goes as diameter squared, so A₁/A₂ = (4.0/2.0)² = 4. The speed in the narrow part is v₂ = 4 × 0.50 = 2.0 m/s.
Bernoulli, same height. P₁ − P₂ = ½ρ(v₂² − v₁²) = ½ × 1000 × (4.0 − 0.25) = 1,875 Pa.
Convert to a height. Δh = ΔP / (ρg) = 1,875 / (1000 × 9.8) = 0.19 m.
The tube over the wide part stands about 19 cm higher. Faster water, lower pressure.
Common mistakes:
Using the diameter ratio instead of the area ratio. That gives v₂ = 1.0 m/s and a pressure difference less than a quarter of the right one.
Subtracting speeds before squaring. ½ρ(v₂ − v₁)² is not the same as ½ρ(v₂² − v₁²).
Using Bernoulli where it fails. Thick, slow flow in a narrow tube loses energy to viscosity, and fast flow can turn turbulent. For when that happens, see our post on Osborne Reynolds and the onset of turbulence.
More practice on these traps is in our guide to fluid mechanics mistakes.
What can a student learn from Bernoulli's way of working?
Carry ideas across subjects. A thesis on breathing and an interest in blood pressure led him to the physics of pipes. Ideas from biology or chemistry can help you see a physics problem, and the reverse.
Reach for a conservation law. His main tool was conservation of energy. When a fluids problem looks messy, ask what is conserved between two points.
Keep working when you are unhappy. MacTutor says he disliked the cold of St Petersburg and felt the loss of his brother there, yet those years were his most productive.
Revise your work before you publish. He reworked Hydrodynamica between 1734 and 1738, mostly in presentation. Rewriting a solution cleanly is a form of revision too.
When a formula like this feels like a string of symbols, the Fermi Concept Library covers every concept in physics, chemistry, biology, mathematics and computing, so you can rebuild the idea behind it before you practise.
Today, rework the pipe example with the narrow part 1.0 cm across, and predict the answer before you calculate.
