Euler's Equations and Bernoulli's Theorem

Newton's second law, written for a continuum whose only internal force is pressure, becomes Euler's equations — the equations of ideal fluid flow that Leonhard Euler set down in 1757 and that still govern lift, sound, and water waves. The move from particle to field mechanics costs one subtlety: the acceleration of a fluid element is not $\partial\mathbf{v}/\partial t$ but the material derivative $\mathrm{D}\mathbf{v}/\mathrm{D}t$ , because the element rides through a spatially varying velocity field. That advective term makes the system nonlinear and is the price of the Eulerian description built in the previous article. The inviscid idealization takes the Cauchy stress of the Stress Tensor and Cauchy's Relation article and strips it to its isotropic core, $\sigma_{ij}=-p\,\delta_{ij}$ : no shear, a single scalar carrying all internal force. This is an enormous simplification — it removes the diffusive momentum transport that will return with the Navier–Stokes equations — yet it captures the Venturi effect, the pitot tube, Torricelli's jet, acoustics, and most of the intuition aerodynamicists actually use. Bernoulli's theorem is the first integral of Euler's equation, and it is best understood as energy bookkeeping per unit volume: along a streamline of steady incompressible flow, $p+\tfrac12\rho v^2+\rho gz$ cannot change. Where continuity forces the flow to speed up, pressure must drop — "fast flow, low pressure" is not magic but the pressure gradient doing the accelerating. This article derives Euler's equations from a force balance, integrates them along streamlines, extracts the sound speed by linearization, and flags the vorticity structure whose full story — Kelvin's circulation theorem and potential flow — occupies the next article.

Key takeaways

Euler's equations are Newton's second law for an inviscid fluid element: $\rho\,\mathrm{D}\mathbf{v}/\mathrm{D}t=-\nabla p+\rho\mathbf{g}$ , with the material derivative supplying the nonlinear advective acceleration.
The inviscid stress is pure pressure, $\sigma_{ij}=-p\,\delta_{ij}$ — Cauchy's equation of motion with the shear stripped away.
Along a streamline of steady, incompressible, inviscid flow, $p+\tfrac12\rho v^2+\rho gz$ is constant (Bernoulli); if the flow is also irrotational the constant is global.
Faster flow means lower static pressure — the working principle of the Venturi meter, the pitot tube, and Torricelli's law $v=\sqrt{2gh}$ .
Linearizing Euler plus continuity about rest yields the acoustic wave equation with $c^2=(\partial p/\partial\rho)_s$ — sound is ideal-fluid physics, and it is adiabatic.
Key idea: setting $\rho$ times the material acceleration equal to $-\nabla p+\rho\mathbf{g}$ gives Euler's equations; projecting the steady equation along a streamline integrates it exactly to Bernoulli's invariant $p+\tfrac12\rho v^2+\rho gz$ .

The core insight

The acceleration of a fluid element is the material derivative $\mathrm{D}\mathbf{v}/\mathrm{D}t=\partial_t\mathbf{v}+(\mathbf{v}\cdot\nabla)\mathbf{v}$ ; setting $\rho$ times this equal to $-\nabla p+\rho\mathbf{g}$ gives Euler's equations, and integrating along a streamline gives Bernoulli's invariant $p+\tfrac12\rho v^2+\rho g z$ .

Where this lives today

Every aircraft you have flown on measures its speed with equation (4): pitot–static probes compare stagnation and static pressure and report $v_\infty=\sqrt{2(p_0-p_\infty)/\rho}$ , and redundant pitot systems remain flight-critical hardware in 2026 — their icing failures (Air France 447) reshaped both probe design and pilot training. The same physics runs the Venturi flowmeters in refineries and hospital gas lines, the carburetors on small engines, and the aspirators on lab benches. At the computational frontier, the compressible Euler equations are the workhorse of aerospace design: inviscid solvers sweep thousands of wing-body candidates before Navier–Stokes refinement, and the newest differentiable-CFD and neural-surrogate pipelines are trained against Euler solutions precisely because the equations are cheap enough to generate data at scale. Astrophysics runs the same system with gravity as the body force — supernova blast waves, galactic gas disks, and the intracluster medium are all Euler flows at Reynolds numbers no laboratory can reach, evolved with the shock-capturing finite-volume methods whose conservation-form bookkeeping descends from equation (2).

The physical picture

Pressure in an inviscid fluid is a constraint force, like the normal force from a frictionless track: it acts perpendicular to every surface, transfers energy between kinetic and potential forms, and never rubs. A parcel entering a constriction feels higher pressure behind than ahead — that imbalance, $-\nabla p$ per unit volume, is what accelerates it. By the time it exits fast and lean, it has traded pressure for speed at a fixed energy budget.

Figure

Pipe constriction read three ways — streamlines crowd where continuity forces the flow to speed up, pressure gauges read lower there, and the energy budget per unit volume just shifts between the pressure and kinetic columns

Bernoulli's theorem is nothing deeper than the work–energy theorem written per unit volume and followed along the path a parcel actually takes. The genuinely structural fact is where it can fail to be global: the Lamb form of the acceleration isolates a term $\mathbf{v}\times\boldsymbol{\omega}$ built from the vorticity $\boldsymbol{\omega}=\nabla\times\mathbf{v}$ . Project along the flow and it dies; step across streamlines and it generally does not. Vorticity is the only obstruction between "constant along each streamline" and "one constant everywhere" — the first hint that vorticity organizes ideal flow, which the next article makes precise through Kelvin's circulation theorem. Where the flow is irrotational the fluid behaves like a gradient field, $\mathbf{v}=\nabla\phi$ with $\nabla^2\phi=0$ for incompressible flow — mathematically as clean as electrostatics, and the foundation of potential-flow aerodynamics.

Figure

Pitot-static probe on the stagnation streamline — the nose port stops the flow and reads stagnation pressure while the side ports read static pressure; their difference is the dynamic pressure that encodes the speed

The core equations

The dynamical statement comes first. Take Cauchy's equation of motion $\rho\,\mathrm{D}v_i/\mathrm{D}t=\partial_j\sigma_{ij}+f_i$ , insert the inviscid stress $\sigma_{ij}=-p\,\delta_{ij}$ so that $\partial_j\sigma_{ij}=-\partial_i p$ , and add gravity as the body force. Derivation A rebuilds this from scratch on a small cube:

\rho\frac{\mathrm{D}\mathbf{v}}{\mathrm{D}t}=\rho\left[\frac{\partial\mathbf{v}}{\partial t}+(\mathbf{v}\cdot\nabla)\mathbf{v}\right]=-\nabla p+\rho\mathbf{g} \qquad (1)

where: $\rho$ — mass density; $p$ — pressure, the isotropic inviscid stress; $\mathbf{g}$ — body force per unit mass (gravity $-g\hat{\mathbf{z}}$ ); $-\nabla p$ — net pressure force per unit volume. Read the left side carefully: it is the parcel's true acceleration, nonlinear in $\mathbf{v}$ . Setting $\mathbf{v}=0$ collapses the equation to hydrostatics, $\nabla p=\rho\mathbf{g}$ — one atmosphere gained per ten meters of water. Euler's equation never travels alone. Mass conservation from the previous article closes the kinematic side of the system:

\frac{\partial\rho}{\partial t}+\nabla\cdot(\rho\mathbf{v})=0 \qquad (2)

where: $\rho\mathbf{v}$ — mass flux density. Equations (1)–(2) plus one thermodynamic closure — incompressibility, an equation of state, or isentropy — form the complete ideal-flow system: five unknowns, five equations, and all of inviscid fluid dynamics. The key to integrating equation (1) is a rewriting of the advective term. The vector identity $\nabla(\mathbf{v}\cdot\mathbf{v})=2(\mathbf{v}\cdot\nabla)\mathbf{v}+2\,\mathbf{v}\times(\nabla\times\mathbf{v})$ , solved for the advective term, produces the Lamb form:

\frac{\mathrm{D}\mathbf{v}}{\mathrm{D}t}=\frac{\partial\mathbf{v}}{\partial t}+\nabla\!\left(\tfrac12 v^2\right)-\mathbf{v}\times\boldsymbol{\omega} \qquad (3)

where: $\boldsymbol{\omega}=\nabla\times\mathbf{v}$ — the vorticity; $\tfrac12v^2$ — kinetic energy per unit mass. The advective mess has split into a pure gradient plus one rotational term. Everything integrable about ideal flow lives in the gradient; everything vortical lives in $\mathbf{v}\times\boldsymbol{\omega}$ , which is always perpendicular to $\mathbf{v}$ — poised to vanish under a streamline projection. That projection is Derivation B, and its output is the most famous equation in fluid mechanics — steady, incompressible, inviscid flow admits the exact first integral

p+\tfrac12\rho v^2+\rho g z=\text{const along a streamline} \qquad (4)

where: $p$ — static pressure; $\tfrac12\rho v^2$ — dynamic pressure; $\rho gz$ — gravitational potential energy density at height $z$ . The three terms share units of pressure and of energy per volume: Bernoulli is an energy budget. The constant generally differs between streamlines; only for irrotational flow ( $\boldsymbol{\omega}=0$ ) does one constant rule the whole fluid. Irrotational flow upgrades the theorem even when unsteady. With $\mathbf{v}=\nabla\phi$ every term of the Lamb-form Euler equation becomes a gradient, and one spatial integration gives

\frac{\partial\phi}{\partial t}+\tfrac12 v^2+\frac{p}{\rho}+g z=f(t) \qquad (5)

where: $\phi$ — the velocity potential; $f(t)$ — a spatially uniform function absorbable into $\phi$ . This unsteady Bernoulli equation is the dynamic boundary condition of water-wave theory: applied at the free surface it links $\partial_t\phi$ to the surface elevation, and with the kinematic condition yields the dispersion relation $\omega^2=gk\tanh(kH)$ . For compressible barotropic flow — $\rho=\rho(p)$ , as in isentropic gas dynamics — the pressure term generalizes to the specific enthalpy:

\frac{1}{\rho}\nabla p=\nabla h,\qquad h=\int\frac{\mathrm{d}p}{\rho} \qquad (6)

where: $h$ — specific enthalpy. Bernoulli becomes $\tfrac12v^2+h+gz=\text{const}$ along streamlines, the form used throughout gas dynamics and the launching point for the sound-speed calculation of Derivation C. One conservation law completes the ideal-flow picture. Around any closed loop $C$ that moves with the fluid, the circulation is frozen:

\frac{\mathrm{D}\Gamma}{\mathrm{D}t}=0,\qquad \Gamma=\oint_C\mathbf{v}\cdot\mathrm{d}\boldsymbol{\ell} \qquad (7)

where: $\Gamma$ — circulation around the material loop $C$ . This is Kelvin's circulation theorem, stated here because it belongs to the Euler system; its proof, its Helmholtz-theorem consequences, and its starring role in lift are the subject of the next article on vorticity and potential flow.

Derivation AEuler equation from a force balance on a fluid element

We build Newton's second law for an infinitesimal inviscid parcel, with pressure as the only internal force. The strategy: take a small cube of fluid, sum the pressure pushes on its faces by Taylor expansion, add the body force, and equate to mass times the material acceleration. Watch two things — how the face-by-face difference manufactures a gradient, and why the acceleration must be $\mathrm{D}\mathbf{v}/\mathrm{D}t$ rather than $\partial_t\mathbf{v}$ . This is Cauchy's tetrahedron logic in its simplest costume: with no shear, the cube suffices.

Step 1. Consider a cube of side lengths $\delta x,\delta y,\delta z$ centred at $\mathbf{x}$ , volume $\delta V$ , mass $\rho\,\delta V$ . Pressure pushes inward on every face. The $x$ -faces contribute the net force

F_x=\left[p\!\left(x-\tfrac{\delta x}{2}\right)-p\!\left(x+\tfrac{\delta x}{2}\right)\right]\delta y\,\delta z

Why a difference: equal pressures on opposite faces cancel; only the spatial variation of $p$ leaves a residue.

Step 2. Taylor expand to first order:

F_x\approx-\frac{\partial p}{\partial x}\,\delta x\,\delta y\,\delta z=-\frac{\partial p}{\partial x}\,\delta V

Why no shear terms: the inviscid stress is $\sigma_{ij}=-p\,\delta_{ij}$ — the tangential tractions that a viscous fluid would add are exactly what "ideal" deletes.

Step 3. Repeat for $y$ and $z$ and add the body force $\rho\mathbf{g}\,\delta V$ :

\mathbf{F}_{\text{tot}}=\left(-\nabla p+\rho\mathbf{g}\right)\delta V

Step 4. Newton's second law for the parcel. Its acceleration is the rate of change of its own velocity — the material derivative from the kinematics article:

\rho\,\delta V\,\frac{\mathrm{D}\mathbf{v}}{\mathrm{D}t}=\left(-\nabla p+\rho\mathbf{g}\right)\delta V

Why $\mathrm{D}/\mathrm{D}t$ : Newton's law applies to a material object, and the parcel samples the field along its path; using $\partial_t\mathbf{v}$ would describe a point of space, which is not a mechanical system.

Step 5. Cancel $\delta V$ :

\rho\left[\partial_t\mathbf{v}+(\mathbf{v}\cdot\nabla)\mathbf{v}\right]=-\nabla p+\rho\mathbf{g}

Two quick checks: setting $\mathbf{v}=0$ recovers hydrostatics $\nabla p=\rho\mathbf{g}$ , giving $p=p_0+\rho g\,(\text{depth})$ — 1 atm per 10 m of water; and the equation is exactly Cauchy's equation $\rho\,\mathrm{D}v_i/\mathrm{D}t=\partial_j\sigma_{ij}+f_i$ evaluated on $\sigma_{ij}=-p\,\delta_{ij}$ .

Go deeper · Derivation B · 1 minBernoulli theorem by projecting steady Euler along a streamline

Derivation BBernoulli theorem by projecting steady Euler along a streamline

We integrate Euler's equation exactly, once, under three stated hypotheses: steady flow, constant density, inviscid fluid with conservative body force. The strategy is to rewrite the advective term in Lamb form, gather every gradient into one bracket, and then project along the velocity — the rotational term $\mathbf{v}\times\boldsymbol{\omega}$ is geometrically perpendicular to $\mathbf{v}$ and dies on contact. Keep track of which hypothesis kills which term; each one is load-bearing, and the FAQ dissects what happens when each fails.

Step 1. Start from steady Euler ( $\partial_t\mathbf{v}=0$ ) in Lamb form, with gravity written as a gradient, $\mathbf{g}=-\nabla(gz)$ :

\rho\left[\nabla\!\left(\tfrac12 v^2\right)-\mathbf{v}\times\boldsymbol{\omega}\right]=-\nabla p-\rho\,\nabla(gz)

Step 2. With $\rho$ constant, sweep all gradients to one side:

\nabla\!\left(p+\tfrac12\rho v^2+\rho gz\right)=\rho\,\mathbf{v}\times\boldsymbol{\omega}

Why constant density matters: only then is $\nabla p/\rho$ itself a gradient; for barotropic flow the same step works with the enthalpy $h=\int\mathrm{d}p/\rho$ replacing $p/\rho$ .

Step 3. Define the Bernoulli function $B=p+\tfrac12\rho v^2+\rho gz$ and project along the flow:

\mathbf{v}\cdot\nabla B=\rho\,\mathbf{v}\cdot(\mathbf{v}\times\boldsymbol{\omega})=0

Why the right side vanishes identically: $\mathbf{v}\times\boldsymbol{\omega}$ is perpendicular to $\mathbf{v}$ , so its dot product with $\mathbf{v}$ is zero for any flow whatsoever — no extra assumption spent.

Step 4. $\mathbf{v}\cdot\nabla B$ is the directional derivative of $B$ along a streamline. Its vanishing means

B=p+\tfrac12\rho v^2+\rho gz=\text{const on each streamline}

Step 5. If additionally the flow is irrotational, $\boldsymbol{\omega}=0$ , Step 2 reads $\nabla B=0$ outright, so one constant governs the entire fluid — Bernoulli goes global. Two quick checks: all three terms carry units of energy per unit volume (Pa); applied surface-to-jet the theorem reproduces Torricelli's $v=\sqrt{2gh}$ , and setting $v=0$ throughout recovers hydrostatics.

Go deeper · Derivation C · 1 minSound speed from linearized Euler dynamics

Derivation CSound speed from linearized Euler dynamics

We show that the ideal-fluid system propagates small disturbances as waves and extract their speed. The strategy is standard linearization: perturb about a uniform rest state, drop all products of small quantities, close the system with a thermodynamic relation, and cross-differentiate to a single wave equation. The physical subtlety worth watching is the closure: sound oscillations are fast compared with heat diffusion, so the compression is adiabatic, not isothermal — the point Newton missed and Laplace fixed.

Step 1. Write $\rho=\rho_0+\rho'$ , $p=p_0+p'$ , $\mathbf{v}=\mathbf{v}'$ , with primes small and no gravity. Linearize continuity, dropping $\nabla\cdot(\rho'\mathbf{v}')$ :

\frac{\partial\rho'}{\partial t}+\rho_0\,\nabla\cdot\mathbf{v}'=0

Step 2. Linearize Euler. The advective term $(\mathbf{v}'\cdot\nabla)\mathbf{v}'$ is second order and drops:

\rho_0\frac{\partial\mathbf{v}'}{\partial t}=-\nabla p'

Why linearization is honest here: audible sound has $v'\sim10^{-4}$ – $10^{-2}\ \mathrm{m/s}$ against $c\sim340\ \mathrm{m/s}$ , so quadratic terms are smaller by six to eight orders of magnitude.

Step 3. Close thermodynamically. For fast, non-diffusive oscillations the entropy of each parcel is fixed:

p'=\left(\frac{\partial p}{\partial\rho}\right)_{\!s}\rho'\equiv c^2\rho'

Why adiabatic: the heat-diffusion length over an acoustic period is micrometres at audible frequencies — parcels compress and rarefy with no time to exchange heat.

Step 4. Take $\partial_t$ of Step 1 and $\nabla\cdot$ of Step 2, then subtract and insert Step 3:

\frac{\partial^2\rho'}{\partial t^2}=c^2\,\nabla^2\rho'

Step 5. This is the wave equation of the one-dimensional wave equation article, now in three dimensions, with propagation speed $c=\sqrt{(\partial p/\partial\rho)_s}$ . For an ideal gas $p\rho^{-\gamma}=$ const under isentropy, giving

c=\sqrt{\frac{\gamma p_0}{\rho_0}}=\sqrt{\gamma R T/M}

As a numerical check: air at 20°C with $\gamma=1.40$ , $p_0=1.013\times10^5$ Pa, $\rho_0=1.204\ \mathrm{kg/m^3}$ gives $c=\sqrt{1.40\times1.013\times10^5/1.204}=\sqrt{1.178\times10^5}=343\ \mathrm{m/s}$ , matching the measured value. The isothermal (Newton) choice would give $\sqrt{p_0/\rho_0}=290\ \mathrm{m/s}$ , 15% low.

Limiting cases

Case	Result	Meaning
Hydrostatics	$\mathbf{v}=0\Rightarrow\nabla p=\rho\mathbf{g}$	No motion leaves the pressure balancing gravity: $p=p_0+\rho g\times$ depth, one atmosphere per 10 m of water.
Steady incompressible, along a streamline	$p+\tfrac12\rho v^2+\rho gz=\text{const}$	Bernoulli's invariant; the constant may differ between streamlines.
Irrotational flow	$\boldsymbol{\omega}=0$ , $\mathbf{v}=\nabla\phi$ , $\nabla^2\phi=0$	Bernoulli's constant goes global; incompressible potential flow solves Laplace's equation, as in electrostatics.
Barotropic compressible	$\tfrac12v^2+h+gz=\text{const}$ , $h=\int\mathrm{d}p/\rho$	Enthalpy replaces $p/\rho$ ; the gas-dynamics form of Bernoulli.
Linearized about rest	$\partial_t^2\rho'=c^2\nabla^2\rho'$ , $c^2=(\partial p/\partial\rho)_s$	Sound: ideal-flow physics at small amplitude, propagating at 343 m/s in room air.
Galilean check	$v\ll\sqrt{p/\rho}$ scales	Dynamic pressure $\tfrac12\rho v^2\ll p$ : flow barely perturbs the pressure field — the quasi-static regime.

Worked examples

Example 1 — Venturi flowmeter (confidence builder). Water ( $\rho=1000\ \mathrm{kg/m^3}$ ) flows through a horizontal pipe of diameter $5.0\ \mathrm{cm}$ that narrows to a throat of diameter $2.5\ \mathrm{cm}$ . A differential gauge reads $\Delta p=p_1-p_2=7.2\ \mathrm{kPa}$ between pipe and throat. Find the volumetric flow rate.

Step 1. Areas: $A_1=\pi(0.025)^2=1.963\times10^{-3}\ \mathrm{m^2}$ , $A_2=\pi(0.0125)^2=4.909\times10^{-4}\ \mathrm{m^2}$ . Continuity gives $v_2=(A_1/A_2)v_1=4v_1$ .

Step 2. Bernoulli on the central streamline (horizontal, so the $\rho gz$ terms cancel): $p_1+\tfrac12\rho v_1^2=p_2+\tfrac12\rho v_2^2$ , hence $v_2^2-v_1^2=2\Delta p/\rho=2\times7200/1000=14.4\ \mathrm{m^2/s^2}$ .

Step 3. Insert $v_2=4v_1$ : $16v_1^2-v_1^2=15v_1^2=14.4$ , so $v_1=\sqrt{0.96}=0.98\ \mathrm{m/s}$ and $v_2=3.92\ \mathrm{m/s}$ .

Step 4. Flow rate: $Q=A_1v_1=1.963\times10^{-3}\times0.98=1.92\times10^{-3}\ \mathrm{m^3/s}$ .

Step 5. $Q\approx1.9\ \mathrm{L/s}$ , with $v_1=0.98\ \mathrm{m/s}$ in the pipe and $3.9\ \mathrm{m/s}$ in the throat. Notice the design logic: a 4:1 area ratio converts a modest flow into a 15-fold difference of dynamic pressure — Venturi meters amplify their own signal.

Example 2 — draining a water tank through a small hole (exam level). A cylindrical tank of cross-section $A=2.0\ \mathrm{m^2}$ is filled with water to depth $h_0=4.0\ \mathrm{m}$ . A hole of area $a=1.0\ \mathrm{cm^2}$ opens at the bottom. Find the initial exit speed, and the time to drain completely (treat the flow as quasi-steady, ideal).

Step 1. Torricelli from Bernoulli, surface to jet: at the free surface $p=p_{\text{atm}}$ , $v\approx0$ (since $a/A=5\times10^{-5}\ll1$ ), height $h$ ; at the jet $p=p_{\text{atm}}$ , height $0$ . So $\rho gh=\tfrac12\rho v^2$ , giving $v=\sqrt{2gh}$ . Initially $v_0=\sqrt{2\times9.81\times4.0}=\sqrt{78.5}=8.9\ \mathrm{m/s}$ .

Step 2. Quasi-steady mass bookkeeping: the outflow lowers the surface, $A\,\mathrm{d}h/\mathrm{d}t=-a\,v=-a\sqrt{2gh}$ .

Step 3. Separate variables: $\mathrm{d}t=-\dfrac{A}{a}\dfrac{\mathrm{d}h}{\sqrt{2g}\,\sqrt{h}}$ , and integrate from $h_0$ to $0$ :

T=\frac{A}{a}\,\frac{2\sqrt{h_0}}{\sqrt{2g}}=\frac{A}{a}\sqrt{\frac{2h_0}{g}}

Step 4. Numbers: $A/a=2.0/1.0\times10^{-4}=2.0\times10^{4}$ and $\sqrt{2\times4.0/9.81}=\sqrt{0.8155}=0.903\ \mathrm{s}$ , so $T=2.0\times10^4\times0.903=1.81\times10^4\ \mathrm{s}$ .

Step 5. The jet starts at $8.9\ \mathrm{m/s}$ and the tank drains in $1.8\times10^4\ \mathrm{s}\approx5.0$ hours. Sanity checks: $T$ equals the initial volume divided by initial flow rate, times 2 — the factor 2 because the jet weakens as the level falls; and a real tank drains slightly slower still (the vena contracta reduces the effective hole area by ~40%, a viscous-and-geometry effect outside ideal theory).

Origins · 1 minDaniel Bernoulli and how this was discovered

Daniel Bernoulli

1700–1782 · Basel and St. Petersburg

Daniel Bernoulli wrote Hydrodynamica (1738) while holding the mathematics chair at St. Petersburg, deriving the speed–pressure trade-off from the living force (kinetic energy) balance a generation before Euler's field equations existed. The book's publication became one of science's ugliest priority disputes: his father Johann issued Hydraulica, backdated it to 1732, and claimed the results — Daniel called it being "robbed of the fruits of ten years' labor," and the two barely reconciled. History sided with the son: the theorem, and the whole energy view of flow it embodies, carries his name. Fittingly, it was his lifelong friend Euler — the two shared rooms and problems in St. Petersburg — who gave Daniel's energy integral its proper home as the first integral of the momentum equations, the derivation this article follows.

History

1644

— Torricelli relates the speed of efflux to the height of fall, the first quantitative jet law

1738

— Daniel Bernoulli's Hydrodynamica derives the pressure–speed trade-off from the kinetic-energy balance

1743

— Johann Bernoulli's backdated Hydraulica ignites the priority feud; d'Alembert's Traité follows in 1744

1757

— Euler publishes the general equations of ideal fluid motion, making Bernoulli's theorem their exact first integral

1816

— Laplace corrects Newton's isothermal sound speed with the adiabatic closure, fixing the 15% discrepancy

1869

— Kelvin proves the circulation theorem, completing the conservation structure of Euler flow

Going further

References. Landau & Lifshitz, Fluid Mechanics (Vol. 6), §§1–5, 64 · Batchelor, An Introduction to Fluid Dynamics, §§3.5, 6.2 · Goldstein, Classical Mechanics, 3rd ed., Ch. 13 · Taylor, Classical Mechanics, Ch. 16 Key numerical results to keep in mind: the Lamb identity $(\mathbf{v}\cdot\nabla)\mathbf{v}=\nabla(\tfrac12 v^2)-\mathbf{v}\times\boldsymbol\omega$ and the vanishing of $\mathbf{v}\cdot(\mathbf{v}\times\boldsymbol\omega)$ underpin the Bernoulli derivation; Example 1 yields $v_1=0.98\ \mathrm{m/s}$ and $Q=1.92\times10^{-3}\ \mathrm{m^3/s}$ ; Example 2 yields $v_0=8.9\ \mathrm{m/s}$ and $T=1.81\times10^4\ \mathrm{s}$ ; the adiabatic sound speed gives $c=\sqrt{\gamma p_0/\rho_0}=343\ \mathrm{m/s}$ for air at 20°C, against the isothermal value of 290 m/s. The vena-contracta area coefficient for a sharp orifice is approximately 0.6, cited qualitatively only. The full story — compressible gas dynamics, shocks, boundary layers, real airfoil theory — lives in the Fluid Mechanics track (migrates later as its own subject); this chapter keeps only the ideal-flow core the continuum thread needs.

Builds on: Ideal Fluid Kinematics and the Continuity Equation · Stress Tensor and Cauchy's Relation · Strain Tensor and Deformation of Continua Leads to: Vorticity, Circulation, and Potential Flow · The Navier–Stokes Equations and Viscous Flow · Elastic Waves in Solids · Hamiltonian Density and Canonical Field Momentum

Is Bernoulli's theorem just energy conservation? If so, why derive it from the momentum equation?

It is energy conservation — but proving that from first principles requires the momentum route. The work–energy theorem per unit volume says kinetic energy changes at the rate pressure and gravity do work; integrating that bookkeeping along a parcel's path gives exactly $p+\tfrac12\rho v^2+\rho gz$ . The momentum derivation earns two things the hand-waving energy argument cannot: it identifies precisely the hypotheses (steady, inviscid, constant density or barotropic, conservative body force), and it exposes the failure term $\mathbf{v}\times\boldsymbol{\omega}$ , telling you the constant is per-streamline and goes global only for irrotational flow. Energy language explains the theorem; momentum language delimits it.

Does Bernoulli explain why airplanes fly? I have heard the equal-transit-time story is false.

The equal-transit-time story is false — air over the top does not need to "catch up" with air below, and measured transit times over a lifting wing differ from that story dramatically. What is true: air over the upper surface moves faster, and by Bernoulli its pressure is lower, and integrating the pressure over the wing gives the lift. But Bernoulli is only the energy half; the dynamical question is why the flow is faster on top, and the answer is circulation, established by the Kutta condition at the sharp trailing edge — the subject of the next article. Bernoulli without circulation predicts zero lift (d'Alembert's paradox); circulation without Bernoulli cannot convert speeds into forces. You need both halves.

If pressure does no work in an ideal fluid, how does it accelerate anything?

Careful with the bookkeeping level. Pressure at a point does no dissipative work — there is no rubbing — but pressure differences do reversible work on parcels all the time: a parcel entering a constriction has higher pressure on its rear face than its front face, and that net push accelerates it, converting pressure energy to kinetic energy at fixed total. The analogy is the normal force on a frictionless roller coaster: it never heats the cart, yet it redirects and (through the track's shape) mediates the exchange between height and speed. "Constraint force" means workless in the dissipative sense, not powerless.

Why exactly does fast flow have low pressure? It feels backwards — should not fast-moving fluid hit harder?

You are conflating static pressure with stagnation pressure. The static pressure $p$ is what the fluid exerts sideways, on a surface aligned with the flow, and that is what Bernoulli says is low where flow is fast — because the parcel got fast by being pushed from high pressure toward low, and the ledger $p+\tfrac12\rho v^2$ balances. What "hits harder" is the stagnation pressure $p_0=p+\tfrac12\rho v^2$ : bring the fast fluid to rest against your hand and the kinetic term converts right back. A pitot tube reads $p_0$ on its nose and $p$ on its side ports; their difference is the speed. Same fluid, two different questions.

The hypotheses say inviscid — but every real fluid has viscosity. Why does Bernoulli work so well in practice?

Because in high-Reynolds-number flow, viscosity's writ runs only in thin boundary layers near surfaces and in wakes. Outside those regions the velocity gradients are gentle, the viscous stress $\sim\mu\nabla^2\mathbf{v}$ is negligible against inertia and pressure, and the flow is ideal to excellent accuracy — this is where pitot tubes and Venturi meters live, and why they routinely achieve percent-level accuracy. Apply Bernoulli through a boundary layer, a wake, or a separated region and it fails immediately: there the shear stress extracts mechanical energy and the Bernoulli function decays downstream. Knowing where the ideal region ends is most of practical aerodynamics — and the Navier–Stokes article makes the boundary layer precise.

What does it mean that pressure becomes enthalpy for compressible flow? Where did thermodynamics sneak in?

Thermodynamics enters through the closure: five equations (mass plus momentum) hold seven fields ( $\rho$ , $p$ , $\mathbf{v}$ , plus temperature implicitly), so you must specify how $p$ and $\rho$ relate. For isentropic flow $\rho=\rho(p)$ , and then $\nabla p/\rho$ is the gradient of $h=\int\mathrm{d}p/\rho$ — which for an ideal gas is precisely the specific enthalpy $c_pT$ . Bernoulli becomes $\tfrac12v^2+h+gz=$ const: a moving gas parcel trades speed against enthalpy, not bare pressure. This is why air accelerating through a nozzle cools — jet-engine and rocket-nozzle design is this equation used in anger.

Can I use Bernoulli in a rotating frame, like for weather or a centrifuge?

Yes, with the frame's fictitious forces added to the potential. In a frame rotating at $\Omega$ , the centrifugal force is conservative and joins gravity in the potential: $B=p+\tfrac12\rho v_{\text{rel}}^2+\rho gz-\tfrac12\rho\Omega^2 r_\perp^2$ is constant along streamlines of steady relative flow. The Coriolis force needs no potential — like the magnetic force it is perpendicular to the velocity, so it drops out of the streamline projection exactly as $\mathbf{v}\times\boldsymbol{\omega}$ did. For large-scale weather, though, the Coriolis term dominates the cross-streamline balance (geostrophy), so Bernoulli along streamlines coexists with rotation-dominated dynamics across them.

Is the sound-speed derivation circular? We assumed small perturbations to derive the speed, but shocks are not small.

Not circular, but definitely limited — and the limit is instructive. Linearization is a controlled approximation: it delivers the exact propagation speed of infinitesimal disturbances, $c^2=(\partial p/\partial\rho)_s$ , and real audible sound sits deep inside its validity range. What it cannot describe is its own breakdown: finite-amplitude waves steepen (crests, being slightly hotter, travel faster), and the steepening runs away into a shock — a discontinuity where the ideal-fluid description itself fails and entropy is produced. The full nonlinear Euler system handles shocks via jump conditions; the linear theory knows only that it has left its jurisdiction. Same pattern as the tetrahedron argument: smooth fields give the clean law, discontinuities need the integral balances.