Euler's Equation Along a Streamline

Newton's second law for a fluid that has no viscosity — the stepping-stone to Bernoulli.

The Energy Equation: Bernoulli & BeyondMechanical EngineeringFree preview
⏱️ About 18 min

Hold a sheet of paper flat and blow across the top: the paper rises. Something about faster moving air is tied to lower pressure — but why? The answer is not magic, and it is not energy conservation alone. It is Newton's second law, written for a fluid. That single force balance, applied to a frictionless fluid element sliding along a streamline, is Euler's equation — and it is where the whole Bernoulli story actually begins.

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The big idea: Euler's equation is Newton's second law (a momentum balance) for an inviscid fluid. It says that a fluid element accelerates along a streamline whenever a net force acts on it — and the only two forces available in a frictionless fluid are the pressure difference pushing it from behind and gravity pulling it down. Integrating that force-along-the-path statement gives the relation between pressure, speed, and elevation that the next lesson names the Bernoulli equation.
🎯 By the end, you'll be able to
  • Derive Euler's equation by applying Newton's second law to a small fluid element along a streamline
  • Interpret each term: convective acceleration, the pressure-gradient force, and the gravity force
  • Integrate Euler's equation along a streamline for a steady, incompressible, inviscid fluid
  • Distinguish this momentum-based result from the thermodynamic energy equation introduced later in the module
📎 Helpful to know first

Newton's second law, applied to a blob of fluid

Take a tiny cylindrical blob of fluid, just big enough to see, sitting on a streamline. Call its length ds and its end-face area dA, so its volume is dV = ds · dA and its mass is ρ dV. We are going to assume the fluid has no viscosity — no internal friction, no shear stress. That is the defining idealization of Euler's equation: the only surface force that can act on our blob is pressure, which always pushes straight in on every face.

Now run F = ma on the blob, but resolved along the streamline (the s-direction). Two forces have a component along s: the pressure on the two end faces, and the blob's own weight. The pressure on the upstream face (call it p) pushes the blob forward; the pressure on the downstream face (p + dp) pushes it back. Because the downstream face is at higher pressure, there is a net pressure force −(dp/ds) ds dA pushing the blob along — in words, the blob accelerates wherever pressure falls in the flow direction. Gravity contributes its component along the streamline, −ρg (dz/ds) dV, where z is elevation. Setting the net force equal to mass times the acceleration along the streamline gives Euler's equation.

\[ \rho V\frac{dV}{ds} = -\frac{dp}{ds} - \rho g\frac{dz}{ds} \]
Euler's equation along a streamline (steady, inviscid). The left side is mass per volume times the convective acceleration along the streamline; the right side is the net force per volume from the pressure gradient and gravity. s is distance along the streamline, V is speed, p is pressure, z is elevation.
A curved streamline carrying a small fluid element: pressure forces push on its two end faces, its weight acts downward, and it travels at speed V along the streamline at elevation z above a datum datum (z = 0) streamline fluid element, length ds V p p + dp weight ρg dV z A net force (pressure difference + gravity along the streamline) accelerates the element.
The free body that gives Euler's equation: a frictionless fluid element on a streamline, pushed by the pressure difference between its faces and pulled by gravity, accelerating along the path.
🔑 Euler's equation is a momentum result — not an energy result

Notice the starting point: F = ma. That makes Euler's equation a momentum balance through and through — it tracks forces and accelerations, not energy. This matters enormously for the rest of this module. The Bernoulli equation (next lesson) is simply Euler's equation integrated, so it inherits the same momentum ancestry. The thermodynamic Energy Equation in Lesson 4 is a different object entirely — it comes from the First Law of Thermodynamics. Keep the two lineages separate in your head: momentum here, thermodynamics there.

What the pressure gradient actually does

The pressure-gradient term −dp/ds is the engine of inviscid flow. Wherever pressure decreases along the streamline (dp/ds < 0), the term is positive and the fluid speeds up — a favourable pressure gradient, like air accelerating into the low pressure above a wing or water speeding into the throat of a nozzle. Where pressure increases along the streamline (dp/ds > 0), the fluid decelerates — an adverse pressure gradient, which a real (viscous) fluid can only climb for so long before its boundary layer separates. Euler's equation, being frictionless, simply records the speed change that any given pressure change demands.

Dividing Euler's equation through by ρ and multiplying by ds turns the rate-along-the-streamline form into a tidy differential relation between the changes in pressure, speed, and elevation as you move a small step ds along the path.

\[ \frac{dp}{\rho} + V\,dV + g\,dz = 0 \]
The differential form obtained from Euler's equation along a streamline. Each term has units of energy per unit mass (J/kg) — but remember it was derived from a momentum balance, not from an energy balance.
✨ Those J/kg units are a coincidence of algebra, not a lineage

The three terms happen to carry units of energy per unit mass, which tempts people to call this 'an energy equation.' Resist that. The terms came out of F = ma; they look energy-shaped only because force-times-distance has units of energy. We will meet the real energy equation — derived from the First Law of Thermodynamics — in Lesson 4, and it is a more general object that also accounts for heat, internal energy, and friction. For now, stay in the momentum lane.

🎮 Interactive: speed change from a pressure drop LIVE
Predict first: If you triple the pressure drop along a horizontal streamline, does the speed change triple too — or something else?

An interactive slider tool computing the downstream fluid speed from a pressure drop along a horizontal streamline, using the integrated Euler equation.

Integrating Euler's equation along a horizontal streamline (dz = 0): V₂² = V₁² + 2(p₁ − p₂)/ρ. Slide the inlet speed, the pressure drop, and the density to see the downstream speed update live.
📝 Worked example: Water (ρ = 1000 kg/m³) flows along a horizontal streamline (dz = 0). At one point the speed is V₁ = 4 m/s and the pressure is p₁ = 200 kPa. Further along, the pressure has fallen to p₂ = 150 kPa. Find the downstream speed V₂.
  1. Integrate Euler's equation along the horizontal streamline (the gravity term vanishes because dz = 0): V₂²/2 − V₁²/2 = (p₁ − p₂)/ρ.
  2. Rearrange: V₂² = V₁² + 2(p₁ − p₂)/ρ.
  3. Substitute: V₂² = (4)² + 2(200{,}000 − 150{,}000)/1000 = 16 + 2(50) = 16 + 100 = 116.
  4. Take the square root: V₂ = √116 = 10.77 m/s. (Check the units: Pa/(kg/m³) = m²/s², so V₂² is in m²/s² as required.)
✓ V₂ ≈ 10.77 m/s — the falling pressure has accelerated the fluid by more than 2.5×.
✏️ Practice: Water (ρ = 1000 kg/m³) moves along a horizontal streamline. The speed is V₁ = 3 m/s where p₁ = 180 kPa, and the pressure falls to p₂ = 130 kPa downstream. Find the downstream speed V₂, in m/s.
m/s
Solution
  1. V₂² = V₁² + 2(p₁ − p₂)/ρ = (3)² + 2(180{,}000 − 130{,}000)/1000 = 9 + 100 = 109.
  2. V₂ = √109 = 10.44 m/s.
✏️ Practice: Now the streamline is not horizontal: the fluid climbs so that z₂ − z₁ = 2 m, but the pressure is the same at both points (p₁ = p₂). The inlet speed is V₁ = 8 m/s, ρ = 1000 kg/m³, g = 9.81 m/s². Find the downstream speed V₂, in m/s. (The fluid should slow as it climbs — confirm it does.)
m/s
Solution
  1. With dp = 0, Euler's integrated form reduces to V₂²/2 − V₁²/2 = −g(z₂ − z₁).
  2. V₂² = V₁² − 2g(z₂ − z₁) = (8)² − 2(9.81)(2) = 64 − 39.24 = 24.76.
  3. V₂ = √24.76 = 4.98 m/s — slower than 8 m/s, as expected for fluid climbing against gravity.

Check your understanding

1. Euler's equation along a streamline is best described as:
Euler's equation comes straight from F = ma applied to a frictionless fluid element — it is a momentum result. The thermodynamic energy equation is a separate, First-Law-derived object met later in this module.
2. Along a streamline where pressure falls in the flow direction (a favourable gradient), the fluid:
A falling pressure (dp/ds < 0) makes the pressure-gradient term positive, so the fluid accelerates — exactly the mechanism behind a nozzle or a wing's suction surface.
3. What assumption is essential to Euler's equation that is NOT true of any real fluid?
Euler's equation drops viscosity entirely. No real fluid is truly inviscid; Euler is an idealization whose integrated form (Bernoulli) is accurate where viscous effects are small.
🔎 Related lessons
The Bernoulli Equation The Reynolds Transport Theorem · soon
✅ Key takeaways
  • Euler's equation is Newton's second law for an inviscid fluid — a momentum balance, not an energy balance.
  • Along a streamline, a fluid element accelerates where pressure falls and decelerates where pressure rises.
  • Integrating Euler's equation relates changes in pressure, speed, and elevation along the path.
  • This momentum-thread result is the direct ancestor of the Bernoulli equation in the next lesson.
➡️ Euler's equation is the momentum story in differential form. Integrate it along a streamline for a steady, incompressible, inviscid fluid and you get a single, famous, conserved quantity — the Bernoulli equation. That is the next lesson.
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