The differential equation of steady inviscid motion along a streamline, dp/ρ + g dz + v dv = 0, is attributed to which scientist?

Introduction / Context:
Bernoulli’s equation is foundational in fluid mechanics, typically obtained by integrating the Euler equation of motion along a streamline. Many learners know Bernoulli, but the underlying differential form and its attribution to Euler are crucial historical and conceptual points.

Given Data / Assumptions:

• Steady flow along a streamline.
• Inviscid (no viscosity) and incompressible fluid.
• Body force due to gravity only.

Concept / Approach:
The Euler equation expresses Newton’s second law for a fluid element neglecting viscosity. Along a streamline it reduces to the differential form: dp/ρ + g dz + v dv = 0. Integrating this relation yields Bernoulli’s constant: p/ρ + g z + v^2/2 = constant.

Step-by-Step Solution:
Start with momentum balance for inviscid flow (Euler’s equation).Project along a streamline to obtain dp/ρ + g dz + v dv = 0.Integrate to recover Bernoulli’s energy equation.

Verification / Alternative check:
All standard derivations of Bernoulli’s equation reference Euler’s equation as the starting point for inviscid flows, confirming the attribution to Leonhard Euler, not Bernoulli or others listed.

Why Other Options Are Wrong:
Bernoulli: associated with the integrated result, not the differential equation form.Cauchy–Riemann: complex analysis relations, unrelated to fluid momentum balance.Laplace: potential equation, not the momentum equation used here.

Common Pitfalls:

• Attributing everything about Bernoulli’s theorem to Bernoulli, overlooking Euler’s contribution.
• Confusing potential flow equations with momentum equations.

Final Answer:
Leonhard Euler