Euler’s equation for steady inviscid flow along a streamline — select the correct differential relation.

Introduction / Context:
Euler’s equation is the inviscid (zero-viscosity) form of the momentum equation for fluid flow. When applied along a streamline under steady conditions for an incompressible fluid, it leads directly to the differential form that integrates to Bernoulli’s equation.

Given Data / Assumptions:

• Steady flow, inviscid fluid, along a streamline.
• Body force due to gravity acts in the −z direction (z positive upward).
• Density ρ is constant (incompressible) for the differential form used here.

Concept / Approach:

Express the momentum balance per unit mass along a streamline. The pressure term appears as dp/ρ, elevation as g * dz, and the kinetic term as v * dv. The sum of these differential energy terms equals zero in the absence of losses or work input, yielding the familiar integrable form.

Step-by-Step Solution:

Verification / Alternative check (if short method exists):

Dimensional analysis confirms dp/ρ has units of specific energy (m²/s²), same as g * dz and v * dv; options with ρ * dp or v² * dv are dimensionally inconsistent in this specific context.

Why Other Options Are Wrong:

(b) and (c) multiply the wrong terms by ρ; (d) uses v² * dv instead of v * dv; “none” is incorrect because (a) is the standard relation.

Common Pitfalls (misconceptions, mistakes):

Forgetting that the equation is per unit mass; confusing the integrated Bernoulli form with the differential Euler form.

Final Answer:

dp/ρ + g * dz + v * dv = 0