Bernoulli’s equation assumptions — which set of assumptions underlies the standard Bernoulli relation along a streamline?

Introduction / Context:
Bernoulli’s equation is a cornerstone of fluid mechanics linking pressure, elevation head, and velocity head. It applies under a specific set of idealizing assumptions; knowing them prevents misuse in real flows where viscosity, unsteadiness, or energy addition/removal is significant.

Given Data / Assumptions:

• Incompressible, homogeneous fluid (constant density).
• Inviscid behavior (neglect viscosity and shear dissipation).
• Steady flow conditions.
• Evaluation along a streamline (or across streamlines in irrotational flow).

Concept / Approach:

Under these assumptions, the mechanical energy per unit weight remains constant along a streamline: p/(rho * g) + v^2/(2 * g) + z = constant. Deviations require head-loss terms or pump/turbine head corrections in the extended Bernoulli equation.

Step-by-Step Solution:

Verification / Alternative check (if short method exists):

Deriving from Euler’s equation for steady inviscid flow and integrating along a streamline directly yields Bernoulli’s relation under the stated assumptions.

Why Other Options Are Wrong:

Picking any subset omits necessary conditions; “none” contradicts fundamental theory.

Common Pitfalls (misconceptions, mistakes):

Applying Bernoulli across a pump/turbine without accounting for added/removed head; ignoring head losses in viscous, turbulent pipes.

Final Answer:

all the above