Bernoulli principle – What stays constant under ideal assumptions? According to Bernoulli's equation for steady, incompressible, inviscid flow with no shaft work or heat transfer, which quantity remains constant along a streamline?

Introduction:
Bernoulli's equation provides an energy balance for fluid flow, forming the basis for pitot tubes, venturimeters, and many engineering estimates. It defines a conserved sum of energy heads along a streamline under specific ideal conditions.

Given Data / Assumptions:

• Steady flow along a streamline.
• Incompressible fluid, negligible viscosity (no head loss).
• No pumps or turbines adding or extracting shaft work; negligible heat interaction.

Concept / Approach:

The Bernoulli head (total head) is H = p/(rho * g) + v^2/(2 * g) + z, the sum of pressure head, velocity head, and elevation head. Under the stated assumptions, H is constant along a streamline. In real systems, additional terms account for pump head, turbine head, and head loss.

Step-by-Step Solution:

Verification / Alternative check:

Venturimeter analysis directly applies Bernoulli between inlet and throat, confirming that changes in one head term are offset by the others when losses are negligible.

Why Other Options Are Wrong:

p + rho * v, p * v, or p/rho − g z: Do not represent the conserved mechanical energy per unit weight.v + z: Lacks pressure term and wrong dimensions for energy conservation.

Common Pitfalls:

Applying Bernoulli across a pump/turbine without adding the corresponding head term, or across regions with significant viscous dissipation without head-loss correction.

Final Answer:

p/(rho * g) + v^2/(2 * g) + z (total head)