Fluid Mechanics — Energy Conservation for Ideal Incompressible Flow For a perfect (non-viscous) incompressible liquid flowing steadily along a continuous stream/streamline, the total head (pressure head + velocity head + datum head) of a fluid particle remains constant from one point to another. This statement is called:

Introduction:Bernoulli's equation expresses conservation of mechanical energy for steady flow of an ideal (non-viscous, incompressible) fluid along a streamline. It relates pressure head, velocity head, and elevation head (datum head).

Given Data / Assumptions:

• Fluid is incompressible.
• Flow is steady along a streamline.
• Viscous losses and shaft work are neglected (ideal conditions).
• No significant heat transfer; gravitational field is uniform.

Concept / Approach:Bernoulli's statement: p/(rhog) + V^2/(2g) + z = constant along a streamline for ideal flow. Here p is pressure, rho is density, V is speed, g is gravitational acceleration, and z is elevation (datum).

Step-by-Step Solution:Identify the heads: pressure head = p/(rhog), velocity head = V^2/(2g), datum head = z.For an ideal, steady, incompressible flow, the sum of these heads remains constant along a streamline.Therefore the statement in the question exactly matches Bernoulli's equation (or Bernoulli's theorem).

Verification / Alternative check:If friction existed, the total head would drop; that situation is handled by the extended energy equation with head loss h_f. The question explicitly assumes a perfect incompressible liquid, so head loss is zero and Bernoulli applies directly.

Why Other Options Are Wrong:Continuity equation: conserves mass (A*V constant) and does not state energy conservation.Pascal's law: pressure in a static fluid transmits equally in all directions; not an energy relation.Archimede's principle: buoyancy in static fluids; unrelated to flowing energy balance.

Common Pitfalls:Confusing continuity (mass) with Bernoulli (energy); applying Bernoulli across pumps/turbines or across different streamlines without proper conditions; ignoring head losses in real fluids.

Final Answer:Bernoulli's equation