For steady, incompressible, non-viscous fluid flow along a streamline through variable cross section between different levels. Show how pressure, kinetic energy, and potential energy are related.

**OR**

Derive the expression for pressure at a depth (h) inside a liquid of densit

Question

For steady, incompressible, non-viscous fluid flow along a streamline through variable cross section between different levels. Show how pressure, kinetic energy, and potential energy are related.

**OR**

Derive the expression for pressure at a depth (h) inside a liquid of density (ρ) under gravity. Explain why an air bubble in water rises from the bottom to top and grow in size. (4+1=5)

Accepted Answer

>

> [!TLDR]
> For steady, incompressible, non-viscous flow, Bernoulli's equation states that P + (1/2)ρv² + ρgh = constant along a streamline — pressure energy, kinetic energy, and gravitational potential energy per unit volume together add up to the same value at every point.
>
> (This is the primary part of the question; the alternative — pressure at depth in a liquid, and why a rising air bubble grows — is not required since the primary is fully answerable.)
>
> Consider an ideal fluid (incompressible, non-viscous) in steady flow through a tube of varying cross-section and varying height, flowing along a streamline. Let subscripts 1 and 2 denote two cross-sections of the tube, at heights h₁ and h₂, areas A₁ and A₂, pressures P₁ and P₂, and flow speeds v₁ and v₂.
>
> By the work-energy theorem applied to the fluid element moving from section 1 to section 2, the net work done on the fluid element by the pressure forces at the two ends equals the change in its kinetic energy plus the change in its gravitational potential energy (since the fluid is ideal, no energy is lost to friction/viscosity):
>
> Work done by pressure = (P₁A₁)(v₁Δt) − (P₂A₂)(v₂Δt) = (P₁ − P₂)ΔV (since A₁v₁Δt = A₂v₂Δt = ΔV, the volume of fluid pushed through in time Δt, by the continuity equation/incompressibility)
>
> This work goes into changing the kinetic energy and potential energy of that fluid element:
>
> (P₁ − P₂)ΔV = [(1/2)ρΔV·v₂² − (1/2)ρΔV·v₁²] + [ρΔV·g·h₂ − ρΔV·g·h₁]
>
> Dividing throughout by ΔV:
>
> P₁ − P₂ = (1/2)ρv₂² − (1/2)ρv₁² + ρgh₂ − ρgh₁
> …

Situation	What happens	Why
Airplane wing	Air moves faster over the curved top surface, slower under the flat bottom. Lower pressure above, higher below → lift.	Bernoulli + angle of attack
Atomiser / spray bottle	Fast air stream over a vertical tube reduces pressure at the top, sucking liquid up.	Low pressure from high speed

Q.For steady, incompressible, non-viscous fluid flow along a streamline through variable cross section between different levels. Show how pressure, kinetic energy, and potential energy are related. OR Derive the expression for pressure at a depth

Concept understanding — Bernoulli Principle

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Common Misconception to Avoid

Real-Life Examples (Exam Favourites)

Q.For steady, incompressible, non-viscous fluid flow along a streamline through variable cross section between different levels. Show how pressure, kinetic energy, and potential energy are related. OR Derive the expression for pressure at a depth

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