Figures 9.20(a) and (b) refer to the steady flow of a (non-viscous) liquid. Which of the two figures is incorrect? Why? ![Fig 9.20 -- Two diagrams of a horizontal pipe with a constriction, each with two open vertical tubes over the wide section and the throat, showing different liquid-column heights](/figures/mechanical-properties-of-fluids/bernoulli-constriction-pipe-liquid-columns-fig-9-20.svg)

> [!TLDR] > For steady flow through a constriction, the equation of continuity makes the fluid speed largest at the narrow throat, and Bernoulli's principle then makes the pressure smallest there. A vertical side-tube rises to a height set by the local pressure, so the column at the throat must be the SHORTEST. Diagram (b) obeys this; diagram (a) shows the tallest column at the throat and is therefore incorrect. ### Concept Two ideas govern an ideal (non-viscous, incompressible) fluid in steady flow: 1. **Equation of continuity:** $A_1 v_1 = A_2 v_2$. Where the pipe is narrow (small $A$), the speed $v$ is large. 2. **Bernoulli's principle** (same height): $P + \tfrac{1}{2}\rho v^2 = \text{constant}$. Where the speed $v$ is large, the pressure $P$ is small. ### Why this fixes the tube heights Each open vertical tube acts as a manometer: the liquid climbs until the static column balances the local pressure in the pipe, so a taller column means a higher local pressure. ### Steps - At the wide section: area large $\Rightarrow$ speed small $\Rightarrow$ pressure **high** $\Rightarrow$ **tall** column. - At the narrow throat: area small $\Rightarrow$ speed large $\Rightarrow$ pressure **low** $\Rightarrow$ **short** column. - Therefore the physically correct picture has the column over the throat SHORTER than the column over the wide part. …

Figures 9.20(a) and (b) refer to the steady flow of a (non-viscous) liquid. Which of the two figures is incorr

Exercises · 9.15

Q. Figures 9.20(a) and

(b) refer to the steady flow of a (non-viscous) liquid. Which of the two figures is incorrect? Why?

Fig 9.20 -- Two diagrams of a horizontal pipe with a constriction, each with two open vertical tubes over the wide section and the throat, showing different liquid-column heights — Figure 9.20

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Concept understanding — Bernoulli Principle

The Intuition: Why Does a Shower Curtain Bulge Inward?

Stand in a running shower, and you'll notice the curtain bulges toward you, not away. That seems backwards — the water is pushing air, so shouldn't the curtain be pushed out? What's actually happening is that the fast-moving water drags the air next to it, making that air move faster than the still air in the rest of the bathroom. And here's the key: faster moving fluid exerts less sideways pressure. The slower air outside the shower pushes harder, so the curtain moves into the low-pressure region.

That's the Bernoulli principle in action: where speed is high, pressure is low.

The Precise Statement

P + \frac{1}{2} \rho v^2 + \rho g h = \text{constant along a streamline}

This is Bernoulli's equation for an ideal fluid (incompressible, non-viscous, steady flow). The three terms are:

$P$ — static pressure (the usual pressure you feel)
$\frac{1}{2} \rho v^2$ — dynamic pressure (pressure due to motion)
$\rho g h$ — gravitational pressure (height effect)

The principle says: if the speed of a fluid increases, its pressure decreases (assuming no change in height). That's the core idea.

Breaking It Down: Why Does This Happen?

Imagine a fluid particle moving along a streamline. As it enters a narrower section of a pipe, it must speed up to keep the same flow rate (continuity equation: $A_1 v_1 = A_2 v_2$ ). To accelerate, a net force must act on it — that force comes from a pressure difference. The pressure behind the particle is higher than the pressure ahead, so the particle speeds up. Result: higher speed, lower pressure.

Note

Bernoulli's equation is really just conservation of energy per unit volume for a fluid. The $\frac{1}{2} \rho v^2$ term is kinetic energy density, $\rho g h$ is potential energy density, and $P$ is the work done by pressure forces. Total energy per volume stays constant.

Common Misconception to Avoid

Watch out

Many students think Bernoulli's principle means "fast air always has low pressure." That's not true — it's a relationship between speed and pressure along a single streamline. If you force air to move fast by blowing it, you're adding energy, not lowering pressure. The principle applies to a fluid moving on its own under conservation of energy, not to a jet from a fan.

Real-Life Examples (Exam Favourites)

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

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For steady flow through a constriction, the equation of continuity makes the fluid speed largest at the narrow throat, and Bernoulli's principle then makes the pressure smallest there. A vertical side-tube rises to a height set by the local pressure, so the column at the throat must be the SHORTEST. Diagram (b) obeys this; diagram (a) shows the tallest column at the throat and is therefore incorrect.

Concept

Two ideas govern an ideal (non-viscous, incompressible) fluid in steady flow:

Equation of continuity: $A_1 v_1 = A_2 v_2$ . Where the pipe is narrow (small $A$ ), the speed $v$ is large.
Bernoulli's principle (same height): $P + \tfrac{1}{2}\rho v^2 = \text{constant}$ . Where the speed $v$ is large, the pressure $P$ is small.

Why this fixes the tube heights

Each open vertical tube acts as a manometer: the liquid climbs until the static column balances the local pressure in the pipe, so a taller column means a higher local pressure.

Steps

At the wide section: area large $\Rightarrow$ speed small $\Rightarrow$ pressure high $\Rightarrow$ tall column.
At the narrow throat: area small $\Rightarrow$ speed large $\Rightarrow$ pressure low $\Rightarrow$ short column.
Therefore the physically correct picture has the column over the throat SHORTER than the column over the wide part. …

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Q.Figures 9.20(a) and

Concept

Why this fixes the tube heights

Steps

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