Water flows through a pipe — velocity at narrow end using Bernoulli

Question

Water flows through a horizontal pipe that narrows from cross-sectional area $A_1 = 8\text{ cm}^2$ to $A_2 = 2\text{ cm}^2$ . If the velocity at the wider end is $v_1 = 3\text{ m/s}$ , find the velocity at the narrower end. Also explain why the pressure decreases at the narrow section.

Solution — Step by Step

For an incompressible fluid (like water) in steady flow, the equation of continuity states:

A_1 v_1 = A_2 v_2

This is a statement of mass conservation — the same volume of water must pass through any cross-section per unit time.

v_2 = \frac{A_1 v_1}{A_2} = \frac{8 \times 3}{2} = \frac{24}{2} = 12\text{ m/s}

The velocity at the narrow end is 12 m/s — four times faster, because the area is four times smaller.

Bernoulli’s equation for steady horizontal flow:

P_1 + \frac{1}{2}\rho v_1^2 = P_2 + \frac{1}{2}\rho v_2^2

Since $v_2 > v_1$ (fluid speeds up at the narrow section), the kinetic energy term $\frac{1}{2}\rho v_2^2$ is larger at the narrow section. For the sum to remain constant, $P_2$ must be smaller than $P_1$ .

Pressure is lower where velocity is higher. This is the Bernoulli principle.

Why This Works

The continuity equation comes from conservation of mass. The Bernoulli equation comes from conservation of energy for a flowing fluid — it says the total mechanical energy per unit volume (pressure + kinetic + potential) remains constant along a streamline for inviscid, incompressible flow.

The key insight: when the pipe narrows, the fluid must accelerate to maintain the same mass flow rate. This acceleration requires a net force — provided by the pressure difference between the wider (higher pressure) and narrower (lower pressure) sections. So the pressure drop is not mysterious; it’s the mechanism that accelerates the fluid.

Alternative Method

We can also verify using Bernoulli directly. Taking $\rho_{water} = 1000\text{ kg/m}^3$ :

P_1 - P_2 = \frac{1}{2}\rho(v_2^2 - v_1^2) = \frac{1}{2}(1000)(144 - 9) = 500 \times 135 = 67500\text{ Pa}

The pressure at the narrow section is 67,500 Pa (about 0.67 atm) lower than at the wide section.

Bernoulli applications to remember: venturimeter (measures flow rate), pitot tube (measures aircraft airspeed), atomizer/spray, aerofoil lift, and flow through a narrow pipe. All involve the same principle: faster flow → lower pressure. For JEE, Bernoulli combined with continuity is a near-certain question type.

Common Mistake

Students often reason intuitively: “at the narrow section, fluid is being squeezed together, so pressure should be higher.” This is wrong. Pressure is NOT higher in the narrow section — it’s lower. The fluid is moving faster there (same mass flowing through a smaller area per unit time), and Bernoulli tells us faster flow means lower pressure. Trust the continuity + Bernoulli derivation over intuition here.

Question

Solution — Step by Step

Why This Works

Alternative Method

Common Mistake

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