Fluid dynamics — continuity equation and Bernoulli's principle applications

Question

What are the continuity equation and Bernoulli’s principle? How do we apply them to solve fluid flow problems like velocity of efflux, venturimeter readings, and lift on an aerofoil?

(JEE Main, NEET — Bernoulli’s principle applications appear frequently, especially Torricelli’s theorem)

Solution — Step by Step

For an incompressible fluid flowing through a pipe of varying cross-section:

A_1 v_1 = A_2 v_2

where $A$ = cross-sectional area and $v$ = fluid velocity.

Meaning: Where the pipe narrows ( $A$ decreases), the fluid speeds up ( $v$ increases). This is why water from a garden hose goes faster when you partially cover the opening with your thumb.

This follows directly from conservation of mass — the mass of fluid entering one end per second must equal the mass leaving the other end per second.

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

Along a streamline in steady, incompressible, non-viscous flow.

Three terms represent:

$P$ = pressure energy per unit volume
$\frac{1}{2}\rho v^2$ = kinetic energy per unit volume
$\rho g h$ = potential energy per unit volume

Key insight: When velocity increases (narrow section), pressure decreases. This is counterintuitive but follows directly from energy conservation — kinetic energy gain must come at the expense of pressure energy.

A tank filled with liquid to height $h$ has a small hole at the bottom. Find the velocity of water coming out.

Apply Bernoulli’s equation between the top surface and the hole:

Top: $P_{atm} + 0 + \rho g h$ (velocity $\approx 0$ for large tank)
Hole: $P_{atm} + \frac{1}{2}\rho v^2 + 0$

P_{atm} + \rho g h = P_{atm} + \frac{1}{2}\rho v^2

v = \sqrt{2gh}

This is the same as the speed of a freely falling object from height $h$ — hence called Torricelli’s theorem.

A venturimeter measures flow rate by using a constriction. At the narrow section, velocity increases and pressure decreases. The pressure difference (measured by a manometer) gives the flow velocity:

v_1 = A_2\sqrt{\frac{2(P_1 - P_2)}{\rho(A_1^2 - A_2^2)}}

The same principle explains: why roofs blow off in storms (high wind velocity above = low pressure above, normal pressure below = net upward force), why a spinning ball curves (Magnus effect), and why two ships moving in parallel are pulled toward each other.

flowchart TD
    A["Fluid dynamics problem"] --> B{"What is known?"}
    B -->|"Pipe areas and one velocity"| C["Use continuity: A₁v₁ = A₂v₂"]
    B -->|"Heights and pressures"| D["Use Bernoulli: P + ½ρv² + ρgh = const"]
    B -->|"Both"| E["Use continuity to relate velocities,<br/>then Bernoulli to find pressure/velocity"]
    D --> F["Torricelli: v = √(2gh) for tank efflux"]
    D --> G["Venturimeter: pressure difference gives flow rate"]
    D --> H["Lift: faster air above = lower pressure = upward force"]

Why This Works

Bernoulli’s equation is just the work-energy theorem applied to a fluid element. As a fluid parcel moves from one point to another, work is done on it by pressure forces and gravity, changing its kinetic energy. The equation simply states that the total mechanical energy per unit volume remains constant along a streamline.

The continuity equation ensures that fluid does not pile up or disappear — it is conservation of mass in disguise. Together, these two equations solve almost every ideal fluid flow problem.

Common Mistake

Bernoulli’s equation is often misapplied to viscous fluids or turbulent flows. It is valid ONLY for: (1) steady flow, (2) incompressible fluid, (3) non-viscous (ideal) fluid, (4) along a streamline. In real pipes with friction, we need to add a head loss term. If the problem mentions “viscous fluid” or “viscosity,” Bernoulli alone is not sufficient — you need Poiseuille’s equation instead.

For Torricelli’s theorem problems, remember that the velocity of efflux depends only on the height of liquid ABOVE the hole, not on the density of the liquid or the size of the hole (as long as the hole is small compared to the tank). $v = \sqrt{2gh}$ is universal for ideal conditions.

Question

Solution — Step by Step

Why This Works

Common Mistake

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