Bernoulli’s Principle

The equation of continuity is based on the principle of conservation of mass. It states: $\rho_1 A_1 v_1 = \rho_2 A_2 v_2$ For incompressible fluids (constant $\rho$): $A_1 v_1 = A_2 v_2$

The volume flow rate (or volume flux) is: $Q = A v$ Measured in $\text{m}^3/\text{s}$, it represents the volume of fluid passing through a cross-section per unit time.

1. Compressible flow: Aerospace engineering, gas turbines, internal combustion engines
2. Viscosity: Biomedical fluid dynamics, lubrication theory, inkjet printing
3. Turbulent flow: Meteorology, hydraulics, pollution dispersion modeling
4. Continuity equation: Blood flow analysis, fluid networks in mechanical and civil engineering

Why water exits a hose faster when the opening is smaller

When you press your thumb over the end of a garden hose, the opening becomes narrower, forcing the water to exit at a much higher speed. Why? The answer lies in Bernoulli's Principle, which connects pressure, speed, and elevation in flowing fluids.

Bernoulli's Theorem

Bernoulli's theorem is derived from the conservation of energy and applies to an incompressible, irrotational, non-viscous fluid in steady flow. It states: $P + \frac{1}{2} \rho v^2 + \rho g h = \text{constant}$

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

Derivation of Bernoulli's Equation

Consider a fluid moving steadily through a pipe of varying diameter and elevation. Let:

• Point 1: pressure $P_1$, velocity $v_1$, height $h_1$
• Point 2: pressure $P_2$, velocity $v_2$, height $h_2$

From the work-energy theorem applied to a fluid element, we derive: $P_1 + \frac{1}{2} \rho v_1^2 + \rho g h_1 = P_2 + \frac{1}{2} \rho v_2^2 + \rho g h_2$ This is Bernoulli's equation in its full form.

Applications of Bernoulli's Principle

Fluid flow from a wide tank

Consider a tank with liquid. Let:

• Point A: inside the tank, very large area (velocity $v_1 \approx 0$)
• Point B: outlet or orifice, height $h$ below point A

Bernoulli's equation simplifies to: $\rho g h = \frac{1}{2} \rho v^2 \Rightarrow v = \sqrt{2gh}$ This result is known as Torricelli's Theorem, which says that the velocity of efflux of a fluid under gravity from an orifice equals the velocity gained by a body in free fall from the same height.

Applications of Bernoulli's Principle

1. Flow from a tank: Describes how liquid jets form under gravity
2. Aerofoil lift: Faster air over the curved wing surface reduces pressure, creating lift
3. Venturi meter: Measures flow speed via pressure difference in a narrowing pipe
4. Atomizers: Use pressure drop to draw liquids into an air stream
5. Pitot tube: Measures fluid speed by comparing static and dynamic pressures

Aerofoil lift

$P_1 - P_2 = \frac{1}{2} \rho (v_2^2 - v_1^2)$

If A and a are the cross sectional area at Y and Q respectively, then from equation (5.2): $Av_1 = av_2 \quad \Rightarrow \quad v_1 = \frac{a}{A}v_2$

Substitute equation (5.13) into (5.12), hence: $P_1 - P_2 = \frac{1}{2}\rho v_1^2 \left(\frac{A^2 - a^2}{a^2}\right)$

Since $h$ in Figure 5.7 is the liquid column height difference between the two arms of the U-shaped tube, then: $P_1 - P_2 = \rho' g h$

Equating equations (5.14) and (5.15) and solving for $v_1$ gives: $v_1 = \frac{a}{A} \sqrt{\frac{2gh\rho'}{\rho (A^2 - a^2)/a^2}}$

Knowing the values for densities, cross section areas and height, the speed of flowing liquid past a point Y can be determined. But, volume flux at Y is: $\frac{V}{t} = Av_1 = Aa \sqrt{\frac{2gh\rho'}{\rho (A^2 - a^2)}}$

Atomizer or sprayer

The atomizer or sprayer (Figure 5.8) is a device that is used to spray paint or an insecticide. When the rubber ball of the atomizer is squeezed, air rushes through the narrow neck of the device. In so doing, the pressure in the narrow channel, at A is reduced. Once the pressure at A is reduced, atmospheric pressure pushes the insecticide up the tube towards the narrow channel. The insecticide is then pushed outwards into a fine spray of droplets.

Pitot tube

The pitot tube (flow meter) is a device used to measure the velocity of a moving fluid. It is very often used in airplanes to measure their relative speed. The schematic diagram of a pitot tube is shown in Figure 5.9. The fluid enters the tube through C and it is immediately brought to stagnation. Hence the pressure at A is sometimes called stagnant pressure. Applying Bernoulli principle: $P + h \rho g + \frac{1}{2}\rho v^2 = P_T$

The static component is determined from tube B and given as $P + h\rho g$, or $P$ if the flow is horizontal (i.e. $h = 0$). The dynamic component is: $\frac{1}{2} \rho v^2$

Hence, the total pressure in tube A is: $P_T = P + \frac{1}{2} \rho v^2$

Therefore, the pressure difference: $\Delta P = P_T - P = \frac{1}{2} \rho v^2 \Rightarrow v = \sqrt{\frac{2 \Delta P}{\rho}}$