# Applications of Bernoulli's Equation Bernoulli's equation describes the relationship between the speed, pressure, and potential energy of a fluid in motion. This principle finds numerous practical applications in engineering and everyday devices. ## 1. Speed of Efflux (Torricelli’s Theorem) Suppose a large tank has an orifice (small hole) at a depth $h$ below the surface of the liquid. Let $v_1$ be the velocity of the liquid at the top and $v_2$ be the velocity with which it leaves the orifice. Since the area of the tank is much larger than the area of the orifice, $v_1 \approx 0$. Applying Bernoulli's equation: $P_1 + \frac{1}{2}\rho v_1^2 + \rho gh_1 = P_2 + \frac{1}{2}\rho v_2^2 + \rho gh_2$ Since both the top and the orifice are open to the atmosphere, $P_1 = P_2 = P_a$. $\rho gh_1 = \frac{1}{2}\rho v_2^2 + \rho gh_2$ $v_2 = \sqrt{2g(h_1 - h_2)} = \sqrt{2gh}$ This is known as **Torricelli's Theorem**, which states that the speed of efflux is equal to the velocity gained by a fluid falling freely through a height $h$. ## 2. Venturi Relation (Venturi Meter) The Venturi effect is the reduction in fluid pressure that results when a fluid flows through a constricted section of a pipe. For a horizontal pipe ($h_1 = h_2$): $P_1 - P_2 = \frac{1}{2}\rho (v_2^2 - v_1^2)$ A **Venturi meter** is a device used to measure the speed of liquid flow. By measuring the pressure difference, the flow velocity can be calculated. ## 3. Filter Pump A filter pump is a device used to circulate and purify liquids by removing suspended impurities. It is essential in systems such as swimming pools, aquariums, and industrial water treatment plants. ### Working Principle The operation of a filter pump relies on creating a pressure difference using Bernoulli's Equation. **Creating Flow:** The pump forces water to move through a pipe. To increase the speed of water, it is channeled through a narrower section (a venturi section). **Pressure Drop:** As the water's velocity increases in the narrow section, its pressure decreases according to Bernoulli's principle. **Suction Effect:** This low-pressure area creates a suction effect at the inlet, which continuously draws water from the source (such as a pool or tank) into the filtration system. **Filtration:** Once drawn in, the water is pushed through a filter medium that traps debris, and the clean water is then returned to the system. | Location | Fluid Velocity | Pressure | | :--- | :--- | :--- | | **Wide Inlet Pipe** | Lower | Higher | | **Narrow Pump Section** | Higher | Lower | ## 4. Atomizer An atomizer is a device that converts a stream of liquid into a fine mist or spray. This mechanism is used in perfume bottles, spray paint cans, and medical nebulizers. ### Working Principle Atomizers use Bernoulli's principle to break a liquid into tiny droplets. **High-Velocity Airflow:** When a bulb is squeezed or a trigger is pressed, a jet of air is forced to move at high speed across the top of a small tube (nozzle). **Pressure Reduction:** This fast-moving stream of air creates a region of low pressure directly above the tube, based on Bernoulli's principle. **Drawing Liquid Up:** The atmospheric pressure inside the container, which is now higher than the pressure above the tube, pushes the liquid up the tube. **Mist Formation:** As the liquid reaches the top, the high-velocity air stream shatters it into millions of tiny droplets, forming a fine mist. ## 5. Lift on an Aerofoil The wing of an airplane is shaped as an **aerofoil**. It is designed such that air moves faster over the top surface than the bottom surface. - **Top Surface:** Higher velocity $\rightarrow$ Lower pressure. - **Bottom Surface:** Lower velocity $\rightarrow$ Higher pressure. The pressure difference creates an upward force called **lift**. ## 6. Swing of a Ball When a cricket ball or tennis ball is spun, it drags a layer of air with it. On one side of the ball, the air velocity due to spin and the air velocity due to the ball's motion are in the same direction (higher velocity). On the other side, they are in opposite directions (lower velocity). The resulting pressure difference causes the ball to curve or "swing" toward the low-pressure side. ## 7. Blood Flow The flow of blood in our arteries can be understood via Bernoulli's principle. If an artery becomes constricted (due to plaque), the blood velocity increases at the constriction, leading to a drop in internal pressure. If the external pressure is high enough, the artery may collapse momentarily.

Applications Of Bernoulli’s Equation | Fluid Mechanics | Physics 11