கொஞ்சம் நோட்ஸ் - EP.1

Bernoulli's Theorem, also known as Bernoulli's Principle, is a fundamental principle in fluid dynamics. It's named after Daniel Bernoulli, a Swiss mathematician and physicist. The theorem describes the behavior of a moving fluid and is essential in understanding various phenomena in aerodynamics, hydraulics, and other fields. Here's a detailed explanation:

Bernoulli's Theorem

Statement: Bernoulli's Theorem states that in a steady, incompressible flow of an ideal fluid (with no viscosity), the total mechanical energy per unit volume remains constant. The mechanical energy is composed of the fluid's kinetic energy, potential energy, and pressure energy.

Mathematical Formulation:

The mathematical expression of Bernoulli's Theorem is:

$$\mathrm{<span\; style="font-family:\; Playwrite\; GB\; J;">P</span>}<span\; style="font-family:\; Playwrite\; GB\; J;">+</span>\frac{\mathrm{<span\; style="font-family:\; Playwrite\; GB\; J;">1</span>}}{\mathrm{<span\; style="font-family:\; Playwrite\; GB\; J;">2</span>}}\mathrm{<span\; style="font-family:\; Playwrite\; GB\; J;">\rho </span>}{\mathrm{<span\; style="font-family:\; Playwrite\; GB\; J;">v</span>}}^{\mathrm{<span\; style="font-family:\; Playwrite\; GB\; J;">2</span>}}<span\; style="font-family:\; Playwrite\; GB\; J;">+</span>\mathrm{<span\; style="font-family:\; Playwrite\; GB\; J;">\rho </span>}\mathrm{<span\; style="font-family:\; Playwrite\; GB\; J;">g</span>}\mathrm{<span\; style="font-family:\; Playwrite\; GB\; J;">h</span>}<span\; style="font-family:\; Playwrite\; GB\; J;">=</span>\text{<span style="font-family: Playwrite GB J;">constant</span>}P\; +\; \backslash frac\{1\}\{2\}\; \backslash rho\; v^2\; +\; \backslash rho\; gh\; =\; \backslash text\{constant\}$$

Where:

• $PP$ is the fluid pressure.

• $\rho \backslash rho$ (rho) is the fluid density.

• $vv$ is the fluid velocity.

• $gg$ is the acceleration due to gravity.

• $hh$ is the height above a reference level.

Components:

1. Pressure Energy:

   • The term $PP$ represents the pressure energy per unit volume of the fluid.

2. Kinetic Energy:

   • The term $\frac{1}{2}\rho v^2\backslash frac\{1\}\{2\}\; \backslash rho\; v^2$ represents the kinetic energy per unit volume of the fluid, where $\frac{1}{2}{v}^2\backslash frac\{1\}\{2\}\; v^2$ is the specific kinetic energy.

3. Potential Energy:

   • The term $\rho gh\backslash rho\; gh$ represents the potential energy per unit volume of the fluid, where $ghgh$ is the specific potential energy (gravitational).

Interpretation:

• Bernoulli's Principle can be interpreted to mean that an increase in the fluid's velocity leads to a decrease in its pressure and vice versa.

• The principle is applicable to various scenarios, such as the lift generated by an airplane wing, the behavior of a fluid flowing through a pipe, and the operation of a venturi meter.

Applications:

1. Aerodynamics:

   • Lift on an Airplane Wing: The shape of an airplane wing (airfoil) causes the air to flow faster over the top surface than the bottom surface. According to Bernoulli's Principle, the pressure on the top surface is lower than on the bottom, creating lift.

2. Hydraulics:

   • Venturi Effect: In a venturi meter, fluid flows through a constricted section of a pipe, increasing its velocity and decreasing its pressure. This principle is used to measure the flow rate of the fluid.

3. Medical Devices:

   • Venturi Masks: These are used to deliver a controlled oxygen concentration to patients. The mask uses the venturi effect to mix oxygen with room air.

Limitations:

• Bernoulli's Theorem assumes that the fluid is ideal (incompressible and non-viscous) and that the flow is steady. Real-world fluids often exhibit viscosity, compressibility, and turbulence, which can affect the applicability of the theorem.

I hope this gives you a comprehensive understanding of Bernoulli's Theorem! If you have any specific questions or need further clarification, feel free to ask.