ANS:A - Bernoulli's theorem.

Bernoulli's theorem is a fundamental principle in fluid mechanics that describes the relationship between the velocity, pressure, and elevation of a fluid in steady, inviscid (non-viscous) flow conditions. Named after the Swiss mathematician Daniel Bernoulli, who formulated it in the 18th century, Bernoulli's theorem is widely used in various engineering applications to analyze fluid flow behavior. Here's a detailed explanation of Bernoulli's theorem:

1. Statement of Bernoulli's Theorem: Bernoulli's theorem states that in a steady flow of an inviscid fluid (a fluid with negligible viscosity), the sum of the pressure energy, kinetic energy, and potential energy per unit volume along a streamline remains constant.
2. Mathematical Representation: Mathematically, Bernoulli's theorem can be expressed as: P+1/2​ρv^2+ρgh=constant Where:
   • P is the pressure of the fluid,
   • ρ is the density of the fluid,
   • v is the velocity of the fluid,
   • g is the acceleration due to gravity,
   • ℎ is the elevation above a reference point.
3. Interpretation of Terms:
   • The first term P represents the pressure energy per unit volume of the fluid. It accounts for the pressure exerted by the fluid.
   • The second term 1/2ρv^2 represents the kinetic energy per unit volume of the fluid. It accounts for the energy associated with the fluid's motion.
   • The third term ρgh represents the potential energy per unit volume of the fluid. It accounts for the energy associated with the fluid's elevation above a reference point.
4. Conservation of Energy: Bernoulli's theorem is derived from the principle of conservation of energy. It states that the total mechanical energy (pressure energy, kinetic energy, and potential energy) of the fluid remains constant along a streamline in the absence of external work or heat transfer.
5. Applications: Bernoulli's theorem has numerous practical applications in engineering, including:
   • Analysis of fluid flow in pipes, channels, and ducts.
   • Design of hydraulic systems, such as pipelines, pumps, and turbines.
   • Understanding aerodynamic principles in aviation and aerospace engineering.
   • Design of ventilation systems in buildings and HVAC (heating, ventilation, and air conditioning) systems.
   • Calculation of fluid velocities and pressures in water distribution networks, including water supply systems and irrigation systems.
6. Limitations: Bernoulli's theorem is based on several assumptions, including steady flow, inviscid fluid, and negligible friction losses. In real-world applications, these assumptions may not always hold true, leading to deviations from the predictions of Bernoulli's theorem.

In summary, Bernoulli's theorem is a fundamental principle in fluid mechanics that describes the conservation of energy in a flowing fluid. It provides valuable insights into the behavior of fluids and is widely used in engineering for the analysis and design of various fluid systems and devices.