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Bernoulli’s theorem is a fundamental principle in fluid dynamics that describes the behavior of an ideal fluid in steady, incompressible flow. Here are the key aspects of Bernoulli’s theorem:

**Description:**

- Bernoulli’s theorem states that in a streamline flow of an ideal fluid (non-viscous and incompressible), the sum of the pressure energy, kinetic energy, and potential energy per unit volume remains constant along any streamline.
- Mathematically, it can be expressed as
: \[ P + \frac{1}{2} \rho v^2 + \rho gh = \text{constant} \] where:\( P \) is the pressure of the fluid,\( \rho \) is the density of the fluid,\( v \) is the velocity of the fluid,\( g \) is the acceleration due to gravity,\( h \) is the height above a reference point.

**Assumptions:**

- Bernoulli’s theorem assumes steady flow (no changes over time), incompressible flow (density remains constant), and non-viscous flow (no internal friction).
- It also assumes that the fluid is along a streamline, meaning there is no crossing or mixing of flow lines.

**Applications:**

- Bernoulli’s theorem has numerous applications in fluid dynamics, aerodynamics, and engineering:
- It explains the lift force on aircraft wings.
- It helps in understanding fluid flow in pipes and channels.
- It is used in designing hydraulic systems, such as pumps and turbines.
- It is applicable in understanding blood flow dynamics in arteries and veins.

**Limitations:**

- Bernoulli’s theorem applies strictly to ideal fluids under ideal conditions. Real fluids may deviate from ideal behavior due to factors such as viscosity, compressibility, turbulence, and boundary effects.
- The theorem does not account for energy losses due to friction or heat transfer within the fluid.

Bernoulli’s theorem is a cornerstone in fluid mechanics, providing insights into the distribution of energy within flowing fluids and serving as a basis for practical applications in engineering and science.

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