# visualize the inverse relationship between fluid speed and pressure along a streamline-Bernoulli's p

The interactive visualization powerfully demonstrates Bernoulli's Principle, proving that the total energy per unit mass along a steady streamline remains constant: $$\frac{P}{\rho}+\frac{1}{2} v^2+g h=$$ Constant. The animation, through synchronized movement and real-time numerical labels, shows the compulsory energy trade-offs: as the fluid parcel moves into a narrowing section, its velocity increases due to mass conservation, forcing a corresponding conversion of Pressure Energy ( $$\frac{P}{\rho}$$ ) into Kinetic Energy ( $$\frac{1}{2} v^2$$ ). Conversely, as the parcel moves to a higher elevation, its Gravitational Potential Energy ( $$g h$$ ) increases, immediately requiring a drop in both Kinetic and Pressure Energy to ensure that the total height of the stacked energy bar never fluctuates. This constant sum confirms the principle as a fundamental statement of energy conservation for ideal fluids.

### Brief audio

{% embed url="<https://youtu.be/9tR8NYFSVi4>" %}

### Condensed Notes

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