Bernoulli’s Equation

An ugly af equation that’s mostly useful, yet mostly incomprehensible without the help of a guiding physicist (or fellow physics student). Bernoulli’s relation (for incompressible ideal fluids (i.e. those with zero internal resistance a.k.a. Viscosity)) relates a fluid’s Pressure at two points in a Steady Flow type situation.

Stated in the more understandable, personal fave and defs preferred way: {\Delta}p = p_1 - p_2 = \frac{1}{2}{\rho}{(v_2^2 - v_1^2)}  - {\rho}g{(y_2 - y_1)}

Where ρ is the density of the fluid in question.

With this form of the equation, we can see that there are two sources of pressure difference.
1. The first source of pressure difference results from a difference of Flow Velocity.
2. The second source results from a difference in Gravitational Potential.

The version of this equation that I hate (but my textbooks seem to have a real hard on for it) is the expanded version: p_1 - {\frac{1}{2}}{\rho}{v_1^2} - {\rho}g{y_1} = p_2 - {\frac{1}{2}}{\rho}{v_2^2} - {\rho}g{y_2}
Or you could simply state: p - {\frac{1}{2}}{\rho}{v^2} - {\rho}{gy} = constant ,, whatever floats your boat (more on that here: Buoyancy)
There’ll be a derivation coming soon.


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