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Academy
Bernoulli Models
Level 1 - Physics topic page in Fluids.
Principle
Along a streamline, steady incompressible flow trades pressure, speed, and height.
Notation
\(p\)
fluid pressure
\(\mathrm{Pa}\)
\(\rho\)
fluid density
\(\mathrm{kg\,m^{-3}}\)
\(v\)
flow speed
\(\mathrm{m\,s^{-1}}\)
\(y\)
vertical position
\(\mathrm{m}\)
\(g\)
gravitational field strength
\(\mathrm{m\,s^{-2}}\)
Method
Bernoulli's equation is a work-energy statement for a moving fluid element along one streamline.
Pressure work
\[p_1\Delta V-p_2\Delta V\]
Kinetic change
\[\Delta K=\frac{1}{2}\rho\Delta V\left(v_2^2-v_1^2\right)\]
Potential change
\[\Delta U=\rho g\Delta V\left(y_2-y_1\right)\]
Bernoulli result
\[p_1+\frac{1}{2}\rho v_1^2+\rho gy_1=p_2+\frac{1}{2}\rho v_2^2+\rho gy_2\]
At the same height, a higher speed must come with a lower pressure if the model assumptions hold.
Rules
These are the compact Bernoulli forms for steady incompressible nonviscous flow.
Bernoulli equation
\[p+\frac{1}{2}\rho v^2+\rho gy=\text{constant}\]
Two-point form
\[p_1+\frac{1}{2}\rho v_1^2+\rho gy_1=p_2+\frac{1}{2}\rho v_2^2+\rho gy_2\]
Horizontal flow
\[p+\frac{1}{2}\rho v^2=\text{constant}\]
Efflux speed
\[v=\sqrt{2gh}\]
Examples
Question
Water flows horizontally from a wide section at
\[v_1=1.5\,\mathrm{m\,s^{-1}}\]
into a narrow section where continuity gives \[v_2=6.0\,\mathrm{m\,s^{-1}}\]
Find \[p_1-p_2\]
Answer
At the same height,
\[p_1+\frac{1}{2}\rho v_1^2=p_2+\frac{1}{2}\rho v_2^2\]
so \[p_1-p_2=\frac{1}{2}\rho\left(v_2^2-v_1^2\right)=\frac{1}{2}(1000)(36-2.25)=1.69\times10^4\,\mathrm{Pa}\]
Checks
- Bernoulli applies along a streamline for steady incompressible flow with negligible viscous loss.
- At the same height, higher speed means lower pressure.
- Use the same pressure reference on both sides of the equation.
- Pumps, turbulence, and large viscous losses break the simple model.