12.5: Bernoulli's Law
Objective
The student will:
- Understand Bernoulli’s principle and be able to discuss its implications.
Vocabulary
- Bernoulli’s Principle: A key principle connecting velocity and air. At those points in space where the velocity of a fluid is high, the pressure is low. At those points in space where the velocity of a fluid is low, the pressure is high.
Introduction
Flying has always been one of the great goals of human engineering. The key question is, what moves the force upward on an airplane wing? We can measure air speed and show that the shape of a wing makes the air go faster over the top of the wing than below the wing. How does this create lift, though?
A key principle connecting velocity and air was expressed by the Swiss mathematician Daniel Bernoulli (1700-1782). Bernoullo's Principle states:
At those points in space where the velocity of a fluid is high, the pressure is low. At those points in space where the velocity of a fluid is low, the pressure is high.
The mathematical equation that Bernoulli derived based on this principle is stated below.
Bernoulli’s Equation: \begin{align*}P_1 + \frac{1}{2} \rho v{_1}^2 + \rho gh_1 = P_2 + \frac{1}{2} \rho v{_2}^2 + \rho gh_2 = \text{constant}\end{align*}
where \begin{align*}P\end{align*} is the pressure of the fluid, \begin{align*}\rho\end{align*} is the density of the fluid, \begin{align*}v\end{align*} is the velocity of the fluid, and \begin{align*}h\end{align*} is the height of the fluid. Though each term in the equation has units of pressure, Bernoulli derived the equation based on the conservation of energy. The equation is, in fact a statement of the conservation of energy.
A demonstration of this principle can be seen with a device called a venture tube, Figure below.
Notice in the Figure above that where the fluid has a greater velocity, the vertical tube as a lower water level. This is the result of a lower internal pressure in the fluid at this point. If you have ever placed your finger partly over the opening of a garden hose, you’ve probably seen water velocity increase. The water sprays out! In the same way, in Figure above, equal volumes of fluid must flow through the large cross sectional area \begin{align*}(A_1)\end{align*} and small cross sectional area \begin{align*}(A_2)\end{align*} of the tube. Therefore, the flow through the \begin{align*}A_2\end{align*} must have a greater velocity than the flow \begin{align*}A_1\end{align*}.
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