Bernoulli's Equation
- Fluid Mechanics 1.5
Bernoulli’s Equation
Let’s recall the Euler equation for an ideal fluid:
\[ \pdv{\b{v}}{t} - \b{v} \times (\curl \b{v}) = -\grad \left( w + \frac{1}{2} v^2 \right) + \b{a} \]
where $w$ is the specific enthalpy and $\b{a}$ represents body forces per unit mass (e.g., gravity). For steady flow, the time derivative term vanishes. If the body forces are conservative, i.e. $\b{a} = -\grad \Phi$, we can rewrite the equation as:
\[ \b{v} \times (\curl \b{v}) = \grad \left( w + \frac{1}{2} v^2 + \Phi \right) \]
Now let’s imagine a fluid particle moving along a particular streamline. We’ll think the velocity direction component of the vector equation above. The left-hand side becomes zero because the cross product of the velocity vector with itself is zero. Thus, we have:
\[ \pdv{}{l} \left( w + \frac{1}{2} v^2 + \Phi \right) = 0 \]
where $l$ is the variable parameterizing the streamline. This implies that the quantity inside the parentheses is constant along a streamline:
\[ w + \frac{1}{2} v^2 + \Phi = \text{const.} \]
This is known as Bernoulli’s equation for ideal fluids. It states that the sum of the specific enthalpy, kinetic energy per unit mass, and potential energy per unit mass remains constant along a streamline. We can also write it as:
\[ \frac{1}{2} v^2 + \varepsilon + \frac{p}{\rho} + \Phi = \text{const.} \]
For gravity, $\Phi = gz$ where $z$ is the height above a reference level.
