Bernoulli's Equation
Steady Flow
Spatial derivative along a streamline.
Kinetic energy.
\( (\vec{\boldsymbol{v}} \cdot \boldsymbol{\nabla}) \)
\( ( \)
\( \dfrac{\rho v^2}{2} \)
\( + \)
\( P\)
\( + \)
\( \rho\varphi \)
\( ) \)
\( = 0 \qquad \)
where \( \; \vec{f} = \boldsymbol{\nabla} \varphi \)
Pressure.
Body forces (e.g. gravity).
(Steady Flow Form)
By integrating, we get the following equation for any arbitrary streamline. It's extremely important to recognize its only constant on a streamline.
Kinetic energy.
\( \dfrac{\rho v^2}{2} \)
\( + \)
\( P\)
\( + \)
\( \rho\varphi \)
\( = constant \qquad \)
where \( \; \vec{f} = \boldsymbol{\nabla} \varphi \)
Pressure.
Body forces (e.g. gravity).
(Steady Flow Form)
Potential Flow
Spatial derivative.
Unsteady velocity term.
Kinetic energy.
\( \boldsymbol{\nabla} \)
\( (\)
\( \rho \dfrac{\partial\phi}{\partial t} \)
\( + \)
\( \dfrac{\rho v^2}{2} \)
\( + \)
\( P \)
\( + \)
\( \rho\varphi \)
\() = 0 \qquad \)
where
\( \; \vec{f} = \boldsymbol{\nabla} \varphi \; \)
\( \; \vec{v} = \boldsymbol{\nabla} \phi \)
Pressure.
Body forces (e.g. gravity).
(Potential Flow Form)
By integrating, we get the following equation. Unlike the steady flow bernoulli's equation, the potential flow bernoulli's equation holds true for any point in space, not just on a streamline.
Unsteady velocity term.
Kinetic energy.
\( \rho \dfrac{\partial\phi}{\partial t} \)
\( + \)
\( \dfrac{\rho v^2}{2} \)
\( + \)
\( P \)
\( + \)
\( \rho\varphi \)
\( = constant \qquad \)
where
\( \; \vec{f} = \boldsymbol{\nabla} \varphi \; \)
\( \; \vec{v} = \boldsymbol{\nabla} \phi \)
Pressure.
Body forces (e.g. gravity).
(Potential Flow Form)