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19: Appendix H- Bernoulli's Equation

  • Page ID
    18046
  • Recall the momentum equation for a homogeneous, inviscid fluid, written in gravity-aligned coordinates:

    \[\frac{D \vec{u}}{D t}=-g \hat{e}^{(z)}-\vec{\nabla} \frac{p}{\rho_{0}}.\]

    Using the vector identity

    \[[\vec{u} \cdot \vec{\nabla}] \vec{u} \equiv(\vec{\nabla} \times \vec{u}) \times \vec{u}+\frac{1}{2} \vec{\nabla}(\vec{u} \cdot \vec{u}),\]

    we can rewrite this as

    \[\frac{\partial \vec{u}}{\partial t}+\vec{\omega} \times \vec{u}+\frac{1}{2} \vec{\nabla}(\vec{u} \cdot \vec{u})=-g \hat{e}^{(z)}-\vec{\nabla} \frac{p}{\rho_{0}},\]

    where \(\vec{\omega} = \vec{\nabla} \times\vec{u}\) is the vorticity. Next, note that the vertical unit vector is the gradient of the vertical coordinate: \(\hat{e}^{(z)}\) = \(\vec{\nabla}_z\). We now substitute this and collect all of the terms that can be expressed as gradients:

    \[\frac{\partial \vec{u}}{\partial t}+\vec{\omega} \times \vec{u}=-\vec{\nabla}\left(\frac{1}{2}(\vec{u} \cdot \vec{u})+g z+\frac{p}{\rho_{0}}\right),\]

    or

    \[\frac{\partial \vec{u}}{\partial t}+\vec{\omega} \times \vec{u}=-\vec{\nabla} B,\]

    where

    \[B=\frac{1}{2}(\vec{u} \cdot \vec{u})+g z+\frac{p}{\rho_{0}}\]

    is called the Bernoulli function1.

    Now assume that the flow is in steady state, i.e., \(\partial\vec{u}/\partial t\) = 0

    \[\vec{\nabla} B=\vec{u} \times \vec{\omega}.\]

    This tells us that the gradient of the Bernoulli function is perpendicular to both \(\vec{u}\) and \(\vec{\omega}\), and therefore that \(B\) does not vary in the direction of either of those vectors. In other words, in steady flow of a homogeneous, inviscid fluid,

    • B is uniform along a vortex filament, and
    • a fluid particle maintains a constant value of \(B\) (since \(\vec{u}\cdot\vec{\nabla}B = DB/Dt = 0\)) as it travels.

    The second point, often called Bernoulli’s Law, famously explains how an airplane flies. Because the upper and lower surfaces of the wing are convex, flow past them is forced to speed up, so that the first term in \(B\) increases. Variation in the second term is negligible, so the third term must decrease to maintain a constant value of \(B\), i.e., the pressure must drop. Wings are designed with the upper surface more convex than the lower, so that the pressure drop is greater. The resulting pressure difference exerts a net upward force (“lift”) on the wing.