Categories: Fluid dynamics, Fluid mechanics, Physics.

Bernoulli’s theorem

For inviscid fluids, Bernuilli’s theorem states that an increase in flow velocity \(\va{v}\) is paired with a decrease in pressure \(p\) and/or potential energy. For a qualitative argument, look no further than one of the Euler equations, with a material derivative:

\[\begin{aligned} \frac{\mathrm{D} \va{v}}{\mathrm{D} t} = \pdv{\va{v}}{t} + (\va{v} \cdot \nabla) \va{v} = \va{g} - \frac{\nabla p}{\rho} \end{aligned}\]

Assuming that \(\va{v}\) and \(\va{g}\) are constant in \(t\), it becomes clear that a higher \(\va{v}\) requires a lower \(p\):

\[\begin{aligned} \frac{1}{2} \nabla \va{v}^2 = \va{g} - \frac{\nabla p}{\rho} \end{aligned}\]

Simple form

For an incompressible fluid with a time-independent velocity field \(\va{v}\) (i.e. steady flow), Bernoulli’s theorem formally states that the Bernoulli head \(H\) is constant along a streamline:

\[\begin{aligned} \boxed{ H = \frac{1}{2} \va{v}^2 + \Phi + \frac{p}{\rho} } \end{aligned}\]

Where \(\Phi\) is the gravitational potential, such that \(\va{g} = - \nabla \Phi\). To prove this theorem, we take the material derivative of \(H\):

\[\begin{aligned} \frac{\mathrm{D} H}{\mathrm{D} t} &= \va{v} \cdot \frac{\mathrm{D} \va{v}}{\mathrm{D} t} + \frac{\mathrm{D} \Phi}{\mathrm{D} t} + \frac{1}{\rho} \frac{\mathrm{D} p}{\mathrm{D} t} \end{aligned}\]

In the first term we insert the Euler equation, and in the other two we expand the derivatives:

\[\begin{aligned} \frac{\mathrm{D} H}{\mathrm{D} t} &= \va{v} \cdot \Big( \va{g} - \frac{\nabla p}{\rho} \Big) + \Big( \pdv{\Phi}{t} + (\va{v} \cdot \nabla) \Phi \Big) + \frac{1}{\rho} \Big( \pdv{p}{t} + (\va{v} \cdot \nabla) p \Big) \\ &= \pdv{\Phi}{t} + \frac{1}{\rho} \pdv{p}{t} + \va{v} \cdot \big( \va{g} + \nabla \Phi \big) + \va{v} \cdot \Big( \frac{\nabla p}{\rho} - \frac{\nabla p}{\rho} \Big) \end{aligned}\]

Using the fact that \(\va{g} = - \nabla \Phi\), we are left with the following equation:

\[\begin{aligned} \frac{\mathrm{D} H}{\mathrm{D} t} &= \pdv{\Phi}{t} + \frac{1}{\rho} \pdv{p}{t} \end{aligned}\]

Assuming that the flow is steady, both derivatives vanish, leading us to the conclusion that \(H\) is conserved along the streamline.

In fact, there exists Bernoulli’s stronger theorem, which states that \(H\) is constant everywhere in regions with zero vorticity \(\va{\omega} = 0\). For a proof, see the derivation of \(\va{\omega}\)’s equation of motion.

References

  1. B. Lautrup, Physics of continuous matter: exotic and everyday phenomena in the macroscopic world, 2nd edition, CRC Press.

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