Categories: Fluid dynamics, Fluid mechanics, Physics.

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 \(p\):

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

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.

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

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