Categories: Fluid dynamics, Fluid mechanics, Physics.

Bernoulli’s theorem

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

DvDt=vt+(v)v=gpρ\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 v\va{v} is constant in tt, it becomes clear that a higher v\va{v} requires a lower pp.

Simple form

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

H=12v2+Φ+pρ\begin{aligned} \boxed{ H = \frac{1}{2} \va{v}^2 + \Phi + \frac{p}{\rho} } \end{aligned}

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

DHDt=vDvDt+DΦDt+1ρDpDt\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:

DHDt=v(gpρ)+(Φt+(v)Φ)+1ρ(pt+(v)p)=Φt+1ρpt+v(g+Φ)+v(pρpρ)\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 g=Φ\va{g} = - \nabla \Phi, we are left with the following equation:

DHDt=Φt+1ρpt\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 HH is conserved along the streamline.

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


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