Categories: Fluid dynamics, Fluid mechanics, Physics.

Bernoulli’s theorem

For inviscid fluids, Bernouilli’s theorem states that an increase in flow velocity v⃗\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:

Dv⃗Dt=∂v⃗∂t+(v⃗⋅∇)v⃗=g⃗−∇pρ\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 v⃗\va{v} is constant in tt, it becomes clear that a higher v⃗\va{v} requires a lower pp.

Simple form

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

H=12v⃗2+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 g⃗=−∇Φ\va{g} = - \nabla \Phi. To prove this theorem, we take the material derivative of HH:

DHDt=v⃗⋅Dv⃗Dt+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=v⃗⋅(g⃗−∇pρ)+(∂Φ∂t+(v⃗⋅∇)Φ)+1ρ(∂p∂t+(v⃗⋅∇)p)=∂Φ∂t+1ρ∂p∂t+v⃗⋅(g⃗+∇Φ)+v⃗⋅(∇pρ−∇pρ)\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 g⃗=−∇Φ\va{g} = - \nabla \Phi, we are left with the following equation:

DHDt=∂Φ∂t+1ρ∂p∂t\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 ω⃗=0\va{\omega} = 0. For a proof, see the derivation of ω⃗\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.