PX285 - J1a - bernoulli's principle from conservation of energy

considering a stream lined flow, with $A_{1}$ and $A_{2}$ as the cross-sectional areas of the flow tube

the rate at which the fluid enters region 1:

ρ_{1} A_{1} u_{1}

work is done on fluid by the pushing from behind (left to right in the sketch):

W = F d = \frac{F u}{t}

the rate at which work is done is:

P_{1} A_{1} u_{1}

the rate at which kinetic energy of the fluid enters region 1:

\frac{1}{2} ρ_{1} A_{1} u_{1} (u_{1})^{2}

and likewise, the gravitational potential energy:

ρ_{1} A_{1} u_{1} g z_{1}

the expressions for region 2 will be identical
from conservation of mass, if flow is incompressible:

ρ_{1} A_{1} u_{1} = ρ_{2} A_{2} u_{2}

considering a constant internal energy, ie. no friction, so the flow is inviscid
from conservation of energy:

ρ A_{1} u_{1} (\frac{u_{1}^{2}}{2} + g z_{1} + \frac{P_{1}}{ρ}) = ρ A_{2} u_{2} (\frac{u_{2}^{2}}{2} + g z_{2} + \frac{P_{2}}{ρ})

the pre-factors cancel out from conservation of mass
therefore, the invariant of the flow along a stream line:

ρ \frac{u^{2}}{2} + ρ g z + P = constant

this is called bernoulli's principle
required conditions:
- incompressible, ie: $\vec{\nabla} \cdot \vec{u} = 0$
- inviscid, ie: $μ = 0$
- steady, ie: $\frac{\partial}{\partial t} = 0$