= Bernoulli's principle
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{title2=$p/\rho+|\mathbf u|^2/2+\Phi=B$}
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= Bernoulli principle
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The <Bernoulli principle> relates <pressure>, <kinetic energy> per unit volume and <Newtonian gravitational potential> along a <streamline> of <steady flow> with <inviscid flow>. For constant <mass density>, the <Bernoulli equation> gives $p/\rho+u^2/2+\Phi$ constant along that <streamline>. Constancy across different <streamlines> requires further hypotheses, such as <irrotational flow>.
Precisely, suppose a <fluid> of constant positive <mass density> obeys the <Euler equations for an inviscid fluid>, its <velocity field> and <pressure> are steady, and the body <force> per unit <mass> is $-\nabla\Phi$. Then
$$
B=\frac{p}{\rho}+\frac{|\mathbf u|^2}{2}+\Phi,\qquad \mathbf u\cdot\nabla B=0.
$$
Every term has units of <energy> per unit <mass>. For a uniform gravitational field, <Newtonian gravitational potential> is $\Phi=gz$, up to an additive constant. The <pressure> term represents work by surrounding <fluid>, so acceleration can exchange <pressure> work with <kinetic energy>. Dotting the steady <Euler equations> with $\mathbf u$ gives the displayed <directional derivative>: this is why constancy follows a <streamline>. https://www.damtp.cam.ac.uk/user/mem2/FLUIDS-IB/dyn.pdf[Cambridge, Fluid Dynamics IB, Dynamics, section 2.4]{ref}
Continue the horizontal water constriction example from <incompressible flow>. Assume negligible viscous losses, <steady flow> and approximately <uniform flow> at both measurement sections, so the <streamline> speeds there equal the mean speeds $0.20$ and $0.80\,\mathrm{m\,s^{-1}}$. Equal height cancels the <Newtonian gravitational potential> terms. The ideal <pressure> drop is
$$
p_1-p_2=\frac{\rho}{2}(U_2^2-U_1^2)=\frac{1000}{2}(0.80^2-0.20^2)=300\,\mathrm{Pa}.
$$
This is the <Venturi effect> in its ideal limit: the gain in <kinetic energy> density is supplied by <pressure> work. Cambridge's treatment uses this relation to connect a measured <pressure> difference to <volumetric flow rate>. https://www.damtp.cam.ac.uk/user/mem2/FLUIDS-IB/dyn.pdf[Cambridge, Fluid Dynamics IB, Dynamics, section 2.5]{ref}
The $300\,\mathrm{Pa}$ result is conditional, rather than a prediction of the actual viscous pipe. <Mass conservation> fixes $Q$ and the ratio of mean speeds even with <viscosity>, but nonuniform <velocity profiles> require an integrated <energy> balance, and wall friction dissipates mechanical <energy>. A pump would supply additional work. In particular, constant mean speed in a straight pipe does not imply constant <pressure>: <Poiseuille flow> needs a continuing <pressure gradient> to overcome <viscosity>. https://uclnatsci.github.io/2021/Electromagnetism-Fluids-and-Waves/fluids/bernoulli.html[UCL, Bernoulli's equation, assumptions and boundary limitations]{ref}
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