= Kinetic energy flux of circular Poiseuille flow
{title2=$P_{\rm kin}=\frac{\pi\Delta p^3a^8}{512\rho^2\nu^3\ell^3}$}
For a parabolic profile $u(s)=U_{\max}(1-s^2/a^2)$ in a pipe of radius $a$, the mean speed is $U_{\max}/2$ and $P_{\rm kin}=\tfrac12\rho\int u^3dA=\pi\rho a^2U_{\max}^3/8$. With <pressure> drop $\Delta p$ over length $\ell$ and <kinematic viscosity> $\nu$, $U_{\max}=\Delta p\,a^2/(4\rho\nu\ell)$, so $P_{\rm kin}=\pi\Delta p^3a^8/(512\rho^2\nu^3\ell^3)$. This is not the pressure-work dissipation $\Delta p\,Q$.
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