Kinetic energy flux of circular Poiseuille flow ($P_{\mathrm{kin}}=\frac{\pi \Delta p^3{a}^8}{512{\rho }^2{\nu }^3{\ell }^3}$)

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 $P_{\mathrm{kin}}=\frac{1}{2}\rho \int u^{3}dA=\pi \rho a^2{U}_{\max }^3/8$. With pressure drop $\Delta p$ over length $\ell$ and kinematic viscosity $\nu$, $U_{\max }=\Delta p{a}^2/(4\rho \nu \ell )$, so $P_{\mathrm{kin}}=\pi \Delta p^3{a}^8/(512{\rho }^2{\nu }^3{\ell }^3)$. This is not the pressure-work dissipation $\Delta pQ$.