# the relationship between numerical modeling and analytical solutions-Poiseuille's Law in fluid mecha

Both the static plot and the dynamic animation of Hagen-Poiseuille flow illustrate the same fundamental principle: the laminar, parabolic velocity profile, $$v(r)=V\_{\text {max }}\left(1-(r / R)^2\right)$$, is the governing characteristic of fluid transport in a capillary viscometer. The static visualization confirms the accuracy of numerical methods (FDM) by showing its convergence to the analytical parabolic curve, thereby establishing FDM's reliability for solving the underlying Navier-Stokes equations, even in its simplest form. The dynamic animation reinforces this insight by showing tracer particles moving fastest at the pipe's center ( $$V\_{\text {max }}$$ ) and exhibiting zero velocity at the wall due to the no-slip condition, visually confirming that flow speed depends only on the radial position, which directly leads to the critical volumetric flow scaling $$Q \propto \Delta P \cdot R^4$$.

### Brief audio

{% embed url="<https://youtu.be/AutAtFY_eyY>" %}

### Condensed Notes

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