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I am reading a derivation of a correction on the Hagen-Poisseuille equation and the goal of the correction is to ignore the differential pressure responsible for the kinetic energy of the fluid in order to take into account just the differential pressure responsible for tackling viscosity of the fluid.

The coefficient of viscosity in the general Hagen-Poisseuille equation is given as $$\eta=\frac{\pi R^4}{8Ql}\Delta p$$ while the correction is basically this $$\eta =\frac{\pi R^4}{8Ql}(\Delta p-\Delta p_1)$$ where $\Delta p_1$ - is the portion of the total differential pressure that is responsible for the kinetic energy of the fluid.

Since this would obviously reduce the value of the viscosity coefficient, I don't quite understand the justification of this reasoning. It would be okay if the rate of flow $Q$ was then measured/calculated as being caused just by the differential pressure responsible for tackling viscosity but that doesn't seem to be the case. Any thoughts on this?

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    $\begingroup$ This does not sound right. Can you give a link to your source? $\endgroup$
    – Thomas
    Sep 30 '16 at 23:48
  • $\begingroup$ Please provide reference. Is the flow, to which correction is being applied, steady? $\endgroup$
    – Deep
    Oct 1 '16 at 5:01
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The Poisseulle equation is the answer of the Navier-Sokes equations in a very special case: laminar (or creeping) flow with no-slip boundary conditions in a cylindrical pipe.

The assumption of laminarity is a crucial one here, as it lets you drop off the convection terms (i.e. $(\vec{v}.\nabla)\vec{v}$ terms, which are second order terms in $\vec{v}$) in the N.S. equations (and of course you must assume the flow has reached its steady state, so the time-derivative terms are also zero).

Now, the $\Delta p_1$ term you mentioned above (Dynamic pressure), is $\frac{1}{2}\rho v^2$, which is second order in $v$. So in the regime that the Poisseulle equation is derived, you should drop this term, and it doesn't make any sense to try to improve the result by adding a term like this. (You can't take this as a perturbation correction to the Poisseulle result, because by considering the second order terms, the N.S. equations will be nonlinear PDEs)
If you want to consider this term, you should go back to the original N.S. equation and solve it when the convection term is present (which would be a very hard task, if not impossible.)

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  • $\begingroup$ I have no knowledge of Navier-Stokes equations to be honest so this answer is a bit tough for me to properly understand. You said that the dynamic pressure term should be dropped since it is second order in $v$, however doesn't the dynamic pressure also constitute the differential pressure that sets the fluid in motion? Don't we underestimate by dropping the dynamic pressure? $\endgroup$
    – bonehead
    Sep 30 '16 at 22:58
  • $\begingroup$ Ok. To put it another way, when we are dealing with Poisseulle flow, we are assuming that the velocity of the fluid is very small (so the second order terms could be neglected), so there are no turbulent effects in the flow, and also the fluid moves very smoothly. $\endgroup$
    – SaMaSo
    Oct 1 '16 at 8:55
  • $\begingroup$ Besides this issue, I think just the external pressure differences can set the fluid in motion, not the pressure of the fluid itself. (the fluid cannot push itself.) The Poisseulle eq. relates the external pressure difference to the volume flow rate and other parameters of the fluid. If there are no external pressure differences, I think the fluid stops moving after a while, as a result of the viscous forces. (and the Poisseulle eq. wouldn't apply in that case) $\endgroup$
    – SaMaSo
    Oct 1 '16 at 9:02

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