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Bernoulli and Poiseuille

I recently read a paper by Cui et al which used Poiseuille's Law and Bernoulli's Principle to model the flow of aqueous humor through an artificial trabeculum drainage system (ATDS), which is essentially a silicone tube. My understanding of Poiseuille's Law and Bernoulli's Principle was a little rusty so I wanted to write a quick explanation of both concepts here. References are listed at the bottom of this post.

Poiseuille's Law

Poiseuille's Law describes how the pressure drops across a long tube for an incompressible, Newtonian fluid undergoing laminar flow. The equation is:

Where:

P is the pressure gradient

mu is the fluid viscosity

L is the tube length

Q is the flow rate

R is the tube radius

Cui et al used this equation to estimate the pressure drop in the eye that they expected to see for a given length of tubing in the ATDS. Notice that Poiseuille's Law is only applicable to laminar flow, but what is laminar flow?

Laminar Flow

Laminar flow is, in layman's terms, the smooth flow of a fluid. All the layers of the fluid are parallel to each other, and there is no disruption between the layers. Fluid that is moving more slowly tends to be laminar - above a certain velocity, most fluids tend to transition to turbulent flow, which has eddies and disruptions and could be considered "rough" flow.

The Reynolds number can be used to characterize whether a fluid flow is laminar or turbulent. The Reynolds number describes the ratio of a fluid's inertial force to its shearing force - in other words, the ratio of how fast the fluid moves compared to how viscous it is. Smaller Reynolds numbers tend to indicate laminar flow (because either the fluid is moving slowly or it is very viscous). Higher Reynolds numbers tend to indicate turbulent flow. The Reynolds number can be calculated as:

Where:

Re is the Reynolds number (unitless)

rho is the fluid density

v is the mean fluid velocity

D_H is the hydraulic diameter of the pipe (the same as the actual diameter for a circular cross-section of pipe)

mu is the dynamic viscosity of the fluid

Bernoulli's Principle

Bernoulli's Principle states that an increase in fluid speed has a corresponding decrease in fluid pressure. It can be written as:

Where:

v is the fluid flow velocity

g is the gravitational constant

z is the height of the fluid above a reference plane

P is the pressure gradient

rho is the fluid density

References

  1. Cui LJ, Li DC, Liu J, Zhang L, Xing Y. Intraocular pressure control of a novel glaucoma drainage device - in vitro and in vivo studies. Int J Ophthalmol 2017; 10(9): 1354-1360.

  2. https://en.wikipedia.org/wiki/Bernoulli%27s_principle

  3. https://en.wikipedia.org/wiki/Hagen%E2%80%93Poiseuille_equation\

  4. https://en.wikipedia.org/wiki/Laminar_flow

  5. https://en.wikipedia.org/wiki/Hydraulic_diameter

  6. https://www.codecogs.com/latex/eqneditor.php


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