Lecture 21

.title[
# Lecture 21
]
.subtitle[
## There Will Be Blood
]
.author[
### Dr. Christopher Kenaley
]
.institute[
### Boston College
]
.date[
### 2023/04/04
]

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<link rel="stylesheet" href="https://cdnjs.cloudflare.com/ajax/libs/font-awesome/5.14.0/css/all.min.css">

- Hagen-Poiseuille law

- Violation of H-P (how it really works)

- Laundry list of other important laws/models

]

]

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# Flow in tubes (inside you!)

Cardiovascular system and elsewhere, too? Where?

]

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# Flow in tubes (inside you!)

`$$\Delta p={\frac{8\mu LQ}{\pi R^{4}}}={\frac{8\pi \mu LQ}{A^{2}}}$$`

]

- The tube is stiff, straight, and uniform

- Blood is Newtonian , i.e., viscosity is constant

- The flow is uniform, laminar and steady, not pulsatile, and the velocity at the wall is zero (no slip at the wall)

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# Flow in tubes (inside you!)

`$$Q=\frac{R^4\Delta P}{8\mu L}$$`
Hagen-Poiseuille law

Valid until the flow is turbulent!
(Re > ~ 2000)
]

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# Flow in tubes (inside you!)

`$$Q=\frac{R^4\Delta P}{8\mu L}$$`
Hagen-Poiseuille law

Effects of ateriosclerosis?
]

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# Flow in tubes (inside you!)

`$$Q=\frac{R^4\Delta P}{8\mu L}$$`
Hagen-Poiseuille law

Non-linear effects of vasodialation and vasoconstriction?
]

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# Flow in tubes (inside you!)

Flow profile is altered by branches:

- Flow entering a side branch results in skewed profile. 
- It takes a certain inlet length before the velocity develops into a parabolic profile again.....

]

`$\frac{l_{inlet}}{D}\approx 0.6Re$`

`$l_{inlet}\approx 0.6ReD$`

]

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# Flow in tubes (inside you!)

Flow profile is altered by branches (non-uniform):

- Aorta mean blood flow is about 6 l/min
- D=3 cm
- Mean velocity is ~ 15 cm/s
- Re~1350

]

`$\frac{l_{inlet}}{D}\approx 0.6Re$`

`$l_{inlet}\approx 0.6ReD$`

]

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# Flow in tubes (inside you!)

Flow is pulsatile, but not everywhere.

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#  Poiseuille, Bernoulli, Leplace, etc. in design

- Number of capillaries (Fick's Law for diffusion)
- Diameter of capillaries (Peclet number for balancing diffusion and convection)
- Relative diameter of other vessels (Murray's law for minimizing circulatory work)
- Relative thickness of vessel walls (Leplace's law for how size sets pressure versus wall tension)
- Relative size of capillaries and alveoli (Diffusion coefficients of O2 in air and water.)
- Concentration of RBCs (cell density vs. viscosity and Poiseuille pressure drop)

]

]

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# Thanks!

Slides created via the R package [**xaringan**](https://github.com/yihui/xaringan).