Lecture 22

.title[
# Lecture 22
]
.subtitle[
## Re: The tiny
]
.author[
### Dr. Christopher Kenaley
]
.institute[
### Boston College
]
.date[
### 2024/04/9
]

---

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

- Being small

- Movement and translocation of the small

- Modeling this

]

![](https://static.wixstatic.com/media/f4f67b_7db070f86706435fafb5acca66f8db69~mv2_d_2420_1330_s_2.png/v1/fill/w_527,h_286,al_c,q_85,usm_0.66_1.00_0.01/ReynoldsNumbersOrganisms.webp)

]

---

#  Life at low Re

- disturbances are manifest over huge relative distances
(boundaries are all important)
- inertia is negligible
-  shape matters considerably less (no wakes)

]

]

---

#  Life at low Re

- disturbances are manifest over huge relative distances.
(boundaries are all important)
- inertia is negligible
-  shape matters considerably less (no wakes)

]

]

.center[
<iframe width="560" height="315" src="https://www.youtube.com/embed/U9g4gRWkFTs" title="YouTube video player" frameborder="0" allow="accelerometer; autoplay; clipboard-write; encrypted-media; gyroscope; picture-in-picture" allowfullscreen></iframe>
]

---

#  Life at low Re

]

]

`$$m\frac{dU}{dt}=6\pi r\mu U$$`
`$$U=U_oe^{-6\pi rt\mu /m}$$`

]

---

#  Life at low Re

speed= 40 μm s^−1

size= 1 μm

`$\textrm{mass=}10^{−12}\textrm{ g}$`

`$\mu$`=1 mPa/s

`$$m\frac{dU}{dt}=6\pi r\mu U$$`
`$$U=U_oe^{-6\pi rt\mu /m}$$`

]

]

---

#  Life at low Re

speed= 40 μm s^−1

size= 1 μm

`$\textrm{mass=}10^{−12}\textrm{ g}$`

`$\mu$`=1 mPa/s

`$$m\frac{dU}{dt}=6\pi r\mu U$$`
`$$U=U_oe^{-6\pi rt\mu /m}$$`

]

```r
u <- function(Uo,m,mu,t,r){
  Uo*exp(-6*pi*r*t*mu/m)
}

t <- seq(0,1,0.001)
Ut <- u(Uo=40*1e-6,m=10^-12,t=t,r=1e-6,mu=1e-3)

p1 <- qplot(t,log(Ut))+theme_classic(20)+ylab("U (m/s)")+xlab("t (s)")
print(p1)
```

]

]

---

#  Life at low Re

]

dist=U x t

]

---

#  Life at low Re

How far will it go?
.pull-left[

]

```
## [1] 4e-08
```

He atom is 1.4 × 10^-10 m

]

---

#  Life at low Re

speed= 10 m s^−1

size= 100 μm

`$\textrm{mass=}2.5^{−10}\textrm{ g}$`

`$\mu$`=.2 mPa/s

In the X . . .
`$$m\frac{dU}{dt}=6\pi r\mu U$$`
In the Y . . .
`$$m\frac{dV}{dt}=6\pi r\mu U-mg$$`

]

]

---

#  Life at low Re

speed= 10 m s^−1

size= 100 μm

`$\textrm{mass=}2.5^{−10}\textrm{ g}$`

`$\mu$`=.2 mPa/s

In the X . . .
`$$m\frac{dU}{dt}=6\pi r\mu U$$`
In the Y . . .
`$$m\frac{dV}{dt}=6\pi r\mu U-mg$$`

]

```r
t2 <- seq(0,100,0.001)
uy <- function(Uo,m,mu,t,r,g=9.8){
  Uo*exp(-6*pi*r*t*mu/m-g)*-1}

Uy <- uy(Uo=10,m=12.5*10^-9,t=t2,r=1e-4,mu=2e-5)
Ux <- u(Uo=10,m=12.5*10^-9,t=t2,r=1e-4,mu=2e-5)

dy <- cumsum(Uy*0.001)
dx <- cumsum(Ux*0.001)
qplot(dx,dy)+ylab("dY (m)")+xlab("dX (m)")+theme_classic(20)
```

]

---

# Thanks!

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