Lecture 15

.title[
# Lecture 15
]
.subtitle[
## Re: Drag
]
.author[
### Dr. Christopher Kenaley
]
.institute[
### Boston College
]
.date[
### 2024/03/12
]

---

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

- Explore a paradox explained by viscosity

- One drag, now two

- Silly example

]

]

---

# Conservation of Energy: Bernoulli

What is the relationship between fluid motion and pressure?

- Potential energy (PE=mgh)

- Kinetic energy ....
`$KE =mu^2/2$`

- Mechanical work (W=Fd=PAd)

- Along a streamline PE + KE + W = constant

]

`$(P_2-P_1)/\rho+(u_2^2-u_1^2)/2=0$`
 
<img src="img/contenergy.png" width="650" />
]

---

# D'Alembert's Paradox

What is the relationship between fluid motion and pressure?

Along a streamline PE + KE + W = constant

- steady
- incompressible
- inviscid*

]

`$(P_2-P_1)/\rho+(u_2^2-u_1^2)/2=0$`
 
<img src="img/paradox.png" width="650" />

No net force?

What can you say intuitively about this situation?
]

---

# D'Alembert's Paradox

What is the relationship between fluid motion and pressure?

Along a streamline PE + KE + W = constant

- steady
- incompressible
- inviscid*

]

`$(P_2-P_1)/\rho+(u_2^2-u_1^2)/2=0$`
 
<img src="img/paradox2.png" width="650" />

No net force? Not when `$\mu\neq0$`
]

---

# D'Alembert's Paradox

What is the relationship between fluid motion and pressure?

Along a streamline PE + KE + W = constant

- steady
- incompressible
- inviscid*

Viscosity robs fluid of its momentum. There is a shear stress exerted on the sphere and energy is dissipated by viscosity

]

`$(P_2-P_1)/\rho+(u_2^2-u_1^2)/2=0$`
 
<img src="img/boundary3.png" width="650" />

No net force? Not when `$\mu\neq0$`
]

---

# D'Alembert's Paradox

What is the relationship between fluid motion and pressure?

Along a streamline PE + KE + W = constant

- steady
- incompressible
- inviscid*

Becomes even more apparent when we consider a shape that results in lots of flow changing velocity quickly (i.e., shape matters!).

]

`$(P_2-P_1)/\rho+(u_2^2-u_1^2)/2=0$`
 
<img src="https://upload.wikimedia.org/wikipedia/commons/thumb/2/23/Form_drag.svg/1200px-Form_drag.svg.png" width="650" />

No net force? Not when `$\mu\neq0$`
]

---

# D'Alembert's Paradox

What is the relationship between fluid motion and pressure?

Along a streamline PE + KE + W = constant

- steady
- incompressible
- inviscid*

What happens when `$P_a<P_p$`?

]

`$(P_2-P_1)/\rho+(u_2^2-u_1^2)/2=0$`
 
<img src="https://upload.wikimedia.org/wikipedia/commons/thumb/2/23/Form_drag.svg/1200px-Form_drag.svg.png" width="650" />

]

---

# Consequences of D'Alembert's Paradox

Flow separation

Where we would always have "skin" drag, with `$\mu\neq 0$`, we now have "pressure" drag.

]

]

---

# Consequences of D'Alembert's Paradox

Viscosity robs fluid of its momentum. There is a shear stress exerted on the sphere and energy is dissipated by viscosity.

Because of viscosity, velocity cannot increase as
much as in the inviscid case.

New stagnation point where the flow separates

**Reynolds number:** Ratio of inertial (pressure) forces to viscous (shear) forces within a fluid which is subjected to relative internal movement due to different fluid velocities.

]

How much does it separate?

`$$\small{\frac{\text{Pressure stress}}{\text{Shear stress}} \rightarrow \frac{P}{\tau} \rightarrow \frac{\rho u^2}{\mu u/L} \rightarrow Re= \frac{\rho uL}{\mu}}$$`

]

---

# Re across size and velocity scales

`$10^{-4}$` ~ Spermatozoa

`$10^{-1}$` ~ Ciliate

1~Smallest Fish

`$10^2$` ~ Blood flow in brain

`$10^3$` ~ Blood flow in aorta

`$10^4$` ~ Birds flying 1

`$2  \textrm{x} 10^5$` ~ Typical pitch in Major League Baseball

`$4 \textrm{x} 10^6$` ~ Human Swimming

`$10^6$` ~ Fastest Fish

`$3 \textrm{x} 10^8$` ~Blue Whale

`$5 \textrm{x} 10^9$` ~A large ship (RMS Queen Elizabeth 2)

]
]

`$$\small{\frac{\text{Pressure stress}}{\text{Shear stress}} = Re= \frac{\rho uL}{\mu}}$$`
.center[
<img src="https://www.researchgate.net/profile/David-Kimmel-2/publication/230049116/figure/fig9/AS:668284278038541@1536342796949/Size-and-Reynolds-number-of-multiple-trophic-levels-The-viscous-world-Reynolds-number.png" width="175" />
]
]

---

# Re across size and velocity scales

The Re measures the relative importance of inertial vs viscous stresses in determining the flow. Conservation of Re implies identical flow patterns

`$10^{-4}$` ~ Spermatozoa

`$4 \textrm{x} 10^6$` ~ Human Swimming

]

What would it feel like to swim like a sperm?

How would you induce flow conditions to experience this?
]

`$$\small{\frac{\text{Pressure stress}}{\text{Shear stress}} = Re= \frac{\rho uL}{\mu}}$$`
.center[
<img src="img/reflows.png" width="300" />

]
]
---

# Thanks!

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