# Lecture Slides

All the things that were used in class or assigned as homework. Sent by email and hard copies scanned and uploaded.

## Lectures 1 - 11: Mathematics of Fluid Mechanics

### Lecture 1: A Topic in Classical Physics

7 pages

**Fluid Definition:** A structure (gas or liquid) where intermolecular forces are moderate to weak. This contrasts with solids where intermolecular forces are strong. Additionally, there is significant molecular thermal motion and disorder in arrangement. The counter example are crystals, where there is periodicity.

**Continuum Hypothesis: **A fluid exhibits no structure, no matter how small it is divided. Where $\lambda$ is the mean free path between molecules. Let $L$ be the characteristic length of interest where

**Time-Space Scale Diagram: **On a power log graph of a bivariate equation

Where $x$ and $t$ correspond to SI length and time scales; the constant $k$ is unity, there distinct regions of length scales separating global, fluids, molecular, and atomic interactions. Which line every 3 orders of magnitude.

Fluid mechanics is a topic in classical physics of fluid is a substance gas or liquid where intermolecular forces are medium to week and there's a significant random movement of molecules and disorder in the arrangement

Continuum hypothesis a fluid is modeled as a Continuum or material that is exhibiting no structure however small is divided

$\lambda$ is the mean free path

the mean free path is a distance between molecular collisions 10-7 M for air at STP greater than a meter at the edge of the atmosphere 10 -10 m in water liquid

Knudsen number

$\text{Kn} = \frac{\lambda}{L} << 1$

but L characteristic length of Interest much greater than Lambda Kn = lambda/L <<< 1 Collision between thermal equilibrium tau is less than 10 to the -8 second and air at STP free continuing Behavior how is much less than the time of Interest which we measure

Is important to recognize that for very small distances approximately within an order of magnitude of the mean free path of the situation at hand

density as a function control volume $\rho = \lim_{\Delta V\rightarrow min} \frac{\Delta m}{\Delta Vol}$

specific volume $ v = 1/\rho$

## Lectures 12 - 22: Physical Flows and Turbulence

### Lecture 12: Vorticity Transport Equation

6 pages

From the equations of motion and the vector identity

Balance or Transport Equation of Vorticity

tilting and the stretching/cromressing of vorticity by velocity gradient

measure of the compressibility of of vorticity

Baroclinic Effect

Viscous Diffusion of Vorticity

Fluid Statics

Body Force Due to Gravitation

Due to Atmospheric Pressure

Focus is on incompressible flow or constant density fluid.

## Lecture 12

I missed this lecture, mom had a bad day and I had a bad call. RIP

### Vorticity Transport Equation

From equations of motion

We find using the vector identity ${\bf V}\cdot\nabla{\bf V} = \nabla \left(\frac{2}\right) -{\bf V}\times{\underline{\omega}}$ then applying $\nabla \times (\cdot)$ of the expression.

for a newtonian fluid the balance of vorticity $\underline{\omega}$. Effect to change in the flow are each of the terms:$\textcolor{red}{\text{screen caps would be nice}}$

$\underline{\omega}\cdot(\nabla{\bf V})$ is the streching

$\underline{\omega} \left(\nabla \cdot{\bf V}\right)$ measure of compressibility and stretching of rotation

$\nabla T\times \nabla s$ boroclinic effect since $s(T, p), \quad \vec{\nabla} s=(\ldots) \vec{\nabla} T+( \ldots) \vec{v} p$

the rest is viscous diffusion

### Fluid statics

If the fluid is not in motion, pressure becomes the dominant effect this is $\bf V=\underline{0}$ and $\nabla {\bf V} = \underline{\underline{0}}$. Equations of motion become $\nabla p = \rho {\bf B}$ the fluid static equation.

Assuming that ${\bf B } = g {\bf e}*z$ and $\rho$ is a constant for most liquids. so $\frac{d p}{dz} = \rho g$ and $p|*{z=0} = p_\text{atm}$. So integrate from the surface atmospheric for the hydrostatic equation $p(z) = \rho g z+p_a $ for $z \geq 0$.

Force and moment but idk if this is right… don’t trust the units...

#### Atmospheric pressure

Use the expression $\frac{dp}{dz} = - \rho g$, assuming a perfect gas $p = \rho RT$ creates the aerostatics equation

Using a given T(z)$\textcolor{red}{\text{he corrected but I still can’t read his correction….}}$

Generally T is const $T(z) = \text{const}= 216.6 \text{ K.}$

Stratosphere $1.1\times 10^{4}<z<2.5\times 10^{4}$ where $z*\text{bot} = 1.1\times 10^{4}$, $\ln (\frac{p}{p*\text{bot}})=\frac{-g}{RT}(z-z_\text{bot})$ is solved straightforward

Focus on liquids where incompressible have $\rho = \text{const}$. for low $\text{Ma}$. of gases and most liquids. Use integral equation of property balance for fixed volume

$\underline{\text{mass:}} \int*{\text{vol}} \frac{\partial}{\partial t}d\text{Vol} + \int*\text{surf} \rho {\bf V}\cdot {\bf n}dS=0$ where usually $\int_\text{surf} {\bf V}\cdot {\bf n}dS=0$

$\underline{momentum}$aaaaayyy $\int*\text{vol}\frac{\partial}{\partial t}(\rho{\bf V})d\text{vol} + \int \rho {\bf V}({\bf V}\cdot {\bf n})dS = \int*\text{vol} \rho {\bf B} d\text{vol} - \int*\text{surf} p {\bf n} dS + \int*\text{surf} \underline{\tau} dS $

or

$\cancelto{0}{\int*\text{vol}\frac{\partial}{\partial t}(\rho{\bf V})d\text{vol} }+ \int \rho {\bf V}({\bf V}\cdot {\bf n})dS = \int*\text{vol} \rho {\bf B} d\text{vol} - \int*\text{surf} p {\bf n} dS + \int*\text{surf} \underline{\tau} dS \\textcolor{red}{\text{i can’t deal with these notes… impossible to transcribe}}$

Ya know, I'm not mathematician here, but even I can see a common denominator ...

from

`u/Synli`

on this post

### unsorted

Lecture 1Specific volume Density Mean free path Fluid property

Fluid mechanics is a topic in classical physics of fluid is a substance gas or liquid where intermolecular forces are medium to week and there's a significant random movement of molecules and disorder in the arrangement Continuum hypothesis a fluid is modeled as a Continuum or material that is exhibiting no structure however small is divided the mean free path is a distance between molecular collisions 10-7 M for air at STP greater than a meter at the edge of the atmosphere 10 -10 m in water liquid but L characteristic length of Interest much greater than Lambda Kn = lambda/L <<< 1 Collision between thermal equilibrium tau is less than 10 to the -8 second and air at STP free continuing Behavior how is much less than the time of Interest which we measure

Is important to recognize that for very small distances approximately within an order of magnitude of the mean free path of the situation at hand Lecture 2

To find the pressure of a hydrostatic fluid look at the unit tetrahedra each of the minor faces has a force orthogonal to eachother. The normal force balancing has to be the balaning them allso that the pressure is the force pre unit area of some arbitrary are Inreality there are viscous effects but are very small for most slightly viscous fluids

Lecture 3Lecture 4Lecture 5Lecture 6Lecture 7Lecture 8Lecture 9Lecture 10Lecture 11Lecture 12

Finite Elements

# Problem 1 What fluid flow and the requirements for the application of fluid continuum mechanicsThermodynamic identity to differential internal energ What is the expression for specific internal energy using thermodynamic identity Use joe’s post in fluid Knudsen number for STP Air 10E-7 m What is vorticity and rotor? What is its relation to spin vector # Problem 2 Critical point something? Not sure right now Find the vorticity of a given field Complete the operation for gradient of a scalar and left hand dot # Problem 3 # Problem 4Write the lagrangian description of the fluid momentum or the equation of motionDefine each component Write the equation for conservation of energy in differential form Write the equation for the viscous stress tensor used in NS equations Will help to know the derivation 10-10 on so lecture 11

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