notes/mechanical/mmme1048_fluid_mechanics.md

---
author: Alvie Rahman
date: \today
title: MMME1048 // Fluid Mechanics
tags: [ uni, nottingham, mechanical, engineering, fluid_mechanics, mmme1048 ]
---

# Lecture 1 // Properties of Fluids (2021-10-06)

## What is a Fluid?

- A fluid may be liquid, vapor, or gas
- No permanent shape
- Consists of atoms in random motion and continual collision
- Easy to deform
- Liquids have fixed volume, gasses fill up container
- **A fluid is a substance for wich a shear stress tends to produce unlimited, continuous
  deformation**

## Shear Forces

- For a solid, application of shear stress causes a deformation which, if not too great (elastic),
  is not permanent and solid regains original positon
- For a fluid, continuious deformation takes place as the molecules slide over each other until the
  force is removed
- **A fluid is a substance for wich a shear stress tends to produce unlimited, continuous
  deformation**

## Density

- Density: $$ \rho = \frac m V $$
- Specific Density: $$ v = \frac 1 \rho $$

### Obtaining Density

- Find mass of a given volume or volume of a given mass
- This gives average density and assumes density is the same throughout

  - This is not always the case (like in chocolate chip ice cream)
  - Bulk density is often used to refer to average density

### Engineering Density

- Matter is not continuous on molecular scale
- For fluids in constant motion, we take a time average
- For most practical purposes, matter is considered to be homogenous and time averaged

## Pressure

- Pressure is a scalar quantity
- Gases cannot sustain tensile stress, liquids a negligible amount

- There is a certain amount of energy associated with the random continuous motion of the molecules
- Higher pressure fluids tend to have more energy in their molecules

### How Does Molecular Motion Create Force?

- When molecules interact with each other, there is no net force
- When they interact with walls, there is a resultant force perpendicular to the surface
- Pressure caused my molecule: $$ p = \frac {\delta{}F}{\delta{}A} $$
- If we want total force, we have to add them all up
- $$ F = \int \mathrm{d}F = \int p\, \mathrm{d}A $$

  - If pressure is constant, then this integrates to $$ F = pA $$
  - These equations can be used if pressure is constant of average value is appropriate
  - For many cases in fluids pressure is not constant

### Pressure Variation in a Static Fluid

- A fluid at rest has constant pressure horizontally
- That's why liquid surfaces are flat
- But fluids at rest do have a vertical gradient, where lower parts have higher presure 

### How Does Pressure Vary with Depth?

![From UoN MMME1048 Fluid Mechanics Notes](./images/vimscrot-2021-10-06T10:51:51,499044519+01:00.png)

Let fluid pressure be p at height $z$, and $p + \delta p$ at $z + \delta z$.

Force $F_z$ acts upwards to support the fluid, countering pressure $p$.

Force $F_z + \delta F_z$acts downwards to counter pressure $p + \delta p$ and comes from the weight
of the liquid above.

Now:

\begin{align*}
F_z &= p\delta x\delta y \\
F_z + \delta F_z &= (p + \delta p) \delta x \delta y \\
\therefore \delta F_z &= \delta p(\delta x\delta y)
\end{align*}

Resolving forces in z direction:

\begin{align*}
F_z - (F_z + \delta F_z) - g\delta m &= 0 \\
\text{but } \delta m &= \rho\delta x\delta y\delta z \\
\therefore -\delta p(\delta x\delta y) &= g\rho(\delta x\delta y\delta z) \\
\text{or } \frac{\delta p}{\delta z} &= -\rho g \\
\text{as } \delta z \rightarrow 0,\, \frac{\delta p}{\delta z} &\rightarrow \frac{dp}{dz}\\
\therefore \frac{dp}{dz} &= -\rho g\\
\Delta p &= \rho g\Delta z
\end{align*}

The equation applies for any fluid.
The -ve sign indicates that as $z$, height, increases, $p$, pressure, decreases.

### Absolute and Gauge Pressure

- Absolute Pressure is measured relative to zero (a vacuum)
- Guage pressure = absolute pressure - atmospheric pressure

  - Often used in industry

- If abs. pressure = 3 bar and atmospheric pressure is 1 bar, then gauge pressure = 2 bar
- Atmospheric pressure changes with altitude

## Compressibility

- All fluids are compressible, especially gasses
- Most liquids can be considered **incompressible** most of the time (and will be in MMME1048, but
  may not be in future modules)

## Surface Tension

- In a liquid, molecules are held together by molecular attraction
- At a boundry between two fluids this creates "surface tension"
- Surface tension usually has the symbol $$\gamma$$

## Ideal Gas

- No real gas is perfect, although many are similar
- We define a specific gas constant to allow us to analyse the behaviour of a specific gas:

  $$ R = \frac {\tilde R}{\tilde m} $$

  (Universal Gas Constant / molar mass of gas)

- Perfect gas law

  $$pV=mRT$$

  or

  $$ p = \rho RT$$

  - Pressure always in Pa
  - Temperature always in K

## Units and Dimentional Analysis

- It is usually better to use SI units
- If in doubt, DA can be useful to check that your answer makes sense