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

# Lecture 2 // Manometers (2021-10-13)

![](./images/vimscrot-2021-10-13T09:09:32,037006075+01:00.png)

$$p_{1,gauge} = \rho g(z_2-z_1)$$

- Manometers work on the principle that pressure along any horizontal plane through a continuous
  fluid is constant
- Manometers can be used to measure the pressure of a gas, vapour, or liquid
- Manometers can measure higher pressures than a piezometer
- Manometer fluid and working should be immiscible (don't mix)

![](./images/vimscrot-2021-10-13T09:14:59,628661490+01:00.png)

\begin{align*}
p_A &= p_{A'} \\
p_{bottom} &= p_{top} + \rho gh \\
\rho_1 &= density\,of\,fluid\,1 \\
\rho_2 &= density\,of\,fluid\,2
\end{align*}

Left hand side:

$$p_A =  p_1 + \rho_1g\Delta z_1$$

Right hand side:

$$p_{A'} = p_{at} + \rho_2g\Delta z_2$$

Equate and rearrange:

\begin{align*}
p_1 + \rho_1g\Delta z_1 &= p_{at} + \rho_2g\Delta z_2 \\
p_1-p_{at} &= g(\rho_2\Delta z_2 - \rho_1\Delta z_1) \\
p_{1,gauge} &= g(\rho_2\Delta z_2 - \rho_1\Delta z_1)
\end{align*}

If $\rho_a << \rho_2$:

$$\rho_{1,gauge} \approx \rho_2g\Delta z_2$$

## Differential U-Tube Manometer

![](./images/vimscrot-2021-10-13T09:37:02,070474894+01:00.png)

- Used to find the difference between two unknown pressures
- Can be used for any fluid that doesn't react with manometer fluid
- Same principle used in analysis

\begin{align*}
p_A &= p_{A'} \\
p_{bottom} &= p_{top} + \rho gh \\
\rho_1 &= density\,of\,fluid\,1 \\
\rho_2 &= density\,of\,fluid\,2
\end{align*}

Left hand side:

$$p_A = p_1 + \rho_wg(z_C-z_A)$$

Right hand side:

$$p_B = p_2 + \rho_wg(z_C-z_B)$$

Right hand manometer fluid:

$$p_{A'} = p_B + \rho_mg(z_B - z_a)$$

\begin{align*}
p_{A'} &= p_2 + \rho_mg(z_C - z_B) + \rho_mg(z_B - zA)\\
       &= p_2 + \rho_mg(z_C - z_B) + \rho_mg\Delta z \\
\\
p_A &= p_{A'} \\
p_1 + \rho_wg(z_C-z_A) &= p_2 + \rho_mg(z_C - z_B) + \rho_mg\Delta z \\
p_1 - p_2 &= \rho_wg(z_C-z_B-z_C+z_A) + \rho_mg\Delta z \\
&= \rho_wg(z_A-z_B) + \rho_mg\Delta z \\
&= -\rho_wg\Delta z + \rho_mg\Delta z 
\end{align*}

## Angled Differential Manometer

![](./images/vimscrot-2021-10-13T09:56:15,656796805+01:00.png)

- If the pipe is sloped then

  $$p_1-p_2 = (\rho_m-\rho_w)g\Delta z + \rho_wg(z_{C2} - z_{C1})$$

- $p_1 > p_2$ as $p_1$ is lower
- If there is no flow along the tube, then

  $$p_1 = p_2 + \rho_wg(z_{C2} - z_{C1})$$