Add notes for mmme1048 fluid mechanics

mechanical/mmme1048_fluid_mechanics.md

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+---
+author: Alvie Rahman
+date: \today
+title: MMME1048 // Fluid Mechanics
+tags: [ uni, nottingham, mechanical, engineering, fluid_mechanics ]
+---
+
+# 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 $F_z$, due to pressure $p$, acts upwards to support the fluid.
+
+Force $F_z + \delta F_z$, due to pressure $p + \delta p$, acts downwards due to weight of fluid
+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