154 lines
4.9 KiB
Markdown
Executable File
154 lines
4.9 KiB
Markdown
Executable File
---
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author: Alvie Rahman
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date: \today
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title: MMME1048 // Fluid Mechanics
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tags: [ uni, nottingham, mechanical, engineering, fluid_mechanics, mmme1048 ]
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---
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# Lecture 1 // Properties of Fluids (2021-10-06)
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## What is a Fluid?
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- A fluid may be liquid, vapor, or gas
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- No permanent shape
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- Consists of atoms in random motion and continual collision
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- Easy to deform
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- Liquids have fixed volume, gasses fill up container
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- **A fluid is a substance for wich a shear stress tends to produce unlimited, continuous
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deformation**
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## Shear Forces
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- For a solid, application of shear stress causes a deformation which, if not too great (elastic),
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is not permanent and solid regains original positon
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- For a fluid, continuious deformation takes place as the molecules slide over each other until the
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force is removed
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- **A fluid is a substance for wich a shear stress tends to produce unlimited, continuous
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deformation**
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## Density
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- Density: $$ \rho = \frac m V $$
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- Specific Density: $$ v = \frac 1 \rho $$
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### Obtaining Density
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- Find mass of a given volume or volume of a given mass
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- This gives average density and assumes density is the same throughout
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- This is not always the case (like in chocolate chip ice cream)
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- Bulk density is often used to refer to average density
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### Engineering Density
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- Matter is not continuous on molecular scale
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- For fluids in constant motion, we take a time average
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- For most practical purposes, matter is considered to be homogenous and time averaged
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## Pressure
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- Pressure is a scalar quantity
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- Gases cannot sustain tensile stress, liquids a negligible amount
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- There is a certain amount of energy associated with the random continuous motion of the molecules
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- Higher pressure fluids tend to have more energy in their molecules
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### How Does Molecular Motion Create Force?
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- When molecules interact with each other, there is no net force
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- When they interact with walls, there is a resultant force perpendicular to the surface
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- Pressure caused my molecule: $$ p = \frac {\delta{}F}{\delta{}A} $$
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- If we want total force, we have to add them all up
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- $$ F = \int \mathrm{d}F = \int p\, \mathrm{d}A $$
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- If pressure is constant, then this integrates to $$ F = pA $$
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- These equations can be used if pressure is constant of average value is appropriate
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- For many cases in fluids pressure is not constant
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### Pressure Variation in a Static Fluid
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- A fluid at rest has constant pressure horizontally
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- That's why liquid surfaces are flat
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- But fluids at rest do have a vertical gradient, where lower parts have higher presure
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### How Does Pressure Vary with Depth?
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Let fluid pressure be p at height $z$, and $p + \delta p$ at $z + \delta z$.
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Force $F_z$ acts upwards to support the fluid, countering pressure $p$.
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Force $F_z + \delta F_z$acts downwards to counter pressure $p + \delta p$ and comes from the weight
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of the liquid above.
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Now:
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\begin{align*}
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F_z &= p\delta x\delta y \\
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F_z + \delta F_z &= (p + \delta p) \delta x \delta y \\
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\therefore \delta F_z &= \delta p(\delta x\delta y)
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\end{align*}
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Resolving forces in z direction:
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\begin{align*}
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F_z - (F_z + \delta F_z) - g\delta m &= 0 \\
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\text{but } \delta m &= \rho\delta x\delta y\delta z \\
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\therefore -\delta p(\delta x\delta y) &= g\rho(\delta x\delta y\delta z) \\
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\text{or } \frac{\delta p}{\delta z} &= -\rho g \\
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\text{as } \delta z \rightarrow 0,\, \frac{\delta p}{\delta z} &\rightarrow \frac{dp}{dz}\\
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\therefore \frac{dp}{dz} &= -\rho g\\
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\Delta p &= \rho g\Delta z
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\end{align*}
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The equation applies for any fluid.
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The -ve sign indicates that as $z$, height, increases, $p$, pressure, decreases.
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### Absolute and Gauge Pressure
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- Absolute Pressure is measured relative to zero (a vacuum)
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- Guage pressure = absolute pressure - atmospheric pressure
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- Often used in industry
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- If abs. pressure = 3 bar and atmospheric pressure is 1 bar, then gauge pressure = 2 bar
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- Atmospheric pressure changes with altitude
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## Compressibility
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- All fluids are compressible, especially gasses
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- Most liquids can be considered **incompressible** most of the time (and will be in MMME1048, but
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may not be in future modules)
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## Surface Tension
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- In a liquid, molecules are held together by molecular attraction
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- At a boundry between two fluids this creates "surface tension"
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- Surface tension usually has the symbol $$\gamma$$
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## Ideal Gas
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- No real gas is perfect, although many are similar
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- We define a specific gas constant to allow us to analyse the behaviour of a specific gas:
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$$ R = \frac {\tilde R}{\tilde m} $$
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(Universal Gas Constant / molar mass of gas)
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- Perfect gas law
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$$pV=mRT$$
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or
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$$ p = \rho RT$$
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- Pressure always in Pa
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- Temperature always in K
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## Units and Dimentional Analysis
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- It is usually better to use SI units
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- If in doubt, DA can be useful to check that your answer makes sense
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