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+ + +### Example 1 + +The thrust of a jet engine on an aircraft at rest can be analysed in terms of the changes in +momentum or the air passing through the engine. + + + +![](./images/vimscrot-2021-11-03T21:51:51,497459693+00:00.png) + +The control volume is drawn far enough in front of the engine that the air velocity entering can +be assumed to be at atmospheric pressurce and its velocity negligible. + +At the exit of the engine the boundary is drawn close where the velocity is known and the air +pressure atmospheric. + +The control volume cuts the material attaching the engine to the aircraft and there will be a force +transmitted across the control volume there to oppose the forces on the engine created by thrust +and gravity. + +The details of the flows inside the control volume do not need to be known as the thrust can be +determined in terms of forces and flows crossing the boundaries drawn. +However, to understand the flows inside the engine in more detail, a more detailed analysis would +be required. + +
+ +## Ideal Fluid + +The actual flow pattern in a fluid is usually complex and difficult to model but it can be +simplified by assuming the fluid is ideal. +The ideal fluid has the following properties: + +- Zero viscosity +- Incompressible +- Zero surface tension +- Does not change phases + +Gases and vapours are compressible so can only be analysed as ideal fluids when flow velocities are +low but they can often be treated as ideal (or perfect) gases, in which case the ideal gas equations +apply. + +## Steady Flow + +Steady flow is a flow which has *no changes in properties with respect to time*. +Properties may vary from place to place but in the same place the properties must not change in +the control volume to be steady flow. + +Unsteady flow does change with respect to time. + +## Uniform Flow + +Uniform flow is when all properties are the same at all points at any given instant but can change +with respect to time, like the opposite of steady flow. + +## One Dimensional Flow + +In one dimensional (1D) flow it is assumed that all properties are uniform over any plane +perpedenciular to the direction of flow (e.g. all points along the cross section of a pipe have +identical properties). + +This means properties can only flow in one direction---usually the direction of flow. + +1D flow is never achieved exactly in practice as when a fluid flows along a pipe, the velocity at +the wall is 0, and maximum in the centre of the pipe. +Despite this, assuming flow is 1D simplifies the analysis and often is accurate enough. + +## Flow Patterns + +There are multiple ways to visualize flow patterns. + +### Streamlines + +A streamline is a line along which all the particle have, at a given instant, velocity vectors +which are tangential to the line. + +Therefore there is no component of velocity of a streamline. + +A particle can never cross a streamline and *streamlines never cross*. + +They can be constructed mathematically and are often shown as output from CFD analysis. + +For a steady flow there are no changes with respect to time so the streamline pattern does not. +The pattern does change when in unsteady flow. + +Streamlines in uniform flow must be straight and parallel. +They must be parallel as if they are not, then different points will have different directions and +therefore different velocities. +Same reasoning with if they are not parallel. + +### Pathlines + +A pathline shows the route taken by a single particle during a given time interval. +It is equivalent to a high exposure photograph which traces the moevement of the particle marked. +You could track pathlines with a drop of injected dye or inserting a buoyant solid particle which +has the same density as the solid. + +Pathlines may cross. + +### Streaklines + +A streakline joins, at any given time, all particles that have passed through a given point. +Examples of this are line dye or a smoke stream which is produced from a continuous supply. + +## Viscous (Real) Fluids + +### Viscosity + +A fluid offers resisistance to motion due to its viscosity or internal friction. +The greater the resistance to flow, the greater the viscosity. + +Higher viscosity also reduces the rate of shear deformation between layers for a given shear stress. + +Viscosity comes from two effects: + +- In liquids, the inter-molecular forces act as drag between layers of fluid moving at different + velocities +- In gases, the mixing of faster and slower moving fluid causes friction due to momentum transfer. + The slower layers tend to slow down the faster ones + +### Newton's Law of Viscosity + +Viscosity can be defined in terms of rate of shear or velocity gradient. + +![](./images/vimscrot-2021-11-17T14:14:05,079195275+00:00.png) + +Consider the flow in the pipe above. +Fluid in contact with the surface has a velocity of 0 because the surface irregularities trap the +fluid particles. +A short distance away from the surface the velocity is low but in the middle of the pipe the +velocity is $v_F$. + +Let the velocity at a distance $y$ be $v$ and at a distance $y + \delta y$ be $v + \delta v$. + +The ratio $\frac{\delta v}{\delta y}$ is the average velocity gradient over the distance +$\delta y$. + +But as $\delta y$ tends to zero, $\frac{\delta v}{\delta y} \rightarrow$ the value of the +differential $\frac{\mathrm{d}v}{\mathrm{d}y}$ at a point such as point A. + +For most fluids in engineering it is found that the shear stress, $\tau$, is directly proportional +to the velocity gradient when straight and parallel flow is involved: + +$$\tau = \mu\frac{\mathrm{d}v}{\mathrm{d}y}$$ + +Where $\mu$ is the constant of proportinality and known as the dynamic viscosity, or simply the +viscosity of the fluid. + +This is Newton's Law of Viscosity and fluids that ovey it are known as Newtonian fluids. + +### Viscosity and Lubrication + +Where a fluid is a thin film (such as in lubricating flows), the velocity gradient can be +approximated to be linear and an estimate of shear stress obtained: + +$$\tau = \mu \frac{\delta v}{\delta y} \approx \mu \frac{v}{y}$$ + +From the shear stress we can calculate the force exerted by a film by the relationship: + +$$\tau = \frac F A$$ + +# Fluid Flow + +## Types of flow + +There are essentially two types of flow: + +- Smooth (laminar) flow + + At low flow rates, particles of fluid are moving in straight lines and can be considered to be + moving in layers or laminae. + +- Rough (turbulent) flow + + At higher flow rates, the paths of the individual fluid particles are not straight but disorderly + resulting in mixing taking place + +Between fully laminar and fully turbulent flows is a transition region. + +## The Reynolds Number + +### Development of the Reynolds Number + +In laminar flow the most influentialfactor is the magnitude of the viscous forces: + +$$viscous\, forces \propto \mu\frac v l l^2 = \mu vl$$ + +where $v$ is a characteristic velocit and $l$ is a characteristic length. + +In turbulent flow viscous effects are not significant but inertia effects (mixing, momentum +exchange, acceleration of fluid mass) are. +Interial forces can be represented by $F = ma$ + +\begin{align*} +m &\propto \rho l^3 \\ +a &= \frac{dv}{dt} \\ +&\therefore a \propto \frac v t \text{ and } t = \frac l v \\ +&\therefore a \propto \frac {v^2} l \\ +&\therefore \text{Interial forces} \propto \rho l^2\frac{v^2} l = \rho l^2v^2 +\end{align*} + +The ratio of internalforces to viscous forces is called the Reynolds number and is abbreviated to +Re: + +$$\Rey = \frac{\text{interial forces}}{\text{viscous forces}} = \frac {\rho l^2v^2}{\mu vl} = \frac {\rho vl} \mu$$ + +where $\rho$ and $\mu$ are fluid properties and $v$ and $l$ are characteristic velocity and length. + +- During laminar flow, $\Rey$ is small as viscous forces dominate. +- During turbulent flow, $\Rey$ is large as intertial forces dominate. + +\textRey is a non dimensional group. +It has no units because the units cancel out. + +Non dimensional groups are very important in fluid mechancics and need to be considered when scaling +experiments. + +If \textRey is the same in two different pipes, the flow will be the same regardless of actual +diameters, densities, or other properties. + +#### \textRey for a Circular Section Pipe + +The characteristic length for pipe flow is the diameter $d$ and the characteristic velocity is +mean flow in the pipe, $v$, so \textRey of a circular pipe section is given by: + +$$\Rey = \frac{\rho vd} \mu$$ + +For flow in a smooth circular pipe under normal engineering conditions the following can be assumed: + +- $\Rey < 2000$ --- laminar flow +- $2000 < \Rey < 4000$ --- transition +- $\Rey > 4000$ --- fully turbulent flow + +These figures can be significantly affected by surface roughness so flow may be turbulent below +$\Rey = 4000$. + +# Euler's Equation + +In a static fluid, pressure only depends on density and elevation. +In a moving fluid the pressure is also related to acceleration, viscosity, and shaft work done on or +by the fluid. + +$$\frac 1 \rho \frac{\delta p}{\delta s} + g\frac{\delta z}{\delta s} + v\frac{\delta v}{\delta s} = 0$$ + +## Assumptions / Conditions + +The Euler euqation applies where the following can be assumed: + +- Steady flow +- The fluid is inviscid +- No shaft work +- Flow along a streamline + +# Bernoulli's Equation + +Euler's equation comes in differential form, which is difficult to apply. +We can integrate it to make it easier + +\begin{align*} +\frac 1 \rho \frac{\delta p}{\delta s} + g\frac{\delta z}{\delta s} + v\frac{\delta v}{\delta s} &= 0 + & \text{(Euler's equation)} \\ +\int\left\{\frac{\mathrm{d}p} \rho + g\mathrm{d}z + v\mathrm{d}v \right\} &= \int 0 \,\mathrm{d}s \\ +\therefore \int \frac 1 \rho \,\mathrm{d}p + g\int \mathrm{d}z + \int v \,\mathrm{d}v &= \int 0 \,\mathrm{d}s \\ +\therefore \int \frac 1 \rho \,\mathrm{d}p + gz + \frac{v^2}{2} &= \text{constant}_1 +\end{align*} + +The first term of the equation can only be integrated if $\rho$ is constant as then: + +$$\int \frac 1 \rho \,\mathrm{d}p = \frac 1 \rho \int \mathrm{d}p = \frac p \rho$$ + +So, if density is constant: + +$$\frac p \rho + gz + \frac{v^2}{2} = \text{constant}_2$$ + +## Assumptions / Conditions + +All the assumptions from Euler's equation apply: + +- Steady flow +- The fluid is inviscid +- No shaft work +- Flow along a streamline + +But also one more: + +- Incompressible flow + +## Forms of Bernoulli's Equation + +### Energy Form + +This form of Bernoulli's Equation is known as the energy form as each component has the units +energy/unit mass: + +$$\frac p \rho + gz + \frac{v^2}{2} = \text{constant}_2$$ + +It is split into 3 parts: + +- Pressure energy ($\frac p \rho$) --- energy needed to move the flow against the pressure + (flow work) +- Potential energy ($gz$) --- elevation +- Kinetic energy ($\frac{v^2}{2}$) --- kinetic energy + +### Elevation / Head Form + +Divide the energy form by $g$: + +$$\frac\rho{\rho g} + z + \frac{v^2}{2g} = H_T$$ + +where $H_T$ is constant and: + +- $\frac{p}{\rho g}$ --- static/pressure haed +- $z$ --- elevation head +- $\frac{v_2}{2g}$ --- dynamic/velocity head +- $H_T$ --- total head + +- Each term now has units of elevations +- In fluids the elevation is sometimes called head +- This form of the equation is also useful in some applications + +### Pressure Form + +Multiply the energy form by $\rho$ to give the pressure form: + +$$p + \rho gz + \frac 1 2 \rho v^2 = \text{constant}$$ + +where: + +- $p$ --- static pressure (often written as $p_s$) +- $\rho gz$ --- elevation pressure +- $\frac 1 2 \rho v^2$ --- dynamic pressure + +- Density is constant +- Each term now has the units of pressure +- This form is useful is we are interested in pressures + +### Comparing two forms of the Bernoulli Equation (Piezometric) + +$$\text{piezometric} = \text{static} + \text{elevation}$$ + +Pressure form: + +\begin{align*} +p_s + \rho gz + \frac 1 2 \rho v^2 &= \text{total pressure} \\ +p_s + \rho gz &= \text{piezometric pressure} +\end{align*} + +Head form: + +\begin{align*} +\frac{p_s}{\rho g} + z + \frac{v^2}{2g} &= \text{total head} \\ +\frac{p_s}{\rho g} + z &= \text{piezometric head} +\end{align*} diff --git a/uni/mmme/1048_thermodynamics_and_fluid_mechanics/fluid_mechanics.md b/uni/mmme/1048_thermodynamics_and_fluid_mechanics/fluid_mechanics.md new file mode 100755 index 0000000..5eb4f1b --- /dev/null +++ b/uni/mmme/1048_thermodynamics_and_fluid_mechanics/fluid_mechanics.md @@ -0,0 +1,506 @@ +--- +author: Alvie Rahman +date: \today +title: MMME1048 // Fluid Mechanics Intro and Statics +tags: [ uni, nottingham, mechanical, engineering, fluid_mechanics, mmme1048, fluid_statics ] +--- + +# Properties of Fluids + +## 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 + +# Fluid Statics + +## Manometers + +![](./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 + + $$p_1 = p_2 + \rho_wg(z_{C2} - z_{C1})$$ + +
+ + +### Exercise Sheet 1 + + + +1. If 4 m$^3$ of oil weighs 35 kN calculate its density and relative density. + Relative density is a term used to define the density of a fluid relative + + > $$ \frac{35000}{9.81\times4} = 890 \text{ kgm}^{-3} $$ + > + > $$1000 - 891.9... = 108 \text{ kgm}^{-3}$$ + +2. Find the pressure relative to atmospheric experienced by a diver + working on the sea bed at a depth of 35 m. + Take the density of sea water to be 1030 kgm$^{-3}$. + + > $$ + > \rho gh = 1030\times 9.81 \times 35 = 3.5\times10^5 + > $$ + +3. An open glass is sitting on a table, it has a diameter of 10 cm. + If water up to a height of 20 cm is now added calculate the force exerted onto the table by + the addition of the water. + + > $$V_{cylinder} = \pi r^2h$$ + > $$m_{cylinder} = \rho\pi r^2h$$ + > $$W_{cylinder} = \rho g\pi r^2h = 1000\times9.8\pi\times0.05^2\times0.2 = 15.4 \text{ N} $$ + +4. A closed rectangular tank with internal dimensions 6m x 3m x 1.5m + high has a vertical riser pipe of cross-sectional area 0.001 m2 in + the upper surface (figure 1.4). The tank and riser are filled with + water such that the water level in the riser pipe is 3.5 m above the + + Calulate: + + i. The gauge pressure at the base of the tank. + + > $$\rho gh = 1000\times9.81\times(1.5+3.5) = 49 \text{ kPa}$$ + + ii. The gauge pressure at the top of the tank. + + > $$\rho gh = 1000\times9.81\times3.5 = 34 \text{ kPa}$$ + + iii. The force exercted on the base of the tank due to gauge water pressure. + + > $$F = p\times A = 49\times10^3\times6\times3 = 8.8\times10^5 \text{ N}$$ + + iv. The weight of the water in the tank and riser. + + > $$V = 6\times3\times1.5 + 0.001\times3.5 = 27.0035$$ + > $$W = \rho gV = 1000\times9.8\times27.0035 = 2.6\times10^5\text{ N}$$ + + v. Explain the difference between (iii) and (iv). + + *(It may be helpful to think about the forces on the top of the tank)* + + > The pressure at the top of the tank is higher than atmospheric pressure because of the + > riser. + > This means there is an upwards force on the top of tank. + > The difference between the force acting up and down due to pressure is equal to the + > weight of the water. + +6. A double U-tube manometer is connected to a pipe as shown below. + + Taking the dimensions and fluids as indicated; calculate + the absolute pressure at point A (centre of the pipe). + + Take atmospheric pressure as 1.01 bar and the density of mercury as 13600 kgm$^{-3}$. + + ![](./images/vimscrot-2021-10-13T10:42:52,999793176+01:00.png) + + > \begin{align*} + P_B &= P_A + 0.4\rho_wg &\text{(1)}\\ + P_C = P_{C'} &= P_B - 0.2\rho_mg &\text{(2)}\\ + P_D = P_{at} &= P_A + 0.4\rho_wg &\text{(3)}\\ + \\ + \text{(2) into (3)} &\rightarrow P_B -0.2\rho_mg-0.1\rho_wg = P_{at} &\text{(4)}\\ + \text{(1) into (4)} &\rightarrow P_A + 0.4\rho_wg - 0.2\rho_mg - 0.1\rho_wg = P_{at} \\ + \\ + P_A &= P_{at} + 0.1\rho_wg + 0.2\rho_mg - 0.4\rho_wg \\ + &= P_{at} + g(0.2\rho_m - 0.3\rho_w) \\ + &= 1.01\times10^5 + 9.81(0.2\times13600 - 0.3\times1000)\\ + &= 124.7\text{ kPa} + > \end{align*} + +
+ +## Submerged Surfaces + +### Prepatory Maths + +#### Integration as Summation + +#### Centroids + +- For a 3D body, the centre of gravity is the point at which all the mass can be considered to act +- For a 2D lamina (thin, flat plate) the centroid is the centre of area, the point about which the + lamina would balance + +To find the location of the centroid, take moments (of area) about a suitable reference axis: + +$$moment\,of\,area = moment\,of\,mass$$ + +(making the assumption that the surface has a unit mass per unit area) + +$$moment\,of\,mass = mass\times distance\,from\,point\,acting\,around$$ + +Take the following lamina: + +![](./images/vimscrot-2021-10-20T10:01:30,080819382+01:00.png) + +1. Split the lamina into elements parallel to the chosen axis +2. Each element has area $\delta A = w\delta y$ +3. The moment of area ($\delta M$) of the element is $\delta Ay$ +4. The sum of moments of all the elements is equal to the moment $M$ obtained by assuing all the + area is located at the centroid or: + + $$Ay_c = \int_{area} \! y\,\mathrm{d}A$$ + + or: + + $$y_c = \frac 1 A \int_{area} \! y\,\mathrm{d}A$$ + + - $\int y\,\mathrm{d}A$ is known as the first moment of area + +
+ + +##### Example 1 + +Determine the location of the centroid of a rectangular lamina. + + + +###### Determining Location in $y$ direction + +![](./images/vimscrot-2021-10-20T10:14:17,688774145+01:00.png) + +1. Take moments for area about $OO$ + + $$\delta M = y\delta A = y(b\delta y)$$ + +2. Integrate to find all strips + + $$M = b\int_0^d\! y \,\mathrm{d}y = b\left[\frac{y^2}2\right]_0^d = \frac{bd^2} 2$$ + + ($b$ can be taken out the integral as it is constant in this example) + + but also $$M = (area)(y_c) = bdy_c$$ + + so $$y_c = \frac 1 {bd} \frac {bd} 2 = \frac d 2$$ + +###### Determining Location in $x$ direction + +![](./images/vimscrot-2021-10-20T10:24:48,372189101+01:00.png) + +1. Take moments for area about $O'O'$: + + $$\delta M = x\delta A = x(d\delta x)$$ + +2. Integrate + + $$M_{O'O'} = d\int_0^b\! x \,\mathrm{d}x = d\left[\frac{x^2} 2\right]_0^b = \frac{db^2} 2$$ + + but also $$M_{O'O'} = (area)(x_c) = bdx_c$$ + + so $$x_c = \frac{db^2}{2bd} = \frac b 2$$ + +
+ +### Horizontal Submereged Surfaces + +![](./images/vimscrot-2021-10-20T10:33:16,783724117+01:00.png) + +Assumptions for horizontal lamina: + +- Constant pressure acts over entire surface of lamina +- Centre of pressure will coincide with centre of area +- $total\,force = pressure\times area$ + +![](./images/vimscrot-2021-10-20T10:36:12,520683729+01:00.png) + +### Vertical Submerged Surfaces + +![](./images/vimscrot-2021-10-20T11:05:33,235642932+01:00.png) + +- A vertical submerged plate does experience uniform pressure +- Centroid of pressure and area are not coincident +- Centroid of pressure is always below centroid of area for a vertical plate +- No shear forces, so all hydrostatic forces are perpendicular to lamina + +![](./images/vimscrot-2021-10-20T11:07:52,929126609+01:00.png) + +Force acting on small element: + +\begin{align*} +\delta F &= p\delta A \\ +&= \rho gh\delta A \\ +&= \rho gh w\delta h +\end{align*} + +Therefore total force is + +$$F_p = \int_{area}\! \rho gh \,\mathrm{d}A = \int_{h_1}^{h_2}\! \rho ghw\,\mathrm{d}h$$ + +#### Finding Line of Action of the Force + +![](./images/vimscrot-2021-10-20T11:15:51,200869760+01:00.png) + +\begin{align*} +\delta M_{OO} &= \delta Fh = (\rho gh\delta A)h \\ +&= \rho gh^2\delta A = \rho gh^2w\delta h \\ +\\ +M_{OO} &= F_py_p = \int_{area}\! \rho gh^2 \,\mathrm{d}A \\ +&= \int_{h1}^{h2}\! \rho gh^2w \,\mathrm{d}h \\ +\\ +y_p = \frac{M_{OO}}{F_p} +\end{align*} + +## Buoyancy + +### Archimedes Principle + +> The resultant upwards force (buoyancy force) on a body wholly or partially immersed in a fluid is +> equal to the weight of the displaced fluid. + +When an object is in equilibrium the forces acting on it balance. +For a floating object, the upwards force equals the weight: + +$$mg = \rho Vg$$ + +Where $\rho$ is the density of the fluid, and $V$ is the volume of displaced fluid. + +### Immersed Bodies + +As pressure increases with depth, the fluid exerts a resultant upward force on a body. +There is no horizontal component of the buoyancy force because the vertiscal projection of the body +is the same in both directions. + +### Rise, Sink, or Float? + +- $F_B = W$ \rightarrow equilirbrium (floating) +- $F_B > W$ \rightarrow body rises +- $F_B < W$ \rightarrow body sinks + +### Centre of Buoyancy + +Buoyancy force acts through the centre of gravity of the volume of fluid displaced. +This is known as the centre of buoyancy. +The centre of buoyancy does not in general correspond to the centre of gravity of the body. + +If the fluid density is constant the centre of gravity of the displaced fluid is at the centroid of +the immersed volume. + +![](./images/vimscrot-2021-12-21T15:08:22,285753421+00:00.png) + diff --git a/uni/mmme/1048_thermodynamics_and_fluid_mechanics/images/vimscrot-2021-10-06T10:51:51,499044519+01:00.png b/uni/mmme/1048_thermodynamics_and_fluid_mechanics/images/vimscrot-2021-10-06T10:51:51,499044519+01:00.png new file mode 100644 index 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