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content/Fluids/Fluid Dynamics/Bernoulli's Equation.md

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+As an [[Ideal Fluid]] flows through a pipe or [[Streamlines and Tubes of Flow#Tubes of Flow|Tube of Flow]], it can change in several ways:
+- The cross sectional area might change.
+- The inlet and outlet may be at different elevations
+- The inlet and outlet pressures may be different.
+
+Using the [[Equations of Continuity|Equation of Continuity]] for fluid flow:
+$$
+A_{1}v_{1}=A_{2}v_{2}
+$$
+We related change in area to changes in velocity. Change in [[pressure]] and elevation are both related to velocity, so each type are not independent of each other.
+### Deriving Bernoulli's Equation
+
+Lets use an example of a pipe:
+![[Pasted image 20240208194137.png|center|500]]
+The pipe has cross-sectional area $A_{1}$ and elevation $y_{1}$ at the inlet and cross-sectional area $A_{2}$ and elevation $y_{2}$ at the end. Because the area changes, the velocity changes from $v_{1}$ to $v_{2}$.
+
+We will use [[Conservation of Energy]] to the [[Systems of Particles|system]] of the fluid between the inlet and outlet.
+
+Lets say that there may be a pressure $p_{1}$ from additional fluid on the left and a pressure $p_{2}$ from additional fluid on the right. This means there are forces $F_{1}=p_{1}A_{1}$ and $F_{2}=p_{2}A_{2}$.
+
+Under both forces and gravity, we will say that the system moves to the right. The figure below shows the system after a time $\delta t$.
+![[Pasted image 20240208194823.png|center|500]]
+In this time, the left side has moved an $\delta x_{1}$ and the right $\delta x_{2}$. These distances are different because of the difference in areas. The same effect would have been reached had we taken out the shaded section of mass $\delta m$ from the inlet and placed it at the outlet.
+
+There are three factors for the net [[Work Done on a System By External Forces|External Work]].
+1. At the inlet, the pressure force is:
+$$
+W_{1}=F_{1}\delta x_{1}=p_{1}A_{1}\delta x_{1}
+$$
+2. At the outlet, the pressure force is:
+$$
+W_{2}=-F_{2}\delta x_{2}=-p_{2}A_{2}\delta x_{2}
+$$
+3. Work done by gravity as a fluid element $\delta m$ moves through vertical displacement $y_{2}-y_{1}$:
+$$
+W_{g}=-\delta m\,g(y_{2}-y_{1})
+$$
+In [[Conservation of Energy]], $\Delta U$ is the [[Potential Energy]] from [[Conservative Force|Conservative Forces]] that act between objects in the system. Since our fluid is ideal, we assume there are no such forces, so $\Delta U=0$.
+
+The net external [[Work]] would be:
+$$
+\begin{align}
+W_{\mathrm{ext}}&=W_{1}+W_{2}+W_{g} \\
+&=p_{1}A_{1}\delta x_{1}-p_{2}A_{2}\delta x_{2}+[-\delta m\,g(y_{2}-y_{1})]
+\end{align}
+$$
+The volume of the shaded fluid element $\delta V$ can be written as $\delta V=A_{1}\delta x_{1}$ and $\delta V=A_{2}\delta x_{2}$, since the fluid is incompressible. The uniform and constant fluid density is $\rho$ and so we can also rewrite our element as $\delta V=\delta m / p$. Substituting:
+$$
+W_{\mathrm{ext}}=(p_{2}-p_{1})(\delta m / p)-\delta m\,g(y_{2}-y_{1})
+$$
+The change in [[Kinetic Energy]] for $\delta m$ is:
+$$
+\Delta K=\frac{1}{2}\delta v\,v^2_{2}-\frac{1}{2}\delta m\,v^2_{1} 
+$$
+Applying conservation of energy:
+$$
+\frac{1}{2}\delta v\,v^2_{2}-\frac{1}{2}\delta m\,v^2_{1} =(p_{2}-p_{1})(\delta m / p)-\delta m\,g(y_{2}-y_{1})
+$$
+And rearranging/cancelling:
+$$
+p_{1}+\frac{1}{2}\rho v^2_{1}+\rho gy_{1}=p_{2}+\frac{1}{2}\rho v^2_{2}+\rho gy_{2}
+$$
+Or:
+$$
+\boxed{p+\frac{1}{2}\rho v^2+\rho gy=\text{constant}}
+$$
+This is Bernoulli's Equation for ideal fluids, which state that the equation above is constant along a streamline.
+
+### Analyzing Bernoulli's Equation
+
+As we've seen, Bernoulli's Equation is a derivation of conservation of energy. We can split it into types of energies:
+
+- $p$ is the Pressure Energy
+- $\frac{1}{2}\rho v^2$ is the Kinetic Energy
+- $\rho gy$ is the Potential Energy
+
+### Special Applications
+
+1. Static [[Pressure]].
+
+Static Pressure is simply a case where velocity is 0.
+$$
+\begin{align}
+p_{1}+\frac{1}{2}\rho v^2_{1}+\rho gy_{1}&=p_{2}+\frac{1}{2}\rho v^2_{2}+\rho gy_{2} \\
+p_{1}+\rho gy_{1}&=p_{2}+\rho gy_{2} \\ p_{2}-p_{1}&=-\rho g(y_{1}-y_{1}) \\
+&=-\rho gh
+\end{align}
+$$
+Which matches the equation we derived in [[Variation of Pressure for Fluids at Rest]].
+<br>
+2. Dynamic Pressure.
+
+Suppose a fluid flowing horizontally, so there is no difference in elevation:
+$$
+\begin{align}
+p_{1}+\frac{1}{2}\rho v^2_{1}+\rho gy_{1}&=p_{2}+\frac{1}{2}\rho v^2_{2}+\rho gy_{2} \\
+p_{1}+\frac{1}{2}\rho v^2_{1}&=p_{2}+\frac{1}{2}\rho v^2_{2}
+\end{align}
+$$
+In this equation, as the speed is large,  pressure must be small, and vice versa. The quantity $\frac{1}{2}\rho v^2$ is called the dynamic pressure.
+<br>
+3. Compressible, viscous flow.
+
+If the fluid is compressible, then its [[Internal Energy in A System of Particles|Internal Potential Energy]] $\Delta U_{\mathrm{int}}$ can change as molecules become closer and further apart.
+
+If the flow is viscous, then the internal kinetic energy $\Delta K_{\mathrm{int}}$ can change, similar to how [[Frictional Work|Frictional]] forces increase the internal energy.
+
+Our complete analysis should include internal energy:
+$$
+\Delta E_{\mathrm{int}} = \Delta U_{\mathrm{int}} + \Delta K_{\mathrm{int}}
+$$
+or:
+$$
+\Delta K+\Delta E_{\mathrm{int}} = W_{\mathrm{ext}}
+$$
+If necessary, Bernoulli's equation could be modified to account for theses other energy transformation.
+
+### Other Applications
+[[The Venturi Meter]]
+[[The Pilot Tube]]
+[[Dynamic Lift]]
+[[Thrust on a Rocket]]

content/Fluids/Fluid Dynamics/Dynamic Lift.md

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+[[Fluid Dynamics|Dynamic]] lift is the force that acts on a body due to its motion through a [[Fluids|Fluid]]. This is not the same as [[Buoyancy]], which uses [[Archimedes' Principle]] in [[Fluid Statics]]. For example, an airplane uses dynamic lift as it moves through the air.
+
+Another example of lift is the flight of balls. Lets analyze how it works.
+
+A ball that is thrown regularly with non-rotational motion has [[Streamlines and Tubes of Flow|Streamlines]] that look like such:
+![[Pasted image 20240221162110.png|center|500]]
+
+A ball that has has only [[Rotational Motion]] has Streamlines that look like such:
+![[Pasted image 20240221162156.png|center|300]]
+Note that without the viscosity of air, the ball would not carry air with it in such a matter. 
+
+A ball with both linear motion and Rotational Motion has Streamlines that look like such:
+![[Pasted image 20240221162401.png|center|500]]
+
+The velocity of the streamlines below is less than the that above the ball. Using [[Bernoulli's Equation]], we can conclude that the pressure of the air below the ball must be greater than that above the ball, so there is a force upwards, leading to the curvature of the thrown ball.
+
+The lift acting on an airplane wing has a similar explanation.
+
+![[Pasted image 20240303192659.png]]
+
+The streamlines above the wing are more bunched up; therefore, the air flow sleep is greater above the wing. Using Bernoulli's Equation, we conclude the pressure above is lesser, and there is a force upwards.
+
+Another explanation is that the streamlines below hit the wing and are deflected downwards, contributing to an upwards lift.

content/Fluids/Fluid Dynamics/Fields of Flow.md

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+Fields of flow can be used to represent a variety of spaces. For example:
+![[Pasted image 20240308102202.png|100%]]
+This field is composed of [[Streamlines and Tubes of Flow|Streamlines]] that represent a homogeneous nonviscous flow.
+![[Pasted image 20240308102351.png]]
+This field of flow represents uniform rotational flow.
+![[Pasted image 20240308102419.png]]
+This field of flow represents a vortex.
+![[Pasted image 20240308102459.png|300]]
+This field of flow represents flow from a linear source. Note that this can be through multiple mediums: fluids, heat, or electrostatics.
+![[Pasted image 20240308102638.png|300]]
+This field of flow represents linear dipole flow. You may recognize it as the electric field of two oppositely charged wires.

content/Fluids/Fluid Dynamics/Fluid Dynamics.md

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+The dynamics of [[Fluids]] in motion.
+
+We can describe the motion of a fluid by dividing the fluid into infinitely small portions, called fluid particles. Using the forces on each fluid particle, we can create equations for velocity and position. ^c18cf6
+
+Another method is to use the density and velocity of the fluid at each point in space at a specific time, which is the method that we will use. We will describe the motion of fluids using density $\rho(x, y, z, t)$ and velocity $\vec{v}(x, y, z ,t)$.
+
+>[!info]-
+>![[General Concepts of Fluid Flow]]
+
+[[Streamlines and Tubes of Flow]]

content/Fluids/Fluid Dynamics/General Concepts of Fluid Flow.md

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+Some general characteristics of fluid flow:
+
+1. **Fluid flow can be steady or nonsteady.** If variables such as pressure, density, and flow velocity are constant in time throughout the fluid, the flow is steady.  If not, it is nonsteady flow. In nonsteady flow, the velocities $\vec{v}$ are functions of time.
+ ^c750db
+2. **Fluid flow can be compressible or incompressible** If density $\rho$ is constant, its flow is incompressible. Liquids are often incompressible, but even for highly compressible gasses changes in density are often insignificant.
+
+3. **Fluid flow can be viscous or nonviscous.** [[Viscosity]] is similar to friction for liquids. [[Kinetic Energy]] with the fluid flow can be transformed into internal energy by viscous forces. The greater the viscosity. the greater the external force needed to maintain flow.
+
+4. **Fluid flow can be rotational or irrotational.** Imagine a leaf carried by the fluid flow. If it is rotated about an axis through its center of mass, it is rotational. Otherwise, it is irrotational.
+

content/Fluids/Fluid Dynamics/Streamlines and Tubes of Flow.md

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+## Streamlines
+In [[General Concepts of Fluid Flow#^c750db|steady flow]], the velocity $\vec{v}$ of a given point is constant in time — meaning that $\vec{v}$ does not change. As a result, every [[Fluid Dynamics#^c18cf6|fluid particle]] arriving at $P$ will move with the same $\vec{v}$ to get to the next point, following a fixed path. This path is called a Streamline.
+
+![[Pasted image 20240205153542.png|center]]
+
+- The magnitude of the velocity vector $\vec{v}$ is always tangential to the streamline.
+- No two streamlines can cross each other. Otherwise, fluid particles may have a choice on whether to go on one path or another path, meaning that it would not be fixed.
+- In principle, we can draw a streamline through every point in the fluid.
+
+## Tubes of Flow
+
+By bundling streamlines together, we can create a region called a tube of flow.
+- No fluid can cross the boundaries of a tube of flow.
+- The fluid that enters at one end must leave the other.
+- Typically when defining a tube of flow, we do it so that the tube is narrow enough so that the velocity is nearly constant over a cross-section.
+
+![[Pasted image 20240205154528.png|center]]
+
+In the image above:
+- Fluid enters at $P$ with cross-sectional area $A_{1}$ and leaves at $Q$ with cross-sectional area $A_{2}$.
+- Fluid particles at $P$ have velocity $v_{1}$ and fluid particles at $Q$ have velocity $v_{2}$.
+- During an amount of time $\delta t$ a fluid element travels distance $v\,\delta t$.
+- The fluid that crosses $A_{1}$ during $\delta t$ has volume $\delta V_{1}$.
+
+If the density at $A_{1}$ is $\rho_{1}$, then the mass of fluid $\delta m_{1}$ is:
+$$
+\delta m_{1}=\rho_{1}\delta V_{1}=\rho_{1}A_{1}v_{1}\,\delta_{1}t
+$$
+We can use this to find *mass flux*, the mass of fluid passing through a cross section per unit time. At $P$ this is approximately: ^b84d6d
+$$
+\frac{\delta m_{1}}{\delta t}=\rho_{1}A_{1}v_{1}
+$$
+To get this value precisely, we must take the limit at $\delta t$ approaches $0$ so that the area $A$ does not change too much.
+
+With the conditions that the flow is steady and there are no other sources or sinks, the fluid mass enters the tube at the same rate that is leaves it. As a result:
+$$
+\rho_{1}A_{1}v_{1}=\rho_{1}A_{2}v_{2}
+$$
+or:
+$$
+\Delta(\rho Av)=0
+$$
+This expresses the [[Law of Conservation of Mass]] in fluid dynamics.
+
+If the fluid is incompressible, then the density remains the same:
+$$
+A_{1}v_{1}=A_{2}v_{2}
+$$
+or defining $R$ as the *volume flow rate*:
+$$
+\Delta R=\Delta (Av)=0
+$$
+$R$ has units $\mathrm{m^3/s}$.
+
+Equations $\Delta (\rho Av) = 0$ and $\Delta R=\Delta (Av)=0$ are called the [[Equations of Continuity]] for one dimensional flow.

content/Fluids/Fluid Dynamics/The Pilot Tube.md

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+A pilot tube is used to measure the flow speed of a gas. A gas flows with density $\rho$ and velocity $\vec{v}_{1}$ parallel to some small openings, which we name point 1.
+![[Pasted image 20240216114211.png|center]]
+The [[Pressure]] in the left arm of the manometer (the thin pipe), which is connected to these openings, is the [[Fluid Statics|Static]] pressure in the gas [[Streamlines and Tubes of Flow|Stream]].
+
+At point 2, the velocity is reduced to 0 as the gas stagnates.
+
+Applying [[Bernoulli's Equation]] to points 1 and 2, we get:
+$$
+p_{1}+\frac{1}{2}\rho v_{1}^2=p_{2}
+$$
+The manometer reads $\rho'gh$ for the difference in pressure, so we can substitute and solve for $v$, giving us:
+$$
+v_{1}=\sqrt{ \frac{2gh\rho'}{\rho} }
+$$

content/Fluids/Fluid Dynamics/The Venturi Meter.md

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+The Venturi Meter is a gauge used to measure the [[Fluid Dynamics|flow]] speed of a fluid in a pipe.
+
+![[Pasted image 20240215213213.png|center|500]]
+
+By applying [[Bernoulli's Equation]] and solving for $v$, we get:
+$$
+v=A_{2}\sqrt{ \frac{2(\rho'-\rho)gh}{\rho(A_{1}^2-A_{2}^2)} }
+$$

content/Fluids/Fluid Dynamics/Thrust on a Rocket.md

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+Using [[Bernoulli's Equation]], we can compute the thrust on a rocket.
+
+Consider a chamber of cross-sectional area $A$ filled with a gas of density $\rho$. There is a small hole at the bottom with cross sectional area $A_{0}$. We want to find the speed at which the gas leaves the chamber, $v_{0}$.
+
+Let us rewrite Bernoulli's Equation accordingly:
+$$
+p-p_{0}=\rho g(p_{0}-y) + \frac{1}{2}\rho(v_{0}^2-v^2)
+$$
+Where $p$ is the pressure inside the chamber and $p_{0}$ is the atmospheric pressure just outside the chamber.
+
+For a gas, the density is usually so small we can neglect variation in pressure due to height. Removing that gives us:
+$$
+p-p_{0}=\frac{1}{2}\rho(v_{0}^2-v^2)
+$$
+Rearranged for $v_{0}$ we get:
+$$
+v_{0}^2=\frac{2(p-p_{0})}{\rho}+v^2
+$$
+We will assume [[Equations of Continuity#^6de4cc|Continuity of mass flow]] so that:
+$$
+Av=A_{0}v_{0}
+$$
+If the hole is so small that $A_{0}$ is much much smaller than $A$, then that means $v_{0}$ is much much larger than $v$. We can neglect $v^2$ in the equation. Thus, $v_{0}$ is:
+$$
+v_{0}=\sqrt{ \frac{2(p-p_{0})}{\rho} }
+$$
+If the chamber is the exhaust change of a rocket, the [[Systems of Variable Mass|thrust on the rocket]] is $v_{0} \frac{ dM }{ dt }$. In a period of time $dt$, the change in mass $dM = \rho A_{0}v_{0}dt$.
+$$
+v_{0} \frac{ dM }{ dt } =v_{0}(\rho A_{0}v_{0})=\rho A_{0}v_{0}^2
+$$
+Using our original equation:
+$$
+\begin{align}
+v_{0}^2 \frac{ dM }{ dt } &=\frac{2(p-p_{0})}{\rho} \rho A_{0}v_{0}^2 \\
+\Aboxed{v_{0}\frac{ dM }{ dt } &= 2A_{0}(p - p_{0})}
+\end{align}
+$$

content/Fluids/Fluid Dynamics/Viscosity.md

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+Viscosity in fluid flow is similar to the concept of friction. If we want to keep a body at a constant velocity, we must have an external force to counteract the friction.
+
+We can apply a similar logic to fluids by envisioning them as a series of plates.
+![[Pasted image 20240308141022.png|100%]]
+
+In the example, a force $\vec{F}$ is applied on the top plate so that it has constant velocity $\vec{v}$ compared to the bottom plate. $\vec{F}$ counteracts the viscous drag not only between the top plate and the fluid, but within the fluid itself. That is why we use the plates to specify that effect.
+
+The speed of each layer differs by a $dv$, making this an example of laminar flow — one that uses [[Streamlines and Tubes of Flow]].
+
+We know that the area of the top plate and each of the layers will have an effect on how much viscous drag there is, and thus the force $\vec{F}$ that we apply. We will thus say that force $F$ is proportional to the area $A$ times change in velocity that results from a change in the height of the layer. In mathematical terms, this is:
+$$
+F \propto A\frac{ dv }{ dy } 
+$$
+or, using $\eta$ as a constant of proportionality:
+$$
+F=\eta A\frac{ dv }{ dy }
+$$
+$\eta$ is the coefficient of velocity and it varies from fluid to fluid.
+
+In the case for the rectangular layers above, the velocity gradient $dv / dy$ is constant, so we would instead write:
+$$
+F=\eta A \frac{v}{D}
+$$
+For a more practice example, we can look at viscosity in pipes.
+![[Pasted image 20240314164513.png|100%]]
+
+In this case, our layers are not rectangular but cylindrical with varying radii. Assuming that the layer next to the walls is at rest, the speed in a shell of radius $r$ is:
+$$
+v=\frac{\Delta p}{4\eta L}(R^2-r^2)
+$$
+Which depends on the pressure difference across the length of the pipe. The speed at the center of the pipe is:
+$$
+v_{0}=\frac{\Delta pR^2}{4\eta L}
+$$
+Using more derivation, we can show that the total [[Streamlines and Tubes of Flow#^b84d6d|mass flux]] is:
+$$
+\frac{ dm }{ dt } =\frac{\rho \pi R^4\Delta p}{8\eta L}
+$$
+This result is Poiseuille's Law.

content/Fluids/Fluid Statics/Archimedes' Principle.md

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+Archimedes' Principle states that:
+
+> A body wholly or partially immersed in a fluid is [[Buoyancy|buoyed]] up by a force equal in magnitude to the weight of the fluid displaced by the body.
+

content/Fluids/Fluid Statics/Buoyancy.md

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+Buoyancy, or the buoyant force, uses Archimedes' Principle:
+
+![[Archimedes' Principle]]
+
+Because of the [[Variation of Pressure for Fluids at Rest]], the [[Pressure]] on the bottom of a submerged object is higher than the pressure at the top of a submerged object, resulting in a buoyant force. When the density of the object is the same as that of the fluid, the object's weight balances with the buoyant force.
+![[Pasted image 20240131224449.png|center|200]]
+When the density of the object is greater than that of the fluid, the volume of water that it displaces is less than its own weight. As a result, it sinks. Its weight $m\vec{g}$ is greater than the buoyant force $\vec{F}_{b}$.
+![[Pasted image 20240131224729.png|center|200]]
+Alternatively, when the density of the object is less than that of the fluid, the volume of water than it displaces is greater than its own weight. As a result, it floats. Its weight $m\vec{g}$ is less than the buoyant force $\vec{F}_{b}$.
+![[Pasted image 20240131224925.png|center|200]]
+The buoyant force acts at the [[Center of Gravity]] of the submerged portion of the object. This point is known as the [[Center of Buoyancy]]. When the Center of Gravity and the Center of Buoyancy are not vertical, the submerged object may tilt.
+![[Pasted image 20240131225824.png|center|200]]

content/Fluids/Fluid Statics/Center of Buoyancy.md

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+The center of buoyancy is the [[Center of Gravity]] of a portion of an object submerged in a fluid. When the Center of Gravity and the Center of Buoyancy are not vertical, a submerged object may tilt.
+![[Pasted image 20240131225824.png|center|200]]