46 lines
1.3 KiB
Markdown
46 lines
1.3 KiB
Markdown
# Fluid Mechanics
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##### Density
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$$\rho = {m \over V}$$
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##### Specific weight
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$$\gamma = {m g \over V} = \rho V$$
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##### Viscosity
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- Resistance to flow
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- Ability to resist shear force
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### Example
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Consider a fluid between two plates a distance, $b$, apart. One is moving with velocity, $\vec{u}$, and the other is fixed in place. The moving plate applies a force, $\vec{p}$ to the fluid. The fluid has continuously changing velocity between the two plates.
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The velocity gradient between the two plates is given by:
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$${du \over dy}$$
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The shearing stress, $T$, is given by:
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$$\vec{T} = {\vec{p} \over A} = \mu {du \over dy}$$
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Where $\mu$ is the absolute viscosity.
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### Pressure in a Fluid
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$$p_2 = p_1 + \rho g z$$
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##### Pascal's Law
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$${F_1 \over A_1} = {F_2 \over A_2}$$
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##### Archemedes Principle
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The upward force of bouyancy, $F_B$, is given by the weight of the displaced fluid.
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### Continuity Equation
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$$A_1 \vec{v}_1 \Delta t = A_2 \vec{v}_2 \Delta t$$
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Conservation of mass principle applied to a steady flow. The mass flow rate $\dot{m}$ is given by:
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$$\dot{m} = \rho_1 A_1 \vec{v}_1 = \rho_2 A_2 \vec{v}_2$$
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Flowrate:
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$$\vec{Q} = A \vec{v}$$
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### Bernoulli's Equation
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$$p_1 + {1\over2}\rho v_1^2 + \rho g h_1 = p_2 + {1\over2}\rho v_2^2 + \rho g h_2$$
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An increase in speed of a fluid occurs with a decrease of static pressure or decrease in potential energy.
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