# Fluid Mechanics

##### Density
$$\rho = {m \over V}$$

##### Specific weight
$$\gamma = {m g \over V} = \rho V$$

##### Viscosity
- Resistance to flow
- Ability to resist shear force

### Example
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.

The velocity gradient between the two plates is given by:
$${du \over dy}$$

The shearing stress, $T$, is given by:
$$\vec{T} = {\vec{p} \over A} = \mu {du \over dy}$$

Where $\mu$ is the absolute viscosity.

### Pressure in a Fluid
$$p_2 = p_1 + \rho g z$$

##### Pascal's Law
$${F_1 \over A_1} = {F_2 \over A_2}$$

##### Archemedes Principle
The upward force of bouyancy, $F_B$, is given by the weight of the displaced fluid.

### Continuity Equation
$$A_1 \vec{v}_1 \Delta t = A_2 \vec{v}_2 \Delta t$$

Conservation of mass principle applied to a steady flow. The mass flow rate $\dot{m}$ is given by:
$$\dot{m} = \rho_1 A_1 \vec{v}_1 = \rho_2 A_2 \vec{v}_2$$

Flowrate:
$$\vec{Q} = A \vec{v}$$

### Bernoulli's Equation
$$p_1 + {1\over2}\rho v_1^2 + \rho g h_1 = p_2 + {1\over2}\rho v_2^2 + \rho g h_2$$

An increase in speed of a fluid occurs with a decrease of static pressure or decrease in potential energy.