# Fluid Dynamics

## Newton's Law of Viscosity

$$ \tau = \mu \frac{du}{dy} $$

Where \( \tau \) is stress. \( \frac{du}{dy} \) is the shear rate and \( \mu \) is the viscosity coefficient.

Pressure on a 3D object varies only with the vertical dimension parallel with gravity.

\( \frac{\partial p}{\partial z} = - \rho g \)

\( \frac{\partial p}{\partial x} = \frac{\partial p}{\partial y} = 0 \)

Gauge pressure:

$$ p_g = \rho g h $$

Where \( h \) is the vertical difference between the measured and gauge positions.

\( p_1 = p_A \)

\( p_2 = p_A + \rho _1 gh_1 \)

\( p_3 = p_2 \)

\( p_4 = p_3 - \rho _3 g h_2 \)

\( p_5 = p_4 - \rho _4 g h_3 \)

\( p_B = p_5 \)

\( p_B = p_A + \rho _1 gh_1 - \rho _3 g h_2 - \rho _4 g h_3 \)

## Bernoulli Equation

$$ p_1 + \frac{1}{2} \rho v_1 ^2 + \rho gz_1 = p_2 + \frac{1}{2} \rho v_2 ^2 + \rho gz_2 = p_{total} $$

Where \( p \) is the static pressure, \( \frac{1}{2} \rho v^2 \) is the dynamic pressure, \( \rho gz \) is the hydrostatic pressure and \( p_{total} \) is the total pressure which is constant along a streamline.

The Bernoulli equation is generally used in three forms, each of which sum to a different streamline constant with different dimensions.

Total pressure:

$$ p + \frac{1}{2} \rho v^2 + \rho gz = p_{pressure} (Pa) $$

Energy per unit mass:

$$ \frac{p}{\rho} + \frac{v^2}{2} + gz = p (J/kg, m^2 s^{-2}) $$

Energy per unit weight (head):

$$ \frac{p}{\rho g} + \frac{v^2}{2g} + z = p_{head} (m) $$