# 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:

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

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