Fluids

Bouyant force:

$$B={\rho }_{fluid}\times V_{displaced}\times g$$

Pressure at depth in fluid:

$$p=p_0+\rho gh$$

Pressure in a hydraulic cylinder:

$$P=\frac{F}{\pi r^2}$$

laminar flow

• a type of fluid flow where a liquid or gas moves in parallel layers, without mixing or disruption between the layers
• “streamlined”
• molecules move together, no big/changing gaps turbulent flow
• little vortices
• speed of fluid / magnitude & direction is constantly changing

continuity equation (relates cross sectional area to velocity at different points) for non-compressible fluid:

$$A_1{v}_1=A_2{v}_2$$

for compressible fluid

$${\rho }_1{A}_1{v}_1={\rho }_2{A}_2{v}_2$$

volume flow rate

$$\frac{dV}{dt}=Av$$

viscosity

• velocity of fluid near edges of pipe will be much slower than in middle of pipe flow rate

$$\frac{dV}{dt}=\int \vec{v}\cdot d\vec{A}=\int v_{\perp }dA$$

• integral over area

mass flow rate:

$$\dot{m}=\frac{dm}{dt}$$

Burnoulli’s equation

$$p_1+\rho g{y}_1+0.5\rho v_1^2=p_2+\rho g{y}_2+0.5\rho v_2^2$$