Physics

Basic Definitions

Density: $ρ = \frac{m}{V}$

• where $m$ is mass, and $V$ is volume
• often in units of $g/cm^3$ or $kg/m^3$

Newtons Laws

Boyle’s law

Pressure and Volume of a gas are inversely related, represented by: $$ P_{1}V_{1} = P_{2}V_{2} $$ where $P$ is pressure and $V$ is volume

Bernoulli’s Principle

For a horizontal flow an increase in velocity must be accompanied by a decrease in pressure

Streamline Equation

$$ P + \frac{ρ}{2}v^2 + ρgh = constant $$ Sum of the 3 terms remain constant. $P$ is static pressure, $\frac{ρ}{2}v^2$ is dynamic pressure, and $ρgh$ is the hydrostatic pressure.

• Static Pressure: $P$ is fluid pressure
• Dynamic Pressure: $ρ$ is density while $v$ is velocity
• Hydrostatic Pressure: $g$ is gravitational acceleration, while $h$ is elevation of the fluid, or heigh above a reference level.

Limitations:

• Deriving the equation assumes flow is laminar and steady, negligible shear forces due to viscosity, and density is constant due to being incompressible

Methods of Derivation:

Conservation of Energy: Work done on a fluid increases its kinetic energy $W = \Delta KE$

Newtons 2nd Law: $F = ma$

Incompressible Flow

A common form of Bernoulli’s equation, valid at any arbitrary point along a streamline, is: $$ {\displaystyle {\frac {v^{2}}{2}}+gz+{\frac {p}{\rho }}={\text{constant}}} $$

where:

• $v$ is the fluid flow speed at a point on a streamline,
• $g$ is the acceleration due to gravity,
• $z$ is the elevation of the point above a reference plane, with the positive z-direction pointing upward – so in the direction opposite to the gravitational acceleration,
• $p$ is the pressure at the chosen point, and
• $ρ$ is the density of the fluid at all points in the fluid.