Fluid Mechanics

Formulas and equations

Density

The density, of a substance is its mass per unit volume.

ρ = \frac{m}{V}

Bernoulli's equation

Bernoulli's principle states that an increase in the speed of a fluid occurs simultaneously with a decrease in static pressure or a decrease in the fluid's potential energy.

p + \frac{1}{2} ρ v^{2} + ρ g h = c o n s t a n t

Pressure at a depth h in a fluid of constant density

p = p_{0} + ρ g h

Change of pressure with height in a constant-density fluid

\frac{d p}{d y} = - ρ g

Absolute pressure

Absolute pressure is a pressure that is relative to the zero pressure in the empty, air-free space of the universe.

p_{a b s} = p_{g} + p_{a t m}

Pascals law

Pascal's law states that if pressure is exerted on a non compressible fluid in an enclosed vessel, this pressure will be distributed evenly throughout the fluid.

\frac{F_{1}}{A_{1}} = \frac{F_{2}}{A_{2}}

Volume flow rate

Volume flow rate is the volume of fluid which passes per unit time

Q = \frac{d V}{d t}

Continuity equation (constant density)

The continuity equation states that in the case of steady flow, the amount of fluid flowing past one point must be the same as the amount of fluid flowing past another point, or the mass flow rate is constant.

A_{1} v_{1} = A_{2} v_{2}

Continuity equation (general form)

ρ_{1} A_{1} v_{1} = ρ_{2} A_{2} v_{2}

Viscosity

Viscosity is a measure of a fluid's resistance to flow. The SI unit of viscosity is poiseiulle (PI). Its other units are newton-second per square metre or pascal-second (Pa s)

η = \frac{F L}{v A}

Poiseuille’s law

The velocity of the steady flow of a fluid through a narrow tube (as a blood vessel or a catheter) varies directly as the pressure and the fourth power of the radius of the tube and inversely as the length of the tube and the coefficient of viscosity.

Q = \frac{(p_{2} - p_{1}) π r^{4}}{8 η l}

Poiseuille’s law for resistance

Resistance in an airway is equal to change in pressure divided by flow rate [Resistance = (Peak Pressure – Plateau Pressure) / Flow L/sec]

R = \frac{8 η l}{π r^{4}}

Reynolds Number

The Reynolds number is the ratio of inertial forces to viscous forces within a fluid which is subjected to relative internal movement due to different fluid velocities.

R e = \frac{ρ u L}{μ}

⇦ Home Key Terms ⇨