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22 formulas

Fluid Mechanics

Bernoulli, viscosity. Every formula below opens into a live, hands-on simulation.

kinematics
A1v1=A2v2A_1 v_1 = A_2 v_2

Continuity Equation

What flows in must flow out — narrow pipes force faster flow.

dynamics
P+12ρv2+ρgh=constP + \tfrac{1}{2}\rho v^2 + \rho g h = \text{const}

Bernoulli's Equation

Pressure, kinetic, and potential energy per unit volume sum to a constant along a streamline.

statics
P=P0+ρghP = P_0 + \rho g h

Hydrostatic Pressure

Pressure grows linearly with depth because of the weight of fluid above.

statics
Fb=ρVgF_b = \rho V g

Archimedes' Principle

Buoyant force equals the weight of fluid displaced.

dynamics
v=2ghv = \sqrt{2 g h}

Torricelli's Law

Falling fluid trades height for speed, just like a dropped ball.

viscous flow
Re=ρvLμRe = \frac{\rho v L}{\mu}

Reynolds Number

Ratio of inertial to viscous forces — high Re means inertia wins, turbulence reigns.

viscous flow
Fd=6πηrvF_d = 6\pi \eta r v

Stokes' Law

Slow, syrupy flow around a tiny sphere produces drag linear in speed.

viscous flow
Q=πr4ΔP8ηLQ = \frac{\pi r^4 \Delta P}{8 \eta L}

Poiseuille's Law

Pipe flow scales with the FOURTH power of radius — narrowing matters massively.

pipe flow
ΔP=fLDρv22\Delta P = f \frac{L}{D} \frac{\rho v^2}{2}

Darcy-Weisbach Equation

Pressure loss in a pipe scales with length, kinetic energy, and a fudge factor for roughness.

statics
F1A1=F2A2\frac{F_1}{A_1} = \frac{F_2}{A_2}

Pascal's Principle

Pressure applied to a confined fluid is transmitted everywhere — undiminished.

rheology
τ=ηdvdy\tau = \eta \frac{dv}{dy}

Newton's Law of Viscosity

Friction inside a fluid is proportional to how fast layers slide past each other.

interfacial
ΔP=2γr\Delta P = \frac{2 \gamma}{r}

Young-Laplace Equation

Curved interfaces compress what's inside — smaller curvature, bigger squeeze.

interfacial
h=2γcos⁡θρgrh = \frac{2 \gamma \cos\theta}{\rho g r}

Jurin's Law (Capillary Rise)

Surface tension pulls liquid up a thin tube until gravity catches up — narrower tube, taller climb.

high re flow
Fd=12CdρAv2F_d = \frac{1}{2} C_d \rho A v^2

Drag Force (Quadratic)

Push air aside fast enough and it pushes back — quadratically.

compressible flow
M=vaM = \frac{v}{a}

Mach Number

Mach number is your speed measured in 'speeds of sound' — at M=1 you outrun your own pressure waves.

governing equations
ρ(∂v∂t+v⋅∇v)=−∇p+μ∇2v+f\rho\left(\frac{\partial \mathbf{v}}{\partial t} + \mathbf{v}\cdot\nabla\mathbf{v}\right) = -\nabla p + \mu\nabla^2\mathbf{v} + \mathbf{f}

Navier-Stokes Equation

Newton's second law written for a fluid parcel: mass times acceleration equals pressure forces plus viscous friction plus body forces.

governing equations
ρDvDt=−∇p+ρg\rho\frac{D\mathbf{v}}{Dt} = -\nabla p + \rho\mathbf{g}

Euler Equation (Inviscid Flow)

With no viscosity, a fluid parcel accelerates purely from pressure differences and gravity.

dynamics
p1−p2=12ρ(v22−v12)p_1 - p_2 = \frac{1}{2}\rho\left(v_2^2 - v_1^2\right)

Venturi Effect

Where a flow speeds up through a constriction, its pressure must drop — Bernoulli in a pipe.

dimensionless numbers
Fr=vgLFr = \frac{v}{\sqrt{gL}}

Froude Number

The ratio of how fast the fluid moves to how fast a gravity wave can travel — it sets whether disturbances can run upstream.

dimensionless numbers
We=ρv2LσWe = \frac{\rho v^2 L}{\sigma}

Weber Number

Whether a moving blob of fluid holds together by surface tension or is torn apart by inertia.

aerodynamics
L′=ρvΓL' = \rho v \Gamma

Kutta-Joukowski Lift Theorem

Lift equals density times speed times circulation — net swirl around the wing forces the air down and the wing up.

open channel
v=1nR2/3S1/2v = \frac{1}{n}R^{2/3}S^{1/2}

Manning's Equation

Open-channel flow speed grows with depth (hydraulic radius) and slope, and falls with roughness.