physica.
34 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.

aerodynamics
FL=ρ v Γ,Γ=2πr2ωF_L = \rho\, v\, \Gamma,\qquad \Gamma = 2\pi r^2 \omega

Magnus Effect

Spin drags a thin layer of air around the ball, speeding the flow on one side and slowing it on the other. By Bernoulli the fast side is low-pressure, so the ball is pushed sideways — the same circulation-times-speed law that gives a wing its lift.

flow measurement
v=2 (p0−p)ρv = \sqrt{\frac{2\,(p_0 - p)}{\rho}}

Pitot Tube Airspeed

Bring the moving air to a dead stop at the tube's nose and all its kinetic energy turns into extra pressure. The size of that extra (dynamic) pressure tells you how fast the air was going.

open channel flow
y2y1=12(1+8 Fr12−1)\frac{y_2}{y_1} = \tfrac{1}{2}\left(\sqrt{1 + 8\,Fr_1^2} - 1\right)

Hydraulic Jump

When shallow water moves faster than its own surface waves (Fr > 1) it can't 'feel' the slower deep water ahead, so it piles up in a sudden standing wall — the liquid analogue of a sonic shock — dumping the excess energy as turbulence.

transient flow
Δp=ρ c Δv\Delta p = \rho\, c\, \Delta v

Water Hammer (Joukowsky)

The moving column of water has momentum; stop it suddenly and that momentum has nowhere to go but into pressure. A compression wave rockets back up the pipe at the speed of sound in water, hammering every fitting it passes.

hydrostatics
GM=IV−BG‾GM = \frac{I}{V} - \overline{BG}

Metacentric Height (Ship Stability)

When a ship heels, the underwater shape shifts the buoyancy force sideways to act through a point called the metacentre. If that point sits above the centre of gravity, buoyancy twists the ship back upright; if below, it capsizes.

everyday aerodynamics
Δp=12ρ (vout2−vin2)\Delta p = \tfrac{1}{2}\rho\,(v_{out}^2 - v_{in}^2)

Why Shower Curtains Billow Inward

The falling spray drags air downward, so air moves fast inside the shower and slow outside. By Bernoulli, fast-moving air has lower pressure, so the higher outside pressure simply pushes the lightweight curtain inward.

biofluid dynamics
Δp=12ρ (v22−v12)\Delta p = \tfrac{1}{2}\rho\,(v_2^2 - v_1^2)

How Prairie-Dog Burrows Self-Ventilate

Wind speeds up with height, so it blows faster over the taller mound. Faster air means lower pressure there (Bernoulli), and the pressure gap between the two openings continuously sucks fresh air through the tunnel — a passive pump.

aerodynamics
FD=12 CD ρ v2A,Re=v DνF_D = \tfrac{1}{2}\,C_D\,\rho\,v^2 A,\qquad Re = \frac{v\,D}{\nu}

Why Golf Balls Have Dimples (Drag Crisis)

A smooth ball's boundary layer separates early, leaving a fat low-pressure wake that drags it back. Dimples deliberately trip the layer turbulent so it clings farther around the ball, shrinking the wake and roughly halving the drag — so the ball flies about twice as far.

vortex dynamics
v r=const  ⇒  v=Γ2πrv\,r = \text{const}\;\Rightarrow\; v = \frac{\Gamma}{2\pi r}

Why Draining Water Swirls (Free Vortex)

As a fluid ring spirals inward toward the drain its radius shrinks, and like a skater pulling in their arms it must spin faster to conserve angular momentum. The rising speed near the hole carves the familiar funnel-shaped surface dip.

wake dynamics
f=St vDf = \frac{St\,v}{D}

Kármán Vortex Street

As fluid sweeps past a blunt body it can't cling to the back, so it rolls up into vortices that peel off alternately from each side. That regular left-right shedding pushes the body sideways at frequency f — the hum of a wire and the sway of a chimney.

pressure vessels
σmax=σ(1+2ab)\sigma_{max} = \sigma\left(1 + \frac{2a}{b}\right)

Why Airplane Windows Are Round

Stress flows through a skin like water through a channel; force it around a sharp corner and it bunches up just as water speeds at a constriction. A circle spreads the flow gently (peak 3×), but a square corner spikes the stress until a crack starts.

turbulence
E(k)=C ε2/3 k−5/3E(k) = C\,\varepsilon^{2/3}\,k^{-5/3}

Kolmogorov Turbulent Energy Cascade

Energy fed in at large scales is handed down, eddy by eddy, to ever-smaller whirls without loss, until the smallest eddies are so fine that viscosity finally smears their motion into heat. Across that 'inertial range' only ε matters, fixing the universal −5/3 slope.