A living archive of fifty moving equations — from fluid dynamics to fields and quanta — each explained, simulated, and ready to solve.
A short introduction to fluid dynamics — what it is, the ideas that carry it, and how to read the fifty moving figures below.
Fluid dynamics is the study of how liquids and gases move, and of the forces that make them move. It is one continuous story told in many dialects: pressure and viscosity in a pipe, buoyancy in an ocean, shock waves in air, surface tension in a single droplet. This archive walks that story one figure at a time — each with a live simulation, a tuneable parameter, a practice problem and a tutorial.
What a fluid is
Anything that keeps deforming under shear — water, air, honey, magma, even crowds. Fluids have no fixed shape, only a history of motion.
Conservation first
Every equation here descends from three bookkeeping rules: mass, momentum and energy are never created, only moved around.
Viscosity vs inertia
Thick, slow flows are ruled by friction; fast, thin flows are ruled by momentum. The Reynolds number is the referee between them.
Laminar to turbulent
Push a flow hard enough and neat layers break into eddies. Turbulence is deterministic, yet practically unpredictable.
Boundaries matter
At a solid wall the fluid sticks. That thin boundary layer decides drag, lift, mixing and most of engineering.
Dimensionless thinking
Re, Fr, Ma, We, Pr — ratios that let a bathtub vortex and a hurricane share one equation.
- 01Search or filter by genre to find the flow you care about.
- 02Drag the parameter slider and watch the simulation respond in real time.
- 03Open the problem, write or upload your answer, then unlock the solution.
- 04Stuck? Tap the tiny ? dot for nudges, or open the tutorial video.
Fifty equations of flow, each rendered live and tuneable in real time. Drag a parameter, watch the fluid answer. Bring headphones.
Browse the archive
- 56Interactive figures
- 8Genres
- 27Fluid flow
- 9Wave functions
- 10Vortices & turbulence
- 9Heat & thermodynamics
- 14Pressure & surface
- 8Dimensionless numbers
Showing 56 of 56 figures
Use the Sections (1–56) button, bottom left, to jump to any equation.
Describes the motion of viscous fluid substances, balancing momentum, pressure, and viscous shear forces.
States that mass is conserved in a flow system. For an incompressible fluid, the divergence of velocity is strictly zero.
An increase in the speed of a fluid occurs simultaneously with a decrease in static pressure or fluid potential energy.
The upward buoyant force exerted on a body immersed in a fluid is equal to the weight of the fluid that the body displaces.
Laminar flow of a Newtonian fluid in a cylindrical pipe creates a perfectly parabolic velocity distribution.
Describes the frictional force exerted on spherical objects with very small Reynolds numbers in a viscous fluid.
A dimensionless number comparing inertial and gravitational forces, determining if flow is subcritical or supercritical.
Predicts flow patterns in different fluid situations. Low values indicate laminar flow; high values indicate turbulence.
When an object moves faster than the speed of sound in a fluid, it generates a conical shock wave boundary.
The speed of fluid flowing out of an opening under gravity is the same as if it dropped from the same height in free fall.
The laminar flow of a viscous fluid in the space between two parallel plates, one of which is moving relative to the other.
Occurs when there is velocity shear in a single continuous fluid, or a velocity difference across the interface of two fluids.
An instability at an interface between two fluids of different densities which occurs when the lighter fluid pushes the heavier fluid.
The ability of a liquid to flow in narrow spaces without the assistance of, or even in opposition to, external forces like gravity.
A pressure surge or wave caused when a fluid in motion is forced to stop or change direction suddenly.
The observable phenomenon that is commonly associated with a spinning object moving through a fluid. The path curves due to pressure differences.
The reduction in fluid pressure that results when a fluid flows through a constricted section of a pipe.
The steady 2D laminar boundary layer that forms on a semi-infinite plate held parallel to a constant velocity flow.
An empirical equation that relates the head loss due to friction along a given length of pipe to the average velocity of the fluid.
Estimates the average velocity of a liquid flowing in a conduit that does not completely enclose the liquid (open channel).
A pressure change occurring anywhere in a confined incompressible fluid is transmitted throughout the fluid such that the same change occurs everywhere.
Describes the capillary pressure difference sustained across the interface between two static fluids due to surface tension.
A set of hyperbolic partial differential equations that describe the flow below a pressure surface in a fluid.
A set of quasilinear hyperbolic equations governing adiabatic and inviscid flow. They represent conservation of mass, momentum, and energy.
Fluid flow in the gap between two rotating cylinders. At specific speeds, toroidal Taylor vortices form.
Describes the evolution of the curl of the velocity field. Vortices stretch, tilt, and diffuse over time.
A fundamental theorem that relates the lift generated by an airfoil to the speed of the fluid and the circulation around the body.
Calculates the pressure drop in an incompressible and Newtonian fluid in laminar flow flowing through a long cylindrical pipe.
In 2D incompressible flows, the stream function contours represent fluid streamlines, and its gradient gives velocity components.
A dimensionless number used to analyze fluid flows where there is an interface between two different fluids, especially for droplet formation.
The ratio of momentum diffusivity (kinematic viscosity) to thermal diffusivity. Defines boundary layer dominance.
Approximates the ratio of the buoyancy to viscous force acting on a fluid. Drives thermal plume mechanics.
An ordinary differential equation governing the dynamics of a spherical cavitation bubble in an infinite body of liquid.
A structure of currents or winds near a horizontal boundary in which the flow direction rotates as one moves away from the boundary.
Viscous fingering occurs when a less viscous fluid is injected into a porous medium displacing a more viscous fluid.
An eigenvalue equation describing the linear two-dimensional modes of disturbance superimposed on a viscous parallel flow.
When the mean free path of molecules approaches the physical scale of the system, the no-slip boundary condition fails.
Mass transfer along an interface between two fluids due to a gradient of the surface tension (e.g. tears of wine).
The jump conditions across a shock wave in a compressible 1D flow, conserving mass, momentum, and energy.
Describes the flow of a fluid through a porous medium, foundational to groundwater flow and petroleum engineering.
A parabolic partial differential equation that describes the distribution of heat (or variation in temperature) in a given region over time.
A second-order linear partial differential equation for the description of waves, such as sound waves, light waves, or water waves.
A fundamental partial differential equation from fluid mechanics, exhibiting shock wave formation and dissipation.
Describes the process of phase separation, by which two components of a binary fluid spontaneously separate and form pure domains.
The effect by which surface waves entering shallower water change in wave height to conserve energy flux.
A second-order partial differential equation whose solutions are harmonic functions, depicting steady-state heat or ideal fluid flow.
A generalization of Laplace's equation, used when the field has a local source or sink (like charge or mass density).
A type of natural convection occurring in a planar layer of fluid heated from below, forming beautiful rotating cells.
An exact solution to the incompressible Navier-Stokes equations, commonly used to test computational fluid dynamics.
Models the ground state of a quantum system of identical bosons using the pseudopotential interaction approximation (Bose-Einstein condensate).
A conservative system travels the path between times t₁ and t₂ that makes the action — the time integral of the Lagrangian L = T − V — stationary, giving the Euler-Lagrange equations of motion.
In a vacuum the divergence of the electric field at any point is set entirely by the local charge density there, so the flux out of a closed surface counts the charge it encloses.
The total field of a discrete set of point charges is the vector sum of each individual Coulomb field, an exact consequence of the linearity of Maxwell's equations.
The Hamiltonian operator generates the time evolution of the wavefunction Ψ, so the complex amplitude — not a trajectory — carries every prediction the theory can make.
When two coherent states overlap, the probability density picks up a cross term 2Re(Ψ₁*Ψ₂). It has no classical analogue and is exactly what paints the fringes.
Entropy counts possibilities: it is the logarithm of the number of microstates Ω consistent with a macrostate, which is why isolated systems drift toward the roomiest configuration.
Fifty figures, one continuous current. Thanks for scrolling all the way down here.