Viscous Torque & Angular Momentum Transport in an Accretion Disk

Viscosity Parameters

Alpha viscosity α 0.08

Disk aspect ratio h = H/R 0.060

Time speed 1.0×

Live Readouts

Elapsed time0.00 t_ν

Accretion rate Ṁ0.000

R (90% mass)0.00

How it works

A gas disk cannot orbit a central mass forever: turbulent (or magnetic) friction between adjacent rings exerts a viscous torque that transports angular momentum outward while mass drifts inward — this is how accretion actually happens. For a Keplerian disk with surface density Σ(R,t) and kinematic viscosity ν(R), the governing equation (Lynden-Bell & Pringle, 1974) is:

∂Σ/∂t = (3/R) ∂/∂R[ R^(1/2) ∂/∂R( ν Σ R^(1/2) ) ]

viscous torque:  G(R) = 2π R³ Σ ν |dΩ/dR|
alpha viscosity: ν(R) = α · c_s · H = α h² √(GM·R)   (Shakura & Sunyaev, 1973)

This simulator integrates that PDE on a radial grid every frame, starting from a narrow ring of gas. Outward-transported angular momentum spreads the outer edge of the ring while a shrinking inner tail drains onto the central object — exactly the mechanism that powers real accretion in protoplanetary disks, X-ray binaries, and AGN.

α — dimensionless turbulent viscosity efficiency; higher α spreads and accretes the ring faster.
h = H/R — disk thickness-to-radius ratio, which sets the local sound speed and viscosity through ν = α h² √(GM R).
Time speed — playback rate; the physics step size is chosen automatically each frame to stay numerically stable.
Orbiting points sample the current Σ(R) profile and drift radially at the local viscous velocity v_R derived from the same equation — their inward/outward motion is not decorative, it is the solved flow field.

Gas parcels (viscous drift) Σ(R) density profile Central object

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