Lava Lamp Model

April 1, 2026·8 min read·By Mohamed Yassine Hemissi

Model

Overview

This project does not solve full fluid dynamics. The lava motion is modeled as a small set of blobs moving inside a bounded lamp volume under a simplified, physics-inspired update rule.

Each blob carries:

position: $x_i(t) \in \mathbb{R}^3$
velocity: $v_i(t) \in \mathbb{R}^3$
temperature: $T_i(t) \in \mathbb{R}$
mass: $m_i > 0$

The simulator then:

updates blob temperatures
computes forces on each blob
integrates velocity and position with a fixed timestep
rebuilds a scalar field for rendering

The main implementation files are:

PhysicsSimulator.ts
physicsSimulatorTimeStep.ts
LavaLampRenderer.ts
lampRenderer.ts

State

For blob i, the state is:

s_i(t) = (x_i(t), v_i(t), T_i(t), m_i)

The mean temperature is:

\bar{T}(t) = \frac{1}{N} \sum_{i=1}^{N} T_i(t)

The normalized height of a blob inside the lamp bounds is:

h_i(t) = \mathrm{clamp}\left( \frac{x_{i,y}(t)-y_{\min}}{y_{\max}-y_{\min}}, 0, 1 \right)

Temperature Evolution

The temperature model is:

\frac{dT_i}{dt} = H_{\mathrm{global}} + H_{\mathrm{bottom}}(h_i) + H_{\mathrm{stochastic}}(i,t) - C_{\mathrm{cool}}(T_i,h_i) - D_{\mathrm{diff}}(T_i,\bar{T})

with:

Global Heating

H_{\mathrm{global}} = G

where G = globalHeating.

Bottom Heating Bias

H_{\mathrm{bottom}}(h_i) = (1-h_i)B

where B = bottomHeatingBias.

Top Cooling Bias

C_{\mathrm{top}}(h_i) = \max(h_i - h_{\mathrm{cool}}, 0)\,C_t

where:

h_cool = topCoolingThreshold
C_t = topCoolingBias

Cooling Toward Ambient

C_{\mathrm{cool}}(T_i,h_i) = (C + C_{\mathrm{top}}(h_i))(T_i - T_a)

where:

C = coolingRate
T_a = ambientTemperature

Diffusion Toward Mean Temperature

D_{\mathrm{diff}}(T_i,\bar{T}) = D(T_i - \bar{T})

where D = diffusionRate.

Stochastic Thermal Forcing

The stochastic term is deterministic per blob id but time-varying:

H_{\mathrm{stochastic}}(i,t) = A\,S(i,t)

where:

A = stochasticAmplitude
S(i,t) is a weighted trigonometric mixture

The code uses:

S(i,t) = 0.88\,P(i,t) + 0.24\,\sin(0.21\omega t + 0.63\phi_i) + 0.17\,\sin(0.47\omega t - 0.31\phi_i) + 0.08\,\sin(1.16\omega t + 1.37\phi_i)

where:

omega = stochasticFrequency
phi_i is a stable phase derived from the blob id
P(i,t) is a plateau-shaped version of the primary sine wave

The explicit timestep update is:

T_i^{n+1} = T_i^n + \Delta t \left( G + (1-h_i)B + A\,S(i,t_n) - (C + C_{\mathrm{top}}(h_i))(T_i^n - T_a) - D(T_i^n - \bar{T}^n) \right)

This is implemented in updateBlobTemperature(...) in physicsSimulatorTimeStep.ts.

Forces

The total force on a blob is:

F_i = F_{\mathrm{buoyancy},i} + F_{\mathrm{gravity},i} + F_{\mathrm{drag},i} + F_{\mathrm{boundary},i}

Buoyancy

Buoyancy is driven by temperature difference from ambient:

F_{\mathrm{buoyancy},i} = \begin{bmatrix} 0 \\ m_i \beta (T_i - T_a) g \\ 0 \end{bmatrix}

where:

beta = buoyancyBeta
g = gravity

Gravity

F_{\mathrm{gravity},i} = \begin{bmatrix} 0 \\ -m_i g \\ 0 \end{bmatrix}

Drag

F_{\mathrm{drag},i} = -c_d v_i

where c_d = dragCoefficient.

Boundary Force

Near the lamp wall, the model applies a spring-like restoring force plus damping:

F_{\mathrm{boundary},i} \approx k\,p - c_b\,v_{\perp}

where:

k = stiffness
c_b = damping
p is penetration into the boundary margin
v_perp is the outward velocity component

This is applied axis-by-axis in computeBoundaryForce(...).

Motion Integration

Acceleration:

a_i = \frac{F_i}{m_i}

Velocity update:

v_i^{n+1} = v_i^n + \Delta t\,a_i^n

Position update:

x_i^{n+1} = x_i^n + \Delta t\,v_i^{n+1}

Then the blob is clamped back into the valid bounds:

x_i^{n+1} \leftarrow \Pi_{\Omega}(x_i^{n+1})

and outward velocity is zeroed on collision.

This logic is implemented by advanceBlobs(...) in physicsSimulatorTimeStep.ts.

Time Stepping

The simulator uses a fixed timestep:

\Delta t = \frac{\texttt{fixedDeltaTimeMs}}{1000}

Real frame time is accumulated, and multiple internal substeps may run before rendering. This is handled in step(...) in PhysicsSimulator.ts.

Scalar Field for Rendering

The blobs are not the final visible lava surface. Instead they generate a scalar field:

F(x) = \sum_{i=1}^{N} s_i\,\phi(\|x-x_i\|; r_i)

where:

s_i is blob strength
r_i is blob influence radius
phi is the radial contribution kernel
some blobs are free interior blobs and some are anchored cap blobs that keep small persistent masses at the top and bottom extremes

That field is rebuilt in rebuildFieldSnapshot() and later rendered as a liquid surface.

Runtime Controls

The exposed controls do not change the structure of the model. They rescale parts of it:

temperature scales global heating, bottom heating, top cooling, and stochastic amplitude
buoyancy scales buoyancy and slightly rebalances gravity
damping scales drag and boundary damping
diffusion scales thermal diffusion and field softness

These mappings are implemented in setParameter(...) in PhysicsSimulator.ts.

Interpretation

The current simulation is best described as:

a bounded particle system
with temperature-driven buoyancy
gravity
linear drag
soft wall response
deterministic stochastic thermal forcing

It is intentionally a simplified, controllable demo model rather than a full physical fluid solver.

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Lava Lamp Model

April 1, 2026·8 min read·By Mohamed Yassine Hemissi

Model

Overview

This project does not solve full fluid dynamics. The lava motion is modeled as a small set of blobs moving inside a bounded lamp volume under a simplified, physics-inspired update rule.

Each blob carries:

position: $x_i(t) \in \mathbb{R}^3$
velocity: $v_i(t) \in \mathbb{R}^3$
temperature: $T_i(t) \in \mathbb{R}$
mass: $m_i > 0$

The simulator then:

updates blob temperatures
computes forces on each blob
integrates velocity and position with a fixed timestep
rebuilds a scalar field for rendering

The main implementation files are:

PhysicsSimulator.ts
physicsSimulatorTimeStep.ts
LavaLampRenderer.ts
lampRenderer.ts

State

For blob i, the state is:

s_i(t) = (x_i(t), v_i(t), T_i(t), m_i)

The mean temperature is:

\bar{T}(t) = \frac{1}{N} \sum_{i=1}^{N} T_i(t)

The normalized height of a blob inside the lamp bounds is:

h_i(t) = \mathrm{clamp}\left( \frac{x_{i,y}(t)-y_{\min}}{y_{\max}-y_{\min}}, 0, 1 \right)

Temperature Evolution

The temperature model is:

\frac{dT_i}{dt} = H_{\mathrm{global}} + H_{\mathrm{bottom}}(h_i) + H_{\mathrm{stochastic}}(i,t) - C_{\mathrm{cool}}(T_i,h_i) - D_{\mathrm{diff}}(T_i,\bar{T})

with:

Global Heating

H_{\mathrm{global}} = G

where G = globalHeating.

Bottom Heating Bias

H_{\mathrm{bottom}}(h_i) = (1-h_i)B

where B = bottomHeatingBias.

Top Cooling Bias

C_{\mathrm{top}}(h_i) = \max(h_i - h_{\mathrm{cool}}, 0)\,C_t

where:

h_cool = topCoolingThreshold
C_t = topCoolingBias

Cooling Toward Ambient

C_{\mathrm{cool}}(T_i,h_i) = (C + C_{\mathrm{top}}(h_i))(T_i - T_a)

where:

C = coolingRate
T_a = ambientTemperature

Diffusion Toward Mean Temperature

D_{\mathrm{diff}}(T_i,\bar{T}) = D(T_i - \bar{T})

where D = diffusionRate.

Stochastic Thermal Forcing

The stochastic term is deterministic per blob id but time-varying:

H_{\mathrm{stochastic}}(i,t) = A\,S(i,t)

where:

A = stochasticAmplitude
S(i,t) is a weighted trigonometric mixture

The code uses:

S(i,t) = 0.88\,P(i,t) + 0.24\,\sin(0.21\omega t + 0.63\phi_i) + 0.17\,\sin(0.47\omega t - 0.31\phi_i) + 0.08\,\sin(1.16\omega t + 1.37\phi_i)

where:

omega = stochasticFrequency
phi_i is a stable phase derived from the blob id
P(i,t) is a plateau-shaped version of the primary sine wave

The explicit timestep update is:

T_i^{n+1} = T_i^n + \Delta t \left( G + (1-h_i)B + A\,S(i,t_n) - (C + C_{\mathrm{top}}(h_i))(T_i^n - T_a) - D(T_i^n - \bar{T}^n) \right)

This is implemented in updateBlobTemperature(...) in physicsSimulatorTimeStep.ts.

Forces

The total force on a blob is:

F_i = F_{\mathrm{buoyancy},i} + F_{\mathrm{gravity},i} + F_{\mathrm{drag},i} + F_{\mathrm{boundary},i}

Buoyancy

Buoyancy is driven by temperature difference from ambient:

F_{\mathrm{buoyancy},i} = \begin{bmatrix} 0 \\ m_i \beta (T_i - T_a) g \\ 0 \end{bmatrix}

where:

beta = buoyancyBeta
g = gravity

Gravity

F_{\mathrm{gravity},i} = \begin{bmatrix} 0 \\ -m_i g \\ 0 \end{bmatrix}

Drag

F_{\mathrm{drag},i} = -c_d v_i

where c_d = dragCoefficient.

Boundary Force

Near the lamp wall, the model applies a spring-like restoring force plus damping:

F_{\mathrm{boundary},i} \approx k\,p - c_b\,v_{\perp}

where:

k = stiffness
c_b = damping
p is penetration into the boundary margin
v_perp is the outward velocity component

This is applied axis-by-axis in computeBoundaryForce(...).

Motion Integration

Acceleration:

a_i = \frac{F_i}{m_i}

Velocity update:

v_i^{n+1} = v_i^n + \Delta t\,a_i^n

Position update:

x_i^{n+1} = x_i^n + \Delta t\,v_i^{n+1}

Then the blob is clamped back into the valid bounds:

x_i^{n+1} \leftarrow \Pi_{\Omega}(x_i^{n+1})

and outward velocity is zeroed on collision.

This logic is implemented by advanceBlobs(...) in physicsSimulatorTimeStep.ts.

Time Stepping

The simulator uses a fixed timestep:

\Delta t = \frac{\texttt{fixedDeltaTimeMs}}{1000}

Real frame time is accumulated, and multiple internal substeps may run before rendering. This is handled in step(...) in PhysicsSimulator.ts.

Scalar Field for Rendering

The blobs are not the final visible lava surface. Instead they generate a scalar field:

F(x) = \sum_{i=1}^{N} s_i\,\phi(\|x-x_i\|; r_i)

where:

s_i is blob strength
r_i is blob influence radius
phi is the radial contribution kernel
some blobs are free interior blobs and some are anchored cap blobs that keep small persistent masses at the top and bottom extremes

That field is rebuilt in rebuildFieldSnapshot() and later rendered as a liquid surface.

Runtime Controls

The exposed controls do not change the structure of the model. They rescale parts of it:

temperature scales global heating, bottom heating, top cooling, and stochastic amplitude
buoyancy scales buoyancy and slightly rebalances gravity
damping scales drag and boundary damping
diffusion scales thermal diffusion and field softness

These mappings are implemented in setParameter(...) in PhysicsSimulator.ts.

Interpretation

The current simulation is best described as:

a bounded particle system
with temperature-driven buoyancy
gravity
linear drag
soft wall response
deterministic stochastic thermal forcing

It is intentionally a simplified, controllable demo model rather than a full physical fluid solver.