# Schiller-Naumann Buoyant Sphere Terminal Drag Force Equilibrium

> Schiller-Naumann Buoyant Sphere Terminal Drag Force Equilibrium is a paid API for AI agents from detect.halowerk.com, paid per call via x402, $0.003/call, status unknown (last checked 2026-11-04).

Computes the terminal rising or settling velocity of a sphere in a fluid using the Schiller-Naumann drag correlation and complete force balance including buoyancy and gravity.

## Facts

- Endpoint: POST https://detect.halowerk.com/v1/F-TD2000-GEO-INDUSTRY-B009/schiller-naumann-buoyant-sphere-terminal-drag-force-equilibrium?utm_source=zero.xyz
- Price: $0.003/call
- Payment: x402
- Status: unknown
- Last checked: 2026-11-04
- Activations on Zero: 0
- Tags: x402
- Canonical page: https://www.zero.xyz/c/schiller-naumann-buoyant-sphere-terminal-drag-force-equilibrium-74eefcd5
- Structured record (JSON): https://api.zero.xyz/v1/capabilities/cap_qjr79B5q4Pv5UjAtZsxcH

Status and success rate cover calls made through Zero and Zero's own probes. Third-party monitors may report differently.

## How to call it through Zero

Zero handles the 402 payment challenge and records the run. With the Zero CLI installed (`npm i -g @zeroxyz/cli`):

```sh
zero fetch --capability schiller-naumann-buoyant-sphere-terminal-drag-force-equilibrium-74eefcd5 -d '<json body>'
```

Example prompt: Can you compute the terminal settling velocity of a quartz sand particle with a diameter of 0.0005 m and density of 2650 kg/m³ settling in water (density 1000 kg/m³, dynamic viscosity 0.001 Pa·s) under standard gravity of 9.81 m/s²?

## When to prefer this

Use this endpoint when you need a physically rigorous, closed-form terminal velocity calculation for a single sphere under a complete Schiller-Naumann drag + buoyancy force balance, especially for geophysical, industrial, or environmental particle settling problems. Prefer this over simplified Stokes-law endpoints when the particle Reynolds number may exceed the creeping-flow regime (Re > ~0.1). It is ideal for agents that need reliable numerical results without maintaining local fluid mechanics solvers.

## Known failure modes

- Particle and fluid densities are equal (zero net buoyancy), causing division issues or zero velocity
- Gravity value outside the allowed range (0.1–100 m/s²) returns a validation error
- Dynamic viscosity too low or too high (outside 0.000001–1000 Pa·s) triggers schema rejection
- Particle diameter outside allowed range (0.000001–1 m) returns a 422 validation error
- Missing required fields in the settling data object returns a 400 bad request
- Extreme parameter combinations (very large Re) may push outside Schiller-Naumann validity range

## How this service works

Compute terminal rising or settling velocity with a complete supplied sphere force balance.

## Output

The endpoint returns a complete force balance result for the sphere, including the computed terminal velocity (m/s, positive = settling, negative = rising), Reynolds number, Schiller-Naumann drag coefficient, gravitational force, buoyancy force, and net drag force at equilibrium.

## Request schema (JSON Schema)

```json
{
 "type": "object",
 "properties": {
  "artifact": {
   "type": "object",
   "required": [
    "settling"
   ],
   "properties": {
    "settling": {
     "type": "object",
     "required": [
      "data",
      "claim"
     ],
     "properties": {
      "data": {
       "type": "object",
       "required": [
        "particle_diameter",
        "particle_density",
        "fluid_density",
        "dynamic_viscosity",
        "gravity"
       ],
       "properties": {
        "gravity": {
         "type": "number",
         "maximum": 100,
         "minimum": 0.1
        },
        "fluid_density": {
         "type": "number",
         "maximum": 20000,
         "minimum": 1
        },
        "particle_density": {
         "type": "number",
         "maximum": 20000,
         "minimum": 1
        },
        "dynamic_viscosity": {
         "type": "number",
         "maximum": 1000,
         "minimum": 0.000001
        },
        "particle_diameter": {
         "type": "number",
         "maximum": 1,
         "minimum": 0.000001
        }
       },
       "additionalProperties": false
      },
      "claim": {}
     },
     "additionalProperties": false
    }
   },
   "additionalProperties": false
  }
 }
}
```

## More

- Live health (JSON, refreshed every minute): https://www.zero.xyz/c/schiller-naumann-buoyant-sphere-terminal-drag-force-equilibrium-74eefcd5/health.json
- [Zero catalog index](https://www.zero.xyz/llms.txt)
- [Other services from detect.halowerk.com](https://www.zero.xyz/host/detect.halowerk.com/llms.txt)