{"uid":"cap_qjr79B5q4Pv5UjAtZsxcH","slug":"schiller-naumann-buoyant-sphere-terminal-drag-force-equilibrium-74eefcd5","name":"Schiller-Naumann Buoyant Sphere Terminal Drag Force Equilibrium","description":"Compute terminal rising or settling velocity with a complete supplied sphere force balance.","url":"https://detect.halowerk.com/v1/F-TD2000-GEO-INDUSTRY-B009/schiller-naumann-buoyant-sphere-terminal-drag-force-equilibrium?utm_source=zero.xyz","method":"POST","headers":{},"bodySchema":{"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}}},"responseSchema":null,"example":null,"exampleRequest":null,"tags":["x402"],"displayCostAmount":"0.003","displayCostAsset":"USDC","priceDynamic":false,"priceHint":null,"priceStatus":"priced","priceSource":"probe","requiresHandshake":false,"reviewCount":0,"rating":{"score":"0.00","successRate":"0.00","reviews":0,"stars":null,"state":"unrated"},"availabilityStatus":"unknown","priceObserved":null,"sessionDeposit":null,"pricing":{"kind":"static","summary":"$0.003/call","primary":{"kind":"static","protocol":"x402","network":"base","amountUsd":"0.003","per":"call","confidence":"exact"},"accepted":[{"kind":"static","protocol":"x402","network":"base","amountUsd":"0.003","per":"call","confidence":"exact"}]},"paymentMethods":[{"uid":"pm_jsDGE6v4A1Jf-8Ijs0vOs","protocol":"x402","methodType":"crypto","chain":"base","mode":"charge","costAmount":"0.003","costPer":"request","priority":0,"asset":"0x833589fCD6eDb6E08f4c7C32D4f71b54bdA02913","unit":"request","depositMicros":null,"planRef":null}],"brandName":null,"brandSlug":null,"brandBaseUrl":null,"brandDocsUrl":null,"whatItDoes":"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.","exampleAgentPrompt":"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?","exampleUseCases":[{"title":"Sediment settling rate in river","prompt":"I need the terminal settling velocity for a silt particle 0.00002 m in diameter with a density of 2600 kg/m³ sinking in river water with density 1000 kg/m³, viscosity 0.0012 Pa·s, and gravity 9.81 m/s². What does the Schiller-Naumann force balance give me?"},{"title":"Oil droplet rise in water column","prompt":"How fast will an oil droplet of diameter 0.001 m and density 870 kg/m³ rise through seawater (density 1025 kg/m³, dynamic viscosity 0.00108 Pa·s) under gravity of 9.81 m/s²? Run the full buoyant sphere terminal velocity calculation."},{"title":"Mars dust particle settling","prompt":"Calculate the terminal settling velocity of a Mars dust particle with diameter 0.000005 m and particle density 2000 kg/m³ in the Martian lower atmosphere (fluid density 0.02 kg/m³, dynamic viscosity 0.0000115 Pa·s) under Mars gravity of 3.72 m/s²."}],"resultDescription":"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.","failureModes":["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"],"whenToPreferThis":"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.","instructions":null,"reviewSummary":null,"reviewSummaryHighlights":null,"reviewSummaryConcerns":null,"reviewSummaryGeneratedAt":null,"activationCount":0,"lastUsedAt":null,"lastSuccessfullyRanAt":null,"lastHealthCheckAt":"2026-11-04T06:41:09.627Z","isFirstParty":false,"canonicalSlug":"schiller-naumann-buoyant-sphere-terminal-drag-force-equilibrium-74eefcd5"}