Math & Numbers

Bernoulli Equation Solver (Velocity / Pressure / Elevation)

Solve the Bernoulli equation p/(ρg) + v²/(2g) + z = const for steady, ideal, incompressible flow between two points on a streamline. Supply all three upstream quantities (p₁, v₁, z₁) and any two of the three downstream quantities; the tool solves for the third. Solve velocity: v₂ = √[v₁² + 2(p₁-p₂)/ρ + 2g(z₁-z₂)] — a negative radicand is physically impossible and is rejected. Solve pressure: p₂ = p₁ + ½ρ(v₁²-v₂²) + ρg(z₁-z₂). Solve elevation: z₂ = z₁ + (p₁-p₂)/(ρg) + (v₁²-v₂²)/(2g). g = 9.81 m/s². Density in kg/m³/g/cm³, pressure in Pa/kPa/bar/atm/psi, velocity in m/s, elevation in m/cm/ft. The upstream pressure/velocity/elevation heads and the total head are also reported.

json· JSON resultOpen tool

Quick start

Call this tool from your code in three languages.

cURL

curl -X POST 'https://api.elysiatools.com/en/api/tools/bernoulli-equation-solver' \
  -H 'Content-Type: application/json' \
  -d '{"unknownVar":"velocity","density":1000,"densityUnit":"kg/m3","p1":200000,"v1":2,"z1":0,"pressureUnit":"Pa","velocityUnit":"m/s","lengthUnit":"m","p2":100000,"v2":0,"z2":0,"decimalPlaces":4}'

API reference

Send a POST request with your inputs as JSON. File parameters require a separate upload first.

Endpoint

HTTP

POST https://api.elysiatools.com/en/api/tools/bernoulli-equation-solver

Parameters

Name	Type	Required	Description
unknownVar	select	No	Which downstream quantity to solve for: velocity v₂, pressure p₂, or elevation z₂. Provide the other two downstream quantities.
density	number	Yes	Fluid density ρ. Enter in the selected Density Unit.
densityUnit	select	No	—
p1	number	Yes	Upstream (point 1) pressure p₁. Enter in the selected Pressure Unit.
v1	number	Yes

MCP integration

Add this tool to your Model Context Protocol server so AI agents can list and call it.

Server configuration

Add this block to your MCP client configuration:

mcp.json

{
  "mcpServers": {
    "elysiatools-bernoulli-equation-solver": {
      "name": "bernoulli-equation-solver",
      "description": "Solve the Bernoulli equation p/(ρg) + v²/(2g) + z = const for steady, ideal, incompressible flow between two points on a streamline. Supply all three upstream quantities (p₁, v₁, z₁) and any two of the three downstream quantities; the tool solves for the third. Solve velocity: v₂ = √[v₁² + 2(p₁-p₂)/ρ + 2g(z₁-z₂)] — a negative radicand is physically impossible and is rejected. Solve pressure: p₂ = p₁ + ½ρ(v₁²-v₂²) + ρg(z₁-z₂). Solve elevation: z₂ = z₁ + (p₁-p₂)/(ρg) + (v₁²-v₂²)/(2g). g = 9.81 m/s². Density in kg/m³/g/cm³, pressure in Pa/kPa/bar/atm/psi, velocity in m/s, elevation in m/cm/ft. The upstream pressure/velocity/elevation heads and the total head are also reported.",
      "baseUrl": "https://api.elysiatools.com/mcp/sse?toolId=bernoulli-equation-solver",
      "command": "",
      "args": [],
      "env": {},
      "isActive": true,
      "type": "sse"
    }
  }
}

List available tools

After connecting to the SSE endpoint, list the exposed tools:

tools/list

{
  "jsonrpc": "2.0",
  "id": 1,
  "method": "tools/list"
}

Call this tool

Invoke the tool by its id, passing arguments built from its parameters:

tools/call

{
  "jsonrpc": "2.0",
  "id": 2,
  "method": "tools/call",
  "params": {
    "name": "bernoulli-equation-solver",
    "arguments": {
  "unknownVar": "velocity",
  "density": 1000,
  "densityUnit": "kg/m3",
  "p1": 200000,
  "v1": 2,
  "z1": 0,
  "pressureUnit": "Pa",
  "velocityUnit": "m/s",
  "lengthUnit": "m",
  "p2": 100000,
  "v2": 0,
  "z2": 0,
  "decimalPlaces": 4
}
  }
}