Flux in three dimensions
What we are building to
- When you have a fluid flowing in three-dimensional space, and a surface sitting in that space, the flux through that surface is a measure of the rate at which fluid is flowing through it.
- Flux can be computed with the following surface integral:where
- denotes the surface through which we are measuring flux.
- is a three-dimensional vector field, thought of as describing a fluid flow.
- is a function which gives a unit normal vector at each point on .
- can be thought of as a tiny unit of area on the surface .
Changing fluid mass in a blob
Key question: How much fluid is leaving/entering this blob as the fluid flows along the vector field defined by ?
More rigorous phrasing: What is the rate of change of mass inside the blob, as a function of time? Assume the velocity of each fluid particle is given by the vector , where are the coordinates of the particle. Also assume that the fluid has a uniform density of throughout the surface.
Flow through each tiny piece of the surface
- Step 1: Break up the surface into many, many tiny pieces.
- Step 2: See how much fluid leaves/enters each piece.
- Step 3: Add up all of these amounts with a surface integral.
Step 1: Break up the surface
Step 2: Measure fluid flow through each piece
Note: Orientation matters
Step 3: Add it all together with an integral
- , which gives the velocity of a fluid particle at a point.
- , which gives the outward facing unit normal vector at an arbitrary point on the surface.
- Imagine cutting the surface up into many small pieces, small enough that each piece can be considered flat.
- Compute how much fluid flows through each piece as a function of the area of that piece, the unit normal vector to that piece, and the fluid velocity in that region.
- "Add up" all these flow rates with a surface integral to get the flux as a whole.
- If you change the orientation of your surface by choosing unit normal vectors facing the opposite direction, the sign of this integral will be flipped.