Flux in two dimensions
What we're building to
- Given a region enclosed by a curve , and a fluid flow determined by a vector field , the rate at which fluid is exiting that region (assuming it has density ) can be measured with the following line integral:Here, is some function that returns the outward unit normal vector at every point on the curve .
- This integral is called a flux integral, or sometimes a "two-dimensional flux integral", since there is another similar notion in three dimensions.
- In any two-dimensional context where something can be considered flowing, such as a fluid, two-dimensional flux is a measure of the flow rate through a curve. In general, the curve isn't necessarily a closed loop.
Changing fluid mass in a region
Key question: How can we measure the instantaneous rate of change of the mass of fluid inside the region enclosed by ?
One bit of length at a time
- The base of the parallelogram is the length of our tiny segment. Let's call that .
- The displacement vector of a fluid particle which starts at some point on the tiny segment will be , where is the velocity vector of fluid at that point, and is the amount of time the fluid flowed.
- The height of the parallelogram will be the component of the displacement vector which is perpendicular to the segment. You can extract this by taking the dot product between and a unit normal vector to the curve . Let's name that unit normal vector .
Exiting mass per unit time
Bringing it together with an integral
- The vector field gives the fluid velocity at each point along the curve.
- The term should be considered a function, , which takes in a point on and outputs the unit normal vector to at that point.
- Using the symbol instead of is just to emphasize that the line integral is around a closed loop.
- indicates a tiny change in arc length along the curve. Conceptually it is no different from the term in the previous section, but now we are considering it as an infinitesimal quantity used for integration.
- As you walk along the curve , the value measures how much the fluid is leaving/entering the region enclosed by at each point. It is positive when the fluid is leaving, and negative when the fluid flows in, so the integral as a whole will return the total mass leaving the region enclosed by per unit time.
Example: Flux through a circle
- Parameterize the circle.
- Find a function for on the circle.
Computing a unit normal vector
- In any context where something can be considered flowing, such as a fluid, two-dimensional flux is a measure of the flow rate through a curve. The flux over the boundary of a region can be used to measure whether whatever is flowing tends to go into or out of that region.
- The flux through a curve can be measured with the line integralwhere
- defines the vector field which indicates the flow rate.
- is some function which return the outward unit normal vector at every point on the curve .