Flux in three dimensions

Kontekst

• Vector fields
• Surface integrals
• Unit normal vector of a surface

• Two-dimensional flux

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:

  $\begin{aligned} \iint_\redE{S} \blueE{\textbf{F}} \cdot \greenE{\hat{\textbf{n}}}\, \redE{d\Sigma} \end{aligned}$

  where
  • $\redE{S}$start color #bc2612, S, end color #bc2612 denotes the surface through which we are measuring flux.
  • $\blueE{F(x, y,z)}$start color #0c7f99, F, left parenthesis, x, comma, y, comma, z, right parenthesis, end color #0c7f99 is a three-dimensional vector field, thought of as describing a fluid flow.
  • $\greenE{\hat{\textbf{n}}(x, y, z)}$start color #0d923f, start bold text, n, end bold text, with, hat, on top, left parenthesis, x, comma, y, comma, z, right parenthesis, end color #0d923f is a function which gives a unit normal vector at each point on $\redE{S}$start color #bc2612, S, end color #bc2612.
  • $\redE{d\Sigma}$start color #bc2612, d, \Sigma, end color #bc2612 can be thought of as a tiny unit of area on the surface $\redE{S}$start color #bc2612, S, end color #bc2612.

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 $\blueE{F(x, y, z)}$start color #0c7f99, F, left parenthesis, x, comma, y, comma, z, right parenthesis, end color #0c7f99?

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 $\blueE{\textbf{F}(x, y, z)}$start color #0c7f99, start bold text, F, end bold text, left parenthesis, x, comma, y, comma, z, right parenthesis, end color #0c7f99, where $(x, y, z)$left parenthesis, x, comma, y, comma, z, right parenthesis are the coordinates of the particle. Also assume that the fluid has a uniform density of $1 \, \text{kg}/\text{m}^3$1, start text, k, g, end text, slash, start text, m, end text, cubed throughout the surface.

Flow through each tiny piece of the surface

• Krok 1: Break up the surface $S$S into many, many tiny pieces.
• Krok 2: See how much fluid leaves/enters each piece.
• Krok 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

Wybierz 1 odpowiedź:

Wybierz 1 odpowiedź:

• (Choice A)  

  $\begin{aligned} \big(\blueE{\textbf{F}}(x_0, y_0, z_0) \Delta t\big) \big(d\Sigma\big) \end{aligned}$

  A

  $\begin{aligned} \big(\blueE{\textbf{F}}(x_0, y_0, z_0) \Delta t\big) \big(d\Sigma\big) \end{aligned}$
• (Choice B)  

  $\begin{aligned} \big(\blueE{\textbf{F}}(x_0, y_0, z_0) \cdot \greenE{\hat{\textbf{n}}}\big) \big(\Delta t\big) \big(d\Sigma\big) \end{aligned}$

  B

  $\begin{aligned} \big(\blueE{\textbf{F}}(x_0, y_0, z_0) \cdot \greenE{\hat{\textbf{n}}}\big) \big(\Delta t\big) \big(d\Sigma\big) \end{aligned}$

Sprawdź

[Odpowiedź]

Note: Orientation matters

Step 3: Add it all together with an integral

• $\blueE{\textbf{F}}(x, y,z)$start color #0c7f99, start bold text, F, end bold text, end color #0c7f99, left parenthesis, x, comma, y, comma, z, right parenthesis, which gives the velocity of a fluid particle at a point.
• $\greenE{\hat{\textbf{n}}}(x, y, z)$start color #0d923f, start bold text, n, end bold text, with, hat, on top, end color #0d923f, left parenthesis, x, comma, y, comma, z, right parenthesis, which gives the outward facing unit normal vector at an arbitrary point on the surface.

Flux

Podsumowanie

• 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 $\redE{d\Sigma}$start color #bc2612, d, \Sigma, end color #bc2612 of that piece, the unit normal vector $\greenE{\hat{\textbf{n}}}$start color #0d923f, start bold text, n, end bold text, with, hat, on top, end color #0d923f to that piece, and the fluid velocity $\blueE{\textbf{F}}$start color #0c7f99, start bold text, F, end bold text, end color #0c7f99 in that region.
• "Add up" all these flow rates with a surface integral to get the flux as a whole.

  $\begin{aligned} \underbrace{ \iint_\redE{S} \blueE{\textbf{F}} \cdot \greenE{\hat{\textbf{n}}}\, \redE{d\Sigma} }_{\text{Flux through $\redE{S}$}} \end{aligned}$

• 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.