divergence theorem - mass flux

[reb659]

1. The problem statement, all variables and given/known data
Water in an irrigation ditch of width w = 3.0 m and depth d = 2.0 m
flows with a speed of 0.40 m/s. For each case, sketch the situation,
then find the mass flux through the surface: (a) a surface of area wd,
entirely in the water, perpendicular to the flow; (b) a surface with area
3wd/2, of which wd is in the water, perpendicular to the flow; (c) a
surface of area wd/2, entirely in the water, perpendicular to the flow;
(d) a surface of area wd, half in the water and half out, perpendicular
to the flow; (e) a surface of area wd, entirely in the water, with its
normal 30 from the direction of the flow.


2. Relevant equations



3. The attempt at a solution

The section we are learning is the divergence theorem, but I don't really see the relation between that and this problem. How can I go about approaching this?

[HallsofIvy]

Those are all basically arithmetic problems! For (a), If water is flowing at 4 m/s, in one second, it will have moved a distance (of course!) of 4 m. The part that is flowing through a 3 m by 2 m rectangle will form a solid 3 m by 2 m by 4 m. What is the volume of that rectangle? For (b), the fact that the entire rectangle is "3wd/2" is irrelevant. Only the part that is in the water has any flow through it- and that is exactly the same as in (a).

The only "difficult" one is (e) where the rectangle is tilted. Draw a right triangle with the length of the rectangle as hypotenuse and one leg perpendicular to the water flow. What is the length of that leg?

[reb659]

Ahh, I see now. I'm still a bit confused on part e though - if the area is wd, wouldn't the answer just be the same as a) because equally areas are completely submerged in water?

[HallsofIvy]

Suppose you had the rectangle turned so the flow was along the length of the rectangle would the flow through the rectangle be the same as if it were horiontal?

Another way to think about this is to break the vector velocity of the water in two components: one tangent to the rectangle and the other perpendicular to it. Only the component perpendicular to the rectangle contributes to flow through the rectangle. Think of it as two flows: one perpendicular and so through the rectangle, the other parallel to the rectangle. That second "flow" does not go through the rectangle.

There is a slight ambiguity but it doesn't affect the answer. In 3 dimensions, there are many directions at "30 degrees" from a single direction. If the length w is at 30 degrees the "projection" to the plane perpendicular to the flow is w cos(30). If it is the length d cos(30). But since you would then multiply by the other to find the area, it is wd cos(30).