# Divergence theorem - mass flux

## Homework Statement

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.

## 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
Homework Helper
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?

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