Calculating the average divergence of the flow of fluid in a box.

[lovepiano25]

Homework Statement

A cube of side 5cm is placed in a fluid. For the following velocity values, calculate the average divergence of the flow. Is there a source or sink within this box? The velocities are given in cm/s. The cube is oriented so that the +z axis exits the front face, the +x axis exits the right face and the +y axis exits the top face.

Front: V = 30i + 60 j - 30 k

Right V = 20 i - 20 j + 20 k

Left V = -100 i + 200 j - 300 k

Top V = 5 i + 50 j - 10 k

Bottom V = 0

Back V = 250 k

Homework Equations

The divergence Theorem

(surface integral) ∫ ∇ F dA = (surface integral) ∫ F n ds

The Attempt at a Solution

I know how to calculate the normal vector but i can not figure out how to go about finding the surface integral.

I would appreciate one example

Thank you

 

[DimReg]

Have you tried taking the dot product of your normal vector and the flow? You should get something that is easy to integrate. This is what appears on the right side of the divergence theorem under the integral sign.

If you got that far and are still confused, what do you think ∫dA should be? What does that imply about ∫100dA?

Note also that on the right side of the divergence theorem, you have a surface integral, but on the left you have a volume integral.