Flux across a line segment

Consider a two-dimensional flow of a fluid with the  velocity field  $$\mathbf{v}=-y\, \mathbf{i} +x\,\mathbf{j}.$$

The aim of this activity is to investigate the physical meaning of the flux and circulation of $\mathbf{v}$ across a line segment.


(a) Calculate the flux of $\mathbf{v}$ across the following line segments:

1. $C_1$: from $A=(0,-1)$ to $B=(0,1)$.
2. $C_2$: from $A=(0,3)$ to $B=(-4,0)$.
3. $C_3$: from $A=(0,-1)$ to $B=(0,2)$.
4. $C_4$: from $A=(-2,0)$ to $B=(0,2)$.

Confirm your answers using the following applet: 


(b) In part (a) you should have found that in some cases the flux was equal to  zero. Use the same applet (above) to investigate where you need to put the endpoints of the line segment in order to obtain a flux equal to zero. For example, use the applet to define the line segment from $A=(-2,-3)$ to $B=(2,3)$ or from $A=(-5,2)$ to $B=(2,5)$, and observe what happens to the flux in each case. 

1. Describe a general condition required for the flux across the line segment to be zero.

2. Explain what happens physically.