Flux in 2D (Part 2)

3. Outward flux across a closed plane curve

Recall that:

If $C$ is a piecewise-smooth, simple closed curve, the net outward flux of a vector field $\mathbf v =v_1(x,y)\,\mathbf i +v_2(x,y)\,\mathbf  j$ across $C$ is given by
$$\text{Net outward flux}=\oint_C \mathbf v\cdot \mathbf n\, dS$$
where $\mathbf n$ is a unit vector normal to $C$, directed outward from the region bounded by $C$.

  
Example 1. Suppose that $C$ is a circle of radius $r$, centre the origin. Let $\mathbf v=x\,\mathbf i+y\,\mathbf j$. In this case, the unit normal vector to $C$ is in the same direction as $\mathbf v$. Thus
$$\mathbf v \cdot \mathbf n =|\mathbf v|=r.$$
It follows that
$$\text{Net outward flux}=\oint_C \mathbf v\cdot \mathbf n dS=\oint_C r\,dS=r\oint_C dS=2\pi r^2.$$

The following applet shows a dynamic view of Example 1. Change the radius of the circle to check the general solution.


Example 2. Consider the same circle $C$ of radius $r$, centre the origin. But now we have $\mathbf v=-y\,\mathbf i+x\,\mathbf j$. In this case, $\mathbf v$ is perpendicular to the unit normal vector $\mathbf n$. Thus
$$\mathbf v \cdot \mathbf n =0.$$
Therefore
$$\text{Net outward flux}=\oint_C \mathbf v\cdot \mathbf n\, dS=0.$$

The following applet shows a dynamic view of Example 2. Change the radius of the circle to check the general solution.


Observation: The previous examples make sense physically. In the Example 1, the fluid is spewing out the origin, and thus flowing through $C$. On the other hand, in Example 2, the fluid is spinning around the origin. None of it crosses $C$.

Flux form of Green's theorem

If $C$ is a positively oriented simple closed curve enclosing a region $D$ and $\mathbf v =v_1(x,y)\,\mathbf i +v_2(x,y)\,\mathbf  j$ then
$$\text{Net outward flux}=\oint_C \mathbf v\cdot \mathbf n \,dS=\iint_D\text{div}(\mathbf v )\,dA$$
where $\text{div} (\mathbf v)=\dfrac{\partial v_1}{\partial x}+\dfrac{\partial v_2}{\partial y}$.


Example 3. Consider again the example of a circle $C$ of radius $r$, centre at the origin, and the velocity vector $\mathbf v =x\, \mathbf i +y\, \mathbf j$. Since
$$\text{div}(\mathbf v)=1+1=2,$$
then
$$\text{Net outward flux}=\oint_C \mathbf v\cdot \mathbf n \,dS=\iint_D\text{div}(\mathbf v )\,dA=\iint_D 2\,dA=2\pi r^2.$$

If we move the circle away from the origin, the flux is still $2\pi r^2$. However, computing $\oint_C \mathbf v \cdot \mathbf n \,dS$ becomes quite hard.

The following applet shows a dynamic representation of Example 3. Move the circle around. Observe that the flux remains constant. You can also observe the flux for a semicircle by dragging the slider for the partial path.

Key Concept

In the context of fluids the divergence measures how much fluid is being added (or taken away); these are known as sources (or sinks). For $\mathbf v =x\, \mathbf i +y\, \mathbf j$, fluid is being added everywhere (imagine rain falling on the ground and then flowing from the origin). For $\mathbf v =-y\, \mathbf i +x\, \mathbf j$ the divergence is zero. No fluid is being added or removed, there are no sources or sinks.