Divergence and Circulation Density

The divergence of a three-dimensional vector field is the extent to which the vector field flow behaves like a source at a given point. It is a local measure of its "out-going-ness"–-the extent to which there is more exiting an infinitesimal region of space than entering it. If the divergence is non-zero at some point, then there must be a source or sink at that position. ^[1]

Let us say we have a velocity field which depicts the fluid flowing into a rectangular region (as in the image), and is defined by the following expression: \[\mathbf F(x,y)=F_x(x,y)\hat{\mathbf x}+F_y\hat{\mathbf y}.\]

Fluid flowing into a rectangular region

As in the image, we have the length and breadth of the rectangle to be \(\Delta x\) and \(\Delta y\). Thus, the fluid flow rates leaving a particular side of the rectangle will be: \[(\text{Velocity of the fluid at the respective corner})\times (\text{length of the side}).\]

According to this, the fluid flow rates at each of the sides of the rectangle would be \[\begin{align} \text{Left} : \quad &\mathbf F(x,y)\cdot (-\hat{\mathbf x})\Delta y \quad\quad =-F_x(x,y)\Delta y\\ \text{Right} : \quad &\mathbf F(x+\Delta x,y)\cdot \hat{\mathbf x}\Delta y \quad =F_x(x+\Delta x,y)\Delta y\\ \text{Top} : \quad &\mathbf F(x,y+\Delta y)\cdot \hat{\mathbf y}\Delta x \quad =F_y(x,y+\Delta y)\Delta x\\ \text{Bottom} : \quad &\mathbf F(x,y)\cdot (-\hat{\mathbf y})\Delta x \quad\quad =-F_y(x,y)\Delta x. \end{align}\]

Now, to find the whole outflow, we need to sum up the fluid flow rates at each of the sides of the rectangle, and we arrive at \[\begin{align} \text{Left + Right} : \quad &-F_x(x,y)\Delta y + F_x(x+\Delta x,y)\Delta y = \left(\dfrac{\partial F_x}{\partial x}\Delta x\right)\Delta y\\ \text{Top + Bottom} : \quad &F_y(x,y+\Delta y)\Delta x -F_y(x,y)\Delta x \quad = \left(\dfrac{\partial F_y}{\partial y}\Delta y\right)\Delta x. \end{align}\]

Finally, adding the resulting values and dividing this sum by the area of the rectangle gives the outflow which is nothing but the divergence of the fluid: \[\dfrac{\left(\dfrac{\partial F_x}{\partial x}\Delta x\right)\Delta y + \left(\dfrac{\partial F_y}{\partial y}\Delta y\right)\Delta x}{\Delta x \Delta y}=\left(\dfrac{\partial F_x}{\partial x}\right) + \left(\dfrac{\partial F_y}{\partial y}\right). \]

The divergence is the flux density of a vector field \(\mathbf F = F_x\hat{\mathbf x} + F_y\hat{\mathbf y}\) at the point \((x,y),\) given by

\[\text{div }\mathbf F = \dfrac{\partial F_x}{\partial x}+\dfrac{\partial F_y}{\partial y}.\]

Now, we devise a method to find the rate at which the fluid is circulating along the region, and again we cansider the case of the rectangle with the same velocity vector: \[\mathbf F(x,y)=F_x(x,y)\hat{\mathbf x}+F_y\hat{\mathbf y}.\]

Fluid flowing into a rectangular region, and the direction of the circulation

[as the formulas are similar I'm copying and pasting the same with subtle changes in signs, text will be added soon]

[text]

\[\begin{array}{rll} \text{Left} : &\mathbf F(x,y)\cdot (-\hat{\mathbf y})\Delta y &=-F_y(x,y)\Delta y\\ \text{Right} : &\mathbf F(x+\Delta x,y)\cdot \hat{\mathbf y}\Delta y &=F_y(x+\Delta x,y)\Delta y\\ \text{Top} : &\mathbf F(x,y+\Delta y)\cdot (-\hat{\mathbf x})\Delta x &=-F_x(x,y+\Delta y)\Delta x\\ \text{Bottom} : &\mathbf F(x,y)\cdot \hat{\mathbf x}\Delta x &=F_x(x,y)\Delta x. \end{array}\]

[more text]

\[\begin{align} \text{Left + Right} : \quad &-F_y(x,y)\Delta y + F_y(x+\Delta x,y)\Delta y = \left(\dfrac{\partial F_y}{\partial x}\Delta x\right)\Delta y\\ \text{Top + Bottom} : \quad &-F_x(x,y+\Delta y)\Delta x -F_x(x,y)\Delta x \quad = -\left(\dfrac{\partial F_x}{\partial y}\Delta x\right)\Delta y\\ \end{align}\]

[more text]

\[\dfrac{\left(\dfrac{\partial F_y}{\partial x}\Delta x\right)\Delta y - \left(\dfrac{\partial F_x}{\partial y}\Delta x\right)\Delta y}{\Delta x \Delta y}=\left(\dfrac{\partial F_y}{\partial x}\right) -\left(\dfrac{\partial F_x}{\partial y} x\right).\]

The circulation density of a vector field \(\mathbf F = F_x\hat{\mathbf x} + F_y\hat{\mathbf y}\) at the point \((x,y)\) is given by:

\[(\text{curl }\mathbf F)\cdot \hat{\mathbf z}= \dfrac{\partial F_x}{\partial x}-\dfrac{\partial F_y}{\partial y}.\]

The curl expression is the \(\mathbf z\) component of the of the curl. For more information on the curl vector, visit: stokes' theorem

1. Jess Brewer, H. DIVERGENCE of a Vector Field. Retrieved from http://musr.phas.ubc.ca/~jess/hr/skept/Gradient/node4.html