Consider the transport of a species A with concentration $c_A,$ which is advected by a velocity field $v.$ For example, suppose you put some colored dye into water flowing in a pipe: The species tends to move with the local mean velocity, $v,$ but may also diffuse relative to the local mean flow with a mass flux $j=-Dgrad{c}_A,$ where $D$ is the diffusion coefficient. The linear relation between mass flux and the concentration gradient is called Fick's law of diffusion.

Alternatively, one may equivalently state that the net rate at which A crosses a fixed surface in the fluid is $c_{A}v+j.$ Also, allow for the possibility that species $A$ is produced or consumed, by chemical reactions at a rate $r_A$ per unit volume.

$($a$)$ Select a material volume that deforms with the fluid, as in the Reynolds Transport Theorem. Draw a diagram to visualize the system. Derive the following species conservation equation

$\frac{del{c}_A}{\text{d}\text{e}\text{l}\text{t}}+v*grad{c}_A=Dgra{d}^2{c}_A+r_A,$

clearly stating all the assumptions you used to arrive at the result.

$($b$)$ Repeat part $($a$)$ using a using control volume that is fixed in the fluid. You should get the same answer.