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Advection-diffusion equation

I am having trouble with the advection-diffusion equation and the proposed solution to it stated in the link above. If it were further expanded and simplified, this would be very helpful.

Steve

Thank you for your feedback.

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Instead of copying and pasting a text version of the web page, you should just provide a link to the original question and answer: math.stackexchange.com/questions/184902/… –  Robert Israel Sep 3 '12 at 20:23

1 Answer 1

The advection-diffusion equation was $$\frac{\partial c}{\partial t} = \nabla \cdot (D \nabla c) - \nabla \cdot (v c) + R $$ where $c$ is a scalar (the concentration) and $v = (v_1, v_2, v_3)$ is a vector (the velocity). I assumed everything could be a function of the three space coordinates $x_1, x_2, x_3$ and the time $t$. Then the equation becomes

$$ \eqalign{\dfrac{\partial c}{\partial t} &= D \left( \dfrac{\partial^2 c}{\partial x^2} +\dfrac{\partial^2 c}{\partial y^2} + \dfrac{\partial^2 c}{\partial z^2} \right) + \dfrac{\partial D}{\partial x} \dfrac{\partial c}{\partial x} + \dfrac{\partial D}{\partial y} \dfrac{\partial c}{\partial y} + \dfrac{\partial D}{\partial z} \dfrac{\partial c}{\partial z} \cr &- \dfrac{\partial}{\partial x} (c v_1) - \dfrac{\partial}{\partial y}(c v_2) - \dfrac{\partial}{\partial z}(c v_3) + R\cr &= D \left( \dfrac{\partial^2 c}{\partial x^2} +\dfrac{\partial^2 c}{\partial y^2} + \dfrac{\partial^2 c}{\partial z^2} \right) + \left(\frac{\partial D}{\partial x} - v_1\right) \frac{\partial c}{\partial x}+ \left(\frac{\partial D}{\partial y} - v_2\right) \frac{\partial c}{\partial y}\cr & +\left(\frac{\partial D}{\partial z} - v_3\right) \frac{\partial c}{\partial z} - c \left(\frac{\partial v_1}{\partial x} + \frac{\partial v_2}{\partial y} + \frac{\partial v_3}{\partial z} \right) + R}$$

I don't know what else you want. Of course if e.g. $D$ is constant you can set the partial derivatives of $D$ to $0$, or if the flow is incompressible you can set $\dfrac{\partial v_1}{\partial x} + \dfrac{\partial v_2}{\partial y} + \dfrac{\partial v_3}{\partial z}$ to $0$. But no such assumptions were made in the original question.

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