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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.


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
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1 Answer

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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