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The advection-diffusion equation I am working with has the form

         ∂c/∂t  =  ∇. (D∇c) - ∇.  (vc) + R , where :

                   c = concentration
                   t = time
                   D = Diffusivity
                  ∇ = Gradient
                  ∇.= Divergence
                   R = Source of quantity c

I am unsure of what to do with the del operators. Could you help me set up the equation for Integration?

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1 Answer 1

Assuming everything is a function of spatial coordinates $x,y,z$ and time $t$, $$\eqalign{\nabla \cdot (D \nabla c) &= \dfrac{\partial}{\partial x} \left(D \dfrac{\partial c}{\partial x}\right) + \dfrac{\partial}{\partial y} \left(D \dfrac{\partial c}{\partial y}\right) + \dfrac{\partial}{\partial z} \left(D \dfrac{\partial c}{\partial z}\right)\cr &= D \left( \dfrac{\partial^2 c}{\partial x^2} +\dfrac{\partial^2 c}{\partial y^2} + \dfrac{\partial^2 c}{\partial z^2} \right) + \dfrac{\partial D}{\partial x} \dfrac{\partial c}{\partial x} + \dfrac{\partial D}{\partial y} \dfrac{\partial c}{\partial y} + \dfrac{\partial D}{\partial z} \dfrac{\partial c}{\partial z} }$$ and similarly if $v = (v_1, v_2, v_3)$ $$\nabla . (c v) = \dfrac{\partial}{\partial x} (c v_1) + \dfrac{\partial}{\partial y}(c v_2) + \dfrac{\partial}{\partial z}(c v_3)$$

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