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In Water Quality Modeling, there is a basic differential equation describing concentration $s(x,t)$ in an advective nondispersive stream. Its formula and solution can be seen in Fig.1.

I tried to examine this solution by caculating reversely, that is: $$\frac{\partial s}{\partial x}=\frac{W(t-t^*)}{Q}\cdot (-\frac{K}{U})\cdot exp(-\frac{Kx}{U})$$

Then the right part of equation became:$$-U\frac{\partial s}{\partial x}-Ks=...=0$$

This is in conflict with the left part ($\frac{\partial s}{\partial t}$), which is not zero.

Could someone tell me what's wrong?

T X
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1 Answers1

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Terrific catch. Note that the text (Fischer?) states "only the waste load is assumed to be varying in time." Presumably, slowly enough that the downstream concentration can be regarded as being in a steady state and that, as you surmised, $\frac{\partial s}{\partial t} =0$.

Grant M
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  • I am still puzzled. Because the waste load is assumed to be varying in time, then ∂s/∂t is a formula with a "∂W(t−t$^∗$)/∂t" part, which is not "0"? – T X Mar 18 '18 at 10:38
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    The solution is showing the steady-state balance between the waste load and the exponential decay of s as it moves downstream. It is common in these advection problems to either set $ \frac{\partial}{\partial t} $ to 0 or look at the Lagrangian problem via the material derivative $\frac{D}{Dt}=\frac{\partial}{\partial t}+\textbf{u}\cdot \nabla$. Does he say what he means by $t^*$? – Grant M Mar 19 '18 at 01:23
  • It is also common to look at a function as the sum of a mean quantity (which might be slowly varying) and a varying residual such that $s=\bar{s}+s'$ Advection/ diffusion PDE problems don't usually have analytic solutions unless you start using these sorts of tricks. – Grant M Mar 19 '18 at 01:31