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I am studying the non-dimensional heat equation, and was wondering what the quantity $$-\frac{\partial\theta}{\partial x}$$ represents. $\theta$ and $x$ are non-dimensional versions of temperature and distance respectively, e.g. $x=X/L$ where $X$ is normal distance, and $L$ is the length of the rod.

What does this quantity bein $0$ or $1$ or some other number actually mean?

EDIT: I asked this question on physics stack exchange in search for a more in depth answer. It is linked here.

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$-\partial \theta/\partial x = 0$ means that if you go along on the rod there is no change in temperature or, in other words, that the temperature is constant w.r.t. to your position on the rod, indicated by $x$.

The number being nonzero you would have a linear decrease in temperature along the rod since

$$-\frac{\partial \theta}{\partial x} = k$$

where $k>0$ implies that

$$\theta = -kx$$

EDIT: In the case $k<0$ you would of course have a linear increase.

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  • $\begingroup$ So does this mean that $k=1$ is no more special than say $k=10$? $\endgroup$ – John Doe Apr 24 '17 at 12:28
  • $\begingroup$ What do you mean by "more special"? $k=10$ is a faster decrease than $k=1$, faster in a sense that a smaller change of position will lead to the same change of temperature, or the same change of position to a greater change in temperature. $\endgroup$ – Jakob Elias Apr 24 '17 at 12:31
  • $\begingroup$ Often in non-dimensional problems, $1$ is sort of like a state of equilibrium - but not in this case? $\endgroup$ – John Doe Apr 24 '17 at 12:33
  • $\begingroup$ This I don't know unfortunately $\endgroup$ – Jakob Elias Apr 24 '17 at 15:41

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