What if we change one of Fourier's law of heat conduction

Question

I'm studying PDE heat diffusion on 1-D rod using the textbook. It states four intuitions leading to Fourier's law of heat conduction $\phi=-K_0\frac{\partial u}{\partial x}$, where $\phi$ is the heat flow, $K_0$ is the thermal conductivity, $u$ is the temperature and $x$ is the distance.

The four intuitions are

1. If the temperature is constant in a region, no heat energy flows.
2. If there are temperature differences, the heat energy flows from the hotter region to the colder region.
3. The greater the temperature differences (for the same material), the greater is the flow of heat energy.
4. The flow of heat energy will vary for different materials, even with the same temperature differences.

The third intuition is not easy to understand. Suppose all other intuitions hold, if the third one became the flow is independent of the temperature differences, what should the equation look like?

Moreover, if the greater the temperature difference, the less the heat flow, what should the equation look like? Is $\phi=-K_0[\frac{\partial u}{\partial x}]^{-1}$ a possible candidate mathematically?

I know all the four intuitions are correct, just wonder how they contribute to the equation.

Answer 1

First I would say that these are not "intuitions", these are physical laws, i.e., the observations that people registered so many times that there is basically no doubt in their validity.

In general, another physical law, the law of conservation, implies (in one dimension) that $$ u_t+q_x=0, $$ where $u$ is your quantity and $q$ is the flux of this quantity. After this, playing with different fluxes, you get different equations:

1. Flux is proportional to $-u_x$ (this is called Fourier's law and your #3) implies the heat equation.
2. Flux is proportional to $u$ implies the linear transport equation (solution is a traveling wave and can be considered as an answer to your first question)
3. Flux is proportional to $u^2$ implies the Hopf equation.
4. In modeling chemotaxis one can put the flux is proportional to $1/v$ where $v$ is some other quantity. Moreover, if one takes the sign "-" then the movement will be to the quantity, and if "+" then from the quantity. Look it up, this may clarify your understanding.
5. Finally, your final idea of "the bigger the difference, the smaller the flow" seems to be totally correct, but of course other functional relations can be used. For instance, if you want to play with something like this numerically, you'd like to make sure that your denominator is never zero.