Does heat flow always in the direction of maximum decrease?

I was wondering if at a given gradient temperature heat could flow in any direction satisfying moving from a higher to a lower temperature. Or it can only flow in the direction of gradient? Consider a point $$(x_0,y_0,z_0)$$ where $$T_0=T(x_0,y_0,z_0)$$ and $$T_1$$, $$T_2$$ are the temperatures of two points very close to the $$(x_0,y_0,z_0)$$. It is true that $$T_0>T_1>T_2$$. Would some amount of heat flow to the point with $$T_1$$ and another amount to the point with $$T_2$$ or all the heat would flow to the point with the maximum temperature difference? What if $$T_1=T_2$$?

• In your example, some energy would flow to $T_1$, and some would flow to $T_2$. If $T_1=T_2$, then equal amounts of energy would flow to each point (assuming 1 dimension or isotropic thermal conductivity). Can you please state your question more clearly? – Drew Apr 12 '20 at 21:04
• I think this wikipedia article will help en.wikipedia.org/wiki/Fick%27s_laws_of_diffusion – Ofek Gillon Apr 13 '20 at 12:00
• I asked this because Fourier law is written in terms of the gradient $$\overrightarrow{q}=-k\nabla T$$ and was thinking if the amount of heat that lefts from a point would be distributed over the local points with lower temperatures or will only flow to the local point of lowest temperature (according to gradient)? – Antonios Sarikas Apr 15 '20 at 14:18

In your example, if $$T_1 = T_2$$ points $$1$$ and $$2$$ are at the same level surface, and there is no flow between that points.
One can also think about it in the following way. At point $$0$$, the temperature is higher than at point $$1$$ and $$2$$. What it means is that the molecules of the medium at point $$0$$ are "jiggling" more rapidly than the molecules at points $$1$$ and $$2$$. These rapidly moving molecules at point $$0$$ will exchange momentum with neighboring molecules, which are "jiggling" little slower, making them "jiggle" little faster. This "jiggling" will "propagate" towards point $$1$$ as well as towards point $$2$$, the difference is that "the rate of propagation" of this "jiggling" $$(\propto \nabla T)$$ towards point $$2$$ will be larger than it is towards point $$1$$ because $$T_0>T_1>T_2$$. The "propagation of jiggling" is called heat transfer and "the rate of propagation" is related to the heat flux $$\mathbf{q}$$. Of course to calculate "total amount of heat transfer", one has to consider where the points $$1$$ and $$2$$ lie relative to point $$0$$, are they on the same line? Is point $$1$$ closer to point $$0$$ than the point $$2$$? How much time has elapsed? Are the points maintained at constant temperature? The situation may become even more complicated if the medium is a fluid! Hope this gives the idea and probably now you can guess what happens if $$T_1=T_2$$.