Note: You can make your own terms or use assumptions for your convenience.

It is well known that the thermal conductivity $(k)$ is a property of the fluid. You already know the concept and the definition of Fourier's Law and Newton's cooling law. In Newton's cooling law, the heat transfer coefficient $(h)$ would be most important parameter to determine the convection heat transfer rate, however the heat transfer coefficient is not a property of the fluid.

$(1)$ Explain the Fourier's Law and the Newton's Cooling Law with your plentiful opinion. $($Maybe equations would be needed.$)$

$(2)$ Near the wall, let's have a certain condition: heat would be transferred from the wall to the ambient. It means that the heat transfer mode would be changed from the solid to the fluid as a following figure. Explain why the heat transfer coefficient is not a property of the fluid through your derivation of heat balance equations.

$(1)$ Note: Consider where the criteria of conduction is$.$ Near the wall, the heat transfer mode would be ambiguous between the conduction and the convection.