Foundations of heat transfer in engineering: Fourier's law applies specifically to which mode of heat transfer in solids and stationary fluids?

Introduction / Context:
Heat can move by three fundamental modes: conduction, convection, and radiation. Each mode has its own governing law and constitutive relation. This question tests recognition of which mode is described by Fourier's law, a cornerstone in thermal analysis of solids and quiescent media.

Given Data / Assumptions:

• Steady or unsteady heat flow in a continuum medium.
• Local thermodynamic equilibrium so that temperature is well-defined.
• No bulk fluid motion for the constitutive relation itself (i.e., not a convective transport law).

Concept / Approach:
Fourier's law states that the heat flux vector q is proportional to the negative temperature gradient: q = −k * grad(T). The thermal conductivity k is a material property (possibly temperature-dependent). This law describes conduction only. Convection heat transfer is described by Newton's cooling law (q' = h * ΔT), while radiation follows the Stefan–Boltzmann law and view-factor relations (q' = εσ(T_s^4 − T_sur^4)).

Step-by-Step Solution:
Identify the law: Fourier's law connects heat flux to temperature gradient via conductivity.Recognize that it does not include a convective heat-transfer coefficient h or radiative terms like σ, ε.Therefore, it characterizes conduction in solids and stationary fluids.Select 'Conduction' as the only correct option.

Verification / Alternative check:
In a plane wall, one-dimensional steady conduction reduces Fourier's law to q' = k * (T_hot − T_cold) / L, matching textbook derivations. Neither Newton's law of cooling nor Stefan–Boltzmann relations can produce this form without additional assumptions.

Why Other Options Are Wrong:
Convection depends on fluid motion and boundary layers; it is summarized by Newton's law, not Fourier's.

Radiation depends on T^4 behavior and surface exchange; Fourier's law contains no emissivity or radiative terms.

'All of these' is incorrect because Fourier's law is not the governing relation for convection or radiation.

Common Pitfalls:

• Confusing conduction within a fluid with convective transport across a boundary layer.
• Assuming Fourier's law applies to composite mechanisms without superposition rules.

Final Answer:
Conduction