📢Point heat steady versus transient

The three-dimensional delta function, $\delta^{(3)}(\vec{x})$, is a mathematical tool defined by its integral property, where integrating over a volume $V$ yields a result of 1 only if the origin ($\vec{x}=0$) is contained within $V$, and 0 otherwise. This function is utilized in heat conduction to describe the concentration of an extensive quantity, such as a heat source, precisely located at a single point $\vec{x}_0$, where the source term $\kappa$ is given by $\kappa(\vec{x})=K \delta^{(3)}\left(\vec{x}-\vec{x}_0\right)$ for a source releasing heat at rate $K$. Although using the delta function is typically a simplification, it yields reasonably good results when the source size is indistinguishable from a point relative to measurement precision. Furthermore, the simulation of these point heat sources reveals two fundamentally different physical behaviors: if the source is continuous (steady state), the system achieves equilibrium and the temperature distribution near the source is classically inversely proportional; however, if the source is an instantaneous pulse (unsteady state), the entire system remains transient as the heat spreads outwards as a thermal wave, causing the temperature at every point to rise, peak, and subsequently decay back to the ambient zero level as the heat fully dissipates.

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Point heat steady versus transient | IllustraDemoviadean on Notion