Question

Which of these equations represent 1-D steady state diffusion?

(a) div(Γ grad Φ)+S=0

(b) \(\frac{d}{dx}(\Gamma\frac{d\phi}{dx})+S=0\)

(c) \(\frac{d\phi}{dt}+\frac{d}{dx}(\Gamma\frac{d\phi}{dx})+S=0\)

(d) \(\frac{d\phi}{dt}+div(\Gamma  grad\phi)+S=0\)

This question was posed to me during an online interview.

The question is from FVM for 1-D Steady State Diffusion topic in chapter Diffusion Problem of Computational Fluid Dynamics

Answer 1

The correct option is (b) \(\frac{d}{dx}(\Gamma\frac{d\phi}{dx})+S=0\)

Best explanation: The term div(Γ grad Φ) represents diffusion in all three directions. One-dimensional diffusion is given by the equation \(\frac{d}{dx}(\Gamma\frac{d\phi}{dx}.\frac{d\phi}{dt})\) is the transient term. So, this should not be present in the steady-state equation. Considering all these, the correct equation is

\(\frac{d}{dx}(\Gamma\frac{d\phi}{dx}+S)=0\).