Question

If aPΦP=aEΦP+aWΦW+aNΦN+aSΦS+S is the general form of a 2-D steady-state diffusion problem, what is aE by considering the following stencil?

(a) \(\frac{\Gamma_E A_E}{\delta y_{PE}}\)

(b) \(\frac{\Gamma_E A_E}{\delta y_{PE}}\)

(c) \(\frac{\Gamma_E A_E}{\delta x_{PE}}\)

(d) \(\frac{\Gamma_E A_E}{\delta x_{WP}}\)

The question was asked in unit test.

My doubt is from FVM for Multi-dimensional Steady State Diffusion in section Diffusion Problem of Computational Fluid Dynamics

Answer 1

Right answer is (c) \(\frac{\Gamma_E A_E}{\delta x_{PE}}\)

Explanation: Flux in the eastern direction is given by

\(\Gamma_E A_E\frac{\partial\phi}{\partial x}\Big|_e=\Gamma_E A_E\frac{(\phi _E-\phi _P)}{\delta x_{PE}}\)

\(\Gamma_E A_E\frac{\partial\phi}{\partial x}\Big|_e=\Gamma _e A_E\frac{\phi_E}{\delta x_{PE}}-\Gamma_E a_E\frac{\phi_P}{\delta x_{PE}}\)

Expanding this while forming the general equation, we will get

\(a_E=\frac{\Gamma_E A_E}{\delta x_{PE}}\).