Question

Which of these equations represent the semi-discretized equation of a 2-D steady-state diffusion problem?

(a) \(\int_A(\Gamma A\frac{\partial \phi}{\partial x})dA+\int_A(\Gamma A \frac{\partial\phi}{\partial y})  dA+\int_{\Delta V} S\,dV=0\)

(b) \(\int_A\frac{\partial}{\partial x}(\Gamma A \frac{\partial\phi}{\partial x})dA+\int_A\frac{\partial}{\partial y}(\Gamma A\frac{\partial\phi}{\partial y})dA+\int_{\Delta V}S\, dV=0\)

(c) \(\int_A(\Gamma A\frac{d\phi}{dx})dA+\int_A(\Gamma A \frac{d\phi}{dy})dA+\int_{\Delta V}S\, dV=0\)

(d) \(\frac{\partial \phi}{\partial t}+\int_A\frac{\partial}{\partial x}(\Gamma A \frac{\partial \phi}{\partial x})  dA+\int_A\frac{\partial}{\partial y}(\Gamma A \frac{\partial \phi}{\partial y})dA+\int_{\Delta V}S\, dV=0\)

This question was addressed to me during an online interview.

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

Answer 1

Right choice is (a) \(\int_A(\Gamma A\frac{\partial \phi}{\partial x})dA+\int_A(\Gamma A \frac{\partial\phi}{\partial y})  dA+\int_{\Delta V} S\,dV=0\)

Easy explanation: The general governing equation for a 2-D steady-state diffusion problem is given by

\(\frac{\partial}{\partial x}(\Gamma\frac{\partial \phi}{\partial x})+\frac{\partial}{\partial y}(\Gamma\frac{\partial \phi}{\partial y})+S=0\)

Here, partial differentiation is used as the variable φ  varies in both x and y directions, but the differentiation is only in the required direction.

Integrating the equation with respect to the control volume,

\(\int_{\delta V}\frac{\partial}{\partial x}(\Gamma\frac{\partial\phi}{\partial x})dV+\int_{\delta V}\frac{\partial}{\partial y}(\Gamma\frac{\partial \phi}{\partial y})dV+\int_{\Delta V} S \,dV=0\)

Applying Gauss Divergence theorem,

\(\int_A(\Gamma A\frac{\partial\phi}{\partial x})dA+\int_A(\Gamma A\frac{\partial\phi}{\partial y})dA+\int_{\Delta V}S \,dV=0\)

This is the semi-discretized form of the equation.