- #1

englisham

- 2

- 1

## Homework Statement

[1] is the one-speed steady-state neutron diffusion equation, where D is the diffusion coefficient, Φ is the neutron flux, Σ

_{a}is the neutron absorption cross-section, and S is an external neutron source. Solving this equation using a 'homogeneous' material allows D to be moved in front of the del, leaving a Laplacian [2]. This equation is easily expanded. In the two-dimensional heterogeneous case, the del operates on this coefficient D(x,y). [3] is the form only including the 'streaming' term. Can anyone help me expand this equation? I am applying a Finite Element discretization and will need to integrate over a rectangular spatial domain with a weighted residual to apply my boundary conditions.

## Homework Equations

-∇⋅D(

**r**)∇Φ(

**r**) + Σ

_{a}(

**r**)Φ(

**r**) = S(

**r**) [1]

-DΔΦ(

**r**) + Σ

_{a}Φ(

**r**) = S(

**r**) [2]

-∇⋅D(x,y)∇Φ(x,y) [3]

## The Attempt at a Solution

-∇⋅D(x,y)∇Φ(x,y) = -∇⋅[D(x,y)(∂Φ/∂x+∂Φ/∂x)] = ?

I cannot figure out the application of the del in this form. Any help or direction to resources is appreciated greatly!