Divergence operator for multi-dimensional neutron diffusion

[englisham]

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_aΦ(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!

 

[pasmith]

$\nabla$ follows the Liebnitz rule for products:
$$\nabla \cdot( f \nabla g) = (\nabla f) \cdot (\nabla g) + f \nabla^2 g$$

 

[englisham]

$\nabla$ follows the Liebnitz rule for products:
$$\nabla \cdot( f \nabla g) = (\nabla f) \cdot (\nabla g) + f \nabla^2 g$$

This is exactly the rule I needed, Thank you so much!
So to start:
-∇⋅D(x,y)∇Φ(x,y) [3]

expands to
- (∇⋅D(x,y))⋅(∇Φ(x,y)) - D(x,y)ΔΦ(x,y) [4]

then
- (∂D/∂x+∂D/∂y)⋅(∂Φ/∂x+∂Φ/∂y) - D(x,y)(∂²Φ/∂x²+∂²Φ/∂y²)] [5]

distributing the lefthand term, removing null terms, and reducing to simpler notation leaves
-[D_xΦ_x + D_yΦ_y + D⋅(Φ_xx + Φ_yy)] [6]

Combining with [1] and simplifying gives
- D⋅(Φ_xx + Φ_yy) - D_xΦ_x - D_yΦ_y + Σ_aΦ - S = 0 [7]

Does this look right?
Thanks!