# Divergence operator for multi-dimensional neutron diffusion

• englisham
In summary, the equation [3] only includes 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.
englisham

## Homework Statement

[1] is the one-speed steady-state neutron diffusion equation, where D is the diffusion coefficient, Φ is the neutron flux, Σa 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) + Σa(r)Φ(r) = S(r) [1]
-DΔΦ(r) + ΣaΦ(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!

dRic2
$\nabla$ follows the Liebnitz rule for products:
$$\nabla \cdot( f \nabla g) = (\nabla f) \cdot (\nabla g) + f \nabla^2 g$$

pasmith said:
$\nabla$ 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
-[DxΦx + DyΦy + D⋅(Φxx + Φyy)] [6]

Combining with [1] and simplifying gives
- D⋅(Φxx + Φyy) - DxΦx - DyΦy + ΣaΦ - S = 0 [7]

Does this look right?
Thanks!

## 1. What is the divergence operator for multi-dimensional neutron diffusion?

The divergence operator for multi-dimensional neutron diffusion is a mathematical operation used in the field of nuclear engineering to describe the rate of change of neutron flux in a nuclear reactor. It is represented by the symbol ∇∙ and is defined as the sum of the partial derivatives of the neutron flux with respect to each spatial dimension.

## 2. How is the divergence operator related to the neutron diffusion equation?

The neutron diffusion equation is a partial differential equation that describes the behavior of neutrons in a nuclear reactor. The divergence operator is used in this equation to represent the source and sink terms that affect the neutron flux, such as neutron production and absorption. In other words, the divergence operator helps to quantify the change in neutron flux due to these sources and sinks.

## 3. Can the divergence operator be applied to any number of dimensions?

Yes, the divergence operator can be applied to any number of dimensions. In nuclear engineering, it is commonly used in three dimensions to model the behavior of neutrons in a reactor core. However, it can also be applied to two or even one dimension to simplify the problem if necessary.

## 4. How is the divergence operator used in practical applications?

The divergence operator is used in practical applications to solve the neutron diffusion equation and predict the behavior of neutrons in a nuclear reactor. It is also used in the design of nuclear reactors, as it helps engineers to optimize the reactor core and ensure safe and efficient operation.

## 5. What other fields use the divergence operator?

The divergence operator is widely used in various fields of science and engineering, including fluid dynamics, electromagnetics, and quantum mechanics. It is a fundamental concept in vector calculus and is essential for understanding and modeling physical systems with multiple dimensions.

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