Notation#

This part of the documentation presents notations for all multipy quantities.

Indexing#

  • \(i\) denotes that a quantity is related to an \(i^{th}\) component of the mixture.

  • \(n\) represents the number of components (species) of the mixture.

Bases and reference frames#

Multicomponent quantities can be written in various bases (e.g. mass or molar) and in reference to different average velocity of the mixture (e.g. mass-averaged or molar-averged). To keep track of a basis and a reference frame related to mixture velocity, we adopted the following notation in naming functions or parameters:

../_images/notation-reference-frames.svg

As an example, Flux.diffusive_mass_molar should be interpreted as a mass diffusive flux relative to a molar-averaged velocity.

Shape of a general multicomponent quantity matrix#

We assume that a matrix \(\pmb{\phi}_i \in \mathcal{R}^{n \times N}\) describes any multicomponent quantity, where \(n\) is the number of components (species) in the mixture and \(N\) is the number of observations of that quantity (for instance spatial positions). Mixture components are thus stored in rows and their observations are stored in columns:

../_images/notation-matrix-shape.svg

Vector calculus primer#

Gradient#

For a scalar function \(f\), the gradient of \(f\) is denoted:

\[\nabla f\]

In three dimensions we can compute this as:

\[\begin{split}\begin{gather} f \begin{bmatrix} \frac{\partial}{\partial x} \\ \frac{\partial}{\partial y} \\ \frac{\partial}{\partial z} \\ \end{bmatrix} = \begin{bmatrix} \frac{\partial f}{\partial x} \\ \frac{\partial f}{\partial y} \\ \frac{\partial f}{\partial z} \\ \end{bmatrix} \end{gather}\end{split}\]

This results in a vector.

For a vector function \(\mathbf{f}\), the gradient of \(\mathbf{f}\) is denoted:

\[\nabla \mathbf{f}\]

In three dimensions we can compute this as:

\[\begin{split}\begin{gather} \begin{bmatrix} \frac{\partial}{\partial x} \\ \frac{\partial}{\partial y} \\ \frac{\partial}{\partial z} \\ \end{bmatrix} \cdot \begin{bmatrix} f_x & f_y & f_z \\ \end{bmatrix} = \begin{bmatrix} \frac{\partial f_x}{\partial x} & \frac{\partial f_y}{\partial x} & \frac{\partial f_z}{\partial x}\\ \frac{\partial f_x}{\partial y} & \frac{\partial f_y}{\partial y} & \frac{\partial f_z}{\partial y}\\ \frac{\partial f_x}{\partial z} & \frac{\partial f_y}{\partial z} & \frac{\partial f_z}{\partial z}\\ \end{bmatrix} \end{gather}\end{split}\]

This results in a tensor.

Divergence#

For a vector function \(\mathbf{f}\), the divergence of \(\mathbf{f}\) is denoted:

\[\nabla \cdot \mathbf{f}\]

In three dimensions we can compute this as:

\[\begin{split}\begin{gather} \begin{bmatrix} \frac{\partial}{\partial x} & \frac{\partial}{\partial y} & \frac{\partial}{\partial z} \end{bmatrix} \cdot \begin{bmatrix} f_x \\ f_y \\ f_z \\ \end{bmatrix} = \frac{\partial f_x}{\partial x} + \frac{\partial f_y}{\partial y} + \frac{\partial f_z}{\partial z} \end{gather}\end{split}\]

This results in a scalar.

Outer product#

The outer product between matrices A and B can be computed using numpy as:

numpy.outer(A,B)

Tensor contraction#

For two tensors, \(\mathbf{A}\) and \(\mathbf{B}\), tensor contraction (scalar product) is denoted:

\[\mathbf{A} : \mathbf{B}\]

The tensor contraction (scalar product) between matrices A and B can be computed using numpy as:

numpy.tensordot(A,B,axes=2)

which achieves the same thing as:

numpy.sum(numpy.multiply(A,B))

Various forms of the divergence theorem#

  • For a scalar field \(\phi\): \(\int_{S(t)} \phi \mathbf{a} dS = \int_{V(t)} \nabla \phi dV\)

  • For a vector field \(\mathbf{q}\): \(\int_{S(t)} \mathbf{q} \cdot \mathbf{a} dS = \int_{V(t)} \nabla \cdot \mathbf{q} dV\)

  • For a tensor field \(\pmb{\tau}\): \(\int_{S(t)} \pmb{\tau} \cdot \mathbf{a} dS = \int_{V(t)} \nabla \cdot \pmb{\tau} dV\)