Mathematical Background

This section on mathematical background covers the various notations and operators used to formulate and define the equations of fluid flow.

Vector or Dyadic Notation

The conservation laws for a continuum medium involve vector and tensor quantities as well as several operators such as gradient and divergence.

In order to have a comprehensive understanding of these equations it is essential to get a good grasp of the notations and operators used in this manual.

In this notation, the governing equation is independent of the choice of the coordinate system. For the purposes of this manual the scalar quantities would be denoted with italicized letters. For example, pressure field is denoted as $p$ . The vector quantities would be denoted with an arrow above the letters, for example velocity field denoted as $\vec{u}$ . Tensor quantities are donated with bold face letters, for example stress tensor represented as $τ$ .

Cartesian Tensor Notation

In this notation, an index subscript is written after the quantity which corresponds to a component of the quantity.

Single subscript is used to denote a component of vector, for example $u_{i}$ for the vector field $\vec{u}$ and two subscripts are used to denote a component of tensor, for example $τ_{i j}$ for the stress tensor $τ$ .

Kroneker Delta

The Kroneker Delta is a second-order isotropic tensor which is defined as: $δ_{i j} = 1 for i = j$ , $δ_{i j} = 0 for i \neq j$ .

Operators

The most frequently used math operator related to flow equations is the $\nabla$ referred to as nabla, grad or del.

The nabla operator operates on the quantity to the right of it and the rules of a derivative of a product still hold. Otherwise the nabla operator behaves like any other vector in an algebraic operation. It is a vector operator and for a Cartesian coordinate system, it is defined as (1)

\nabla = \frac{\partial}{\partial x} \hat{i} + \frac{\partial}{\partial y} \hat{j} + \frac{\partial}{\partial z} \hat{k}

$\nabla$ appears in several different ways when applied to scalar, vector and tensor quantities.

Gradient

When the nabla operator is applied on a scalar quantity

φ

, it is called Gradient and it gives a vector whose components are the partial derivatives. (2)

\nabla φ = \frac{\partial φ}{\partial x} \hat{i} + \frac{\partial φ}{\partial y} \hat{j} + \frac{\partial φ}{\partial z} \hat{k}

The gradient of

φ

points to the direction of greatest change of

φ

and has a magnitude equal to the rate of change of

φ

with respect to distance in that direction.

When gradient is applied to a tensor quantity it produces a tensor one rank higher. When applied to a vector quantity

\vec{u}

it gives a second-order tensor given by (3)

\nabla (\vec{u}) = (\frac{\partial}{\partial x} \hat{i} + \frac{\partial}{\partial y} \hat{j} + \frac{\partial}{\partial z} \hat{k}) (u \hat{i} + v \hat{j} + w \hat{k}) \equiv (\begin{matrix} \frac{\partial u}{\partial x} & \frac{\partial u}{\partial y} & \frac{\partial u}{\partial z} \\ \frac{\partial v}{\partial x} & \frac{\partial v}{\partial y} & \frac{\partial v}{\partial z} \\ \frac{\partial w}{\partial x} & \frac{\partial w}{\partial y} & \frac{\partial w}{\partial z} \end{matrix})

The gradient of a vector quantity represents the gradient of each component of the vector field individually, each of which is a scalar.

Divergence

The divergence of a vector quantity

\vec{u}

is defined as a scalar quantity given by (4)

\nabla \cdot \vec{u} = \frac{\partial u}{\partial x} + \frac{\partial v}{\partial y} + \frac{\partial w}{\partial z}

The divergence of a vector field represents the flux generation per unit volume (flux density) at each point on the field.

Figure 2. Divergence at a Point

The divergence of a tensor gives a tensor which is one rank lower. When applied to a second-order stress tensor

τ

it yields a vector field whose ith component is expressed as (5)

{(\nabla \cdot τ)}_{i} = \frac{\partial τ_{i j}}{\partial x_{j}}

Div

The div operator is defined as

\vec{u} \cdot \nabla

and is expressed as (6)

u \frac{\partial}{\partial x} + v \frac{\partial}{\partial y} + w \frac{\partial}{\partial z}

It maps a vector quantity to a scalar which can then be applied to a scalar, vector or higher order tensors.

Laplacian

The Laplacian of a scalar quantity

φ

is a scalar defined as (7)

\nabla^{2} φ = \frac{\partial^{2} φ}{\partial x^{2}} + \frac{\partial^{2} φ}{\partial y^{2}} + \frac{\partial^{2} φ}{\partial z^{2}}

Double Dot Product (Scalar Product of Two Tensors)

The double dot product is a doubly contracted product between two tensors. It is expressed as (8)

A : B = A_{i j} B_{j i}

Total Derivative

The total derivative, also known as material derivative or substantial derivative, describes the rate of change of a physical quantity of a fluid element with time as it moves along a trajectory in a velocity field. The total derivative of a scalar quantity

φ

is a scalar defined as (9)

\frac{D φ}{D t} = \frac{\partial φ}{\partial t} + \vec{u} \cdot \nabla φ

The total derivative of a quantity can be seen as a sum of local time derivative and convective derivative.