4.1.3. Vector and Tensor Calculus¶

4.1.3.1. What can integral and differential equations describe in fluid mechanics?¶

Differential equations describe:

• A point or particle

Integral equations describe:

• A control volume and fluxes across control surfaces

4.1.3.2. What kind of information can integral and differential equations provide?¶

Differential equations are needed for:

• Field information, i.e. continuous problems

Integral equations are needed for:

• Average quantities, i.e. discontinuous problems

4.1.3.3. What is the degree of an equation?¶

The highest power, i.e. Equation (1) is 2nd degree, also called non-linear.

(1)${{\partial \over {\partial t}} (\rho u)} + {{\partial \over {\partial x}} (\rho u^2)} = -{{\partial p} \over {\partial x}} + \mu {{\partial^2 u} \over {\partial x^2}}$

If the highest power is 1, i.e. Equation (2) is 1st degree, also called linear.

(2)${{\partial u} \over {\partial t}} + c {{\partial u} \over {\partial x}}=0$

4.1.3.4. What is the order of an equation?¶

The highest derivative, i.e. Equation (3) is 2nd order.

(3)${{\partial u} \over {\partial t}} = {{\partial^2 u} \over {\partial x^2}}$

4.1.3.5. What is the difference between ODEs and PDEs?¶

ODEs:

• An equation relating a dependent variable (say x) to one independent variable (say u).
• Uses ordinary derivative notation e.g. $$dx / dt = a$$ and $$du / dt =0$$.
• Suitable for describing fluids in a moving frame of reference.

PDEs:

• An equation relating a dependent variable (say u) to more than one independent variable (say x and t).
• Uses partial derivative notation e.g. $$\partial u / \partial t + a (\partial u / \partial x) = 0$$.
• Suitable for describing fluids in a fixed frame of reference.

4.1.3.6. What is the difference between index and invariant notation?¶

Index notation:

• Coordinate system, e.g. Cartesian, Cylindrical, Spherical ($$x$$, $$y$$, $$z$$) or ($$x_1$$, $$x_2$$, $$x_3$$)

Invariant notation:

• Coordinate free system, e.g. vector notation (div, grad, curl)

4.1.3.7. What is Stokes’ Theorem?¶

Converts line integrals to surface integrals ($$\vec{u}$$ is continuous).

$\Gamma = \oint_C \vec{u} \cdot d \vec{C} = \int_S \nabla \times \vec{u} \cdot d \vec{S} = \int_S \vec{\omega} \cdot d \vec{S}$

(It’s a bit like the Gauss divergence theorem, it goes from a length to an area - Gauss divergence theorem goes from an area to a volume).

4.1.3.8. What is the gradient of a scalar field?¶

$\begin{split}\text{grad } u = \nabla u = \begin{bmatrix} \partial / {\partial x_1} \\ \partial / {\partial x_2} \\ \partial / {\partial x_3} \end{bmatrix} ^T u = {{\partial u} \over {\partial x_1}} \vec{i} + {{\partial u} \over {\partial x_2}} \vec{j} + {{\partial u} \over {\partial x_3}} \vec{k} = {{\partial u} \over {\partial x_j}}\vec{e}_i = {{\partial u_i} \over {\partial x_j}} = {{\partial u} \over {\partial x}} \vec{i} + {{\partial u} \over {\partial y}} \vec{j} + {{\partial u} \over {\partial z}} \vec{k}\end{split}$
• $$i$$ = principal direction
• $$j$$ = number of dimensions
• Always points in the direction of increasing u
• In the Navier-Stokes equations, $$\nabla \vec{u}$$ is sometimes used implying $$\vec{u}$$ is a row vector for each principal direction.

4.1.3.9. What is the gradient of a vector?¶

$\begin{split}\text{grad } \vec{u} = \nabla \vec{u} = \begin{bmatrix} \partial / {\partial x_1} \\ \partial / {\partial x_2} \\ \partial / {\partial x_3} \end{bmatrix} ^T \begin{bmatrix} u_1 \\ u_2 \\ u_3 \end{bmatrix} = {\sum_{j=1}^3} \nabla_j u_i = \begin{bmatrix} {{\partial u_1} \over {\partial x_1}} + {{\partial u_1} \over {\partial x_2}} + {{\partial u_1} \over {\partial x_3}}\\ {{\partial u_2} \over {\partial x_1}} + {{\partial u_2} \over {\partial x_2}} + {{\partial u_2} \over {\partial x_3}} \\ {{\partial u_3} \over {\partial x_1}} + {{\partial u_3} \over {\partial x_2}} + {{\partial u_3} \over {\partial x_3}} \end{bmatrix}\end{split}$

4.1.3.10. What is the curl of a vector field?¶

In 3D:

$\begin{split}\nabla \times \vec{u} = \begin{vmatrix} \vec{i} & \vec{j} & \vec{k} \\ \partial / {\partial x_1} & \partial / {\partial x_2} & \partial / {\partial x_3} \\ u_1 & u_2 & u_3 \end{vmatrix} = \left( {{\partial u_3} \over {\partial x_2}} - {{\partial u_2} \over {\partial x_3}} \right) \vec{i} + \left( {{\partial u_1} \over {\partial x_3}} - {{\partial u_3} \over {\partial x_1}} \right) \vec{j} + \left( {{\partial u_2} \over {\partial x_1}} - {{\partial u_1} \over {\partial x_2}} \right) \vec{k}\end{split}$

In 2D:

$\begin{split}\nabla \times \vec{u} = \begin{vmatrix} \partial / {\partial x_1} & \partial / {\partial x_2} \\ u_1 & u_2 \end{vmatrix} = {{\partial u_2} \over {\partial x_1}} - {{\partial u_1} \over {\partial x_2}}\end{split}$

Method in 3D:

• Cover row and column for each component $$i$$, $$j$$ and $$k$$
• Take determinant of $$2 \times 2$$ matrix

The physical meaning is the amount to which the vector field rotates.

4.1.3.11. What is the divergence of the Kronecker delta?¶

$\begin{split}\nabla_i \cdot \delta_{ij} = \begin{bmatrix} \partial / {\partial x_1} \\ \partial / {\partial x_2} \\ \partial / {\partial x_3} \end{bmatrix}^T \begin{bmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{bmatrix} = \begin{bmatrix} \partial / {\partial x_1} & \partial / {\partial x_2} & \partial / {\partial x_3} \end{bmatrix}\end{split}$

i.e.

$\nabla_i \cdot \delta_{ij} = \nabla_j$

or

${\partial \over {\partial x_i}} {\delta_{ij}} = {\partial \over {\partial x_j}}$

This operation converts a column vector ($$\nabla_i$$) into a row vector ($$\nabla_j$$).

4.1.3.12. What is the divergence of a vector field?¶

$\begin{split}\text{div } \vec{u} = \nabla \cdot \vec{u} = \begin{bmatrix} \partial / {\partial x_1} \\ \partial / {\partial x_2} \\ \partial / {\partial x_3} \end{bmatrix} ^T \cdot \begin{bmatrix} u_1 \\ u_2 \\ u_3 \end{bmatrix} = {{\partial u_1} \over {\partial x_1}} + {{\partial u_2} \over {\partial x_2}} + {{\partial u_3} \over {\partial x_3}} = {{\partial u_i} \over {\partial x_i}}\end{split}$
• The extent to which the vector field is a source (+ve) or sink (-ve)
• If $$div \vec{u} = 0$$ the vector field is divergence free

4.1.3.13. What is the Gauss divergence theorem?¶

$\int_S \phi (\vec{u} \cdot \vec{n}) dS = \int_V \nabla \cdot (\phi \vec{u}) dV$

Conditions for applicability:

• $$\phi$$ should be continuous
• $$\partial \phi / \partial x_i$$ should exist
• $$\partial \phi / \partial x_i$$ should be continuous

4.1.3.14. What is the Hamilton operator in Cartesian coordinates?¶

Del or Nabla or Hamilton operator:

$\begin{split}\nabla = \begin{bmatrix} \partial / {\partial x_1} \\ \partial / {\partial x_2} \\ \partial / {\partial x_3} \end{bmatrix} ^T \cdot \begin{bmatrix} \vec{i} \\ \vec{j} \\ \vec{k} \end{bmatrix}= {\partial \over {\partial x_1}} \vec{i} + {\partial \over {\partial x_2}} \vec{j} + {\partial \over {\partial x_3}} \vec{k} = {\partial \over {\partial x_j}} \vec{e}_j = {\partial \over {\partial x}} \vec{i} + {\partial \over {\partial y}} \vec{j} + {\partial \over {\partial z}} \vec{k}\end{split}$

4.1.3.15. What is the Laplacian operator?¶

$\begin{split}\nabla^2 = \nabla \cdot \nabla = \begin{bmatrix} \partial / {\partial x_1} \\ \partial / {\partial x_2} \\ \partial / {\partial x_3} \end{bmatrix} ^T \cdot \begin{bmatrix} \partial / {\partial x_1} \\ \partial / {\partial x_2} \\ \partial / {\partial x_3} \end{bmatrix}= {\partial^2 \over {\partial x_1^2}} + {\partial^2 \over {\partial x_2^2}} + {\partial^2 \over {\partial x_3^2}} = {{\partial^2} \over {\partial x_i^2}} = {\partial^2 \over {\partial x^2}} + {\partial^2 \over {\partial y^2}} + {\partial^2 \over {\partial z^2}}\end{split}$
• Some authors use $$\Delta$$ for the Laplacian

4.1.3.16. What is the Laplacian of a scalar field?¶

$\begin{split}\nabla^2 u = \begin{bmatrix} \partial / {\partial x_1} \\ \partial / {\partial x_2} \\ \partial / {\partial x_3} \end{bmatrix} ^T \cdot \begin{bmatrix} \partial / {\partial x_1} \\ \partial / {\partial x_2} \\ \partial / {\partial x_3} \end{bmatrix} u = {{\partial^2 u} \over {\partial x_1^2}} + {{\partial^2 u} \over {\partial x_2^2}} + {{\partial^2 u} \over {\partial x_3^2}} = {{\partial^2 u_i} \over {\partial x_j^2}}\end{split}$
• $$i$$ = principal direction
• $$j$$ = number of dimensions
• The extend to which the scalr field represents a source (+ve) or sink (-ve)
• In the Navier-Stokes equations, $$\nabla^2 \vec{u}$$ is sometimes used implying $$\vec{u}$$ is a row vector for each principal direction.

4.1.3.17. What is a rank 0, 1 and 2 tensor?¶

• Rank 0 = scalar e.g. $$p = p$$ (components in no direction)
• Rank 1 = vector e.g. $$\vec{u} = u_j$$ (components in one direction)
• Rank 2 = matrix e.g. $$\overset{\underset{\mathrm{\rightrightarrows}}{}}{\tau} = \tau_{ij}$$ (components in two directions)

Imagine a cube:

• $$i$$ = number of directions (columns)
• $$j$$ = principal direction (rows)
$p = p$
$\begin{split}u_j = \begin{bmatrix} \partial / {\partial u_1} \\ \partial / {\partial u_2} \\ \partial / {\partial u_3} \end{bmatrix} ^T\end{split}$
$\begin{split}\tau_{ij} = \begin{bmatrix} \tau_{11} & \tau_{12} & \tau_{13} \\ \tau_{21} & \tau_{22} & \tau_{23} \\ \tau_{31} & \tau_{32} & \tau_{33} \end{bmatrix}\end{split}$

4.1.3.18. What is the tensor product of two vectors?¶

$\begin{split}\overset{\underset{\mathrm{\rightrightarrows}}{}}{T} = \vec{u} \otimes \vec{u} = \begin{bmatrix} u_1^2 & u_1 u_2 & u_1 u_3 \\ u_2 u_1 & u_2^2 & u_2 u_3 \\ u_3 u_1 & u_3 u_2 & u_3^2 \end{bmatrix} = u_i u_j\end{split}$
• $$i$$ = row
• $$j$$ = column

4.1.3.19. What is the tensor product of the Hamliton operator and a vector?¶

$\begin{split}\overset{\underset{\mathrm{\rightrightarrows}}{}}{T} = \nabla \otimes \vec{u} = \begin{bmatrix} {\partial / \partial x} (u) & {\partial / \partial x} (v) & {\partial / \partial x} (w) \\ {\partial / \partial y} (u) & {\partial / \partial y} (v) & {\partial / \partial y} (w) \\ {\partial / \partial z} (u) & {\partial / \partial z} (v) & {\partial / \partial z} (w) \end{bmatrix} = \nabla_i u_j\end{split}$
• $$i$$ = row
• $$j$$ = column

4.1.3.20. What is the divergence of a rank 2 tensor?¶

$\begin{split}\vec{u} = \nabla \cdot \overset{\underset{\mathrm{\rightrightarrows}}{}}{T} = \begin{bmatrix} {\partial / \partial x_1} (T_{11}) + {\partial / \partial x_2} (T_{12}) + {\partial / \partial x_3} (T_{13}) \\ {\partial / \partial x_1} (T_{21}) + {\partial / \partial x_2} (T_{22}) + {\partial / \partial x_3} (T_{23}) \\ {\partial / \partial x_1} (T_{31}) + {\partial / \partial x_2} (T_{32}) + {\partial / \partial x_3} (T_{33}) \end{bmatrix} = {\partial \over {\partial x_j}} T_{ij}\end{split}$

4.1.3.21. What is the transpose of the tensor product of the Hamilton operator and a vector?¶

$(\nabla \otimes \vec{u})^T = \vec{u} \otimes \nabla$

4.1.3.22. What is the difference between a symmetric and non-symmetric tensor?¶

• Symmetric $$\longrightarrow$$ $$\vec{a} \otimes \vec{b} = (\vec{a} \otimes \vec{b})^T = \vec{b} \otimes \vec{a}$$
• Non-symmetric $$\longrightarrow$$ $$\nabla \otimes \vec{u} = (\nabla \otimes \vec{u})^T \ne \vec{u} \otimes \nabla$$

4.1.3.23. How do grad, div, curl and the tensor product affect the rank?¶

Symbol Name Meaning
$$\nabla_j p$$ Vector (row) Grad operation increases rank by 1
$$\nabla_j u_i$$ Matrix Grad operation increases rank by 1
$$\nabla_j \cdot u_j$$ Scalar Divergence operator reduces rank by 1
$$\nabla_j \cdot T_{ij}$$ Vector (row) Divergence operator reduces rank by 1
$$\nabla \times \vec{u}$$ Vector Curl operator maintains rank if $$\vec{u}$$ is 3D, or reduces rank by 1 if $$\vec{u}$$ is 2D
$$\nabla_i \otimes u_j$$ Matrix Tensor product increases rank by 1