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?

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

4.1.3.22. 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