Appendix

The purpose of this appendix is to present the essential and comprehensive elements required to derive the governing equations for the motion of incompressible Newtonian fluids within general orthogonal coordinate systems.

1. Lower- and upper-case letters are used to represent that the quantities are described on the Cartesian and the general orthogonal coordinate systems, respectively.

2. I do not use the Einstein summation rule throughout this document for clarity; the summation symbol \(\sum\) always precedes when the indices are to be summed.

3. The subscripts are only used to indicate that the quantity is covariant; they are not used to represent exponents.

Prerequisite

• Basis vector
• Transformation
• Christoffel symbol
• Change of basis vector
• Nabla operator

Equation

• Divergence of velocity
• Material derivative
• Strain-rate tensor
• Divergence of tensor
• Energy diffusion and dissipation