Tensor Analysis for Engineers: Transformations-Mathematics-Applications by Mehrzad Tabatabaian

Author:Mehrzad Tabatabaian [Mehrzad Tabatabaian]
Language: eng
Format: epub, mobi
Publisher: Mercury Learning and Information
Published: 2020-10-14T16:00:00+00:00

CHAPTER 13

COORDINATE INDEPENDENT GOVERNING EQUATIONS

Reliable mathematical models, also referred to as governing equations, are important tools for engineering analysis. A reliable and validated mathematical model of a physical phenomenon is a set of algebraic relations among quantities and their various derivatives, such as Newton’s 2nd law of motion, equilibrium equations for momentum flux, Fourier’s law of heat flux, Fick’s law of mass flux, Navier-Stokes equations for flow of fluids, Maxwell’s electromagnetic equations, etc. These governing equations, along with some fundamental principles (like the 2nd law of thermodynamics, conservations of energy, mass, electric charge, etc.) form the foundation of engineering science and its applications.

The quantities related to any physical phenomenon can be represented by tensors of different ranks stated in the relevant governing equations, such as force and acceleration vectors and mass of a body as a scalar, in Newton’s 2nd law of motion; gradient of temperature in Fourier’s law; divergence of velocity vector in fluid flow, etc. Sometimes, for analysis and design purposes, we need to have the relevant governing equations written in coordinate systems other than the Cartesian system. For example, for analyzing mechanical stresses in the wall of a cylindrical pressure vessel we prefer to choose the cylindrical polar coordinate system that is a natural fit to the shape of the vessel. This requirement has encouraged scientists and engineers to define various coordinate systems suitable for solving the governing equations practise. In other words, and in the context of tensor analysis, we would like to use the relations derived and discussed in the previous sections to write down the terms involved in well-known governing equations in engineering in a form that is general enough for application in an arbitrary but well-defined coordinate system.

In this section, we derive some new relations mostly used in engineering, in addition to those discussed in the previous sections. We hope that this helps readers in writing down similar coordinate independent terms involved in equations of their choice for their applications.

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