Application of tensor theory in engineering science
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Tensor theory is extremely useful in advanced engineering theory. It is used to help describe or model many natural phenomena such as: physical forces, potential fields, particle or control element motion, wave propagation, etc.
Constructions notes:
- Ai'j'k' = xi'i xj'j ykk' Aijk
Specific examples are:
- Navier-Stokes equations Presented in partial differential equation form.
- Vorticity is an important quantity in various research, modeling and design calculations regarding lift, drag, and propulsion. It is a tensor quantity defined as: insert gif here when available.
- dynamics of systems of rigid (assumed incompressible) bodies and particles
- stress and strain within elastic bodies
- Tensor equations to model fluid flow can be derived as follows:
- Assume the fluid consists of particles which can be individually tracked as they move in relation to Euclidean 3-space. Thus an individual particle can be tracked as it moves.
- We shall use rectangular cartesian coordinates to describe our Euclidean 3 space .... zr
- In the Lagrangian method, all particles are then described by:
Equation (1) zr=zr(a,t) where a stands for the set of 3 labels representing the 3 dimensions or axis of Euclidean space ... xi,xj,xk.