Vector and Tensor Analysis
 
Introduction
 
Transport Phenomena is the subject which deals with the movement of different physical quantities in any chemical or mechanical process and describes the basic principles and laws of transport. It also describes the relations and similarities among different types of transport that may occur in any system. Transport in a chemical or mechanical process can be classified into three types:
 
  • Momentum transport deals with the transport of momentum in fluids and is also known as fluid dynamics.
  • Energy transport deals with the transport of different forms of energy in a system and is also known as heat transfer.
  • Mass transport deals with the transport of various chemical species themselves.
 

Three different types of physical quantities are used in transport phenomena: scalars (e.g. temperature, pressure and concentration), vectors (e.g. velocity, momentum and force) and second order tensors (e.g. stress or momentum flux and velocity gradient). It is essential to have a primary knowledge of the mathematical operations of scalar, vector and tensor quantities for solving the problems of transport phenomena. In fact, the use of the indicial notation in cartesian coordinates will enable us to express the long formulae encountered in transport phenomena in a concise and compact fashion. In addition, any equation written in vector tensor form is equally valid in any coordinate system.

 

In this course, we will using the following notations for scalar, vector and tensor quantities:
Notations:
a,b,c
scalar quantities
vector quantities
2nd order tensor quantities

 

 
Cartesian coordinates and unit vectors
 
A xyz cartesian coordinate system may also be conventionally written as shown in Fig.1.1 below.
Fig. 1.1 3-dimensional cartesian coordinate system with unit vector

Here, and are the unit vectors in x, y and z direction respectively.
Tensor quantities
 

Most of us might have already encountered scalars and vectors in the study of high-school physics. It was pointed out that the vectors also have a direction associated with them along with a magnitude, whereas scalars only have a magnitude but no direction. Extending this definition, we can loosely define a 2nd order tensor as a physical quantity which has a magnitude and two different directions associated with it. To better understand, why we might need two different directions for specifying a particular physical quantity. Let us take the example of the stresses which may arise in a solid body, or in a fluid. Clearly, the stresses are associated with magnitude of forces, as well as with an area, whose direction is also need to be specified by the outward normal to the face of the area on which a particular force acting. Hence, we will require 32, i.e., 9 components to specify a stress completely in a 3 dimensional cartesian coordinate system. In general, an nth order tensor will be specified by 3n components (in a 3-dimensional system). However, the number of components alone cannot determine whether a physical quantity is a vector or a tensor. The additional requirement is that there should be some transformation rule for obtaining the corresponding tensors when we rotate the coordinate system about the origin. Thus, the tensor quantities can be defined by two essential conditions:

 
  • These quantities should have 3n components. According to this definition, scalar quantities are zero order tensors and have 30= 1 component. Vector quantities are first order tensors and have 31 = 3 components. Second order tensors have 32 = 9 components and third order tensors have 33 = 27 components. Third and higher order tensors are not used in transport phenomena, and are not dealt here.
  • The second necessary requirement of any tensor quantity is that it should follow some transformation rule.
 
Kronecker delta & Alternating Unit Tensor
 
There are two quantities which are quite useful in conveniently and concisely expressing several mathematical operations on tensors. These are the Kronecker delta and the alternating unit tensor.
 
Kronecker delta
 
Kronecker delta or Kronecker’s delta is a function of two index variables, usually integer, which is 1 if they are equal and 0 otherwise.

It is expressed as a symbol δij
δij=1, if i=j

δij=0, if i≠j
Thus, in three dimensions, we may also express the Kronecker delta in matrix form
 
 
Alternating unit tensor
 
The alternating unit tensor εijk is useful when expressing certain results in a compact form in index notation. It may be noted that the alternating unit tensor has three index and therefore 27 possible combinations but it is a scalar quantity .
 
  • εijk=0 if any two of indices i, j, k are equal. For example ε113,ε131,ε111,ε222=0
  • εijk=+1 when the indices i, j, k are different and are in cyclic order (123), For example ε123

  • εijk=-1 when the indices i, j, k are different and are in anti-cyclic order. For example ε321

 
Free indices and Dummy indices
 
Free indices

Free indices are the indices which occur only once in each tensor term. For example, i is the free index in following expression vij wj

In any tensorial equation, every term should have an equal number of free indices. For example, vij wj =cj dj is not a valid tensorial expression since the number of free indices (index i) is not equal in both terms.

Any free indices in a tensorial expression can be replaced by any other indices as long as this symbol has not already occurred in the expression. For example, Aij Bj= CiDjEj is equivalent to Akj Bj= CkDjEj.

 
The number of free indices in an equation gives the actual number of mathematical equations that will arise from it. For example, in equation Aij Bj= CiDjEj corresponds to 31 = 3 equations since there is only one free indices i. It may be noted that each indices can take value i=1, 2 or 3.
 
Dummy indices
Dummy indices are the indices that occur twice in a tensor term. For example, j is the dummy index in Aij Bj.

Any dummy index implies the summation of all components of that tensor term associated with each coordinate axis. Thus, when we write Aiδi, we actually imply .
 

Any dummy index in a tensor term can be replaced by any other symbol as long as this symbol has not already occurred in previous terms. For example, Aijkδjδk= Aipqδpδq.

 

Note: The dummy indices can be renamed in each term separately in a equations but free indices should be renamed for all terms in a tensor equations. For example, Aij Bj= CiDjEj can be replaced by Akp Bp= CkDjEj.

 
Here, i is the free index which has been replaced by k in both terms but j is a dummy index and can be replaced either in one term or both.
 
Summation convention in vector and tensor analysis
According to the summation convention rule, if k is a dummy index which repeats itself in a term then there should be a summation sign associated with it. Therefore, we can eliminate the implied summation sign and can write the expression in a more compact way. For example, using the summation convention
 
can be simply written as εijkεljk . Since j and k are repeating, there is no need to write summation sign over these indices