Fluid Mechanics for MAP/Scalar, Vectors and Tensors

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MotivationEdit

Each scientific field needs an optimised language. Without it, this field can not develope. For example, calculus could not develop with Roman numbers.


 


Without arabic numbers and the decimal system, even the basic operations would not be possible.

Fluid Mechanics deals with the changes of flow and fluid variables in three space dimensions and in time. In order to tackle with this complexity, there is also a need for an optimized language in the field of fluid mechanics. Unfortunately, for the same vector operation there are multiple mathematical notations:

 

Symbols are not unique and, somewhat, confusing, i.e. unified language is beneficial to speed up the development in this field. In this chapter, we will be particularly introduced to such unified mathematical concept, Tensor which is very important for computational fluid dynamics and also many other field of science. This chapter will give a gentle introduction about fascinating Tensor but readers are encouraged to read more from the references.Due to this short introduction,Tensor will not be used any other chapter.


TensorEdit

The concept of tensor could be easily understood if we want to imagine a function which takes vector as input and do some operation with it and then give another vector as output. For example, let's assume a vector which is defined in Euclidean reference co-ordinate system. Now for some unknown reason, the co-ordinate system changed via rotation or translation. To find our previous vector defined in the new co ordinate system, we can assume a magical operator or function which will take the initial components of the vector in old co-ordinate system as input and give us back the new vector component in new co-ordinate system as output. In other way, New position vector= f(Old position vector). It is found that if this operator or function is linear, then this function or operator would be a 2nd rank tensor. A second rank tensor looks like   matrix. For example,

  is a   matrix. If we now multiply this matrix with another matrix with   matrix we will get as an output,   matrix where the element of these matrices could be imagined as the vector components of initial and new vector.
So we can say that a 2nd rank Tensor is capable to keep the map or information of how a vector will change between new to old co-ordinate system. Now, we have small introduction to Tensor, we can present the definition of tensor. A tensor   is a linear transformation from a vector space   to  .[1] Thus, we can write

 

which means tensor A operates on vector u to give new vector v.

This concept is found to be extremely useful where all computational calculations have to be performed in somehow matrix format. A very nice illustration of use of tensor is shown here[2]. Very common examples of Tensor are : Deformation Gradient Tensor, Stress Tensor, Einstein's famous stress-energy tensor. Even dot and cross product of two vectors are also tensor.

Although the concept of linear operation is very useful to understand the application of tensor at first time,it's not all. Let's consider about a cylinder which is stretched along it's length and due to poisson's contraction, we will see contraction in radial direction to keep the volume constant. So we can say that the deformation along different directions is somehow related on the stress applied along different directions.It appears that this relationship is a tensor which is called Stiffness Tensor and this relationship forms like :[Strain Tensor(written in column vector and reduced form)]=[Stiffness Tensor,C][Stress Tensor(same as Strain Tensor)]. The relationship for isotropic material will look like below:
 

This above matrix looks differently for an anisotropic material[3]. So in other way, we can say this particular tensor can help to express how the Stress Tensor will be influencing Strain Tensor and this 'how' is a signature of type of material which is point of interest. Tensor could be envisioned in more easy way. Let us consider a vector which has components of  . This components are essential to express this vector precisely in a given co-ordinate system.These components together will give value of the vector by  . If this co-ordinate system change also, the vector will be represented with 3 components with different value   but   would give the same value as previous which means that something remains as identity of the vector. The components changed keeping something constant. Now lets consider about tensor. Imagine we form an entity where we have 3 vectors as components(meaning actually it has 9 or   components of vectors now) and they are somehow coupled.This means that when the co-ordinate system will change, the individual components of vector(hence the entity) will be determined keeping 'some other value' constant like   in the case of vector. Of course, this 'other value' has to be calculated by using 9 components of that entity. After transformation, this new 9 components will give the same 'other value'. It would be shown later on actually that vector itself is 1st order tensor.

Scalar: Tensor of 0th orderEdit

We may associate some properties of fluid or flow at one point without any preference of direction.


 


Those are examples of such fluid variables.

Summation can be only done between variables having the same unit, i.e.


 


 


Multiplication is valid for variables having different units.


 


Products of variables having different units, have a new physical meaning.


 


 


Power loss (Energy loss per unit time)


 
Pressure drop measurement in a pipe


 


 


 

Watt is the unit for Power.

Vectors: 1st order tensorsEdit

 
A vector

Variables of flow, associated with a point and a direction, like force and velocity, are vectors.

The direction of a vector should be specified in relation with respect to a given frame of reference. This frame of reference is arbitrary as the units. Let us use the Cartesian coordinate system having 3 mutually orthogonal axes.   are components of this vector and can differ with respect to selected coordinate system but the amplitude not. Components allow us to reconstruct the vector in a particular system of reference. One should distinguish vector as an entity from its components.



 
Components of a vector

Repeated indices in SumsEdit

Summation of variables, identified with indices can be shortly written with the summation symbol fro series:

 

In short, we will write

 

In other words, repeated indices means series sum over this index. Thus,

 
 

Now we can introduce the component representation of a vector  :

 

Let's look some examples:

 

whereas

 

indicates no summation over   and   and

 

indicates also no summation. Other examples of summation are:

 
 

Dummy and free indexEdit

In the following expression

 

the repeating index   is also a dummy index, i.e. can be replaced by  . The following equality

 

is valid only when  . Then,   is called free index

  for  

can be expanded as

 

 

 

Einstein summation conventionEdit

Any expression involving a repeated index (sub- or superscript) shall automatically considered for its sum over the index range  .

Remark:

  • All indices have the same range,  , unless stated otherwise.
  • No idex appear more than twice in any given expresion.

Double sumEdit

Sumed over i:

 


Sumed over j:

 

SubstitutionEdit

Let   and  .


Then  . Here we find more than two repeated indices.


  Find duplicated dumy index:  .


  Change dummy index:  .


  Substitude:  .

Kronecker DeltaEdit

 


 


 


 


 


  replaces one of the dummy index with the other one.

Scalar MultiplicationEdit

Remember that,

 


 


 


Addition of VectorsEdit

 


 


  • Commutative:
 


  • Associative:
 

Scalar ProductEdit

 


If m and n are unit vectors in the direction of   and  .

 


 


 


 


Vector ProductEdit

 
Vector product of two vectors

Consider vector products of unit vectors.

 


 


 


 


The symmetric determinant will be the same:

 



Permutation symbolEdit

The permutation symbol is defined as:

 

Hence, the vector product can be written as:

 

Triple Scalar ProductEdit

 

Tensor ProductEdit

Product of two tensors

 

where

 


Similarly,

 


Thus, generally, the product of a tensor having   order and another tensor having   order will result in a tensor having an order of m + n.

Therefore,

 


The order is increased by two.


Substitution

 


 


Consequently if a tensor has a pair of repeated indices, its order reduces by two. And it can be shown that:

 


 


Index j is replaced by index i. This opperation is called substitution. Generally:

 


Contraction

 

The total order is reduced by four.

Or equally:

 


The order of   is reduced by two.

This operation is called contraction with respect to i and j.


Scalar product of tensors

Remember:

 


Contraction of a tensor product of two   order tensors results in scalar product of two vectors.

Hence, the scalar product of two tensors are:

 


Remember:

 


 


Consider,

 


In conclusion, using component notation and basic rules the form of the tensor can be deduced.

Important Relations

 


 


 

(repeated index)


 


 


Contraction with respect to k n.

 


Twice contraction,

 


 


 


and

 

Tripple Vector ProductEdit

 
Triple vector product

Therefore, it can be a linear sum of two vectors.

 


Use component notation.

 


 


 


 


 


Derivative of a TensorEdit

Let A scalar derivative with respect to  :

 


Let A a vector:

 


Generally, the   derivative of a tensor of the   order is a tensor of the   order.

Special case,

 


 


 


Let  :

 


   :


 

ReferenceEdit