Powerful tool For Effective study and to Understand Flow Devices…… P M V Subbarao Professor Mechanical Engineering Department I I T Delhi Tensor Notation for Fluid Flow

Introductory Remarks The tensor Notation is a powerful tool that enables the reader to study and to understand more effectively the fundamentals of fluid mechanics. Enables to recall all conservation laws of fluid mechanics without memorizing any single equation. The quantities encountered in fluid dynamics are redefined as tensors.

Concept of Tensor Representation A physical quantity which has a definite magnitude but not a definite direction exhibits a zeroth-order tensor, which is a special category of tensors. A first-order tensor encompasses physical quantities with a definite magnitude with 3 components and a definite direction that can be decomposed in 3 directions. A second order tensor is a quantity, which has 9 definite components along 9 definite directions.

Definition of A Vector in Euclidian Space According to Einstein's summation convention, it can be written as:

Engineering Quantity as Tensor In 1823, the French mathematician, Augustin Baron Cauchy (1789– 1857) introduced the concept of stress by eliminating the difficulty that.σ is a function of two vectors, at the price that stress became a second- order tensor.

Actions inside a Differential Fluid Volume

Cartesian Fluid Element

Generic Representation Stress Tensor in Fluid Mechanics

Scalar Inventions turned into Tensors !!!!! Temperature is a scalar quantity. Heat flux is defined with direction and Magnitude : A Vector. Mathematically it is possible to have: Using the principles of vector calculus applied to Fourier Law of conduction:

Further Physical Description Why k must be same in all directions? Will k be same in all directions? Why k cannot be different each direction? Why k cannot be a vector? Will mathematics approve this ? What is the most general acceptable behavior of k, approved by both physics and mathematics?

Most General form of Fourier Law of Conduction We are at cross roads !!!!! If

Physically & mathematically Feasible Model Taking both physics and mathematics into consideration, the most feasible model for Fourier’s Law of conduction is: Thermal conductivity of a general material is a tensor.

Surprising Inventions !!!

Fire Resistant Wood Among the assessment properties of wood composite of structural members in building construction, fire performance is important and getting more attention nowadays. A new composite called molded carbon phenolic spheres (CPS), a mixture of sugi wood charcoal powders and phenol formaldehyde resin molded with a hot press is developed by a research group in Japan. The heat due to a fire accident should be thrown out fast outside the building.

Spread of Fire in A Room

Micro-structure of CPS

Thermal Performance

Einstein Notation : 1916 Range convention: Whenever a subscript appears only once in a term, the subscript takes all possible values. E.g. in 3D space: Summation convention: Whenever a subscript appears twice in the same term the repeated index is summed over the index parameter space. E.g. in 3D space:

Scalar Product : Work & Energy Scalar or dot product of two vectors results in a scalar quantity. Apply the Einstein's summation convention to work or energy scalars. Rearrange the unit vectors and the components separately:

Kronecker delta In Cartesian coordinate system, the scalar product of two unit vectors is called Kronecker delta, which is: Using the Kronecker delta,

Vector or Cross Product : Creation of Torque The vector product of two vectors is a vector that is perpendicular to the plane described by those two vectors. Apply the index notation With  ijk as the permutation symbol with the following definition

Using the above definition, the vector product is given by:

Scalar Triple product For every three vectors A, B and C, combination of dot and cross products gives

Non repeated subscripts Non repeated subscripts remain fixed during the summation. E.g. in 3D space one for each i = 1, 2, 3 and j is the dummy index.

Special Notes To avoid confusion between fixed and repeated indices or different repeated indices, etc, special notes are proposed. Note 1: No index can be repeated more than twice. Note 2: Number of free indices shows how many quantities are represented by a single term. Note 3: If the equation looks like this:

Tensors : Vectors of Second Order Second order vectors are more complex constructs. The three projections of this tensor, onto coordinate axes are obtained by inner product. These projections are vectors (not scalars!). In array form the three components of a tensor A are vectors denoted by

where each vector  i has three components therefore  is written in the matrix form