Powerful tool For Effective study and to Understand Flow Devices……

P M V SubbaraoProfessor

Mechanical Engineering Department

I I T Delhi

Vector Notation for Viscous Fluid Flow

Importance

• The tensor analysis is a powerful tool that enables the reader to study and to understand more effectively the fundamentals of fluid mechanics.

• Enables to derive all conservation laws of fluid mechanics without memorizing any single equation.

• The quantities encountered in fluid dynamics are tensors.

• 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 and 9 definite directions.

A Vector in Euclidian Space

3

1332211 ˆˆˆˆ

iiieAeAeAeAA

According to Einstein's summation convention, it can be written as: iieA ˆ

Einstein Notation

Range convention: Whenever a subscript appears only once in a term, the subscript takes all possible values. E.g. in 3D space:

zyxorxxxixi ,, ,,3,2,1 321

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:

3,2,1 332211

3

1

ibababababai

iiii

Scalar Product : The Base of Civilization

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.

WsF

. WseFe iiii

Rearrange the unit vectors and the components separately:

WsFee iiii

Kronecker delta

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

. 0

; 1

jiforee

jiforee

jiij

jiij

Using the Kronecker delta,

WsFsFee jiijiiii

Thomas Savery

• As an English army officer, Thomas Savery was once ejected from the Lord of the Admiralty's office as a lunatic because he proposed a ship that could be propelled by side-mounted wheels rather than by wind or oars.

• He exhibited great fondness for mechanics, and for mathematicians natural philosophy and gave much time to experimenting, to the contriving of various kinds of apparatus, and to invention.

• July 2, 1698, patented the design of the first engine which had the most important advance in actual construction.

•

Watt's Double-Acting Engine, 1784

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.

rF

Apply the index notation

jikijkjjii rFereFerF

With ijk as the permutation symbol with the following definition

kikjjiijk or for 0

npermutatio cyclicfor 1ijk

npermutatio anticyclicfor 1ijk

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

jikijkkk rFee

Non repeated subscripts

Non repeated subscripts remain fixed during the summation. E.g. in 3D space equations. threedenotes ˆ jiji exa

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:

summed.not are indces the,ˆii xu