Complex Mathematics is Due to Empiricism???

P M V SubbaraoProfessor

Mechanical Engineering Department

I I T Delhi

Mathematical View of Tensors in Fluid Mechanics

Tensors : Vectors of Second Order

• Second order vectors are more complex constructs.

tensor.a is

Let

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 321

ie ˆ

321 iii

where each vector i has three components

31

21

11

1

i

therefore is written in the matrix form

ij

333231

232221

131211

32

22

12

2

i

33

23

13

3

i

Pseudo Tensor Product• The tensor product is a product of two or more vectors,

where the unit vectors are not subject to scalar or vector operation.

• Consider the following tensor operation:

jijijjii BAeeBeAeBA

The result of this purely mathematical operation is a second order tensor with nine components:

The operation with any tensor such as the above second order one acquires a engineering use if it is multiplied with a vector (or another tensor).

Einstein's Notation for creation of a tensor

A tensor can be written as dyadic product.

Dyadic product of two vectors is a tensor such that

ba

jijiij baee

Dyadic product has the following properties

abba

bcacbac

Scalar product of Tensor with a vector

. bac

jijikk baeecebac . .

Rearranging the unit vectors and the components separately:

jikjik baceeebac . .

It is important to note that in the above equation, the unit vector ek must be scalar multiplied with the closest unit vector namely ei.

jikj bacebac ki . jiij bacebac .

The result of this operation is a vector with the same direction as vector b. jiij bacebac .

Variants of Scalar product of a vector & Tensor

Different results are obtained if the positions of the terms in a dot product of a vector with a tensor are reversed.

kkjiji cebaeecba . .

kjijki cbaecba .

jjii cbaecba .

kiik cbaecba .

Differential Operator

• The spatial differential operator, (nabla, del) which has a vector character.

• In Cartesian coordinate system, the operator nabla is defined as:

The material acceleration is:

3

32

21

1 xe

xe

xe ii

ii e

xe ,

332211 aeaeaeaea ii

j

iji

ii x

VVe

t

Vea

jijii

i VVet

Vea ,

Scalar Product of and V: Comma convention

• A subscript comma followed by an index indicates partial differentiation with respect to each coordinate.

• Summation and range conventions apply to indices following a comma as well. E.g. in 3D space:

3

3

2

2

1

1, x

V

x

V

x

V

x

VV

i

iii

jjj

i Vex

eV ..

j

iij x

VV

.

iijiij VVV ,,.

Vector Product ×V• This operation is called the rotation or curl of the velocity

vector v.

• Its result is a first-order tensor or a vector quantity. Using the index notation, the curl of v is written as:

jji

i Vex

eV

i

jkijk x

VeV

ijkijk VeV ,

Tensor Product of and V

• This operation is called the gradient of the velocity vector V.

• Its result is a second tensor.

• Using the index notation, the gradient of the vector v is written as:

jji

i Vex

eV

jji

i Vex

eV

i

jji x

VeeV

ijji VeeV ,

The components of Gradient of A vector

ijji VeeV ,

applied to a product of two or more vectors

• Using the Leibnitz's chain rule of differentiation:

FssFFssFsF

An Engineering Domination to Natural ExpertsFlying Faster

???????????????

Isn’t it very easy to do?????

A Research Area, where Group Efforts were Major

Failures……

The Wright brothers may be the most famous people in the history of aviation for the first aeroplane flight in 1903, but the first ever powered and controlled flights were carried out in lighter-than-air craft before either of the Wright brothers was even born.

History of Early Jet Engines

• Sir Isaac Newton in the 18th century was the first to theorize that a rearward-channeled explosion could propel a machine forward at a great rate of speed.

• This theory was based on his third law of motion.

• As the hot air blasts backwards through the nozzle the plane moves forward.

• Henri Giffard built an airship which was powered by the first aircraft engine, a three-horse power steam engine.

• It was very heavy, too heavy to fly.

Giffard’s Air Ship

Performance of Griffard’s Airship

• The airship successfully flew on the 24th September 1852, launching from the Paris Hippodrome and flying 27km (17 miles) to Elancourt, near Trappes.

• The small engine was not very powerful and it could not overcome the prevailing winds to allow Giffard to make the return flight.

• The top speed of Giffard's airship was just six miles per hour.

• However, he did manage to turn the airship in slow circles, proving that in calm conditions controlled flight was possible.

Hiram Maxim -- Triple Biplane

• In 1894, American Hiram Maxim tried to power his triple biplane with two coal fired steam engines.

• Maxim began his aerial experiments at Baldwyns Park, England, leading to the construction in 1893 of his enormous biplane Test-Rig, which weighed about 7,000 pounds.

• The machine's two steam engines each produced 180 h.p. and turned two pusher propellers each 17-1/2 feet in diameter.

• The device was intended to be a test vehicle it was held to a track, preventing it from rising more than a couple of feet.

Samuel Langley:Aerodrome

• Samuel Langley made a model airplanes that were powered by steam engines.

• In 1896, he was successful in flying an unmanned airplane with a steam-powered engine, called the Aerodrome.

• It flew about 1 mile before it ran out of steam.

• Otto in the late 1800's, invented the first gasoline engine.

• He then tried to build a full sized plane, the Aerodrome A, with a gas powered engine.

• In 1903, it crashed immediately after being launched from a house boat.

                                                      <>

The Great Grand Fathers of Modern Flights

Kate Carew Interviews the Wright Brothers• “Are you manufacturing any racing machines?”

• “Not just now, but we intend to.”

• “How much can I buy one for?”

• “Seven thousand five hundred-dollars.”

• “Is that all? It doesn’t seem like an outside price for a perfectly good airship?”

• “Airship!” shouted the Wright brothers indignantly.

• “Is that the wrong word?”

• “An airship,” said Wilbur contemptuously, “is a big, clumsy balloon filled with gas.”

• “Well, I don’t see why your biplane shouldn’t be called an airship, too.”

• “It’s a flying machine,” said Wilbur.

• “The name we prefer is ‘flyer,’” said Orville.

• “An airship would cost $50,000,” said Wilbur.

• “More like $150,000,” said Orville, and they argued the question.

Transformation of an Airship into a Flying Machine

http://www.wrightbrothers.org/History_Wing/Aviations_Attic/Carew_Interview/Carew_Interview.htm

Development of an Ultimate Fluid machine

Can We Identify the Cause?

Which is best for describing how aircraft get the needed lift to fly? Bernoulli's equation or Newton's laws and conservation of momentum? 

Actions inside a Differential Fluid Volume

1F

3F

2F

Cartesian Fluid Element

Scalar Product of and a Second Order Tensor