More Innovative Vector Actions animating but non-alive Fields ……

P M V SubbaraoProfessor

Mechanical Engineering Department

I I T Delhi

Dot to Cross Operations in Fluid Mechanics

Fluid Mechanical Interpretation of Divergence

Fluid Mechanical Interpretation of Divergence

Accounting for Mass Flux Thru a Differential Volume

A vector field whose divergence vanishes identically is called as solenoidal Field.

Non-Solenoid Steady Flows• Incompressible – A vector dominated…..

• Compressible – Both vector and scalar ….

1930’s Flying Story : Cruising at High Altitudes ?!?!?!

• Common engineers defined fast and slow only thru velocity.

• Aircraft were trying to approach high altitudes for a better fuel economy.

• This led to numerous crashes for unknown reasons. • These included:• The rapidly increasing forces on the various surfaces, which led

to the aircraft becoming difficult to control to the point where many suffered from powered flight into terrain when the pilot was unable to overcome the force on the control stick.

• The Mitsubishi Zero was infamous for this problem, and several attempts to fix it only made the problem worse.

• In the case of the Super-marine Spitfire, the wings suffered from low torsional stiffness.

• The P-38 Lightning suffered from a particularly dangerous interaction of the airflow between the wings and tail surfaces in the dive that made it difficult to "pull out“.

• Flutter due to the formation of thin high pressure line on curved surfaces was another major problem, which led most famously to the breakup of de Havilland Swallow and death of its pilot, Geoffrey de Havilland, Jr.

Divergence Theorem

• The divergence theorem says is that the expansion or contraction (divergence or convergence) of material inside a volume is equal to what goes out or comes in across the boundary.

• The divergence theorem is primarily used

– to convert a surface integral into a volume integral.

– to convert a volume integral to a surface integral.

Divergence (Gauss’s) theorem:

vsdvFsdF

..

Dot Product to Vector or Cross Product

• The vector product of two vectors is a vector that is perpendicular to the plane described by those two vectors.

sFW

The components of the torque vector are:

jjii reFerF

• The dot product of two vectors is a scalar, which was easily felt by every common engineer at early stages.

rF

Wild Solutions to Extrasomatism

Creation of Motive Power: Dangerous & Inhuman Technologies by copying

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.

•

The Family of Steam Engines

A Direct Hardware Creations due to the Cross Product…..

Watt's Double-Acting Engine, 1784

Further Use of Gradient for Human Welfare

V

V

•Assume we insert small paddle wheels in a flowing river.•The flow is higher close to the center and slower at the edges.•Therefore, a wheel close to the center (of a river) will not rotate since velocity of water is the same on both sides of the wheel.•Wheels close to the edges will rotate due to difference in velocities.•The curl operation determines the direction and the magnitude of rotation.

Flow Measurement in a River

Curl of a vector field:

Curl of a vector field:

Circulation is the amount of force that pushes along a closed boundary or path. It's the total "push" you get when going along a path, such as a circle.Curl is simply circulation per unit area, circulation density, or rate of rotation (amount of twisting at a single point)Curl is a vector field with magnitude equal to the maximum "circulation" at each point and oriented perpendicularly to this plane of circulation for each point. More precisely, the magnitude of curl is the limiting value of circulation per unit area.

VAd

dsnVVcurl

Ad

ˆ.lim

0