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v ξ - i v η = d w d ς = d w d z d z d ς = v x - i v y d z d ς

This shows that the magnitude of the velocity is changed, in the transformation from the z -plane to the ζ -plane, by the reciprocal of the factor by which linear dimensions of small figures are changed. Thus the kinetic energy of the fluid contained within a closed curve in the z -plane is equal to the kinetic energy of the corresponding flow in the region enclosed by the corresponding in the ζ -plane.

Flow around elliptic cylinder (Batchelor, 1967). The transformation of the region outside of an ellipse in the z -plane into the region outside a circle in the ζ -plane is given by

z = ς + λ 2 ς ς = 1 2 z + 1 2 z 2 - 4 λ 2 1 / 2

where λ is a real constant so that

x = ξ 1 + λ 2 ς 2 , y = η 1 - λ 2 ς 2

This converts a circle of radius c with center at the origin in the - plane into the ellipse

x 2 a 2 + y 2 b 2 = 1

in the z -plane, where

λ = 1 2 a 2 - b 2 1 / 2

If the ellipse is mapped into a circle in the ζ -plane, it is convenient to use polar coordinates ( r , θ ) , especially since the boundary corresponds to a constant radius. The radius that maps to the elliptical boundary is ( ellipse.m in the complex directory)

r o = 1 2 log a + b a - b

The transformation from the polar coordinates to the z - plane is defined by

z = 2 λ cosh ω where ω = r + i θ
Transformation of cylindrical polar coordinates into orthogonal, elliptical coordinates

The polar coordinates ( r , θ ) , transform to an orthogonal set of curves which are confocal ellipses and conjugate hyperbolae.

This transformation can be substituted into the complex potential expression for the flow of a fluid past a circular cylinder.

w = - 1 2 a + b U - i V e ω - r o + U + i V e r o - ω

It is convenient to write - for the angle which the direction of motion of the flow at infinity makes with the x - axis so that

U + i V = U 2 + V 2 1 / 2 e - i α

The complex potential now becomes

w = - U 2 + V 2 1 / 2 a + b cosh ω - r o + i α

The corresponding velocity potential and stream function are

ϕ = - U 2 + V 2 1 / 2 a + b cosh r - r o cos θ + α ψ = - U 2 + V 2 1 / 2 a + b sinh r - r o sin θ + α

The velocity potentials and streamlines are illustrated below for flow past an elliptical cylinder ( fellipse.m in the complex directory). Note the stagnation streamlines on either side of the body. These two stagnation points are regions of maximum pressure and result in a torque on the body. Which way will it rotate?

Flow past an ellipse of an inviscid fluid that is in steady translation at infinity.

Pressure distribution. When an object is in a flow field, one may wish to determine the force exerted by the fluid on the object, or the 'drag' on the object. Since the flow field discussed here has assumed an inviscid fluid, it is not possible to determine the viscous drag or skin friction directly from the flow field. It is possible to determine the 'form drag' from the normal stress or pressure distribution around the object. However, one must be critical to determine if the calculated flow field is physically realistic or if some important phenomena such as boundary layer separation may occur but is not allowed in the complex potential solution.

The Bernoulli theorems give the relation between the magnitude of velocity and pressure. We have assumed irrotational, incompressible flow. If in addition we assume the body force can be neglected, then the quantity, H , must be constant along a streamline.

Questions & Answers

what are the products of Nano chemistry?
Maira Reply
There are lots of products of nano chemistry... Like nano coatings.....carbon fiber.. And lots of others..
Even nanotechnology is pretty much all about chemistry... Its the chemistry on quantum or atomic level
Preparation and Applications of Nanomaterial for Drug Delivery
Hafiz Reply
Application of nanotechnology in medicine
what is variations in raman spectra for nanomaterials
Jyoti Reply
I only see partial conversation and what's the question here!
Crow Reply
what about nanotechnology for water purification
RAW Reply
please someone correct me if I'm wrong but I think one can use nanoparticles, specially silver nanoparticles for water treatment.
yes that's correct
I think
what is the stm
Brian Reply
is there industrial application of fullrenes. What is the method to prepare fullrene on large scale.?
industrial application...? mmm I think on the medical side as drug carrier, but you should go deeper on your research, I may be wrong
How we are making nano material?
what is a peer
What is meant by 'nano scale'?
What is STMs full form?
scanning tunneling microscope
how nano science is used for hydrophobicity
Do u think that Graphene and Fullrene fiber can be used to make Air Plane body structure the lightest and strongest. Rafiq
what is differents between GO and RGO?
what is simplest way to understand the applications of nano robots used to detect the cancer affected cell of human body.? How this robot is carried to required site of body cell.? what will be the carrier material and how can be detected that correct delivery of drug is done Rafiq
analytical skills graphene is prepared to kill any type viruses .
Any one who tell me about Preparation and application of Nanomaterial for drug Delivery
what is Nano technology ?
Bob Reply
write examples of Nano molecule?
The nanotechnology is as new science, to scale nanometric
nanotechnology is the study, desing, synthesis, manipulation and application of materials and functional systems through control of matter at nanoscale
Is there any normative that regulates the use of silver nanoparticles?
Damian Reply
what king of growth are you checking .?
What fields keep nano created devices from performing or assimulating ? Magnetic fields ? Are do they assimilate ?
Stoney Reply
why we need to study biomolecules, molecular biology in nanotechnology?
Adin Reply
yes I'm doing my masters in nanotechnology, we are being studying all these domains as well..
what school?
biomolecules are e building blocks of every organics and inorganic materials.
anyone know any internet site where one can find nanotechnology papers?
Damian Reply
sciencedirect big data base
how did you get the value of 2000N.What calculations are needed to arrive at it
Smarajit Reply
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Source:  OpenStax, Transport phenomena. OpenStax CNX. May 24, 2010 Download for free at http://cnx.org/content/col11205/1.1
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