<< Chapter < Page Chapter >> Page >
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

Do somebody tell me a best nano engineering book for beginners?
s. Reply
what is fullerene does it is used to make bukky balls
Devang Reply
are you nano engineer ?
s.
what is the Synthesis, properties,and applications of carbon nano chemistry
Abhijith Reply
so some one know about replacing silicon atom with phosphorous in semiconductors device?
s. Reply
Yeah, it is a pain to say the least. You basically have to heat the substarte up to around 1000 degrees celcius then pass phosphene gas over top of it, which is explosive and toxic by the way, under very low pressure.
Harper
how to fabricate graphene ink ?
SUYASH Reply
for screen printed electrodes ?
SUYASH
What is lattice structure?
s. Reply
of graphene you mean?
Ebrahim
or in general
Ebrahim
in general
s.
Graphene has a hexagonal structure
tahir
On having this app for quite a bit time, Haven't realised there's a chat room in it.
Cied
what is biological synthesis of nanoparticles
Sanket Reply
what's the easiest and fastest way to the synthesize AgNP?
Damian Reply
China
Cied
types of nano material
abeetha Reply
I start with an easy one. carbon nanotubes woven into a long filament like a string
Porter
many many of nanotubes
Porter
what is the k.e before it land
Yasmin
what is the function of carbon nanotubes?
Cesar
I'm interested in nanotube
Uday
what is nanomaterials​ and their applications of sensors.
Ramkumar Reply
what is nano technology
Sravani Reply
what is system testing?
AMJAD
preparation of nanomaterial
Victor Reply
Yes, Nanotechnology has a very fast field of applications and their is always something new to do with it...
Himanshu Reply
good afternoon madam
AMJAD
what is system testing
AMJAD
what is the application of nanotechnology?
Stotaw
In this morden time nanotechnology used in many field . 1-Electronics-manufacturad IC ,RAM,MRAM,solar panel etc 2-Helth and Medical-Nanomedicine,Drug Dilivery for cancer treatment etc 3- Atomobile -MEMS, Coating on car etc. and may other field for details you can check at Google
Azam
anybody can imagine what will be happen after 100 years from now in nano tech world
Prasenjit
after 100 year this will be not nanotechnology maybe this technology name will be change . maybe aftet 100 year . we work on electron lable practically about its properties and behaviour by the different instruments
Azam
name doesn't matter , whatever it will be change... I'm taking about effect on circumstances of the microscopic world
Prasenjit
how hard could it be to apply nanotechnology against viral infections such HIV or Ebola?
Damian
silver nanoparticles could handle the job?
Damian
not now but maybe in future only AgNP maybe any other nanomaterials
Azam
Hello
Uday
I'm interested in Nanotube
Uday
this technology will not going on for the long time , so I'm thinking about femtotechnology 10^-15
Prasenjit
can nanotechnology change the direction of the face of the world
Prasenjit Reply
At high concentrations (>0.01 M), the relation between absorptivity coefficient and absorbance is no longer linear. This is due to the electrostatic interactions between the quantum dots in close proximity. If the concentration of the solution is high, another effect that is seen is the scattering of light from the large number of quantum dots. This assumption only works at low concentrations of the analyte. Presence of stray light.
Ali Reply
how did you get the value of 2000N.What calculations are needed to arrive at it
Smarajit Reply
Privacy Information Security Software Version 1.1a
Good
Berger describes sociologists as concerned with
Mueller Reply
Got questions? Join the online conversation and get instant answers!
QuizOver.com Reply

Get the best Algebra and trigonometry course in your pocket!





Source:  OpenStax, Transport phenomena. OpenStax CNX. May 24, 2010 Download for free at http://cnx.org/content/col11205/1.1
Google Play and the Google Play logo are trademarks of Google Inc.

Notification Switch

Would you like to follow the 'Transport phenomena' conversation and receive update notifications?

Ask