<< Chapter < Page Chapter >> Page >
2 ϕ x 2 + 2 ϕ y 2 = 0 2 ψ x 2 + 2 ψ y 2 = 0

The Cauchy-Riemann conditions also imply that the gradient of the velocity potential and the stream function are orthogonal.

ϕ ψ = ϕ x ψ x + ϕ y ψ y = 0

If the gradients are orthogonal then the equipotential lines and the streamlines are also orthogonal, with the exception of stagnation points where the velocity is zero.

Since the derivative

d w d z = lim δ z 0 δ w δ z

is independent of the direction of the differential δ z in the ( x , y ) -plane, we may imagine the limit to be taken with δ z remaining parallel to the x-axis ( δ z = δ x ) giving

d w d z = ϕ x + i ψ x = v x - i v y

Now choosing z to be parallel to the y-axis ( δ z = i δ y ) ,

d w d z = 1 i ϕ y + ψ y = - i v y + v x = v x - i v y

These equations are a restatement that an analytical function has a derivative defined in the complex plane. Moreover, we see that the real part of w ' ( z ) is equal to v x and the imaginary part of w ' ( z ) is equal to - v y . If v is written for the magnitude of v and for the angle between the direction of v and the x -axis, the expression for d w / d z becomes

d w d z = v x - i v y = v e - i θ or v x = real d w d z v y = - imaginary d w d z

Flow Fields. The simplest flow field that we can imagine is just a constant translation, w = ( U - i V ) z where U and V are real constants. The components of the velocity vector can be determined from the differential.

w ( z ) = ( U - i V ) z = ( U - i V ) ( x + i y ) = U x + V y + i ( - V x + U y ) = ϕ + i ψ d w d z = ( U - i V ) = v x - i v y v x = U , v y = V ϕ = U x + V y , ψ = - V x + U y

Another simple function that is analytical with the exception at the origin is

w ( z ) = A z n = A r n e i n θ = A r n cos n θ + i A r n sin n θ = ϕ + i ψ thus ϕ = A r n cos n θ ψ = A r n sin n θ d w d z = n A z n - 1 = v x - i v y

where A and n are real constants. The boundary condition at stationary solid surfaces for irrotational flow is that the normal component of velocity is zero or the surface coincides with a streamline. The expression above for the stream function is zero for all r when θ = 0 and when θ = π / n . Thus these equations describe the flow between these boundaries are illustrated below.

Fig. 6.5.1 (Batchelor, 1967) Irrotational flow in the region between two straight zero-flux boundaries intersecting at an angle π / n .

Earlier we discussed the Green's function solution of a line source in two dimensions. The same solution can be found in the complex domain. A function that is analytical everywhere except the singularity at z o is the function for a line source of strength m .

w ( z ) = m 2 π ln ( z - z o ) , line source dw dz = m 2 π 1 ( z - z o ) = v x - i v y

This results can be generalized to multiple line sources or sinks by superposition of solutions. A special case is that of a source and sink of the same magnitude.

w ( z ) = i m i 2 π ln ( z - z i ) , multiple line sources w ( z ) = m 2 π ln z - z o z + z o , source - sink pair

The above flow fields can be viewed with the MATLAB code corner.m, linesource.m, and multiple.m in the complex subdirectory.

Assignment 7.5

Line Source Solution For z o at the origin, derive expressions for the flow potential, stream function, components of velocity, and magnitude of velocity for the solution to the line source in terms of r and θ . Plot the flow potentials and stream functions. Compute and plot the flow potentials and stream function for the superposition of multiple line sources corresponding to the zero flux boundary conditions at y = + 1 and - 1 of the earlier assignment.

Questions & Answers

How we are making nano material?
LITNING Reply
what is a peer
LITNING Reply
What is meant by 'nano scale'?
LITNING Reply
What is STMs full form?
LITNING
scanning tunneling microscope
Sahil
what is Nano technology ?
Bob Reply
write examples of Nano molecule?
Bob
The nanotechnology is as new science, to scale nanometric
brayan
nanotechnology is the study, desing, synthesis, manipulation and application of materials and functional systems through control of matter at nanoscale
Damian
Is there any normative that regulates the use of silver nanoparticles?
Damian Reply
what king of growth are you checking .?
Renato
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
?
Kyle
yes I'm doing my masters in nanotechnology, we are being studying all these domains as well..
Adin
why?
Adin
what school?
Kyle
biomolecules are e building blocks of every organics and inorganic materials.
Joe
anyone know any internet site where one can find nanotechnology papers?
Damian Reply
research.net
kanaga
sciencedirect big data base
Ernesto
Introduction about quantum dots in nanotechnology
Praveena Reply
what does nano mean?
Anassong Reply
nano basically means 10^(-9). nanometer is a unit to measure length.
Bharti
do you think it's worthwhile in the long term to study the effects and possibilities of nanotechnology on viral treatment?
Damian Reply
absolutely yes
Daniel
how to know photocatalytic properties of tio2 nanoparticles...what to do now
Akash Reply
it is a goid question and i want to know the answer as well
Maciej
characteristics of micro business
Abigail
for teaching engĺish at school how nano technology help us
Anassong
How can I make nanorobot?
Lily
Do somebody tell me a best nano engineering book for beginners?
s. Reply
there is no specific books for beginners but there is book called principle of nanotechnology
NANO
how can I make nanorobot?
Lily
what is fullerene does it is used to make bukky balls
Devang Reply
are you nano engineer ?
s.
fullerene is a bucky ball aka Carbon 60 molecule. It was name by the architect Fuller. He design the geodesic dome. it resembles a soccer ball.
Tarell
what is the actual application of fullerenes nowadays?
Damian
That is a great question Damian. best way to answer that question is to Google it. there are hundreds of applications for buck minister fullerenes, from medical to aerospace. you can also find plenty of research papers that will give you great detail on the potential applications of fullerenes.
Tarell
what is the Synthesis, properties,and applications of carbon nano chemistry
Abhijith Reply
Mostly, they use nano carbon for electronics and for materials to be strengthened.
Virgil
is Bucky paper clear?
CYNTHIA
carbon nanotubes has various application in fuel cells membrane, current research on cancer drug,and in electronics MEMS and NEMS etc
NANO
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
Got questions? Join the online conversation and get instant answers!
Jobilize.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