0.6 Solution of the partial differential equations  (Page 9/13)

 Page 9 / 13

The circle theorem. (Batchelor, 1967) The following result, known as the circle theorem (Milne-Thompson, 1940) concerns the complex potential representing the motion of an inviscid fluid of infinite extent in the presence of a single internal boundary of circular form. Suppose first that in the absence of the circular cylinder the complex potential is

${w}^{o}=f\left(z\right)$

and that $f\left(z\right)$ is free from singularities in the region $|z|\le a$ , where $a$ is a real length. If now a stationary circular cylinder of radius $a$ and center at the origin bounds the fluid internally, the flow is modified to the following complex potential:

${w}^{1}=f\left(z\right)+\overline{f}\left({a}^{2}/z\right)$

We show that the surface of the cylinder, $|z|=a$ , is a streamline.

${a}^{2}=z\phantom{\rule{0.166667em}{0ex}}\overline{z}$
$\begin{array}{ccc}\hfill {\left({w}^{1}\left(z\right)|}_{\left|z\right|=a}& =& {\left(f\left(z\right)|}_{\left|z\right|=a}+{\left(\overline{f}\left({a}^{2}/z\right)|}_{\left|z\right|=a}\hfill \\ \hfill \phantom{\rule{1.em}{0ex}}& =& {\left(f\left(z\right)|}_{\left|z\right|=a}+{\left(\overline{f}\left(z\overline{z}/z\right)|}_{\left|z\right|=a}\hfill \\ & =& {\left(f\left(z\right)|}_{\left|z\right|=a}+{\left(\overline{f}\left(\overline{z}\right)|}_{\left|z\right|=a}\hfill \\ & =& 2\phantom{\rule{0.166667em}{0ex}}\mathrm{Real}\left[{\left(f\left(z\right)|}_{\left|z\right|=a}\right]+i\phantom{\rule{0.166667em}{0ex}}0\hfill \\ & =& {\varphi }_{\left|z\right|=a}+i\phantom{\rule{0.277778em}{0ex}}{\psi }_{\left|z\right|=a}\hfill \end{array}$

A complex potential of this form thus has $|z|=a$ as a streamline, $\psi =0$ ; and it has the same singularities outside $|z|=a$ as $f\left(z\right)$ , since if $z$ lies outside $|z|=a$ , ${a}^{2}/z$ lies in the region inside this circle where $f\left(z\right)$ is known to be free from singularities. Consequently the additional term $\overline{f}\left({a}^{2}/z\right)$ in the equation represents fully the modification to the complex potential due to the presence of the circular cylinder. It should be noted that the complex potentials considered, both in the absence and in the presence of the circular cylinder, refer to the flow relative to axis such that the cylinder is stationary.

The simplest possible application of the circle theorem is to the case of a circular cylinder held fixed in a stream whose velocity at infinity is uniform with components $\left(-U,-V\right)$ . In the absence of the cylinder the complex potential is $-\left(U-i\phantom{\rule{0.166667em}{0ex}}V\right)z$ , it is singular at infinity and the circle theorem shows that, with the cylinder present,

$w\left(z\right)=-\left(U-i\phantom{\rule{0.166667em}{0ex}}V\right)z-\left(U+i\phantom{\rule{0.166667em}{0ex}}V\right)\phantom{\rule{0.166667em}{0ex}}{a}^{2}/z$

The potentials and streamlines for the steady translation of an inviscid fluid past a circular cylinder can be viewed with the MATLAB code circle.m .

Conformal Transformation (Batchelor, 1967). We now have the complex potential flow solutions of several problems with fairly simple boundary conditions. These solutions are analytical functions whose real and imaginary parts satisfy the Laplace equation. They also have a streamline that coincides with the boundary to satisfy the condition of zero flux across the boundary. Conformal transformations can be used to obtain solutions for boundaries that are transformed to different shapes. Suppose we have an analytical function $w\left(z\right)$ in the $z=x+iy$ plane. This solution can be transformed to the $\zeta =\xi +i\eta$ plane as another analytical function provided that the relation between these two planes, $\zeta =F\left(z\right)$ is an analytical function. This mapping is a connection between the shape of a curve in the $z$ - plane and the shape of the curve traced out by the corresponding set of points in the $\zeta -plane$ . The solution in the $\zeta -plane$ is analytical, i.e., its derivative defined, because the mapping, $\zeta =F\left(z\right)$ is an analytical function. The inverse transformation is also analytical.

$\begin{array}{c}\frac{dw\left(\varsigma \right)}{d\varsigma }=\frac{d\phantom{\rule{0.166667em}{0ex}}w\left(z\right)}{dz}\phantom{\rule{0.166667em}{0ex}}\frac{dz}{d\varsigma }=\frac{\frac{d\phantom{\rule{0.166667em}{0ex}}w\left(z\right)}{dz}}{\frac{dF\left(z\right)}{dz}}\hfill \\ \frac{d\phantom{\rule{0.166667em}{0ex}}w\left(z\right)}{dz}=\frac{dw\left(\varsigma \right)}{d\varsigma }\phantom{\rule{0.166667em}{0ex}}\frac{dF\left(z\right)}{dz}\hfill \end{array}$

$w\left(\zeta \right)$ is thus the complex potential of an irrotational flow in a certain region of the $\zeta$ -plane, and the flow in the $z$ -plane is said to been 'transformed' into flow in the $\zeta$ -plane. The family of equipotential lines and streamlines in the $z$ -plane given by $\varphi \left(x,y\right)=\text{const.}$ and $\psi \left(x,y\right)=\text{const.}$ transform into families of curves in the $\zeta$ -plane on which $\varphi$ and $\psi$ are constant and which are equipotential lines and streamlines in the $\zeta$ -plane. The two families are orthogonal in their respective plane, except at singular points of the transformation. The velocity components at a point of the flow in the $\zeta$ -plane are given by

are nano particles real
yeah
Joseph
Hello, if I study Physics teacher in bachelor, can I study Nanotechnology in master?
no can't
Lohitha
where we get a research paper on Nano chemistry....?
nanopartical of organic/inorganic / physical chemistry , pdf / thesis / review
Ali
what are the products of Nano chemistry?
There are lots of products of nano chemistry... Like nano coatings.....carbon fiber.. And lots of others..
learn
Even nanotechnology is pretty much all about chemistry... Its the chemistry on quantum or atomic level
learn
da
no nanotechnology is also a part of physics and maths it requires angle formulas and some pressure regarding concepts
Bhagvanji
hey
Giriraj
Preparation and Applications of Nanomaterial for Drug Delivery
revolt
da
Application of nanotechnology in medicine
has a lot of application modern world
Kamaluddeen
yes
narayan
what is variations in raman spectra for nanomaterials
ya I also want to know the raman spectra
Bhagvanji
I only see partial conversation and what's the question here!
what about nanotechnology for water purification
please someone correct me if I'm wrong but I think one can use nanoparticles, specially silver nanoparticles for water treatment.
Damian
yes that's correct
Professor
I think
Professor
Nasa has use it in the 60's, copper as water purification in the moon travel.
Alexandre
nanocopper obvius
Alexandre
what is the stm
is there industrial application of fullrenes. What is the method to prepare fullrene on large scale.?
Rafiq
industrial application...? mmm I think on the medical side as drug carrier, but you should go deeper on your research, I may be wrong
Damian
How we are making nano material?
what is a peer
What is meant by 'nano scale'?
What is STMs full form?
LITNING
scanning tunneling microscope
Sahil
how nano science is used for hydrophobicity
Santosh
Do u think that Graphene and Fullrene fiber can be used to make Air Plane body structure the lightest and strongest. Rafiq
Rafiq
what is differents between GO and RGO?
Mahi
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
Rafiq
if virus is killing to make ARTIFICIAL DNA OF GRAPHENE FOR KILLED THE VIRUS .THIS IS OUR ASSUMPTION
Anam
analytical skills graphene is prepared to kill any type viruses .
Anam
Any one who tell me about Preparation and application of Nanomaterial for drug Delivery
Hafiz
what is Nano technology ?
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?
what king of growth are you checking .?
Renato
Got questions? Join the online conversation and get instant answers!