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U 3 = a 2 4 π R e r f c R 4 t / a 2 , t > 0 R = ( x - x o ) 2 + ( y - y o ) 2 + ( z - z o ) 2 1 / 2
The a 2 .factor has the units of time / L 2 . If time is made dimensionless with respect to a 2 / R o 2 and R with respect to R o , then the factor will disappear from the argument of the erfc.

Assignment 7.4

Plot the profiles of the response to a continuous source in 1, 2, and 3 dimensions using the MATLAB code contins.m and continf.m in the diffuse subdirectory. From the integral of the profiles as a function of time, determine the magnitude, spatial and time dependence of the source. Note: The exponential integral function, expint will give error messages for extreme values of the argument. It still computes the correct values of the function.

Convective-diffusion equation

The convective-diffusion equation in one dimension will be expressed in terms of velocity and dispersion,

u t + v u x = K 2 u x 2 u ( x , ) = 0 , x > 0 u ( 0 , t ) = 1 , t > 0

The independent variables can be transformed from ( x , t ) to a spatial coordinate that translates with the velocity of the wave in the absence of dispersion, ( y , t ) .

y = x - v t

This transforms the equation to the diffusion equation in the transformed coordinates.

u t = K 2 u y 2

To see this, we will transform the differentials from x to y .

y t = - v y x = 1

The total differentials expressed as a function of ( x , t ) or ( y , t ) are equal to each other.

d u = u t x d t + u x t d x d u = u t y d t + u y t d y

The total differentials expressed either way are equal. The partial derivatives in t and x can be expressed in terms of partial derivatives in t and y by equating the total differentials with either d t or d x equal to zero and dividing by the non-zero differential.

u t x = u t y + u y t y t x = u t y - v u y t and u x t = u y t y x t = u y t 2 u x 2 t = 2 u y 2 t

Substitution into the original equation results in the transformed equation. This result could have been derived in fewer steps by using the chain rule but would not have been as enlightening.

The boundary condition at x = 0 is now at changing values of y . We will seek an approximate solution that has the boundary condition u ( y - ) = 1 . A simple solution can be found for the following initial and boundary conditions.

u ( y , 0 ) = 1 , y < 0 1 / 2 , y = 0 0 , y > 0 u ( y - , t ) = 1 u ( y , t ) = 0

This system is a step with no dispersion at t = 0 . Dispersion occurs for t > 0 as the wave propagates through the system. The solution can be found with a similarity transform, which we will discuss later. For now, the approximate solution is given as

u = 1 2 erfc y 4 K t = 1 2 erfc x - v t 4 K t

The boundary condition at x = 0 will be approximately satisfied after a small time unless the Peclet number is very small.

Similarity transformation

In some cases a partial differential equation and its boundary conditions (and initial condition) can be transformed to an ordinary differential equation with boundary conditions by combining two independent variables into a single independent variable. We will illustrate the approach here with the diffusion equation. It will be used later for hyperbolic PDEs and for the boundary layer problems.

The method will be illustrated for the solution of the one-dimensional diffusion equation with the following initial and boundary conditions. The approach will follow that of the Hellums-Churchill method.

Questions & Answers

anyone know any internet site where one can find nanotechnology papers?
Damian Reply
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.
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
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
characteristics of micro business
for teaching engĺish at school how nano technology help us
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
what is fullerene does it is used to make bukky balls
Devang Reply
are you nano engineer ?
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.
what is the actual application of fullerenes nowadays?
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.
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.
is Bucky paper clear?
carbon nanotubes has various application in fuel cells membrane, current research on cancer drug,and in electronics MEMS and NEMS etc
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.
Do you know which machine is used to that process?
how to fabricate graphene ink ?
for screen printed electrodes ?
What is lattice structure?
s. Reply
of graphene you mean?
or in general
in general
Graphene has a hexagonal structure
On having this app for quite a bit time, Haven't realised there's a chat room in it.
what is biological synthesis of nanoparticles
Sanket Reply
what's the easiest and fastest way to the synthesize AgNP?
Damian Reply
types of nano material
abeetha Reply
I start with an easy one. carbon nanotubes woven into a long filament like a string
many many of nanotubes
what is the k.e before it land
what is the function of carbon nanotubes?
I'm interested in nanotube
what is nanomaterials​ and their applications of sensors.
Ramkumar Reply
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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