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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

are nano particles real
Missy Reply
Hello, if I study Physics teacher in bachelor, can I study Nanotechnology in master?
Lale Reply
no can't
where we get a research paper on Nano chemistry....?
Maira Reply
nanopartical of organic/inorganic / physical chemistry , pdf / thesis / review
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
no nanotechnology is also a part of physics and maths it requires angle formulas and some pressure regarding concepts
Preparation and Applications of Nanomaterial for Drug Delivery
Hafiz Reply
Application of nanotechnology in medicine
has a lot of application modern world
what is variations in raman spectra for nanomaterials
Jyoti Reply
ya I also want to know the raman spectra
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
Nasa has use it in the 60's, copper as water purification in the moon travel.
nanocopper obvius
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 .?
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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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