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u t = K 2 u x 2 , t > 0 , x > 0 u ( x , 0 ) = u I C u ( 0 , t ) = u B C

The PDE, IC and BC are made dimensionless with respect to reference quantities.

u * = u - u I C u o t * = t t o x * = x x o u * t * = K t o x o 2 2 u * x * 2 u * ( x * , 0 ) = 0 u * ( 0 , t * ) = u B C - u I C u o = 1 , u o = u B C - u I C

There are three unspecified reference quantities and two dimensionless groups. The BC can be specified to equal unity. However, the system does not have a characteristic time or length scales to specify the dimensionless group in the PDE. This suggests that the system is over specified and the independent variables can be combined to specify the dimensionless group in the PDE to equal 1/4.

K t o x o 2 = 1 4 η = x 4 K t is dimensionless u ( x , t ) = u ( η )

The partial derivatives will now be expressed as a function of the derivatives of the transformed similarity variable.

η x = 1 4 K t η t = - η 2 t u t = d u d η η t = - η 2 t d u d η u x = 1 4 K t d u d η 2 u x 2 = 1 4 K t d 2 u d η 2

The PDE is now transformed into an ODE with two boundary conditions.

d 2 u * d η 2 + 2 η d u * d η = 0 u * ( η = 0 ) = 1 u * ( η ) = 0
Let v = d u * d η d v d η + 2 η v = 0 d v v = d ln v = - 2 η d η v = C 1 e - η 2 u * = C 1 0 η e - η 2 d η + C 2 u * ( η = 0 ) = 1 C 2 = 1 u * ( η ) = 0 = C 1 0 e - η 2 d η + 1 C 1 = - 1 0 e - η 2 d η u * ( η ) = - 0 η e - η 2 d η 0 e - η 2 d η + 1 = erfc ( η ) u * x , t = erfc x 4 K t

Therefore, we have a solution in terms of the combined similarity variable that is a solution of the PDE, BC, and IC.

Complex potential for irrotational flow

Incompressible, irrotational flows in two dimensions can be easily solved in two dimensions by the process of conformal mapping in the complex plane. First we will review the kinematic conditions that lead to the PDE and boundary conditions. Because the flow is irrotational, the velocity is the gradient of a velocity potential. Because the flow is solenoidal, the velocity is also the curl of a vector potential. Because the flow is two dimensional, the vector potential has only one non-zero component that is identified as the stream function. The kinematic condition at solid boundaries is that the normal component of velocity is zero. No condition is placed on the tangential component of velocity at solid surfaces because the fluid must be inviscid in order to be irrotational.

v = ϕ = ϕ x , ϕ y , 0 = v x , v y , v z v = 0 = 2 ϕ v = × A = ( A 3 y , - A 3 x , 0 ) = ψ y , - ψ x , 0 = v x , v y , v z × v = w = 0 = 2 ψ v = ϕ = × A ϕ x , ϕ y , 0 = ψ y , - ψ x , 0 or ϕ x = ψ y and ϕ y = - ψ x

Both functions are a solution of the Laplace equation, i.e., they are harmonic and the last pair of equations corresponds to the Cauchy-Riemann conditions if ϕ and ψ are the real and imaginary conjugate parts of a complex function, w ( z ) .

w ( z ) = ϕ ( z ) + i ψ ( z ) z = x + i y = r e i θ = r cos θ + i sin θ , r = z = x 2 + y 2 1 / 2 , θ = arctan y / x or ϕ ( z ) = real w ( z ) ψ ( z ) = imaginary w ( z )

The Cauchy-Riemann conditions are the necessary and sufficient condition for the derivative of a complex function to exist at a point z o , i.e., for it to be analytical . The necessary condition can be illustrated by equating the derivative of w ( z ) taken along the real and imaginary axis.

d w ( z ) d z = Re w ' ( z ) + i Im w ' ( z ) = lim δ ϕ + i δ ψ δ x + i 0 = ϕ x + i ψ x = lim δ ϕ + i δ ψ 0 + i δ y i i = ψ y - i ϕ y ϕ x = ψ y and ψ x = - ϕ y , Q . E . D .

Also, if the functions have second derivatives, the Cauchy-Riemann conditions imply that each function satisfies the Laplace equation.

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 .?
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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