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Second order PDE. The classification of second order PDEs as elliptic, parabolic, and hyperbolic arise from a transformation of the independent variables. The classification apply to quasilinear (i.e., linear in the highest order derivatives) but we will only discuss linear equations with constant coefficients here. Numerical solutions are needed for quasilinear systems. Again let u denote the dependent variables and t , x , y , z as the independent variables. Examples of the different classes of equations are

0 = 2 u x 2 + 2 u y 2 + 2 u z 2 + ρ = 2 u + ρ , elliptic equation u t = 2 u x 2 + 2 u y 2 + 2 u z 2 + ρ = 2 u + ρ , parabolic equation 2 u t 2 = 2 u x 2 + 2 u y 2 + 2 u z 2 + ρ = 2 u + ρ , hyperbolic equatio n

The ρ term represents sources. When the cgs system of units is used in electrostatics and ρ is the charge density, the source is expressed as 4 π ρ . The factor 4 π is absent with the mks or SI system of units. The parabolic PDEs are sometimes called the diffusion equation or heat equation. In the limit of steady-state conditions, the parabolic equations reduce to elliptic equations. The hyperbolic PDEs are sometimes called the wave equation. A pair of first order conservation equations can be transformed into a second order hyperbolic equation.

Convective-diffusion equation. The above equations represented convection without diffusion or diffusion without convection. When both the first and second spatial derivatives are present, the equation is called the convection-diffusion equation.

u t + u x = 1 N P e 2 u x 2

Usually a dimensionless group such as the Reynolds number, or Peclet number appears as a factor to quantify the relative contribution of convection and diffusion.

Systems described by the poisson and laplace equation

We saw earlier that an irrotational vector field can be expressed as the gradient of a scalar and if in addition the vector field is solenoidal, then the scalar potential is the solution of the Laplace equation.

v = - ϕ , irrotational flow v = Θ = - 2 ϕ v = 0 = - 2 ϕ , incompressible , irrotational flow

Also, if the velocity field is solenoidal then the velocity can be expressed as the curl of the vector potential and the Laplacian of the vector potential is equal to the negative of the vorticity. If the flow is irrotational, then the vorticity is zero and the vector potential is a solution of the Laplace equation.

v = × A , incompressible flow × v = w = - 2 A , for A = 0 2 ψ = - w , in two dimensions × v = 0 = - 2 A , irrotational flow and A = 0 2 ψ = 0 , for two dimensional , irrotational , incompressible flow

Other systems, which are solution of the Laplace equation, are steady state heat conduction in a homogenous medium without sources and in electrostatics and static magnetic fields. Also, the flow of a single-phase, incompressible fluid in a homogenous porous media has a pressure field that is a solution of the Laplace equation.

Systems described by the diffusion equation

Diffusion phenomena occur with viscous flow, thermal conduction, and molecular diffusion. Heat conduction and diffusion without convection are described by the diffusion equation. Convection is always present in fluid flow but its contribution to the momentum balance is neglected for creeping (low Reynolds number) flow or cases where the velocity is perpendicular to the velocity gradient. In this case

Questions & Answers

where we get a research paper on Nano chemistry....?
Maira Reply
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
what is variations in raman spectra for nanomaterials
Jyoti Reply
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 .?
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
yes I'm doing my masters in nanotechnology, we are being studying all these domains as well..
what school?
biomolecules are e building blocks of every organics and inorganic materials.
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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