The unit square is now discretized into
JMAX by
JMAX grid points where the boundary conditions and dependent variables will be evaluated. The grid spacing is
$\delta =1/(JMAX-1)$ . The first and last row and column are boundary values.
The partial derivatives will be approximated by finite differences. For example, the second derivative of vorticity is discretized by a Taylor's series.
The vorticity at the boundary is discretized and expressed in terms of the components of velocity at the boundary, the stream function values on the boundary and a stream function value in the interior grid. (A greater accuracy is possible by using two interior points.) The stream function at the first interior point
$(i=2)$ from the
$x$ boundary is written with a Taylor's series as follows.
The boundary condition on the stream function is specified by the normal component of velocity at the boundaries. Since we have assumed zero normal component of velocity, the stream function is a constant on the boundary, which we specify to be zero.
The stream function at the boundary is calculated from the normal component of velocity by numerical integration using the trapezodial rule, e.g.,
The finite difference equations for the PDE and the boundary conditions are a linear system of equations with two dependent variables. The dependent variables at a
${x}_{i}$ ,
${y}_{j}$ grid point will be represented as a two component vector of dependent variables,
The components of the 2x2 coefficient matrix are the coefficients from the difference equations. The first row is coefficients for the stream function equation and the second row is coefficients for the vorticity equation. The first column is coefficients for the stream function variable and the second column is the coefficients for the vorticity variable. For example, at interior points not affected by the boundary conditions,
The coefficients for the interior grid points adjacent to a boundary are modified as a result of substitution the boundary value of stream function or the linear equation for the boundary vorticity into the difference equations. The stream function equation is coupled to the vorticity with the
${\mathbf{e}}_{ij}$ coefficient and the vorticity equation is coupled to the stream function through the boundary conditions. For example, at a
$x=0$ boundary, the coefficients will be modified as follows.
Questions & Answers
how do you translate this in Algebraic Expressions
In this morden time nanotechnology used in many field .
1-Electronics-manufacturad IC ,RAM,MRAM,solar panel etc
2-Helth and Medical-Nanomedicine,Drug Dilivery for cancer treatment etc
3- Atomobile -MEMS, Coating on car etc.
and may other field for details you can check at Google
Azam
anybody can imagine what will be happen after 100 years from now in nano tech world
Prasenjit
after 100 year this will be not nanotechnology maybe this technology name will be change .
maybe aftet 100 year . we work on electron lable practically about its properties and behaviour by the different instruments
Azam
name doesn't matter , whatever it will be change... I'm taking about effect on circumstances of the microscopic world
Prasenjit
how hard could it be to apply nanotechnology against viral infections such HIV or Ebola?
Damian
silver nanoparticles could handle the job?
Damian
not now but maybe in future only AgNP maybe any other nanomaterials
Azam
Hello
Uday
I'm interested in Nanotube
Uday
this technology will not going on for the long time , so I'm thinking about femtotechnology 10^-15
Prasenjit
can nanotechnology change the direction of the face of the world
At high concentrations (>0.01 M), the relation between absorptivity coefficient and absorbance is no longer linear. This is due to the electrostatic interactions between the quantum dots in close proximity. If the concentration of the solution is high, another effect that is seen is the scattering of light from the large number of quantum dots. This assumption only works at low concentrations of the analyte. Presence of stray light.