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.
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.
Tarell
what is the actual application of fullerenes nowadays?
Damian
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.
Tarell
what is the Synthesis, properties,and applications of carbon nano chemistry
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.
Harper
Do you know which machine is used to that process?