Streamline, Streakline, and Pathline

From Wiki =)

Funny picture

Time dependent field is a key!

The red particle moves in a flowing fluid; its pathline is traced in red; the tip of the trail of blue ink released from the origin follows the particle, but unlike the static pathline (which records the earlier motion of the dot), previously released ink moves up with the flow. (This is a streakline.) The dashed lines represent contours of the velocity field (streamlines), showing the motion of the whole field at the same time.

Trace and pressure tensor

Since the trace of any tensor is independent of any coordinate system, the most complete coordinate-free decomposition of a symmetric tensor is to represent it as the sum of a constant tensor and a traceless symmetric tensor. (Symon (1971) Ch. 10) Thus:

where δ_ij is the Kronecker delta. The first term on the right is the constant tensor, also known as the pressure, and the second term is the traceless symmetric tensor, also known as the shear tensor.

Symon, Keith (1971). Mechanics. Addison-Wesley, Reading, MA. ISBN 0-201-07392-7.

From wiki

Working directory

import os
print os.getcwd()

Data visualization tool

A quick look at six open source graphics utilities

M. Tim Jones (mtj@mtjones.com), Senior Principal Software Engineer, Emulex Corp.

http://www.ibm.com/developerworks/linux/library/l-datavistools/

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This is a good document in that the author explains, compares, and recommend data visualization tools.

Fourier series theorem

Fourier Series theorem is relevant to Ballooning representation.

Theorem 21.2 (Fourier Series Theorem)
.

1. Let  be a function which is piecewise continuous on  .Its Fourier series is given by

at each point  where the one sided derivatives  and  both exist.

2. If  is continuous the result is

Excerpt from http://www.math.ohio-state.edu/~gerlach/math/BVtypset/node27.html#Fourier_series_theorem

Ref. A. Thyagaraja, J. Plasma Physics 59, 367 (1998)

Cygwin Setting only for X11

Following module should be installed :

X11>xorg-server (required, the Cygwin/X X Server)

X11>xinit (required, scripts for starting the X server: xinit, startx, startwin.sh, startxwin.bat (and a shortcut on the Start Menu to run it), startxdmcp.bat )

NET> openssh

Material Derivative(Lagrangian)

This comes from Wiki. I just liked this description for distinction between Eulerian and Lagrangian.

—————————————————————–

This static position derivative is called the Eulerian derivative.

An example of this case is a swimmer standing still and sensing temperature change in a lake early in the morning: the water gradually becomes warmer due to heating from the sun.

If, instead, the path x(t) is not a standstill, the (total) time derivative of φ may change due to the path. For example, imagine the swimmer is in a motionless pool of water, indoors and unaffected by the sun. One end happens to be a constant hot temperature and the other end a constant cold temperature, by swimming from one end to the other the swimmer senses a change of temperature with respect to time, even though the temperature at any given (static) point is a constant. This is because the derivative is taken at the swimmer’s changing location. A temperature sensor attached to the swimmer would show temperature varying in time, even though the pool is held at a steady temperature distribution.

That is, the path follows the fluid current described by the fluid’s velocity field v. So, the material derivative of the scalar φ is:

Method of characteristics (from wiki)

For a first-order PDE, the method of characteristics discovers curves (called characteristic curves or just characteristics) along which the PDE becomes an ordinary differential equation (ODE). Once the ODE is found, it can be solved along the characteristic curves and transformed into a solution for the original PDE.

For the sake of motivation, we confine our attention to the case of a function of two independent variables x and y for the moment. Consider a quasilinear PDE of the form

		( 1)

Suppose that a solution u is known, and consider the surface graph z = u(x,y) in R³. A normal vector to this surface is given by

As a result, equation (1 ) is equivalent to the geometrical statement that the vector field

is tangent to the surface z = u(x,y) at every point. In other words, the graph of the solution must be a union of integral curves of this vector field. These integral curves are called the characteristic curves of the original partial differential equation.

The equations of the characteristic curve may be expressed invariantly by the Charpit-Lagrange equations^[1]

or, if a particular parametrization t of the curves is fixed, then these equations may be written as a system of ordinary differential equations for x(t), y(t), z(t):

These are the characteristic equations for the original system.

2D turbulence

2D turbulence – Inverse cascade, self-organization

ref)

http://www.fluid.tue.nl/WDY/2Dturb/2Dturb.html

 

 

 

Resistivity and Plasma

Sometimes, I thought that plasma is a kind of metal. And when I asked a question to my friend who were studying liquid Lithium metal that why you did experiment with not plasma but lithium, he said, liquid metal is a kind of conducting material like plasma and shows some similar behavior like plasma with regard to MagnetoHydroDynamics(MHD).

I don’t know lithium well, and I’m not sure whether lithium is an exception or not about tendency of resistivity for metal, anyway, but there is a difference of dependence on temperature about resistivity between metal and plasma. 

Before we deal with the subject, if we look at only results, Interestingly, as temperature goes to high, resistivity of metal increases, but resistivity of plasma decreases.

Just want to check the difference of tendency on resistivity between plasma and metal in terms of temperature. In addition, will historically review improvement of developing theory for resistivity of plasma following some lecture notes and books ( Greg Hammett’s Lecture notes , etc… )

 

Resistivity ( Reference : Wikipedia )

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In general, electrical resistivity of metals increases with temperature, while the resistivity of semiconductors decreases with increasing temperature. In both cases, electron-phonon interactions can play a key role. At high temperatures, the resistance of a metal increases linearly with temperature. As the temperature of a metal is reduced, the temperature dependence of resistivity follows a power law function of temperature. Mathematically the temperature dependence of the resistivity ρ of a metal is given by the Bloch-Gruneissen formula:

where ρ(0) is the residual resistivity due to defect scattering, A is a constant that depends on the velocity of electrons at the fermi surface, the Debye radius and the number density of electrons in the metal. Θ_R is the Debye temperature as obtained from resistivity measurements and matches very closely with the values of Debye temperature obtained from specific heat measurements. n is an integer that depends upon the nature of interaction:

1. n=5 implies that the resistance is due to scattering of electrons by phonons (as it is for simple metals)
2. n=3 implies that the resistance is due to s-d electron scattering (as is the case for transition metals)
3. n=2 implies that the resistance is due to electron-electron interaction.

As the temperature of the metal is sufficiently reduced (so as to ‘freeze’ all the phonons), the resistivity usually reaches a constant value, known as the residual resistivity. This value depends not only on the type of metal, but on its purity and thermal history. The value of the residual resistivity of a metal is decided by its impurity concentration. Some materials lose all electrical resistivity at sufficiently low temperatures, due to an effect known as superconductivity.

An even better approximation of the temperature dependence of the resistivity of a semiconductor is given by the Steinhart-Hart equation:

where A, B and C are the so-called Steinhart-Hart coefficients.

This equation is used to calibrate thermistors.

In non-crystalline semi-conductors, conduction can occur by charges quantum tunnelling from one localised site to another. This is known as variable range hopping and has the characteristic form of , where n=2,3,4 depending on the dimensionality of the system.

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