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Assembly of 3D Parts Using Javascript Web Graphics Library 3D Compared to Javascript 3D Graphics 3D Graphics Tutorial Beginnings Proof of Pythagoras a^2+b^2=c^2
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Time Dependent Wave Equation 1D One Dimensional Optical Resonator Modes
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Probability of Birthday Match Random Creation of Probability Function Profiles Two Dies Probability Distribution Three Dies Probability Distribution
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Energy Conduction in a 2D Numerically Modeled Solid Lattice Unbalanced Gas Dynamics:Energy Flow in a Gas
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Hover over the menu bar to pick a physics animation.

Rotating Gas in a 3D Toroid

Here I will examine the dynamics of a gas in a toroid where the gas has an learner selectable initial rotational speed .

Toroid Geometry

A toroid is an axisymmetric object that is characterized by two parameters: 1. A median circle of radius `R_s` (similar to the equator of a sphere). 2. A set of tubular circles of radius `r_s` centered at this median .
To locate any point on the tubular circle one also needs to define the azimuth, `theta` pointing to a point on the median circle as well as the altitude `phi` of the point on the tube. Thus, an equation in Cartesian coordinates that defines the point can be written

`bb "r"_v=[r_x,r_y,r_z]=[ R_s cos theta + r_s cos theta cos phi,\ \ \ R_s sin theta + r_s sin theta cos phi,\ \ \ r_s sin phi] \ \ \ \ \ (0)`

For viewing on the canvas, the toroid can be rotated by multiplying each toroid vertex position vector by a 3`x`3 rotation matrix, M.   Particles are added to the toroid in groups at intervals along the `theta` angles. These Particles can be given a combination of random velocities as well equal drift speeds.   For this animation. the drift speeds will be along the median or `theta` direction (i.e. they will be locally parallel to the toroid median line).  For display only, the particles also need to be rotated with M in order to remain inside the toroid.

The initial velocity vectors of the particles are defined by the rotation speed `v_theta` and the random speed `v_r`.

`bb "v"=v_theta hat bb theta+(v_(rx)hat bb "x"+v_(ry)hat bb "y"+v_(rz)hat bb "z")`

where

`hat bb theta=-sin theta hat bb "x"+cos theta hat bb "y"`

and

`sqrt(v_(rx)^2+v_(ry)^2+v_(rz)^2)=v_r`

and the relative values of the three components are chosen using the random number generator: To start, define:

`bb "v"_t=[v_(xt),v_(yt),v_(zt)]=2[rnd()-1/2,rnd()-1/2,rnd()-1/2]`

where each `rnd()` yields a different random number between 0 and 1. Now divide `bb "v"_t` by its magnitude to make it a unit vector:

`hat bb "v"_t=bb "v"_t/|bb "v"_t|`

finally we have the vector version of `v_r`:

`bb "v"_r=v_r hat bb "v"_t`

Ratio of Rotational Speed to the Speed of Sound

The Moving Gas inside the toroid can be thought of as a wind tunnel similar to what's used to test the lift and drag of aircraft bodies. Most wind tunnels have capabilities for very fast air speeds, `v_t`. The ratio of `v_t` to the speed of sound in still (non-movving) air is called the Mach number, M. The equation for the speed of sound in an ideal gas is usually given as:

`v_(sound)=sqrt(gamma (k T_K)/m)`

Here `gamma=C_P/C_V` is the ratio of the heat capacity at constant pressure to that at constant volume `k` is Boltzmann's constant, `T_K` is the Kelvin (absolute) temperature, and m is the mass of the particle. We have seen for all thermal problems that `k T_K` is exactly the same as the average `random` kinetic energy `k T_K=E_(avera"ge")`. So the equation becomes:

`v_(sound)=sqrt(gamma (E_(avera"ge"))/m)`

where m is the particle mass, taken to be 1 in this program. Now the `random` kinetic energy is the same as the thermal kinetic energy in still air i.e. it does not include the rotational energy. The running rotational energy is a bit smaller than the starting rotational energy because the average particle distance from center increases due to the need to make circles in the toroid. Our toroid Mach number, M, can be written as

`M=v_theta/sqrt(gamma (v_r^2)/2)`

For our simple spherical particles `gamma=5/3`.

x angle: 90 degrees
y angle: 50 degrees
z angle: 0 degrees
Gas Rotation Speed: 3
Gas Random Speed: 1
Toroid Radius Ratio: o.4
Canvas 1:3D Drawing of Toroid Particles and Particle Radial Distribution
Canvas 2:Plots of Energy Distribution and Mean Speed and Distance

Particle Dynamics Inside the Toroid

The particles often collide with the inside of the toroid. When they do they are reflected about the local normal of the inside of the toroid tube without losing kinetic energy or anglular momentum. To find the local normal, one must use some vector algebra.

Hereafter, a bold character denotes a vector which has [x,y,z] components.   The particle's position can be characterized by two vectors: Its azimuth `bb "R"_theta` and its altitude `bb "r"_phi`.   The actual position vector will be called `bb "r"_v` and it will be the sum of the two vectors:

`bb "r"_v=bb "R"_theta+bb "r"_phi \ \ \ \ \ (1)`

To determine whether the particle is close enough to the inner wall to be reflected, we need the magnitude of `bb "r"_phi`. We therefore solve equation 1 for `bb "r"_phi` and obtain:

`bb "r"_phi=bb "r"_v-bb "R"_theta \ \ \ \ (2)`

Now we need the value of `bb "R"_theta`. We know that this vector neets the toroid median line at some angle `theta`. The vector `bb "r"` is defined by both a `theta` and a `phi` as can be seen by examining equation 0. Note that .

`sin phi` =`r_z/|(bb "r")| \ \ \ \ (3)`

From equation 3 we can easily obtain `cos phi=sqrt(1-sin^2 phi)`.   Now we can use equation 0 and `cos phi` to obtain `bb "R"_theta`.

`bb "R"_theta=R/|bb "r"|[r_x/(cos phi),r_y/cos phi,0] \ \ \ \ \ (4)`

From equations 2 and 4 one obtains `bb "r"_phi` and this vector is along the unit normal `hat bb "n"` to the inside of the toroid.

`hat bb "n"=bb "r"_phi/(|bb "r"_phi|) \ \ \ \ \ (5)`

When the particle collides with the inside, the velocity increment:

`delta bb "v"=2 (hat bb "n" cdot bb "v") hat bb "n" \ \ \ \ \ (5)`

is added to the incident velocity in order to conserve angular momentum and energy.

Explanation of the Graphics and Plots

To get a better view of the toroid's angular rotation angle sliders are provided. The nominal axes are the x axis perpendicular to the screen, the z axis vertical and the y axis horizontal. These are the axis directions generally used in physics texts. When all the slider settings are zero, the toroid axis is initially along screen's vertical direction. The conserved quantities are the kinetic energy and the angular momentum. These are listed on Canvas 2 under the plot of the average particles' distance from center. There is a slight variation in the total angular momentum but That is just due to the limited numerical effort to integrate the particle velocities and positions. These can be easily improved by using a 4th order Runge Kutta integration method.

The particles' average distance from center oscillates after startup. Note that the oscillation frequency is a function of the rotational speed of the particles. This is reasonable because the centrifugal force needed to keep the particles inside the toroid is also a function of this speed. So what happens is the gas density increases as the distance from the tube center. Gas density increase causes the pressure to increase likewise. So the increased pressure acts like a spring and the oscillation frequency is a natural function of this spring. Plots of the gas energy probability distribution agree with what is expected for any 3 dimensional monoatomic gas.