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Time Dependent Wave Equation 1D One Dimensional Optical Resonator Modes
Gas Physics Introduction
Gas Physics Introduction Chapter 1:Energy Distribution Evolution Energy Diffusion from Mono-energetic of a Three Dimensional Gas Rotating Gas in a 3D Toroid Gas Velocity Equilibration 2D Gas Velocity Equilibration 3D Circulating Gas in a Wind Tunnel Density Modulated Rotating Gas in a 3D Toroid Position Diffusion (Mixing) of a Three Dimensional Gas Energy Distribution 3D with MathJax Equations and Embedded Canvases Gas Energy Distributions 2D Rigid Rotors Energy Distribution 2D 1D Gas Energy and Momentum Distribution Rotor Energies 2D Rotor Energies 3D H2O Energies 3D CO2 Energies 3D 2D Rigid Arrays with Rotation and Translation 3D Rigid Arrays with Rotation and Translation Gas Energy Distributions 1D, 2D, 3D Chapter 2:Particle Scattering Effects
Inelastic Collision Conversion to Internal Energy Brownian Motion Animation 2D Using Hard Sphere Scattering Large Disc Kinetics 2D Condensation of a Gas on a 2D Solid Maxwell's Demon Energy Seperator 2D Minimize Final Energy 2D and 3D Gas Diffusion 2D Gas Expansion 2D Linear Gas Expansion 2D Constant Pressure Gas Expansion 2D Circular Gas Explosion 2D Circular MagnetoCaloric Effect 2D Convection Loops in a Numerical Gas Chapter 3: Gas Energy Equipartition Energy Equipartition of a Two Dimensional Gas with Two Different Masses Energy Equipartition of a Three Dimensional Gas with Two Different Masses Gas Energy Equipartition 3D
Chapter 2.1
Chapter 2.2
Chapter 2.3
Chapter 3.1
Chapter 3.2
Chapter 3.3
Chapter 4.1
Chapter 4.2
Chapter 4.3
Photoelectric Effect Alpha Particle Emission from a Large Nucleus 2D Black Body Emission from Charged Harmonic Oscillators Quantum Animation of Energy Band Gap 1D Animation of Quantum Wave Packets in a pn Junction 1D Quantum Mirror Focusing by Iterating the Schrodinger Equation Quantum Lens Focusing by Iterating the Schrodinger Equation Quantum Two Mirror Resonator by Iterating the Schrodinger Equation Entangled Quantum Object Propagation in a Parabolic Potential Classic and Quantum Object Propagation in a Parabolic Potential Accurate Wave Packet Propagation in a Parabolic Potential Comparison of Quantum to Classical Physics 1D-Time Dependent Accurate Wave Packet Propagation in a Swaged Potential Series Solution of Wave Packet Propagation in a Infinite Square Potential Propagation of a Wave Packet in a Parabolic Potential Propagation of a 2D Wave Packet in a 2D Parabolic Potential Propagation of an Asymmetric Shaped Wave Packet in a 2D Parabolic Potential Two Slit Diffraction by Iterating the Schrodinger Equation Two Potential Slit Diffraction by Iterating the Schrodinger Equation Propagation of an Electron Wave Packet in an Infinite Square Well (Series Solution) Propagation of a Free Wave Packet (Series Solution) Series and Algebraic Solutions for Propagation of a Wave Packet in a Parabolic Potential Comparison of Quantum to Classical Physics 1D-Time Independent Mexican Hat Stationary Quantum States of a Finite Square Potential Well Various Wave Packet Motion in a Finite Square Potential Well Various Wave Packet Motion in a Swaged Potential Well Comparison of Quantum to Classical Physics 1D-Time Independent Cosine/Parabola Electromagnetic Modes of a 2D Cavity Black Body Radiation Experiment and Theory Emission of the Inner Walls of a Black Body Cavity Multiple Slit Diffraction Statistics of Indistinguishable Dice or Particles Quantum Field Collapse Visualization Matrix Operations and Eigenmodes
Chapter 4
Chapter 5
Chapter 6
Chapter 7
Chapter 8
Chapter 9
Chapter 10
Chapter 11
Chapter 12
Tumbling Block on Inclined Plane New Roller on Inclined Plane Sliding Block on Inclined Plane Classic Objects Motion 1D Generation of Curve for Fastest Time between Two Points Curve for Fastest Time between Two Points Bead Sliding On a Stiff Wire Sound Wave in Two Solid Media Mechanical Properties of a 2D Numerically Modeled Lattice Vibrating Reed Numerical Model 2D Laminar Flow of a Fluid in a 2D Bearing Animation of a Fountain 2D Gravity Leveling of a 2D Liquid Longitudinal and Shear Wave Sound Speed in a 2D Lattice Mechanical Properties of a 2D Numerically Modeled Lattice Vibration Frequencies of a 2D Numerically Modeled Lattice Damped Harmonic Oscillator Motion and Modes of a Particle in a Mexican Hat Potential Forced Oscillator Roller Bearing Cam and Roller Follower Simple Hybrid Gear Train Ratchet Animation Child's Swing Height Increase Mechanics Roller Chain Animation Elasticity of 3D Gas Driving a Piston Free to Forced Harmonic Oscillator Transition Physics of Mounting a Tire on a Rim Physics of a Pendulum Clock Escapement Physics of a Gas Pressure Gauge
Chapter 4
Chapter 5
Chapter 6
Chapter 7
Chapter 8
Chapter 9
Chapter 10
Chapter 11
Chapter 12
Physics of a Gas Pressure Gauge Ion Drift in Neutral Particle Sea Diffusion Equation Solved by Finite Difference Method Linear Particle Diffusion Heat Flow 1D with Time Dependent Boundary Solid Cube in Liquid Column Drag on a Large Sphere Due to a Digital Gas Animation of a Gas Centrifuge Laminar Flow of a Gas in an Annular Space Laminar Flow of a Fluid in a 2D Bearing Incompressible Fluid Flow through a Constriction Fluid Flow Due To Pressure Gradients
Chapter 4
Chapter 5
Chapter 6
Chapter 7
Chapter 8
Chapter 9
Chapter 10
Chapter 11
Chapter 12
Parallel Resistors Gas Discharge Animation Oscilloscope Animation Permanent Magnet in Solenoid Cross Product of Two Vectors Magnetic Braking with Cylindrical Geometry Stopping a Falling Magnet by Eddy Currents Magnetic Brake Magnetic Field of a Long Solenoid Coil Light Propagation in a Transparent Dielectric Material Dipole Array Animations Angular Distribution of Emission of Arrays of Excited Dipoles Geometric Derivation of Snell's Refraction Angle Law Electromagnetic (EM) Wave Response to a Row of Dipoles Electromagnetic (EM) Wave Response to a Single Dipole Optics of a Transparent Plate by Iterating the Maxwell Wave Equation Diffraction by Iterating the Maxwell or Schrodinger Wave Equations Lens Focusing by Iterating the Maxwell or Schrodinger Wave Equations Optical Wave Packet Propagation by Finite Difference Matrix 1D Potential(x,y) Using Laplace's Equation Div(gradV)=0 with Variable Boundary Geometry Potential(x,y) Using Laplace's Equation Div(epsilon*gradV)=0 Laplace's Equation Div(gradV)=0 with Mixed BCs Potential(x,y) for Surface Electrodes with Dielectric Slab Dipolar Media Field Plots LCR Oscillator Animation Beam Trajectory of a Moving Laser Beam Trajectory in a Square Ring Laser Doppler Effect Converting Magnetic Forces to Electric Forces Converting Permanent Magnet Orbital Moments to Solenoid Currents Electron Beam Collimation by Axial Magnetic Field Electron Buncher for Klystron Charged Particle Spatial and Speed Bunching Electrical Conduction-Particle Picture Electromagnetic Bell Triode Vacuum Tube
Chapter 4
Chapter 5
Chapter 6
Chapter 7
Chapter 8
Chapter 9
Chapter 10
Chapter 11
Chapter 12
E=mc^2 and Mexican Jumping Bean Radar Pulse Return Time Interval Experiment to Determine Gamma Current Loop with a Rotating Test Particle Moving Fabry-Perot Interferometer Space Time Invariant Intervals of a Moving Clock Relativity Transformations by Requiring Simulataneity Accelerated Space Trip Clock Times and Rates Minguzzi Invariant Inertial Frame Time Accelerated Space Trip with Light Pulse Signaling
Chapter 4
Chapter 5
Chapter 6
Chapter 7
Chapter 8
Chapter 9
Chapter 10
Chapter 11
Chapter 12
Solar Power Generation Planet Formation Illumination of Planets Change of Orbit due to Asteroid Impact Earth Time Zones Day Lengths Astronomical Aberration Kepler Planetary and Sun Orbits Model of Ocean Tides Gravity Course Iteration Spacecraft Speed Assistance by Gravity Hohmann Transfer from One Circcular Orbit to Another Star Distances Using the Parallax Method
Chapter 4
Chapter 5
Chapter 6
Chapter 7
Chapter 8
Chapter 9
Chapter 10
Chapter 11
Chapter 12
Semiconductor Electron Energies and Band Gap 1D Semiconductor Electron Energies and Band Gap 2D Simple Fermi Surface p-nJunctionDiode Depletion Width Temperature Dependence of 2D Fermi-Dirac Particles
Chapter 4
Chapter 5
Chapter 6
Chapter 7
Chapter 8
Chapter 9
Chapter 10
Chapter 11
Chapter 12
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
Chapter 4
Chapter 5
Chapter 6
Chapter 7
Chapter 8
Chapter 9
Chapter 10
Chapter 11
Chapter 12
Multiple Slider Animation Template for Plotting with HTML Enhanced Sliders or Ranges Collapsible Dropdown Menu Example Arrow Functions Fast Fourier Transform (FFT) Demonstration
Animation of a Progress Bar TextBox Array Created Using Javascript Template for Scientific Plots Chapter 5
Chapter 6
Chapter 7
Chapter 8
Chapter 9
Chapter 10
Chapter 11
Chapter 12
Time Dependent Wave Equation 1D One Dimensional Optical Resonator Modes
Chapter 4
Chapter 5
Chapter 6
Chapter 7
Chapter 8
Chapter 9
Chapter 10
Chapter 11
Chapter 12
Probability of Birthday Match Random Creation of Probability Function Profiles Two Dies Probability Distribution Three Dies Probability Distribution
Chapter 4
Chapter 5
Chapter 6
Chapter 7
Chapter 8
Chapter 9
Chapter 10
Chapter 11
Chapter 12
Light Rays in a Transparent Isosceles Prism Light Ray in a Series of Wedges of Various Index of Refraction Snells Law of Refraction Refraction Due to Dipoles in Two Media Refraction and Reflection Two Media Animation of Concave Mirror Reflection Animation of Convex Lens Refraction Prism Refraction-Wave Picture Thick Lens Ray Trace Monochromator (Czerny-Turner Type) Wavelets from Concave Mirror> Thick Lens Focusing
Chapter 4
Chapter 5
Chapter 6
Chapter 7
Chapter 8
Chapter 9
Chapter 10
Chapter 11
Chapter 12
Sagnac Effect Fiber Optical Gyro (FOG) Ring Laser Gyro (RLG)
Chapter 4
Chapter 5
Chapter 6
Chapter 7
Chapter 8
Chapter 9
Chapter 10
Chapter 11
Chapter 12
Paticle Motion in a Venturi: Wall Reflections Paticle Motion in a Venturi Potential Elliptical Ion Trap Using Boundary Potential Distribution Prediction of Hard Disc Collision Time and Position Rigid Rotor Collisions
Chapter 4
Chapter 5
Chapter 6
Chapter 7
Chapter 8
Chapter 9
Chapter 10
Chapter 11
Chapter 12
Acoustic Wave with Built-in Pressure Acoustic Waves in 2D Numerically Modeled Solid Wave 1D on a Linear Array Waves on Discrete Model of Vibrating String Linear Acoustic Wave in a 3D Gas Spherical Acoustic Wave in a 3D Gas Laterally Perturbed Two Dimensional Gas with Lennard Jones Repelling Forces Longitudinally Perturbed Two Dimensional Gas with Lennard Jones Repelling Forces
Chapter 4
Chapter 5
Chapter 6
Chapter 7
Chapter 8
Chapter 9
Chapter 10
Chapter 11
Chapter 12
Bicep and Forearm Animation Blood Pressure Measurement Multiple Dose Residual Medicine Vs Time The Eye and its Resolution Breathing Animation Scoliosis Animation Muscle Animation
Chapter 4
Chapter 5
Chapter 6
Chapter 7
Chapter 8
Chapter 9
Chapter 10
Chapter 11
Chapter 12
Scattering from a Crystalline Target 2D Scattering from a Crystalline Target 3D
Chapter 4
Chapter 5
Chapter 6
Chapter 7
Chapter 8
Chapter 9
Chapter 10
Chapter 11
Chapter 12
Energy Conduction in a 2D Numerically Modeled Solid Lattice Unbalanced Gas Dynamics:Energy Flow in a Gas
Chapter 4
Chapter 5
Chapter 6
Chapter 7
Chapter 8
Chapter 9
Chapter 10
Chapter 11
Chapter 12

Hover over the menu bar to pick a physics animation.

Laminar Flow of a Gas at a Rough Surface

This animation will show particles moving in an annular space between two concentric corrugated cylinders. The cylinders, since they are sinusoidally corrugated, will be "rough" on the surfaces where the particles contact. The roughness will simulate the relatively rough surfaces, compared to molecular sizes, that man can achieve. Both cylinders can be rotating, at different speeds, about their centers. By the expression "laminar flow" we expect that the boundary particles' rotational speeds will be almost the same as its adjacent rough cylinder. We also expect that the particle speed variation between the stationary and rotating cylinder will be roughly linear with respect to the radial position of the particle. These sentences are the definition of laminar flow. Usually one would expect a laminar flow demonstration to have a solid object with a slightly rough surface be immersed in an infinite bath of a flowing fluid, either a liquid or a gas. Because this animation adheres to laws of physics and one cannot express an infinite gas by a computer I chose to demonstrate laminar flow in a gas of particles between two rotating concentric cylinders.

As is well known from wind tunnel experience, moving the air over a wing airfoil gives exactly the same physical results as moving the wing through the air and that is what can be expected here.

Handling the Collisions at the Corrugated Surfaces

In order to correctly simulate the interaction of the particles with the surfaces, we need to compute the way the particles are reflected from the corrugated cylinders. Since the particle collisions do not change the motion of the rotating surface, we can use the same expressions as we would use for the particle collision with a particle of infinite radius and of infinite mass. Then the expression for the reflection will involve only the corrugation surface normal and the velocity of the corrugation. So we need an expression for the reflection from a moving oblique surface.

Particle Reflection at a Moving Oblique Surface

Hereafter please note that bold characters like r are vectors and bold characters with caret hats like ˆn are unit vectors.

Let the surface be moving at velocity

vs=vsxˆx+vsyˆy

. and let it be rotated about the z axis so that its normal vector is.

n=nxˆx+nyˆy

.   Let the particle velocity be

vp=vpxˆx+vpyˆy

where vpx>0.   To compute the final velocity of the particle we first need the difference in the velocities

dv=(vpx-vsx)ˆx+(vpy-vsy)ˆy

The velocity of the surface will not change during the collision because it is assumed to be of infinite mass. The velocity of the particle will become (in vector form)

vp=vp0-2(n⋅dv)n    (1)

where vp0 is the particle velocity before collision. Note that if the surface velocity equals the particle velocity, then dv=0, so then, as expected, the particle velocity will not change as a result of the collision.

Particle Reflection from our Corrugated Cylinder

For the present problem, the expression for the corrugation radius from center is:

r(θ)=rm+Acos(npθ)

where θ is the angle of vector r from center, rm is the median radius, A is the amplitude of the corrugation, and integer np is the number of periods of the corrugation. When the corrugation rotates at rate ω the expression for r(θ,t) becomes:

r(θ,t)=rm+Acos(np(θ-ωt)

where t is time. The vector expression for the radial vector to angle θ is

r(t)=r(θ,t)(cos(θ-ωt)ˆx-sin(θ-ωt)ˆy)

Then the expression for the surface velocity vector vs is:

vs=ωr(θ,t)(-sin(θ-ωt)ˆx+cos(θ-ωt)ˆy)

For those same parameters, the local slope, tanψ, of the corrugated surface is expressed as:

tanψ=drdc=drd(θ)d(θ)dc=-Anpsin[npθ]r tanψ=-Ad(npθ)dcsin[np(θ-ωt)]=-Anprsin[np(θ-ωt)]

where the element dc=rdθ is an increment along the corrugation circumference. We now compute sin and cos of ψ as

cosψ=1√1+tan2ψ

sinψ=tanψ√1+tan2ψ

To get the surface normals, nx and ny we need to rotate the local slope parameters by angle θ:

nx=cosθcosψ+sinθsinψ=cos(θ-ψ)

ny=cosθsinψ-sinθcosψ=sin(θ-ψ)

Then:

ˆns=nxˆx+nyˆy

Now, to use equation 1, we take the dot product of vp-vs with ns.

ns⋅vs=ωr(θ)(-sinθcos(θ-ψ)-cosθsin(θ-ψ))=-ωr(θ)sinψ

and

ns⋅vp=nxvpx+nyvpy

Then the expression for the particle velocity after collision is:

vp=vp0-2(nxvpx+nyvpy-ωr(θ)sinψ)ˆns

Particle Intercept with a Moving Straight Line

We would like to predict where a particle with given position and velocity will intercept the moving corrugation. As a first approximation we will compute the intercept with a moving straight line. If the particle is already close to the corrugation, then the corrugation will not curve much between the present particle location and its intercept so straight line approximation will be reasonably accurate. The y coordinates of the line's points can be described by the equation

y=y20+v2yt+s2(x-v2xt-x20)

where (x20,y20) is a fixed point on the moving line,vx2 and vy2 are the velocity components of this point, s2 is the slope of the line, and t is time.

The particle's motion can be described by

x=x10+v1xt
y=y10+v1yt

To get the time of intersection we combine the above three equations and obtain

y10+v1yt=y20+v2yt+s2(x10+v1xt-v2xt-x20)

Separating out the time dependent parts of this equation we have

(v1y-v2y)t-s2(v1x-v2x)t=y20-y10+s2(x10-x20)

t=y20-y10+s2(x10-x20)v1y-v2y-s2{v1x-v2x)

Having found t, we obtain the intersection (x,y) as p style='text-Align:center'> x=x10+v1xt
y=y10+v1yt

Using the Particle Intercept

When the particle intercepts the boundary of the corrugation, then we must reflect it. Since the corrugation is curved, the intersection with the straight line is an approximation of the intersection with the corrugation. It is a good approximation for particles that are close to the corrugation boundary and for corrugation amplitudes that are reasonably small. So first, before we use the intercept, we should find cases where the particle is close to the boundary. Then we use the intercept coordinates to find the angle θ and from that we can obtain the normal vector of the corrugation bound. Then we use equation 1 to "reflect" the particle from the corrugation.

Explanation of the Controls

The controls are sliders and check boxes. The labeling of the check boxes is self explanatory so I will describe the sliders. The sliders adjust the dimensions of the corrugation and the initial speeds of the particles. The initial speed of all particles is the same but the directions of their initial velocity are random. So the initial energy of each particle is just E=mv22. Since the directions are random, I call the initial speed the "Thermal Speed". The actual thermal energy distribution is a Maxwell-Boltzmann distribution but the initial mono-energetic distribution that I've used will not affect the results of the laminar flow demonstration animated here. An adjustable integer parameter is the number of corrugation periods in a complete circle. In addition, the tangential speeds of the inner and outer corrugations can be chosen.

The amplitudes of the inner and outer sine wave corrugations can also be chosen. If the amplitudes are small, then the times for the inner bound and outer bound particle speeds to approach the speeds of their adjacent corrugation will be longer. Another parameter that affects these times is the number of periods of corrugation.

Explanation of the Numerical Result and Plots of the Animation

As discussed in the first paragraph, we expect that the average tangential speed of the particles will be approximately equal to the speed of the adjacent corrugated cylinder. That fact is demonstrated in the printed results on Canvas 1. The printed inner bound particle speed agrees quite well with the speed of the inner corrugation. The printed outer bound particle speed also agrees quite well with the speed of the outer corrugation. These are partial definitions of laminar flow. The remaining part of the definition is seen in the green histogram (bar graph) and is that the the speed variation between the two cylinders is roughly linear in the radial distance between the cylinders. The histogram variance form frame to frame is quite large. That is to be expected because the particles can quickly move from bin radius to bin radius. As one might expect, the average particle energy increases as the currugations rotate. Essentially the corrugations act like "fans" that continuously increase the both the rotational energy of the particles and their random direction (thermal) energy.

In order to keep the particles within the cylinder annulus, the animation stops when the average particle energy becomes greater than 0.35 which corresponds to a particle speed of 0.84 pixels per second. Occasionally particles briefly escape the annulus and the number of escapees is also printed on Canvas 1.

Canvas 1: Numerical Results and Histogram Bar Graphs
Canvas 2: Particle and Corrugation Animation