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| 1 | +{ |
| 2 | + "cells": [ |
| 3 | + { |
| 4 | + "cell_type": "markdown", |
| 5 | + "id": "1533bd8e", |
| 6 | + "metadata": {}, |
| 7 | + "source": [ |
| 8 | + "Names: " |
| 9 | + ] |
| 10 | + }, |
| 11 | + { |
| 12 | + "cell_type": "code", |
| 13 | + "execution_count": 22, |
| 14 | + "id": "8be05185", |
| 15 | + "metadata": {}, |
| 16 | + "outputs": [], |
| 17 | + "source": [ |
| 18 | + "# your import statements\n" |
| 19 | + ] |
| 20 | + }, |
| 21 | + { |
| 22 | + "cell_type": "markdown", |
| 23 | + "id": "c17b4ce4", |
| 24 | + "metadata": {}, |
| 25 | + "source": [ |
| 26 | + "Consider a long hallway (length $L$) in an office building. If we assume that any\n", |
| 27 | + "cigarette smoke mixes across the width of the hallway and vertically\n", |
| 28 | + "through the depth of the hallway much faster than it mixes along the\n", |
| 29 | + "hallway, we can write the diffusion of cigarette smoke as an equation\n", |
| 30 | + "$$\\frac {\\partial S} {\\partial t}\n", |
| 31 | + "= \\frac {\\partial \\kappa \\partial S}{\\partial x^2}$$\n", |
| 32 | + "where $S$ is the concentration of smoke, $\\kappa$ is the rate of diffusion of\n", |
| 33 | + "smoke, t is the time and x is\n", |
| 34 | + "distance along the hallway. At the centre of the hallway is the smoker and here the smoke is 5 su (smoke units). At one end of the hallway (x=0) is an open window, here the smoke is 0. At the other end of the hallway (x=L) is a closed window (no smoke through the closed window)." |
| 35 | + ] |
| 36 | + }, |
| 37 | + { |
| 38 | + "cell_type": "markdown", |
| 39 | + "id": "9eb3d006", |
| 40 | + "metadata": {}, |
| 41 | + "source": [ |
| 42 | + "**Question 1**\n", |
| 43 | + "\n", |
| 44 | + "Under what conditions can you move $\\kappa$ outside the derivative? Make that assumption here, and also make the steady state assumption. What is your new differential equation?" |
| 45 | + ] |
| 46 | + }, |
| 47 | + { |
| 48 | + "cell_type": "markdown", |
| 49 | + "id": "5f110569", |
| 50 | + "metadata": {}, |
| 51 | + "source": [] |
| 52 | + }, |
| 53 | + { |
| 54 | + "cell_type": "markdown", |
| 55 | + "id": "ace733e0", |
| 56 | + "metadata": {}, |
| 57 | + "source": [ |
| 58 | + "**Question 2**\n", |
| 59 | + "\n", |
| 60 | + "Using a centre-difference scheme, separating your hallway into $N=10$ divisions (so $N+1=11$ grid points) write down linear equations for the 1th to $N/2-1=4$th grid points and the $N/2+1=6$th to $N-1=9$th grid points. Note that the 0th and $N=10$th grid points are boundary points and there is a special condition at the $N/2=5$th grid point." |
| 61 | + ] |
| 62 | + }, |
| 63 | + { |
| 64 | + "cell_type": "markdown", |
| 65 | + "id": "b0f3282e", |
| 66 | + "metadata": {}, |
| 67 | + "source": [] |
| 68 | + }, |
| 69 | + { |
| 70 | + "cell_type": "markdown", |
| 71 | + "id": "6bcb3435", |
| 72 | + "metadata": {}, |
| 73 | + "source": [ |
| 74 | + "**Question 3**\n", |
| 75 | + "\n", |
| 76 | + "What type of boundary condition is at $x=0$ (0th grid point)? Write it as linear equation." |
| 77 | + ] |
| 78 | + }, |
| 79 | + { |
| 80 | + "cell_type": "markdown", |
| 81 | + "id": "1abcc6cf", |
| 82 | + "metadata": {}, |
| 83 | + "source": [] |
| 84 | + }, |
| 85 | + { |
| 86 | + "cell_type": "markdown", |
| 87 | + "id": "305884ee", |
| 88 | + "metadata": {}, |
| 89 | + "source": [ |
| 90 | + "**Question 4**\n", |
| 91 | + "\n", |
| 92 | + "What type of boundary condition is at $x=L$ (N=10th grid point)? Write it as linear equation." |
| 93 | + ] |
| 94 | + }, |
| 95 | + { |
| 96 | + "cell_type": "markdown", |
| 97 | + "id": "9f3f791b", |
| 98 | + "metadata": {}, |
| 99 | + "source": [] |
| 100 | + }, |
| 101 | + { |
| 102 | + "cell_type": "markdown", |
| 103 | + "id": "3ffd3078", |
| 104 | + "metadata": {}, |
| 105 | + "source": [ |
| 106 | + "**Question 5**\n", |
| 107 | + "\n", |
| 108 | + "Consider the condition at the centre of the hallway ($x=L/2$, $N/2=5$th grid point) Write it as linear equation." |
| 109 | + ] |
| 110 | + }, |
| 111 | + { |
| 112 | + "cell_type": "markdown", |
| 113 | + "id": "648910b3", |
| 114 | + "metadata": {}, |
| 115 | + "source": [] |
| 116 | + }, |
| 117 | + { |
| 118 | + "cell_type": "markdown", |
| 119 | + "id": "68450247", |
| 120 | + "metadata": {}, |
| 121 | + "source": [ |
| 122 | + "**Question 6**\n", |
| 123 | + "\n", |
| 124 | + "Put all your equations into a matrix equation and solve the matrix equation. Plot the solution." |
| 125 | + ] |
| 126 | + }, |
| 127 | + { |
| 128 | + "cell_type": "code", |
| 129 | + "execution_count": null, |
| 130 | + "id": "577152a7", |
| 131 | + "metadata": {}, |
| 132 | + "outputs": [], |
| 133 | + "source": [] |
| 134 | + } |
| 135 | + ], |
| 136 | + "metadata": { |
| 137 | + "kernelspec": { |
| 138 | + "display_name": "Python 3 (ipykernel)", |
| 139 | + "language": "python", |
| 140 | + "name": "python3" |
| 141 | + }, |
| 142 | + "language_info": { |
| 143 | + "codemirror_mode": { |
| 144 | + "name": "ipython", |
| 145 | + "version": 3 |
| 146 | + }, |
| 147 | + "file_extension": ".py", |
| 148 | + "mimetype": "text/x-python", |
| 149 | + "name": "python", |
| 150 | + "nbconvert_exporter": "python", |
| 151 | + "pygments_lexer": "ipython3", |
| 152 | + "version": "3.10.1" |
| 153 | + } |
| 154 | + }, |
| 155 | + "nbformat": 4, |
| 156 | + "nbformat_minor": 5 |
| 157 | +} |
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