{
 "cells": [
  {
   "cell_type": "markdown",
   "id": "e431b092-84fd-4f3b-81a2-ffdaa41c38b4",
   "metadata": {},
   "source": [
    "# Hugoniot equation for a calorically perfect gas"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "c161010c-7b40-4ead-86ce-6d9b95f78c67",
   "metadata": {},
   "source": [
    "The Hugoniot equation relates thermodynamic quantities (pressure $p$, specific volume $v$ and internal energy $e$) across a shock wave. \n",
    "\n",
    "The general form (shown below) holds for a perfect gas, chemically reacting gas, real gas, etc."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "id": "cc5df44f-34c8-422e-9505-aae5f39f9288",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle - e_{1} + e_{2} = \\left(\\frac{p_{1}}{2} + \\frac{p_{2}}{2}\\right) \\left(v_{1} - v_{2}\\right)$"
      ],
      "text/plain": [
       "Equation(-e1 + e2, (p1/2 + p2/2)*(v1 - v2))"
      ]
     },
     "execution_count": 1,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "from sympy import *\n",
    "from sympy_equation import (\n",
    "    Eqn,\n",
    "    table_of_expressions,\n",
    "    process_arguments_of_add,\n",
    "    divide_term_by_term,\n",
    ")\n",
    "\n",
    "e1, e2, p1, p2, v1, v2 = symbols(\"e1, e2, p1, p2, v1, v2\", real=True, positive=True)\n",
    "h = Eqn(e2 - e1, (p1 + p2) / 2 * (v1 - v2))\n",
    "h"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "7d74b2e1-87eb-4eec-b015-fc08e9581933",
   "metadata": {},
   "source": [
    "For a calorically perfect gas, $e = C_{v} T$ and $T = p v / R$, where:\n",
    "\n",
    "* $C_{v}$: specific heat at constant volume.\n",
    "* $T$: temperature.\n",
    "* $R$: specific gas constant.\n",
    "\n",
    "Also note that $\\gamma$ is the ratio of specific heats.\n",
    "\n",
    "Find a new form of the Hugoniot equation for a calorically perfect gas, which shows the relationship between the pressure ratio and the specific volume ratio."
   ]
  },
  {
   "cell_type": "markdown",
   "id": "0b613e8e-b1b4-46bd-a723-5d5c7a4d2941",
   "metadata": {},
   "source": [
    "Let's define a few new equations:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "id": "d4f8586f-b879-42d4-8b57-ad0c34f009d5",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle C_{v} = \\frac{R}{\\gamma - 1}$"
      ],
      "text/plain": [
       "Equation(C_v, R/(gamma - 1))"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "text/latex": [
       "$\\displaystyle e_{1} = C_{v} T_{1}$"
      ],
      "text/plain": [
       "Equation(e1, C_v*T1)"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "text/latex": [
       "$\\displaystyle e_{2} = C_{v} T_{2}$"
      ],
      "text/plain": [
       "Equation(e2, C_v*T2)"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "text/latex": [
       "$\\displaystyle T_{1} = \\frac{p_{1} v_{1}}{R}$"
      ],
      "text/plain": [
       "Equation(T1, p1*v1/R)"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "text/latex": [
       "$\\displaystyle T_{2} = \\frac{p_{2} v_{2}}{R}$"
      ],
      "text/plain": [
       "Equation(T2, p2*v2/R)"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "Cv, R, T1, T2, gamma = symbols(\"C_v, R, T1, T2, gamma\", real=True, positive=True)\n",
    "eq1 = Eqn(Cv, R / (gamma - 1))\n",
    "eq2 = Eqn(e1, Cv * T1)\n",
    "eq3 = Eqn(e2, Cv * T2)\n",
    "eq4 = Eqn(T1, p1 * v1 / R)\n",
    "eq5 = Eqn(T2, p2 * v2 / R)\n",
    "display(eq1, eq2, eq3, eq4, eq5)"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "1818e969-c185-4db2-ba8f-a8722201da45",
   "metadata": {},
   "source": [
    "Substitutes them into the Hugoniot equation:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "id": "19056549-244a-43ff-a414-c136b60d1cba",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle - \\frac{p_{1} v_{1}}{\\gamma - 1} + \\frac{p_{2} v_{2}}{\\gamma - 1} = \\left(\\frac{p_{1}}{2} + \\frac{p_{2}}{2}\\right) \\left(v_{1} - v_{2}\\right)$"
      ],
      "text/plain": [
       "Equation(-p1*v1/(gamma - 1) + p2*v2/(gamma - 1), (p1/2 + p2/2)*(v1 - v2))"
      ]
     },
     "execution_count": 3,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# multi step substitutions in order to achieve the desired form\n",
    "e = h.subs(eq2, eq3).subs(eq4, eq5).subs(eq1)\n",
    "e"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "b5894029-511c-4d4d-920d-31f03667853c",
   "metadata": {},
   "source": [
    "Let's move $\\gamma - 1$ from the denominator of the LHS to the numerator of the RHS:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "id": "50bee31e-ed99-4dcd-8624-c37f18448d0a",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle - p_{1} v_{1} + p_{2} v_{2} = \\left(\\gamma - 1\\right) \\left(\\frac{p_{1}}{2} + \\frac{p_{2}}{2}\\right) \\left(v_{1} - v_{2}\\right)$"
      ],
      "text/plain": [
       "Equation(-p1*v1 + p2*v2, (gamma - 1)*(p1/2 + p2/2)*(v1 - v2))"
      ]
     },
     "execution_count": 4,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "e = e.dolhs.collect(1 / (gamma - 1)) * (gamma - 1)\n",
    "e"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "5113dda6-4592-4d76-9787-b2f74187d340",
   "metadata": {},
   "source": [
    "Now we can start working with pressure ratios and specific volume ratios. First, let's expand the equation so it is easier to achieve the next steps:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "id": "f7e8704f-69a1-48ff-b3ad-174dd8501042",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle - p_{1} v_{1} + p_{2} v_{2} = \\frac{\\gamma p_{1} v_{1}}{2} - \\frac{\\gamma p_{1} v_{2}}{2} + \\frac{\\gamma p_{2} v_{1}}{2} - \\frac{\\gamma p_{2} v_{2}}{2} - \\frac{p_{1} v_{1}}{2} + \\frac{p_{1} v_{2}}{2} - \\frac{p_{2} v_{1}}{2} + \\frac{p_{2} v_{2}}{2}$"
      ],
      "text/plain": [
       "Equation(-p1*v1 + p2*v2, gamma*p1*v1/2 - gamma*p1*v2/2 + gamma*p2*v1/2 - gamma*p2*v2/2 - p1*v1/2 + p1*v2/2 - p2*v1/2 + p2*v2/2)"
      ]
     },
     "execution_count": 5,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "e = e.expand()\n",
    "e"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "072f70f8-f8ca-4325-a728-e68e2062fb20",
   "metadata": {},
   "source": [
    "Let's divide both sides by $p_{1}$:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "id": "60cd9b21-e2b6-499b-b64d-6364eb5633b4",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle - v_{1} + \\frac{p_{2} v_{2}}{p_{1}} = \\frac{\\gamma v_{1}}{2} - \\frac{\\gamma v_{2}}{2} + \\frac{\\gamma p_{2} v_{1}}{2 p_{1}} - \\frac{\\gamma p_{2} v_{2}}{2 p_{1}} - \\frac{v_{1}}{2} + \\frac{v_{2}}{2} - \\frac{p_{2} v_{1}}{2 p_{1}} + \\frac{p_{2} v_{2}}{2 p_{1}}$"
      ],
      "text/plain": [
       "Equation(-v1 + p2*v2/p1, gamma*v1/2 - gamma*v2/2 + gamma*p2*v1/(2*p1) - gamma*p2*v2/(2*p1) - v1/2 + v2/2 - p2*v1/(2*p1) + p2*v2/(2*p1))"
      ]
     },
     "execution_count": 6,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "e = (e / p1).expand()\n",
    "e"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "f44e28d3-4fc9-4443-990e-ae22832b8186",
   "metadata": {},
   "source": [
    "and then by $v_{1}$:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "id": "656b3d48-1aaf-4f4e-a773-53503d7ce19b",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle -1 + \\frac{p_{2} v_{2}}{p_{1} v_{1}} = \\frac{\\gamma}{2} - \\frac{\\gamma v_{2}}{2 v_{1}} + \\frac{\\gamma p_{2}}{2 p_{1}} - \\frac{\\gamma p_{2} v_{2}}{2 p_{1} v_{1}} - \\frac{1}{2} + \\frac{v_{2}}{2 v_{1}} - \\frac{p_{2}}{2 p_{1}} + \\frac{p_{2} v_{2}}{2 p_{1} v_{1}}$"
      ],
      "text/plain": [
       "Equation(-1 + p2*v2/(p1*v1), gamma/2 - gamma*v2/(2*v1) + gamma*p2/(2*p1) - gamma*p2*v2/(2*p1*v1) - 1/2 + v2/(2*v1) - p2/(2*p1) + p2*v2/(2*p1*v1))"
      ]
     },
     "execution_count": 7,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "e = (e / v1).expand()\n",
    "e"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "b0491055-2be4-4109-828d-e49ff53941ca",
   "metadata": {},
   "source": [
    "Let's move the -1 constant from the LHS to the RHS:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "id": "1181339a-1f9a-470b-be10-717a56a82575",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle \\frac{p_{2} v_{2}}{p_{1} v_{1}} = \\frac{\\gamma}{2} - \\frac{\\gamma v_{2}}{2 v_{1}} + \\frac{\\gamma p_{2}}{2 p_{1}} - \\frac{\\gamma p_{2} v_{2}}{2 p_{1} v_{1}} + \\frac{1}{2} + \\frac{v_{2}}{2 v_{1}} - \\frac{p_{2}}{2 p_{1}} + \\frac{p_{2} v_{2}}{2 p_{1} v_{1}}$"
      ],
      "text/plain": [
       "Equation(p2*v2/(p1*v1), gamma/2 - gamma*v2/(2*v1) + gamma*p2/(2*p1) - gamma*p2*v2/(2*p1*v1) + 1/2 + v2/(2*v1) - p2/(2*p1) + p2*v2/(2*p1*v1))"
      ]
     },
     "execution_count": 8,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "e = e + 1\n",
    "e"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "90356d90-99e4-4cb5-bdf5-6b29747e2841",
   "metadata": {},
   "source": [
    "Now, let's bring the following term from the RHS to the LHS:\n",
    "\n",
    "* $p_{2} v_{2} / (2 p_{1} v_{1})$\n",
    "* $\\gamma p_{2} v_{2} / (2 p_{1} v_{1})$\n",
    "\n",
    "In order to avoid typing errors, we are going to select those expressions from the arguments that compose the RHS:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "id": "959c6e23-4584-4f92-8552-986d93bfd284",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/markdown": [
       "| idx | args |\n",
       "|:-----:|:------|\n",
       "| 0 | $\\frac{1}{2}$ |\n",
       "| 1 | $\\frac{\\gamma}{2}$ |\n",
       "| 2 | $\\frac{v_{2}}{2 v_{1}}$ |\n",
       "| 3 | $- \\frac{p_{2}}{2 p_{1}}$ |\n",
       "| 4 | $\\frac{\\gamma p_{2}}{2 p_{1}}$ |\n",
       "| 5 | $- \\frac{\\gamma v_{2}}{2 v_{1}}$ |\n",
       "| 6 | $\\frac{p_{2} v_{2}}{2 p_{1} v_{1}}$ |\n",
       "| 7 | $- \\frac{\\gamma p_{2} v_{2}}{2 p_{1} v_{1}}$ |\n"
      ],
      "text/plain": [
       "<IPython.core.display.Markdown object>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "toe = table_of_expressions(e.rhs)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "id": "183da512-16ec-4c79-98d7-b88b5158bd36",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle \\frac{\\gamma p_{2} v_{2}}{2 p_{1} v_{1}} + \\frac{p_{2} v_{2}}{2 p_{1} v_{1}} = \\frac{\\gamma}{2} - \\frac{\\gamma v_{2}}{2 v_{1}} + \\frac{\\gamma p_{2}}{2 p_{1}} + \\frac{1}{2} + \\frac{v_{2}}{2 v_{1}} - \\frac{p_{2}}{2 p_{1}}$"
      ],
      "text/plain": [
       "Equation(gamma*p2*v2/(2*p1*v1) + p2*v2/(2*p1*v1), gamma/2 - gamma*v2/(2*v1) + gamma*p2/(2*p1) + 1/2 + v2/(2*v1) - p2/(2*p1))"
      ]
     },
     "execution_count": 10,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "e = e - toe[6] - toe[7]\n",
    "e"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "e8fc4022-54a7-474b-a0c6-ee8dc2e5c199",
   "metadata": {},
   "source": [
    "Let's get rid of the constant 2 at denominators:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "id": "0db47673-a381-43b4-9084-96618c80b347",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle \\frac{\\gamma p_{2} v_{2}}{p_{1} v_{1}} + \\frac{p_{2} v_{2}}{p_{1} v_{1}} = \\gamma - \\frac{\\gamma v_{2}}{v_{1}} + \\frac{\\gamma p_{2}}{p_{1}} + 1 + \\frac{v_{2}}{v_{1}} - \\frac{p_{2}}{p_{1}}$"
      ],
      "text/plain": [
       "Equation(gamma*p2*v2/(p1*v1) + p2*v2/(p1*v1), gamma - gamma*v2/v1 + gamma*p2/p1 + 1 + v2/v1 - p2/p1)"
      ]
     },
     "execution_count": 11,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "e = (e * 2).expand()\n",
    "e"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "a6ba4f2d-e815-420c-bd07-aaeb3c1f607e",
   "metadata": {},
   "source": [
    "At this point, it would be nice to factor terms sharing the same ratio. For example:\n",
    "\n",
    "$$\n",
    "\\begin{aligned}\n",
    "\\gamma \\frac{p_{2}v_{2}}{p_{1}v_{1}} + \\frac{p_{2}v_{2}}{p_{1}v_{1}} &= \\frac{p_{2}v_{2}}{p_{1}v_{1}} (\\gamma + 1) \\\\\n",
    "\\gamma \\frac{p_{2}}{p_{1}} - \\frac{p_{2}}{p_{1}} &= \\frac{p_{2}}{p_{1}} (\\gamma - 1) \\\\\n",
    "-\\gamma \\frac{v_{2}}{v_{1}} + \\frac{v_{2}}{v_{1}} &= -\\frac{v_{2}}{v_{1}} (\\gamma - 1) \\\\\n",
    "\\end{aligned}\n",
    "$$\n",
    "\n",
    "First, let's work on the LHS, which is an addition of two terms. If we were to explore its arguments with `e.lhs.args`, we would see that the index of the first term is 0, and the index of the second term is 1."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "id": "bca04a7a-e7f2-416d-8ecf-75eef8527eee",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle \\frac{p_{2} v_{2} \\left(\\gamma + 1\\right)}{p_{1} v_{1}} = \\gamma - \\frac{\\gamma v_{2}}{v_{1}} + \\frac{\\gamma p_{2}}{p_{1}} + 1 + \\frac{v_{2}}{v_{1}} - \\frac{p_{2}}{p_{1}}$"
      ],
      "text/plain": [
       "Equation(p2*v2*(gamma + 1)/(p1*v1), gamma - gamma*v2/v1 + gamma*p2/p1 + 1 + v2/v1 - p2/p1)"
      ]
     },
     "execution_count": 12,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "e = e.applylhs(factor)\n",
    "e"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "62e591a0-b0c2-42d3-a135-6e28379ba01e",
   "metadata": {},
   "source": [
    "On the RHS we can simply collect terms:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 13,
   "id": "1151ec3d-062e-4569-b8c0-70cbbccf3511",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle \\frac{p_{2} v_{2} \\left(\\gamma + 1\\right)}{p_{1} v_{1}} = \\gamma + 1 + \\frac{v_{2} \\left(1 - \\gamma\\right)}{v_{1}} + \\frac{p_{2} \\left(\\gamma - 1\\right)}{p_{1}}$"
      ],
      "text/plain": [
       "Equation(p2*v2*(gamma + 1)/(p1*v1), gamma + 1 + v2*(1 - gamma)/v1 + p2*(gamma - 1)/p1)"
      ]
     },
     "execution_count": 13,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "e = e.dorhs.collect(p2 / p1).collect(v2 / v1)\n",
    "e"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "92cc0530-a704-442d-b86e-f39ece4eabea",
   "metadata": {},
   "source": [
    "Note, however, that we have two opposite terms: $(1 - \\gamma)$ and $(\\gamma - 1)$. It would be preferable to change $(1 - \\gamma) = -(\\gamma - 1)$."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 14,
   "id": "966b8f39-9639-4896-a805-78c730d529e9",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle \\frac{p_{2} v_{2} \\left(\\gamma + 1\\right)}{p_{1} v_{1}} = \\gamma + 1 - \\frac{v_{2} \\left(\\gamma - 1\\right)}{v_{1}} + \\frac{p_{2} \\left(\\gamma - 1\\right)}{p_{1}}$"
      ],
      "text/plain": [
       "Equation(p2*v2*(gamma + 1)/(p1*v1), gamma + 1 - v2*(gamma - 1)/v1 + p2*(gamma - 1)/p1)"
      ]
     },
     "execution_count": 14,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "e = e.subs(1 - gamma, Mul(-1, gamma - 1, evaluate=False))\n",
    "e"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "b4e4ad6e-cb41-40de-88cb-a411df34d496",
   "metadata": {},
   "source": [
    "We can now divide both sides by $\\gamma - 1$:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 15,
   "id": "3db49892-510c-4a46-a2a8-28295702c52e",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle \\frac{p_{2} v_{2} \\left(\\gamma + 1\\right)}{p_{1} v_{1} \\left(\\gamma - 1\\right)} = \\frac{\\gamma + 1 - \\frac{v_{2} \\left(\\gamma - 1\\right)}{v_{1}} + \\frac{p_{2} \\left(\\gamma - 1\\right)}{p_{1}}}{\\gamma - 1}$"
      ],
      "text/plain": [
       "Equation(p2*v2*(gamma + 1)/(p1*v1*(gamma - 1)), (gamma + 1 - v2*(gamma - 1)/v1 + p2*(gamma - 1)/p1)/(gamma - 1))"
      ]
     },
     "execution_count": 15,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "e = e / (gamma - 1)\n",
    "e"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "93909776-9cf0-4fef-9ad2-239634e3b7a4",
   "metadata": {},
   "source": [
    "There is a small problem to be solved on the RHS: the division was not performed term by term. Right now, the RHS is a multiplication instead of an addition:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 16,
   "id": "df7d4013-5a12-443b-ad92-b70b147ac7a1",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "sympy.core.mul.Mul"
      ]
     },
     "execution_count": 16,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "e.rhs.func"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 17,
   "id": "d365b463-c0f9-4339-8454-56051203f645",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/markdown": [
       "| idx | args |\n",
       "|:-----:|:------|\n",
       "| 0 | $\\frac{1}{\\gamma - 1}$ |\n",
       "| 1 | $\\gamma + 1 - \\frac{v_{2} \\left(\\gamma - 1\\right)}{v_{1}} + \\frac{p_{2} \\left(\\gamma - 1\\right)}{p_{1}}$ |\n"
      ],
      "text/plain": [
       "<IPython.core.display.Markdown object>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "text/plain": [
       "<sympy_equation.utils.table_of_expressions object at 0x72207f313ec0>"
      ]
     },
     "execution_count": 17,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "table_of_expressions(e.rhs)"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "cace8a3a-52ef-4ad1-a344-4f8a761aceb8",
   "metadata": {},
   "source": [
    "Let's evaluate that division term by term using `divide_term_by_term`, exposed by this module. This function automatically detects the deonominator:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 18,
   "id": "cac67b83-6ccf-4c85-a66e-e2d94a0e6658",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle \\frac{p_{2} v_{2} \\left(\\gamma + 1\\right)}{p_{1} v_{1} \\left(\\gamma - 1\\right)} = \\frac{\\gamma}{\\gamma - 1} + \\frac{1}{\\gamma - 1} - \\frac{v_{2}}{v_{1}} + \\frac{p_{2}}{p_{1}}$"
      ],
      "text/plain": [
       "Equation(p2*v2*(gamma + 1)/(p1*v1*(gamma - 1)), gamma/(gamma - 1) + 1/(gamma - 1) - v2/v1 + p2/p1)"
      ]
     },
     "execution_count": 18,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "e = e.applyrhs(divide_term_by_term)\n",
    "e"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "a4619cdd-12f2-41dc-9736-f7257d12beaf",
   "metadata": {},
   "source": [
    "Now we can combine the terms on the RHS where the denominator is $\\gamma - 1$. \n",
    "\n",
    "This is the perfect opportunity to showcase another function exposed by this module: `process_arguments_of_add`, which is going to replace the specified terms of an addition with the results of some user-provided function. First, let's get the indices of the terms we'd like to combine:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 19,
   "id": "dd07ad0c-7b14-49c4-8bc0-bd949dc39b51",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/markdown": [
       "| idx | args |\n",
       "|:-----:|:------|\n",
       "| 0 | $\\frac{1}{\\gamma - 1}$ |\n",
       "| 1 | $\\frac{\\gamma}{\\gamma - 1}$ |\n",
       "| 2 | $\\frac{p_{2}}{p_{1}}$ |\n",
       "| 3 | $- \\frac{v_{2}}{v_{1}}$ |\n"
      ],
      "text/plain": [
       "<IPython.core.display.Markdown object>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "text/plain": [
       "<sympy_equation.utils.table_of_expressions object at 0x72207f57e180>"
      ]
     },
     "execution_count": 19,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "table_of_expressions(e.rhs, mode=\"args\")"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "a19cd907-0a57-4c5c-a7a4-41cb6a1dff26",
   "metadata": {},
   "source": [
    "The terms to be replaced are located at index 0 and 1. We are going to replace them with the result of SymPy's `together`:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 20,
   "id": "b9a0771f-1446-4a35-8606-11b68ad7a8dc",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle \\frac{p_{2} v_{2} \\left(\\gamma + 1\\right)}{p_{1} v_{1} \\left(\\gamma - 1\\right)} = \\frac{\\gamma + 1}{\\gamma - 1} - \\frac{v_{2}}{v_{1}} + \\frac{p_{2}}{p_{1}}$"
      ],
      "text/plain": [
       "Equation(p2*v2*(gamma + 1)/(p1*v1*(gamma - 1)), (gamma + 1)/(gamma - 1) - v2/v1 + p2/p1)"
      ]
     },
     "execution_count": 20,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "e = e.applyrhs(lambda expr: process_arguments_of_add(expr, [0, 1], together))\n",
    "e"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "1efea972-de2d-4582-ab2d-e6f7f277c481",
   "metadata": {},
   "source": [
    "Let's now bring the pressure ratio to the LHS:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 21,
   "id": "f3ce7f81-83b2-4be1-bcea-9ab8288df01d",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle - \\frac{p_{2}}{p_{1}} + \\frac{p_{2} v_{2} \\left(\\gamma + 1\\right)}{p_{1} v_{1} \\left(\\gamma - 1\\right)} = \\frac{\\gamma + 1}{\\gamma - 1} - \\frac{v_{2}}{v_{1}}$"
      ],
      "text/plain": [
       "Equation(-p2/p1 + p2*v2*(gamma + 1)/(p1*v1*(gamma - 1)), (gamma + 1)/(gamma - 1) - v2/v1)"
      ]
     },
     "execution_count": 21,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "e = e - p2 / p1\n",
    "e"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "0f0bd0c6-159a-4ed1-886d-555b84cdab57",
   "metadata": {},
   "source": [
    "and collect it:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 22,
   "id": "43e7ef1e-6518-496e-bad4-d28d3c775526",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle \\frac{p_{2} \\left(-1 + \\frac{v_{2} \\left(\\gamma + 1\\right)}{v_{1} \\left(\\gamma - 1\\right)}\\right)}{p_{1}} = \\frac{\\gamma + 1}{\\gamma - 1} - \\frac{v_{2}}{v_{1}}$"
      ],
      "text/plain": [
       "Equation(p2*(-1 + v2*(gamma + 1)/(v1*(gamma - 1)))/p1, (gamma + 1)/(gamma - 1) - v2/v1)"
      ]
     },
     "execution_count": 22,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "e = e.dolhs.collect(p2 / p1)\n",
    "e"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "161a487a-25d8-45cc-be82-9ad3fc068bec",
   "metadata": {},
   "source": [
    "Finally, we can divide both side by the term in the parenthesys of the LHS. We can easily select it with:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 23,
   "id": "9f559d15-6ffb-40f4-9d55-afb9c6c04743",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/markdown": [
       "| idx | args |\n",
       "|:-----:|:------|\n",
       "| 0 | $p_{2}$ |\n",
       "| 1 | $\\frac{1}{p_{1}}$ |\n",
       "| 2 | $-1 + \\frac{v_{2} \\left(\\gamma + 1\\right)}{v_{1} \\left(\\gamma - 1\\right)}$ |\n"
      ],
      "text/plain": [
       "<IPython.core.display.Markdown object>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "toe = table_of_expressions(e.lhs)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 24,
   "id": "76bb015c-2dfa-4b8f-8d47-59862c3bd2ed",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle \\frac{p_{2}}{p_{1}} = \\frac{\\frac{\\gamma + 1}{\\gamma - 1} - \\frac{v_{2}}{v_{1}}}{-1 + \\frac{v_{2} \\left(\\gamma + 1\\right)}{v_{1} \\left(\\gamma - 1\\right)}}$"
      ],
      "text/plain": [
       "Equation(p2/p1, ((gamma + 1)/(gamma - 1) - v2/v1)/(-1 + v2*(gamma + 1)/(v1*(gamma - 1))))"
      ]
     },
     "execution_count": 24,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "e = e / toe[2]\n",
    "e"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "6b5fa4f1-030e-4909-bd36-f8169b8dfc52",
   "metadata": {},
   "outputs": [],
   "source": []
  }
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