{
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   "id": "ca8508c0-dfaa-43a8-ae0d-728696106584",
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   "outputs": [],
   "source": [
    "\n"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "06f676ad-f84e-49df-afa9-8e4846df53cb",
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   "source": [
    "Downstream Mach number:\n",
    "M22=1+γ−12M12γM12−γ−12\n",
    "M22​=γM12​−2γ−1​1+2γ−1​M12​​\n",
    "Pressure ratio:\n",
    "P2P1=1+2γγ+1(M12−1)\n",
    "P1​P2​​=1+γ+12γ​(M12​−1)\n",
    "Temperature ratio:\n",
    "T2T1=(1+γ−12M12)(2γM12−(γ−1))(γ+1)2M12\n",
    "T1​T2​​=(γ+1)2M12​(1+2γ−1​M12​)(2γM12​−(γ−1))​\n",
    "Density ratio:\n",
    "ρ2ρ1=(γ+1)M122+(γ−1)M12\n",
    "ρ1​ρ2​​=2+(γ−1)M12​(γ+1)M12​​"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "id": "f2363adc-cdd6-491d-a403-65c7182c12a7",
   "metadata": {},
   "outputs": [
    {
     "ename": "SyntaxError",
     "evalue": "invalid character '−' (U+2212) (4049633316.py, line 2)",
     "output_type": "error",
     "traceback": [
      "\u001b[0;36m  Cell \u001b[0;32mIn[2], line 2\u001b[0;36m\u001b[0m\n\u001b[0;31m    M22=1+γ−12M12γM12−γ−12\u001b[0m\n\u001b[0m           ^\u001b[0m\n\u001b[0;31mSyntaxError\u001b[0m\u001b[0;31m:\u001b[0m invalid character '−' (U+2212)\n"
     ]
    }
   ],
   "source": [
    "Downstream Mach Number\n",
    "M22=1+γ−12M12γM12−γ−12\n",
    "M22​=γM12​−2γ−1​1+2γ−1​M12​​\n",
    "\n",
    "Pressure Ratio\n",
    "P2P1=1+2γγ+1(M12−1)\n",
    "P1​P2​​=1+γ+12γ​(M12​−1)\n",
    "\n",
    "Temperature Ratio\n",
    "T2T1=(1+γ−12M12)(2γM12−(γ−1))(γ+1)2M12\n",
    "T1​T2​​=(γ+1)2M12​(1+2γ−1​M12​)(2γM12​−(γ−1))​\n",
    "\n",
    "Density Ratio\n",
    "ρ2ρ1=(γ+1)M122+(γ−1)M12\n",
    "ρ1​ρ2​​=2+(γ−1)M12​(γ+1)M12​​\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "fdf90d22-be4d-4862-9b23-76434d2e2305",
   "metadata": {},
   "outputs": [],
   "source": []
  }
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