11 "The isothermal two-fluid model\n",
12 "==============================\n",
14 "The model consists in the phasic mass and momentum balance equations.\n",
16 "The main unknowns are $\\alpha$, $P$, $\\vec{u}_g$, $\\vec{u}_l$. The model uses stiffened gas laws $p_g(\\rho_g)$ and $p_l(\\rho_l)$ valid around the saturation points $(P=1 bar, T=373K)$ or $(P=155 bars, T=618K)$ depending on the value of the enum \\ref pressureEstimate.\n",
18 "The subscript $k$ stands for $l$ for the liquid phase and $g$ for the gas phase. The common\n",
19 "averaged pressure of the two phases is denoted by $p$. \n",
21 "In our model, pressure equilibrium between the two phases is postulated, and the resulting system to solve is:\n",
24 "\\begin{array}{ccll}\n",
25 " \\frac{\\partial m_g}{\\partial t}& +& \\nabla \\cdot \\vec{q}_g &= 0,\\\\[1.5ex]\n",
26 "\\frac{\\partial m_l}{\\partial t} &+ &\\nabla \\cdot \\vec{q}_l &= 0,\\\\[1.5ex]\n",
27 "\\frac{\\partial \\vec{q}_g}{\\partial t}& +& \\nabla \\cdot (\\vec{q}_g\\otimes\\frac{\\vec{q}_g}{m_g})+ \\alpha_g \\vec\\nabla p&\\\\[1.5ex] \n",
28 " &+&\\Delta p \\nabla \\alpha_g -\\nu_g\\Delta \\vec{u}_g &= m_g\\vec{g}-K\\rho_m||\\vec{u}_g-\\vec{u}_l||(\\vec{u}_g-\\vec{u}_l)-K_s\\delta(x)m_g||\\vec{u}_g||\\vec{u}_g\\\\[1.5ex]\n",
29 "\\frac{\\partial \\vec{q}_l}{\\partial t}& +& \\nabla \\cdot (\\vec{q}_l\\otimes\\frac{\\vec{q}_l}{m_l})+ \\alpha_l \\vec\\nabla p&\\\\[1.5ex]\n",
30 "&+&\\Delta p \\nabla \\alpha_l -\\nu_l\\Delta \\vec{u}_l &= m_l\\vec{g}-K\\rho_m||\\vec{u}_l-\\vec{u}_g||(\\vec{u}_l-\\vec{u}_g)-K_s\\delta(x)m_l||\\vec{u}_l||\\vec{u}_l\\\\\n",
36 "- $\\nu_k$ is the viscosity of phase $k$, set by [IsothermalTwoFluid](../../../Models/inc/IsothermalTwoFluid.hxx)::setViscosity\n",
37 "- $\\Delta p$ denotes the pressure default $p-p_k$ between the bulk average pressure and the interfacial average pressure.\n",
38 "- $\\vec g$ the gravity vector ([IsothermalTwoFluid](../../../Models/inc/IsothermalTwoFluid.hxx)::setGravity)\n",
39 "- $K$ the interphase friction coefficient ([IsothermalTwoFluid](../../../Models/inc/IsothermalTwoFluid.hxx)::setDragCoeffs),\n",
40 "- $K_s(\\vec x)$ the singular friction function, $\\delta_s(\\vec x)$ the Dirac delta function with support on the set $s$ ([IsothermalTwoFluid](../../../Models/inc/IsothermalTwoFluid.hxx)::setPressureLossField).\n",
44 " \\left\\{\\begin{array}{clc}\n",
45 "\t\\alpha_g +\\alpha_l &=& 1 \\\\[1.5ex]\n",
46 "\tm_k &=& \\alpha_k \\rho_k \\\\[1.5ex]\n",
47 "\t\\vec{q}_k &=& \\alpha_k \\rho_k \\vec{u}_k\n",
48 "\t\\end{array}\\right..\n",
52 "The class : [IsothermalTwoFluid](../../../Models/inc/IsothermalTwoFluid.hxx) implements the isentropic two-fluid model \n",
54 "\\subpage ExampleIsothermalPage \"Here are C and Python example scripts using the isothermal two-fluid model\"\t\n",
58 "<IPython.core.display.Markdown object>"
62 "output_type": "display_data"
66 "from IPython.display import display, Markdown\n",
67 "display(Markdown('IsothermalPage.md'))"
73 "display_name": "Python 2",
82 "file_extension": ".py",
83 "mimetype": "text/x-python",
85 "nbconvert_exporter": "python",
86 "pygments_lexer": "ipython2",