Flow Equations

The governing flow equations for the variable-area formulation of the compressible Euler equations are as follows:

var element = document.getElementById("moose-equation-547e0420-637d-45fa-afcd-2a2ce3ece1dd");katex.render("\\pd{\\rho A}{t} + \\pd{\\rho u A}{x} = 0 \\,,", element, {displayMode:true,throwOnError:false,macros:{"\\eqc":"\\,,","\\eqp":"\\,.","\\pd":"\\frac{\\partial #1}{\\partial #2}","\\pr":"\\left(#1\\right)","\\ddt":"\\frac{d #1}{d t}"}});

var element = document.getElementById("moose-equation-8b202991-98dc-4d6c-9c2e-0b4b5fd09782");katex.render("\\pd{\\rho u A}{t} + \\pd{(\\rho u^2 + p) A}{x} = p \\pd{A}{x} - F A + \\rho g_x A \\,,", element, {displayMode:true,throwOnError:false,macros:{"\\eqc":"\\,,","\\eqp":"\\,.","\\pd":"\\frac{\\partial #1}{\\partial #2}","\\pr":"\\left(#1\\right)","\\ddt":"\\frac{d #1}{d t}"}});

var element = document.getElementById("moose-equation-a47711cf-9323-46df-ad68-e6575d8182b6");katex.render("\\pd{\\rho E A}{t} + \\pd{u (\\rho E + p) A}{x} - \\pd{}{x}\\pr{k \\pd{T}{x} A} = \\rho u g_x A + q''' A \\,,", element, {displayMode:true,throwOnError:false,macros:{"\\eqc":"\\,,","\\eqp":"\\,.","\\pd":"\\frac{\\partial #1}{\\partial #2}","\\pr":"\\left(#1\\right)","\\ddt":"\\frac{d #1}{d t}"}});

where

$var element = document.getElementById("moose-equation-c74a5332-637b-4b97-841d-24e12aadc676");katex.render("t", element, {displayMode:false,throwOnError:false,macros:{"\\eqc":"\\,,","\\eqp":"\\,.","\\pd":"\\frac{\\partial #1}{\\partial #2}","\\pr":"\\left(#1\\right)","\\ddt":"\\frac{d #1}{d t}"}});$ is time,
$var element = document.getElementById("moose-equation-4e0c2fbf-efce-4999-ba46-79c6d943720c");katex.render("x", element, {displayMode:false,throwOnError:false,macros:{"\\eqc":"\\,,","\\eqp":"\\,.","\\pd":"\\frac{\\partial #1}{\\partial #2}","\\pr":"\\left(#1\\right)","\\ddt":"\\frac{d #1}{d t}"}});$ is the spatial position along the chosen axis,
$var element = document.getElementById("moose-equation-9fb90692-7fb8-4543-8d4e-ad3f8df7c1bd");katex.render("A", element, {displayMode:false,throwOnError:false,macros:{"\\eqc":"\\,,","\\eqp":"\\,.","\\pd":"\\frac{\\partial #1}{\\partial #2}","\\pr":"\\left(#1\\right)","\\ddt":"\\frac{d #1}{d t}"}});$ is the cross-sectional area,
$var element = document.getElementById("moose-equation-a8009b53-aa08-428a-9a9a-e681ba2e9c6f");katex.render("\\rho", element, {displayMode:false,throwOnError:false,macros:{"\\eqc":"\\,,","\\eqp":"\\,.","\\pd":"\\frac{\\partial #1}{\\partial #2}","\\pr":"\\left(#1\\right)","\\ddt":"\\frac{d #1}{d t}"}});$ is the density,
$var element = document.getElementById("moose-equation-157eab2a-714c-4294-b939-fb033aed9d99");katex.render("u", element, {displayMode:false,throwOnError:false,macros:{"\\eqc":"\\,,","\\eqp":"\\,.","\\pd":"\\frac{\\partial #1}{\\partial #2}","\\pr":"\\left(#1\\right)","\\ddt":"\\frac{d #1}{d t}"}});$ is the axial velocity,
$var element = document.getElementById("moose-equation-fc97f323-9953-48dd-95bc-6f0a7230bc29");katex.render("E", element, {displayMode:false,throwOnError:false,macros:{"\\eqc":"\\,,","\\eqp":"\\,.","\\pd":"\\frac{\\partial #1}{\\partial #2}","\\pr":"\\left(#1\\right)","\\ddt":"\\frac{d #1}{d t}"}});$ is the specific total energy,
$var element = document.getElementById("moose-equation-7a403e68-67e7-4645-bb16-60a4f9f82b38");katex.render("p", element, {displayMode:false,throwOnError:false,macros:{"\\eqc":"\\,,","\\eqp":"\\,.","\\pd":"\\frac{\\partial #1}{\\partial #2}","\\pr":"\\left(#1\\right)","\\ddt":"\\frac{d #1}{d t}"}});$ is the pressure,
$var element = document.getElementById("moose-equation-7e528c24-972c-4711-9b2b-34509aa847c3");katex.render("T", element, {displayMode:false,throwOnError:false,macros:{"\\eqc":"\\,,","\\eqp":"\\,.","\\pd":"\\frac{\\partial #1}{\\partial #2}","\\pr":"\\left(#1\\right)","\\ddt":"\\frac{d #1}{d t}"}});$ is the temperature,
$var element = document.getElementById("moose-equation-65997dbc-35e9-43ba-a91b-baf884663ced");katex.render("k", element, {displayMode:false,throwOnError:false,macros:{"\\eqc":"\\,,","\\eqp":"\\,.","\\pd":"\\frac{\\partial #1}{\\partial #2}","\\pr":"\\left(#1\\right)","\\ddt":"\\frac{d #1}{d t}"}});$ is the thermal conductivity,
$var element = document.getElementById("moose-equation-4a93a2b6-6e30-4ce5-b208-0bdf9025ff02");katex.render("F", element, {displayMode:false,throwOnError:false,macros:{"\\eqc":"\\,,","\\eqp":"\\,.","\\pd":"\\frac{\\partial #1}{\\partial #2}","\\pr":"\\left(#1\\right)","\\ddt":"\\frac{d #1}{d t}"}});$ is the viscous drag force density,
$var element = document.getElementById("moose-equation-4c5d3762-146f-47f9-a953-9bb852eacb3d");katex.render("g_x", element, {displayMode:false,throwOnError:false,macros:{"\\eqc":"\\,,","\\eqp":"\\,.","\\pd":"\\frac{\\partial #1}{\\partial #2}","\\pr":"\\left(#1\\right)","\\ddt":"\\frac{d #1}{d t}"}});$ is the component of acceleration due to gravity in the axial direction, and
$var element = document.getElementById("moose-equation-1015bcdf-370b-4822-a2aa-9e495c8a29ae");katex.render("q'''", element, {displayMode:false,throwOnError:false,macros:{"\\eqc":"\\,,","\\eqp":"\\,.","\\pd":"\\frac{\\partial #1}{\\partial #2}","\\pr":"\\left(#1\\right)","\\ddt":"\\frac{d #1}{d t}"}});$ is the heat source rate density.

The viscous drag force density is computed as

var element = document.getElementById("moose-equation-64894b7b-1c84-4dda-8dde-796cb39c30db");katex.render("F = \\frac{f \\rho u |u| A}{2 D_h} \\,,", element, {displayMode:true,throwOnError:false,macros:{"\\eqc":"\\,,","\\eqp":"\\,.","\\pd":"\\frac{\\partial #1}{\\partial #2}","\\pr":"\\left(#1\\right)","\\ddt":"\\frac{d #1}{d t}"}});

where $var element = document.getElementById("moose-equation-798e7701-20dc-44fc-b6f7-3d39fe0c9da8");katex.render("f", element, {displayMode:false,throwOnError:false,macros:{"\\eqc":"\\,,","\\eqp":"\\,.","\\pd":"\\frac{\\partial #1}{\\partial #2}","\\pr":"\\left(#1\\right)","\\ddt":"\\frac{d #1}{d t}"}});$ is the Darcy friction factor and $var element = document.getElementById("moose-equation-4a68eb86-ebdb-45cb-8d63-8f8d9b52b1b7");katex.render("D_h", element, {displayMode:false,throwOnError:false,macros:{"\\eqc":"\\,,","\\eqp":"\\,.","\\pd":"\\frac{\\partial #1}{\\partial #2}","\\pr":"\\left(#1\\right)","\\ddt":"\\frac{d #1}{d t}"}});$ is the hydraulic diameter.

Passive Transport

Optionally, equations may be added to model the passive transport of scalar quantities $var element = document.getElementById("moose-equation-63404535-daf9-4f30-974e-9f8e3eec016e");katex.render("y", element, {displayMode:false,throwOnError:false,macros:{"\\eqc":"\\,,","\\eqp":"\\,.","\\pd":"\\frac{\\partial #1}{\\partial #2}","\\pr":"\\left(#1\\right)","\\ddt":"\\frac{d #1}{d t}"}});$ , corresponding to some amount per unit volume, where "amount" may be a mass or a number of particles, for example:

var element = document.getElementById("moose-equation-e8c3d762-1180-47da-a222-331098151edb");katex.render("\\pd{y A}{t} + \\pd{y u A}{x} = 0 \\,.", element, {displayMode:true,throwOnError:false,macros:{"\\eqc":"\\,,","\\eqp":"\\,.","\\pd":"\\frac{\\partial #1}{\\partial #2}","\\pr":"\\left(#1\\right)","\\ddt":"\\frac{d #1}{d t}"}});

Each scalar quantity $var element = document.getElementById("moose-equation-ded6c331-c38d-4453-9443-477531b0aa42");katex.render("y", element, {displayMode:false,throwOnError:false,macros:{"\\eqc":"\\,,","\\eqp":"\\,.","\\pd":"\\frac{\\partial #1}{\\partial #2}","\\pr":"\\left(#1\\right)","\\ddt":"\\frac{d #1}{d t}"}});$ advects with the carrier fluid velocity $var element = document.getElementById("moose-equation-abb8a3e9-85c3-4635-a8de-c874cf23a2bc");katex.render("u", element, {displayMode:false,throwOnError:false,macros:{"\\eqc":"\\,,","\\eqp":"\\,.","\\pd":"\\frac{\\partial #1}{\\partial #2}","\\pr":"\\left(#1\\right)","\\ddt":"\\frac{d #1}{d t}"}});$ without providing any feedback to the flow.

Flow Equations
Passive Transport