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-a4cf7bf2-ce6f-4090-8de4-793818e68f88");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-1927e07f-61a0-4ac6-a597-6eedc4c87743");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-25de674a-3ca1-487e-b22a-abe0dc73d4dc");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

$var element = document.getElementById("moose-equation-55377325-7840-42cf-800c-f06823792c24");katex.render("t", element, {displayMode:false,throwOnError:false,macros:{"\\eqc":"\\,,","\\eqp":"\\,.","\\pd":"\\frac{\\partial #1}{\\partial #2}","\\pr":"\\left(#1\\right)","\\ddt":"\\frac{d #1}{d t}"}});$ is time,
$var element = document.getElementById("moose-equation-17fcc5fb-6e68-4175-8750-91a6a1c3e808");katex.render("x", element, {displayMode:false,throwOnError:false,macros:{"\\eqc":"\\,,","\\eqp":"\\,.","\\pd":"\\frac{\\partial #1}{\\partial #2}","\\pr":"\\left(#1\\right)","\\ddt":"\\frac{d #1}{d t}"}});$ is the spatial position along the chosen axis,
$var element = document.getElementById("moose-equation-bd7d0db1-cfeb-4d09-aab0-48b25dd915a1");katex.render("A", element, {displayMode:false,throwOnError:false,macros:{"\\eqc":"\\,,","\\eqp":"\\,.","\\pd":"\\frac{\\partial #1}{\\partial #2}","\\pr":"\\left(#1\\right)","\\ddt":"\\frac{d #1}{d t}"}});$ is the cross-sectional area,
$var element = document.getElementById("moose-equation-294e0e02-7bdc-440f-ac00-ac9894e1093f");katex.render("\\rho", element, {displayMode:false,throwOnError:false,macros:{"\\eqc":"\\,,","\\eqp":"\\,.","\\pd":"\\frac{\\partial #1}{\\partial #2}","\\pr":"\\left(#1\\right)","\\ddt":"\\frac{d #1}{d t}"}});$ is the density,
$var element = document.getElementById("moose-equation-664e0cc9-d622-45d3-b358-74605c9fcb80");katex.render("u", element, {displayMode:false,throwOnError:false,macros:{"\\eqc":"\\,,","\\eqp":"\\,.","\\pd":"\\frac{\\partial #1}{\\partial #2}","\\pr":"\\left(#1\\right)","\\ddt":"\\frac{d #1}{d t}"}});$ is the axial velocity,
$var element = document.getElementById("moose-equation-50f40ca0-c463-4133-bae4-687093595533");katex.render("E", element, {displayMode:false,throwOnError:false,macros:{"\\eqc":"\\,,","\\eqp":"\\,.","\\pd":"\\frac{\\partial #1}{\\partial #2}","\\pr":"\\left(#1\\right)","\\ddt":"\\frac{d #1}{d t}"}});$ is the specific total energy,
$var element = document.getElementById("moose-equation-bfb928e1-014f-42a6-8630-39cfe29505b5");katex.render("p", element, {displayMode:false,throwOnError:false,macros:{"\\eqc":"\\,,","\\eqp":"\\,.","\\pd":"\\frac{\\partial #1}{\\partial #2}","\\pr":"\\left(#1\\right)","\\ddt":"\\frac{d #1}{d t}"}});$ is the pressure,
$var element = document.getElementById("moose-equation-7a038e08-08fd-472e-aa13-a9e995c82abe");katex.render("T", element, {displayMode:false,throwOnError:false,macros:{"\\eqc":"\\,,","\\eqp":"\\,.","\\pd":"\\frac{\\partial #1}{\\partial #2}","\\pr":"\\left(#1\\right)","\\ddt":"\\frac{d #1}{d t}"}});$ is the temperature,
$var element = document.getElementById("moose-equation-26199d0a-3146-4743-aa5a-a7b96670181b");katex.render("k", element, {displayMode:false,throwOnError:false,macros:{"\\eqc":"\\,,","\\eqp":"\\,.","\\pd":"\\frac{\\partial #1}{\\partial #2}","\\pr":"\\left(#1\\right)","\\ddt":"\\frac{d #1}{d t}"}});$ is the thermal conductivity,
$var element = document.getElementById("moose-equation-6fa99f8f-be97-41aa-bcec-5da95af2e861");katex.render("F", element, {displayMode:false,throwOnError:false,macros:{"\\eqc":"\\,,","\\eqp":"\\,.","\\pd":"\\frac{\\partial #1}{\\partial #2}","\\pr":"\\left(#1\\right)","\\ddt":"\\frac{d #1}{d t}"}});$ is the viscous drag force density,
$var element = document.getElementById("moose-equation-331d7ba0-3aaf-4228-bfb3-027f125fbb44");katex.render("g_x", element, {displayMode:false,throwOnError:false,macros:{"\\eqc":"\\,,","\\eqp":"\\,.","\\pd":"\\frac{\\partial #1}{\\partial #2}","\\pr":"\\left(#1\\right)","\\ddt":"\\frac{d #1}{d t}"}});$ is the component of acceleration due to gravity in the axial direction, and
$var element = document.getElementById("moose-equation-9bc2183a-3718-460e-82cb-c4fe31cef35f");katex.render("q'''", element, {displayMode:false,throwOnError:false,macros:{"\\eqc":"\\,,","\\eqp":"\\,.","\\pd":"\\frac{\\partial #1}{\\partial #2}","\\pr":"\\left(#1\\right)","\\ddt":"\\frac{d #1}{d t}"}});$ is the heat source rate density.

The viscous drag force density is computed as

var element = document.getElementById("moose-equation-8556a815-71cb-40d1-b004-60b46fb18cd9");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 $var element = document.getElementById("moose-equation-47dc9beb-a45c-4086-9626-70454360b1bd");katex.render("f", element, {displayMode:false,throwOnError:false,macros:{"\\eqc":"\\,,","\\eqp":"\\,.","\\pd":"\\frac{\\partial #1}{\\partial #2}","\\pr":"\\left(#1\\right)","\\ddt":"\\frac{d #1}{d t}"}});$ is the Darcy friction factor and $var element = document.getElementById("moose-equation-2fbb076e-f738-4d52-baa0-5270b129fde3");katex.render("D_h", element, {displayMode:false,throwOnError:false,macros:{"\\eqc":"\\,,","\\eqp":"\\,.","\\pd":"\\frac{\\partial #1}{\\partial #2}","\\pr":"\\left(#1\\right)","\\ddt":"\\frac{d #1}{d t}"}});$ is the hydraulic diameter.

Passive Transport

Optionally, equations may be added to model the passive transport of scalar quantities $var element = document.getElementById("moose-equation-c352c265-63cb-4eb2-90e7-21a9a2274fe4");katex.render("y", element, {displayMode:false,throwOnError:false,macros:{"\\eqc":"\\,,","\\eqp":"\\,.","\\pd":"\\frac{\\partial #1}{\\partial #2}","\\pr":"\\left(#1\\right)","\\ddt":"\\frac{d #1}{d t}"}});$ , 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-62139644-8d05-4205-a3e0-576239d971df");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 $var element = document.getElementById("moose-equation-535dbb9d-70a8-4ddc-b415-b6bd7dd87265");katex.render("y", element, {displayMode:false,throwOnError:false,macros:{"\\eqc":"\\,,","\\eqp":"\\,.","\\pd":"\\frac{\\partial #1}{\\partial #2}","\\pr":"\\left(#1\\right)","\\ddt":"\\frac{d #1}{d t}"}});$ advects with the carrier fluid velocity $var element = document.getElementById("moose-equation-302f708a-ff7c-462e-9d78-2c19abd3f18f");katex.render("u", element, {displayMode:false,throwOnError:false,macros:{"\\eqc":"\\,,","\\eqp":"\\,.","\\pd":"\\frac{\\partial #1}{\\partial #2}","\\pr":"\\left(#1\\right)","\\ddt":"\\frac{d #1}{d t}"}});$ without providing any feedback to the flow.

Flow Equations
Passive Transport