Mass transport in a Stokes flow in a pipe

Table of Contents

1. Running the model
2. Data files
- 2.1. Json file
- 2.2. Cfg file
3. Geometry
- 3.1. Model & Toolbox
4. Input parameters
5. Numerical Experiments

1. Running the model

The command line to run this pipestokes case

mpirun -np 4 feelpp_toolbox_fluid --case "github:{repo:toolbox,path:examples/modules/heatfluid/examples/pipestockes_mass}"

2. Data files

2.1. Json file

JSON file - [Edit the file]

2.2. Cfg file

CFG file - [Edit the file]

3. Geometry

3.1. Model & Toolbox

We consider a 2D model representative of a laminar incompressible flow around an obstacle. The flow domain, named $\Omega_f$ , is contained into the rectangle $\lbrack 0,Long \rbrack \times \lbrack 0,Haut \rbrack$ . It is characterised, in particular, by its dynamic viscosity $\mu_f$ and by its density $\rho_f$ .

The goal of this benchmark is to couple the Stockes equations and the Concentration equations.
we remind that the Stokes equation are

{\begin{matrix} μ Δ u + \nabla p = f \nabla \cdot u = 0 \end{matrix}

$\left\{ \begin{aligned} \boldsymbol{\mu}\boldsymbol{\Delta u}+\boldsymbol{\nabla p}=f\\ \boldsymbol{\nabla}\cdot\boldsymbol{u}=0 \end{aligned} \right.$

with $\boldsymbol{\mu}$ is the dynamic viscosity, $\boldsymbol{p}$ is the pressure , $f$ the source and u the velocity.

And the Concentration equations is

\begin{matrix} \frac{\partial C}{\partial t} + u \cdot \nabla C - \nabla \cdot (D_{p} \nabla C) = 0, in Ω \end{matrix}

$\begin{array}[c]{rl} \frac{\partial C}{\partial t} + \boldsymbol{u} \cdot \nabla C - \nabla \cdot \left( D_{p} \nabla C \right) = 0, \quad \text{ in } \Omega \end{array}$

With $D_{p}$ the diffusion coefficient on the plasma.

We used the heat fluid toolbox, we replaced the temperature by the Concentration, k by $D_{p}$ , and we posed $\rho C_{p}=1$ to have the same kind of equations.

4. Input parameters

The following table displays the various fixed and variables parameters of this test-case.

Table 1. Fixed and Variable Input Parameters
Name	Description	Units
$u$	fluid velocity	$m/s$
$\rho$	density of the fluid	$kg/m^3$
$\nu$	dynamic viscosity	$kg/(m×s)$
$p$	pression	$Pa$
$f$	source term	$kg/(m^3×s)$
$C_p$	thermal capacity	$J/(kg∗K)$
$T$	Temperature	$K$
$Q$	heat source	$W.m^{-3}$
$D_{p}$	the diffusion coefficient on the plasma	$\mu m²/s$

4.1. initial condition

For the fluid:

We use a parabolic velocity profile, in order to describe the flow inlet by $\Gamma_{in}$ , which can be express by

$v_{inlet}=D y(height-y)$

To determine $D$ , we know that for $y=\frac{height}{2}$ we have the maximal velocity, so

$D=\frac{v_{max}}{\frac{height}{2}\left(height-\frac{height}{2}\right)}$

For the Concentration:

We give as source this Concentration

$C_{in}=300*(y>0.15)*(y<0.5)+(293.15*(y<(0.15-1e-9)))+(293.15*(y>(0.5-1e-9)))$

4.2. Materials

   "Materials":
    {
        "Fluid":{
            "rho":"1.0",
            "mu":"1.0",
            "k":"2400.e-6",
            "Cp":"1.0"

        }
    }

4.3. Boundary conditions

For the fluid:

We set

On $\Gamma_{in}$ , an inflow Dirichlet condition : $\boldsymbol{u}_f=(v_{in},0)$
On $\Gamma_{wall}$ and $\Gamma_{obst}$ , a homogeneous Dirichlet condition : $\boldsymbol{u}_f=\boldsymbol{0}$
On $\Gamma_{out}$ , a Neumann condition : $\boldsymbol{\sigma}_f\boldsymbol{ n }_f=\boldsymbol{0}$

For the Concentration:

On $\Gamma_{in}$ , an inflow Dirichlet condition : $\boldsymbol{C}_f=C_{in}$

     "BoundaryConditions":
    {
        "velocity":
        {
            "Dirichlet":
            {
                "inlet":
                {
                   "expr":"{D*y*(height-y),0}:y:height:D"
                },
                "wall1":
                {
                    "expr":"{0,0}"
                },
                "wall2":
                {
                    "expr":"{0,0}"
                }
            }
        },
        "fluid":
        {
            "outlet":
            {
                "outlet":
                {
                    "expr":"0"
                }
            }
        },
        "temperature":
        {
            "Dirichlet":
            {
                "inlet":
                {
                    "expr":"300*(y>0.15)*(y<0.5)+(293.15*(y<(0.15-1e-9)))+(293.15*(y>(0.5-1e-9))):y"
                }
            }
        }
    }

4.4. Fields

We are intersting in the visualisation of the three fields : the velocity, the pressure and the temperature of the fluid

    "Exports":
    {
        "fields":["fluid.velocity","fluid.pressure","heat.temperature","fluid.pid"]
    }

4.5. Measures

the pressure is measured on two points to see the behavior of the pressure as a function of time

          "Measures":
            {
                "Forces":"wall2",
                "Points":
                {
                    "pointA":
                    {
                        "coord":"{0.6,0.2,0}",
                        "fields":"pressure"
                    },
                    "pointB":
                    {
                        "coord":"{0.15,0.2,0}",
                        "fields":"pressure"
                    }
                }
            }

5. Numerical Experiments

We run this model, using the command labeled at the top, we have the following results.

For The temperature: