# Non-Newtonian Flow Through Expansion Channel

## Overview

The purpose of this numerical simulation is to validate the flow velocity profile for a Non-Newtonian fluid via the Power-Law model.

The numerical simulation were carried out using the Reynolds-Averaged Navier–Stokes (RANS) approach at laminar flow conditions. The results of SimScale simulation runs were compared to the analytical results shown in [1] [2]. The flow regime selected for the study has a Reynolds number of Re=40

$Re=40$

.

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## Geometry

The geometry of the study is a 2 dimensional channel with a 3:1 expansion (see Fig.1.). A brief description of the dimensions are provided by the table below.

• with respect to expansion step.

## Domain and Analysis type

The domain is the internal region of the geometry with the domain extents same as the geometrical dimensions. For this study a full hexahedral structured mesh was created with the “blockMesh” open-source tool (see Fig.2.). The Mesh was refined at the expansion step in both horizontal and vertical directions. Two meshes, mesh M_2 (intermediate) and mesh M_3 (fine) were used for the study to check for mesh independence of results. The details of the mesh are listed in the following table:

Mesh and Element types :

Mesh type Number of cells Type
blockMesh 30.7 – 77.6 thousand 3D hexahedral

The numerical analysis performed is detailed as follows:

Tool Type : OPENFOAM®

Turbulence Model : Laminar

Non-Newtonian model : Power-Law

## Simulation Setup

Fluid:

Non-Newtonian

• normalized Consistancy Index (by density): k[m2/s]
$k\left[{m}^{2}/s\right]$

based on Re=40

$Re=40$

• Flow/Power Index n=0.52

Boundary Conditions:

For the inlet boundary, a fixed velocity condition was applied, while a pressure boundary condition was applied at the outlet. No-slip condition was applied for the walls and empty condition for the side faces. The following table provides the further details.

Boundary type Velocity Pressure
Inlet laminar Fixed Value: 0.5 ms1

Outlet Zero Gradient Fixed Value: 0 Pa

Wall no-slip Fixed Value: 0.0 ms1

Custom 2D Empty 2D Empty

## Results

The numerical simulation results for the various Power Index ranging from n=0.52

$n=0.5-2$

for a Re=40

$Re=40$

are compared with analytical formulation given by Tanner [1] and shown by [2]. The formulations for the Generalized Reynolds number for non-Newtonian flows for this case is as follows:

Generalized Reynolds number: Re=ρ V2nHnK

where, ρ

$\rho$

is the density, V

$V$

is the flow inlet velocity, H

$H$

is the inlet height, n

$n$

is the power index and K

$K$

is the consistency factor. All quantities here are in standard S.I units.

Two meshes M_2 and M_3 were analyzed. As no further improvement was observed for mesh M_3 the results presented are for mesh M_2. A comparison of the velocity profile in the fully developed region downstream of the expansion at X=28

$X=28$

is shown in fig.3. The figure shows the normalized velocity profile along the normalized channel height for n>1

$n>1$

shear thickening or dilatant fluid (e.g corn starch water), n=1

$n=1$

Newtonian fluid and n<1

$n<1$

Shear thinning or pseudoplastic fluids (e.g ketchup, paint, blood).

The Velocity contours with streamlines and corresponding profiles for Power Index n=0.5,1.0,2.0

$n=0.5,1.0,2.0$

are shown in the figure Fig.4 below.

## References

 [1] (1, 2) Tanner, “Engineering Rheology”, Oxford University Press, Oxford, 1992.
 [2] (1, 2) MANICA, A.L. de BORTOLI, “Simulation of Incompressible Non-Newtonian Flows Through Channels with Sudden Expansion Using the Power-Law Model”.

## Disclaimer

This offering is not approved or endorsed by OpenCFD Limited, producer and distributor of the OpenFOAM software and owner of the OPENFOAM® and OpenCFD® trade marks. OPENFOAM® is a registered trade mark of OpenCFD Limited, producer and distributor of the OpenFOAM software.