# 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 $$Re = 40$$.

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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.

Fig.1. Geometry of the 3:1 Expansion channel

• 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

Fig.2. Expansion channel mesh used for the SimScale case

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): $$k [m^2/s]$$ based on $$Re=40$$
• Flow/Power Index $$n = 0.5 - 2\$$

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: $$0.5\ ms^{-1}$$ Zero Gradient
Outlet Zero Gradient Fixed Value: $$0\ Pa$$
Wall no-slip Fixed Value: $$0.0\ ms^{-1}$$ Zero Gradient
Custom 2D Empty 2D Empty

## Results¶

The numerical simulation results for the various Power Index ranging from $$n = 0.5 - 2$$ for a $$Re=40$$ 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: $$Re=\frac{\rho\ V^{2-n} H^{n} }{K}$$

where, $$\rho$$ is the density, $$V$$ is the flow inlet velocity, $$H$$ is the inlet height, $$n$$ is the power index and $$K$$ 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 $$X = 28$$ is shown in fig.3. The figure shows the normalized velocity profile along the normalized channel height for $$n > 1$$ shear thickening or dilatant fluid (e.g corn starch water), $$n = 1$$ Newtonian fluid and $$n < 1$$ Shear thinning or pseudoplastic fluids (e.g ketchup, paint, blood).

Fig.3. Result comparison of numerical and analytical data. Normalized velocity profile along channel height at x=28.

The Velocity contours with streamlines and corresponding profiles for Power Index $$n = 0.5, 1.0, 2.0$$ are shown in the figure Fig.4 below.

Fig.4. Mean velocity contours with streamlines and profiles for n = 0.5, 1.0 and 2.0.

## 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.