Non-Newtonian Flow
Introduction
This model shows the influence of shear rate dependent viscosity on the flow of a linear polystyrene solution. For this type of flow, you can u the Carreau viscosity model. Due to rotational symmetry, it is possible to reduce the model dimensions from 3D to axisymmetric 2D (e Figure 1).
Model Definition
For non-Newtonian flow, µ denotes the viscosity (kg/(m·s)), u the velocity (m/s), ρ the density of the fluid (kg/m 3) and p the pressure (Pa). The equations to solve are the momentum and continuity equations.
(1)
In the Carreau model, the viscosity depends on the shear rate, , which for an axisymmetric model in cylindrical coordinates is defined according to Equation 2:
(2)
The viscosity is given by
(3)
where µ∞ is the infinite shear rate viscosity, µ0 is the zero shear rate viscosity, λ is a parameter with units of time, and n is a dimensionless parameter. A solution of linear polystyrene in 1-chloronaphthalene has the properties listed in Table 1 (Ref. 1).TABLE 1: PROPERTIES OF A SOLUTION OF LINEAR POLYSTYRENE IN 1-CHLORONAPHTALENE.PARAMETER VALUE
µ∞0
µ0166 Pa·s
λ 1.73·10-2 s
ρt ∂∂u
∇µ∇u ∇u ()T +()⋅–ρu ∇⋅u ∇p ++0=∇u ⋅0
=γ·γ·12--2u r ()22u z v r +()22v z ()24u
r ---⎝⎠⎛⎞2
+++⎝⎠⎛⎞=µµ∞µ0µ∞–()1λγ·()2+[]n 1–()
2-----------------
+=
The model domain is depicted in Figure 1.Figure 1: Model domain. The geometry can be simplified assuming axisymmetry.The boundary conditions at the inlet and the outlet are t to fixed pressures and vanishing viscous stress:
(4)and
(5)
To study the effect on viscosity at different inlet pressures, the model makes u of the parametric solver to vary p in from 10 kPa to 210 kPa. The axis of rotation requires the symmetry condition:
(6)
while all other boundaries impo the no-slip condition:
(7) n
0.538 ρ450 kg/m 3
TABLE 1: PROPERTIES OF A SOLUTION OF LINEAR POLYSTYRENE IN 1-CHLORONAPHTALENE.
PARAMETER VALUE
No slip
Inlet
No slip
Symmetry Outlet
p p in
=p 0
=n η∇u ∇u ()T +()[]⋅0=u n ⋅0
=u 0=
Results and Discussion
Figure 2 shows that the velocity distribution is more pronounced at the inlet compared to the outlet. This is becau the cross-ction is greater at the outlet. The figure also shows that the region with greatest velocity graditnd is in the contraction, which means that the shear rate will be largest there.
Figure 2: Velocity field throughout the modeling domain.
Becau the fluid is shear thinning, the viscosity depends on the shear rate and is shown in Figure 3. It reaches its lowest value clo to the wall in the contraction between the piston and the wall.
Figure 3: Viscosity distribution in the model domain. The lowest viscosity occurs at the wall
in the contraction region.
Showing the result of a parametric study of the inlet pressure, Figure 4 contains a cross-ctional plot of the viscosity across the contraction (indicated by a red line in Figure 3). Sweeping through a range of inlet pressures impos greater velocities on the non-Newtonian fluid. As the velocity increas, the shear rate also increas and the viscosity decreas. An optimal condition is to have as flat a viscosity profile as possible. This is hindered by also wanting to put through as high a flow rate as possible.
Figure 4: Parametric study of the process, sweeping the inlet pressure from 10 kPa to 210 kPa, while investigating a cross-ctional viscosity plot. The greater the inlet pressure (and pressure differential) the less the viscosity and more varied its distribution through the cross ction.
Reference
1. R.B. Bird, W.E. Stewart, and E.N. Lightfoot, Transport Phenomena , John Wiley & Sons, 1960.
Model Library path: CFD_Module/Tutorial_Models/non_newtonian_flow Modeling Instructions
M O D E L W I Z A R D
1Go to the Model Wizard window.
2Click the 2D axisymmetric
button.
