1. No category

V Conferência Nacional de Mecânica dos Fluidos, Termodinâmica e Energia
MEFTE 2014, 11–12 Setembro 2014, Porto, Portugal
© APMTAC, 2014
Development length in channel flows of inelastic fluids described by
the Sisko viscosity model
LL Ferrás1, C Fernandes1, O Pozo1, AM Afonso2, MA Alves2, JM Nóbrega1, FT Pinho3
1
Instituto de Polímeros e Compósitos/I3N, Universidade do Minho, Campus de Azurém 4800-058 Guimarães,
Portugal [email protected], [email protected], [email protected], [email protected]
2
Departamento de Engenharia Química, Centro de Estudos de Fenómenos de Transporte, Faculdade de Engenharia da
Universidade do Porto, Rua Dr. Roberto Frias s/n, 4200-465, Porto, Portugal [email protected], [email protected]
3
Centro de Estudos de Fenómenos de Transporte, Faculdade de Engenharia da Universidade do Porto, Rua Dr.
Roberto Frias s/n, 4200-465, Porto, Portugal [email protected]
ABSTRACT: This work presents a numerical study regarding the dimensionless development length,
=L/H required for fully-developed channel flow of inelastic non-Newtonian fluids described by the
generalized Newtonian model with the Sisko viscosity equation. The simulations were carried out for
generalized Reynolds numbers in the range 0  Re gen  100 , for a flow behaviour index, n , in the range
0.25  n  2 and for an infinite dimensionless shear rate viscosity,  , varying in the range 0    5 .
A new non-linear relationship between  and Re was derived and for specific values of n, new exact
solutions are also presented for the velocity profile under fully-developed flow conditions.
KEY-WORDS: Development length, Sisko model, channel flow.
1
INTRODUCTION
The length required to achieve a fully developed flow in pipes and channels has been for a long time a
subject of interest (see [1] and [2] and the references therein). Durst et al. [1] derived a correlation for the
development length of Newtonian fluids in pipe and channel flows of Newtonian fluids, and Poole and
Ridley [2], extended the work of Durst et al. [1] to the pipe flow of inelastic fluids described by the
power-law model. Common to these works is that the development length is determined on the basis of
the velocity profile, specifically when the velocity at the center of the channel reaches 99% of the fullydeveloped value.
In this work we devise a correlation for the development length of fluids described by the Sisko model [3]
in planar channels.
2
GOVERNING EQUATIONS
The governing equations for steady, incompressible, laminar and isothermal flows are the conservation of
mass equation
 u  0
(1)
and the momentum equation,
u

   uu   p    τ ,
 t

 
(2)
where u is the velocity vector,  is the fluid density, p is the pressure and τ is the extra stress tensor,
that is given by,
τ      u   u T   2   D
(3)
where D is the symmetric rate of strain tensor and     is the shear viscosity model that follows the
Sisko model,
      K  n 1
(4)
where K is the flow consistency index,  is the infinite shear rate viscosity, n is the flow behaviour
index and   2 tr  D2  ( tr   is the trace of a tensor) is the shear rate.
3
NUMERICAL PROCEDURE
The exact velocity profiles for the fully developed flow, required to identify the fully developed
conditions, can only be determined analytically for specific values of the exponent, n= 1/3, 1/2, 1 and 2,
but, the exact shear rate profile can be obtained for a wider range of flow index, namely n=1/4, 1/3, 1/2,
2/3, 3/4, 1, 3/2.
When an analytical solution is not possible for fully-developed flow, the shear rate value was used to
compute numerically the pressure gradient and the velocity at the centre of the channel, through a
numerical method.
For the numerical solution of the system of governing equations ((1)-(4)) we used the Finite Volume
Method together with the SIMPLE procedure of Patankar [4] to couple the velocity and pressure fields
[5]. The numerical computations were performed with the opensource OpenFOAM software [6].
4
RESULTS AND DISCUSSION
The development length  was defined as the axial distance required for the velocity to reach 99% of
the calculated maximum value. The Reynolds number adopted is the generalized version usually
employed for power-law fluids, cf. [7],
n
6 U 2  n H n  4n  2 
(4)
Re gen 

 ,
K
 n 
and the dimensionless infinite shear rate viscosity,  , is defined as    /( UH ) , where U is the
flow average velocity and H is the channel width.
1
semi-analytical solution
numerical results
0.5
0
2
  2.5
1.5
u/U
1
0.1
0.2
0.3
0.4
0.5
0
0
0.1
0.2
y/H
0.3
0.4
  2.5
u/U
1.3
y=0.99u analytical
numerical results
6
x/H
8
10
0.3
0.4
0.5
  5
1.4
1.3
1.2
1.2
1.2
4
0.2
1.5
1.4
u/U
1.4
2
0.1
y/H
1.5
 1
0
0
0.5
y/H
1.5
1.3
1
0.5
0.5
0
0
  5
1.5
u/U
u/U
2
 1
u/U
2
1.5
0
2
4
6
x/H
8
10
0
2
4
6
8
10
x/H
Figure 1: (Top) Comparison between the semi-analytical and the numerical solutions for three different
values of  . (Bottom) Centreline velocity profile along the channel length.
In Fig. 1 (Top) we show a good agreement between the numerical and the analytical results, for the
fully-developed velocity profile. Fig. 1 (Bottom) presents the Centreline velocity profile along the
channel length for three different values of  , with Re gen  1 and n  0.5 . For   1 , 2.5 and 5 we
obtained a development length, =L/H, of 0.835,0.746 and 0.747 , respectively. This shows that the
development length decreases with increase of  , which was expected due to the growth of momentum
diffusion along the channel thickness promoted by the increase of  .
REFERENCES
[1] F Durst, S Ray, B Unsal, and OA Bayoumi, (2005). The Development Lengths of Laminar Pipe and Channel
Flows, ASME J. Fluids Eng., 127, 1154-1160.
[2] RJ Poole, and BS Ridley, (2007). Development Length Requirements for Fully-Developed Laminar Pipe Flow
of Inelastic Non-Newtonian Liquids, ASME J. Fluids Eng., 129, 1281–1287.
[3] A.W. Sisko, (1958). The flow of lubricating greases. Ind. Engng Chem. 50, 1789–1792.
[4] S.V. Patankar, (1980), Numerical Heat Transfer and Fluid Flow, Hemisphere, Washington D.C..
[5] PJ Oliveira, FT Pinho, and GA Pinto, (1998). Numerical simulation of non-linear elastic flows with a general
collocated finite-volume method, J. Non-Newtonian Fluid Mech., 79, 1–43
[6] The OpenFOAM Foundation, www.openfoam.org
[7] P. Ternik, (2009) . Planar sudden symmetric expansion flows and bifurcation phenomena of purely viscous shear
thinning fluids, J. Non-Newtonian Fluid Mech. 157, 15–25.