A numerical method for incompressible non-Newtonian fluid flows based on the lattice Boltzmann method
Introduction
During the last two decades, the lattice Boltzmann method (LBM) [1], [2], [3], [4], [5] has been developed into an alternative and promising numerical scheme for simulating viscous fluid flows and multicomponent and multiphase fluid flows (see reviews [6], [7], [8], [9]). In particular, the LBM has been successfully applied to various kinds of complex flows mainly for Newtonian fluids. On the other hand, non-Newtonian fluid flows are also of great importance in many science and engineering applications, such as complex flows containing surfactants and/or colloids [10], [11] and plastic flows including carbon fiber [12], [13] in industrial devices.
One of the advantages of the LBM is that the shear tensor can be computed locally, with no need of taking space derivatives of the velocity field [14]. Hence, the LBM is considered to offer excellent possibilities for simulating non-Newtonian fluid flows. As for previous studies, Aharonov and Rothman [15] first introduced a lattice Boltzmann model for power-law fluids [16], which are non-Newtonian fluids based on the power-law model. Rakotomalala et al. [17] proposed an LB model for non-Newtonian shear-thickening and shear-thinning fluids whose viscosity depends on local shear rate. Giraud et al. [18], [19] also developed two- and three-dimensional LB models of simple viscoelastic fluid flows. Then, Boek et al. [20] showed the validity of the early model by Aharonov and Rothman in two-dimensional fundamental flow problems. Gabbanelli et al. [21] also proposed similar but another LBM for the truncated power-law fluids by setting lower and upper cut-off values of the fluid viscosity. More recently, Sullivan et al. [22] have extended such LBM to a three-dimensional model, and applied it to non-Newtonian flow problems in a porous structure.
In most of the above-mentioned techniques, a relaxation time in the BGK collision term is varied as a function of the shear stress at each time step to give the correct local viscosity. Indeed, such practices of varying the relaxation time are very common in LB modeling of turbulent flows (e.g. see Ref. [23]) as well as of non-Newtonian fluid flows. However, the relaxation time has influence on the numerical stability of the LB scheme, that is, the LBM with the BGK model becomes unstable for the relaxation time close to [24], [25](corresponding to small viscosities). In particular, for non-Newtonian fluid flows, the relaxation time is also related to the local shear rate, so that numerical instability can occur despite relatively large zero-shear rate viscosity (which is the viscosity when the shear rate tends to zero). To avoid this difficulty, for example, Gabbanelli et al. set the lower and upper bounds on the viscosity in Ref. [21]. For practical use, however, it is desirable to develop a new LB model which is applicable in a wide range of shear-dependent viscosity.
In recent years, Inamuro [26] has proposed the lattice kinetic scheme for Newtonian viscous fluid flows as an extension scheme of the LBM. In this scheme, the relaxation time is set to unity so that the numerical stability can be obtained for relatively high Reynolds number flows; nevertheless one is able to determine the fluid viscosity using a constant parameter appearing in the additional term of the equilibrium distribution function. Thus, taking advantage of such a heuristic approach, one can construct a new LB model for non-Newtonian fluid flows by regarding the constant parameter as a variable parameter dependent on the local shear rate.
The paper is organized as follows. In Section 2 we give preliminary description of non-Newtonian fluids with shear-dependent viscosity. In Section 3 we propose a numerical method for incompressible non-Newtonian fluid flows based on the LBM. In Section 4 we investigate the appropriateness and accuracy of the method in a channel flow. We present numerical examples, flow in a reentrant corner geometry and flow in a three-dimensional porous structure, in Section 5. In two problems, flow characteristics and local shear rate are calculated. Also, in the latter problem the relation between pressure drops and fluid flux is examined. Finally, concluding remarks are given in Section 6.
Section snippets
Background
An incompressible viscous fluid is assumed in the present study. For a non-Newtonian fluid, effective viscosity is found to vary with local shear rate . Here the shear rate is related to the second invariant of the symmetric strain rate tensor as follows:withwhere is the fluid velocity, and subscripts and represent Cartesian coordinates and the summation convention is used hereafter.
A commonly used model for non-Newtonian fluids is the power-law, or
Numerical method
Hereafter, we use non-dimensional variables defined by a characteristic length H, a characteristic particle speed c, a characteristic time scale where U is a characteristic flow speed, and a reference density [9]. In the LBM, a modeled fluid composed of identical particles whose velocities are restricted to a finite set of N vectors is considered. Although the 15-velocity model () is used in the following descriptions, they can be directly applied to other velocity
Appropriateness and accuracy
To verify the appropriateness and accuracy of the method, we calculate non-Newtonian fluid flows between parallel walls. Here we use the two-dimensional nine-velocity model () for simplicity. The nine-velocity model has the following velocity vectors: , , and . The basic idea and formulation for the nine-velocity model are the same as those for the 15-velocity model except that the values
Flow in a reentrant corner geometry
In contrast to the previous unidirectional flow, we now consider a more demanding geometry, namely, the reentrant corner geometry presented in Fig. 3. The domain is divided into square lattice so that . Initial conditions are and in the whole domain. The periodic boundary condition with pressure difference is used at the inlet and outlet, and the no-slip boundary condition is used on the walls. The parameter ranges from to . Other parameters are fixed at
Concluding remarks
We propose a numerical method for incompressible non-Newtonian fluid flows based on the LBM. In the simulation of power-law fluid flows between parallel walls, the calculated velocity profiles are in good agreement with theory for the power-law exponents of . The error analysis is carried out and the method is more accurate than the standard approach of making the relaxation time a function of local shear rate. Using the present method, we simulate power-law fluid flows in a reentrant
Acknowledgments
The authors would like to thank a reviewer for valuable comments and helpful suggestions on the manuscript. This research was supported by the Grant-in-Aid (No. 18760121) and by the CLUSTER of Ministry of Education, Culture, Sports, Science and Technology, Japan.
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Present address: Tokai Rika Co. Ltd., Aichi 480-0195, Japan.