© 2015 The Korean Society of Rheology and Springer 105 Korea-Australia Rheology Journal, 27(2), 105-111 (May 2015) DOI: 10.1007/s13367-015-0011-2 www.springer.com/13367 pISSN 1226-119X eISSN 2093-7660 An implicit rheological model for numerical simulation of generalized Newtonian fluids Mohsen Lashkarbolok 1 * , Shukoofeh Izadi 2 , Hadi Alemi 2 and Sita Drost 3 1 Department of Engineering, Golestan University, Golestan, Iran 2 Islamic Azad University of Azadshahr, Azadshahr, Iran 3 Department of Process and Energy, Delft University of Technology, 2628 CD Delft, Netherlands (Received October 6, 2014; final revision received March 26, 2015; accepted March 27, 2015) Fitting an explicit curve over some discrete data extracted from a rheometer is the usual way of writing a rheological model for generalized Newtonian fluids. These explicit models may not match totally with the extracted data and may ignore some features of the rheological behavior of the fluids. In this paper, a cubic- spline curve fitting is used to fit a smooth curve from discrete rheological data. Spline interpolation avoids the problem of Runge's phenomenon, which occurs in interpolating using high degree polynomials. The for- mulation for applying presented rheological model is described in the context of least squares meshfree technique. One problem is solved to show validity of the scheme: a fluid with rather complex rheology model is considered and solved by both conventional explicit and proposed implicit models to show the advantages of the presented method. Keywords: cubic spline, collocated discrete least squares, meshless method, radial point interpolation 1. Introduction The relation between the rate of the strain and apparent viscosity shows the basic characteristic of the deformation behavior of a generalized Newtonian fluid. There are some well known models which are used to describe the mentioned relation explicitly. In the power-law model (Blair et al., 1939), the apparent viscosity varies as a power of strain rate. This model can be used to simulate both shear thinning and shear thickening behavior of a fluid by tuning its parameter. When there are significant deviations from the power-law model at very high and very low shear rates, it is necessary to use a model which takes account of the limiting values of viscosities. Based on the molecular network considerations, Carreau (1972) presented a rheology model which limits the value of the apparent viscosity at extreme values of shear rate (or rate of strain). Cross (1965) presented a four-parameter model, which also considers limiting values for the viscosity at low and high values of the shear rates. The Bingham plas- tic model (Bingham, 1916) is the simplest equation, which describes the flow of a fluid with a yield stress. Herschel– Bulkley model by Herschel and Bulkley (1926) uses a three constant explicit equation to describe the nonlinear- ity of a fluid with plastic behavior. Casson model was used by Blair (1959) to describe the rheological behavior of many fluids in food industry, blood and biological flu- ids. Comprehensive reviews on rheological models were presented by Bird (1987) and Macosko (1994). All of the mentioned models use an explicit formulation with some adjustable parameters to fit an appropriate curve over some scattered given data. In this paper, an implicit cubic-spline method is used to fit a curve over the shear rate-apparent viscosity data. Spline interpolation avoids the problem of Runge's phenomenon. The problem of oscillatory results of interpolation with polynomials of high degree, near the ends of an interval, calls Runge's phenomenon. In the present study, this implicit model is linked to the governing equations to be used in the calculation of the viscosity and its derivatives at any point in the domain of a problem. A least squares based meshfree method named collocated discrete least squares (CDLS) is used in the numerical solution of the problems. The method presented by Afshar and Lashkarbolok (2008) is integral free, simple in formulation, and flexible in domain discretization. Here, the governing equations are considered to be solved implicitly. It means that the evolution of the pressure, velocity and stresses are computed simultaneously at each time step. To connect the pressure to the continuity equa- tion, conventional artificial compressibility technique is used. The presented rheology model is limited to isother- mal problems but with some modifications, it can consider the temperature effect on the viscosity. In this case, two dimensional version of cubic spline may be used to obtain required viscosity at given shear rate and temperature at any point in the domain of the problem. In addition, energy equation must be added to the governing equations to find the temperature distribution at any point in the fluid. In this paper, we focus on the isothermal problems. Although the problems are assumed to be steady state, the governing equations are solved in time to the point that a steady-state solution is obtained. To validate the proposed method, a benchmark problem of cavity flow *Corresponding author; E-mail: mlbolok@iust.ac.ir