Chemical Engineering Science 63 (2008) 3898--3908 Contents lists available at ScienceDirect Chemical Engineering Science journal homepage: www.elsevier.com/locate/ces Completed double layer boundary element method for periodic fibre suspension in viscoelastic fluid Huy Nguyen-Hoang a , Nhan Phan-Thien a, b , Boo Cheong Khoo a, b, ∗ , Xi-Jun Fan c , Hua-Shu Dou d a Department of Mechanical Engineering, National University of Singapore, Kent Ridge, Singapore 117576, Singapore b Singapore-MIT Alliance, National University of Singapore, Singapore 117576, Singapore c School of Aerospace, Mechanical and Mechatronic Engineering, The University of Sydney, Sydney, NSW 2006, Australia d Temasek Laboratories, National University of Singapore, Singapore 117508, Singapore ARTICLE INFO ABSTRACT Article history: Received 17 March 2007 Received in revised form 22 April 2008 Accepted 29 April 2008 Available online 8 May 2008 Keywords: Fibre suspension Viscoelastic matrix Periodic Viscosity CDLBEM A numerical method to simulate the periodic fibre suspension in viscoelastic fluid is developed with the completed double layer boundary element method (CDLBEM). The periodic summations that arise in the formulation were well handled by Ewald summation technique to speed up the convergence rate in the computation. The formulation for velocity field in periodic fibre suspension in viscoelastic fluid is derived and is used to simulate multiple fibres suspended in a viscoelastic shear flow. Simulations are carried out for various fibre aspect ratios and volume fractions ranging from dilute to concentrated regimes. Numeri- cal results of macroscopic rheological properties of the system are compared to available experiments on viscoelastic fibre suspensions, and are found to agree reasonably well with the experimental data. © 2008 Elsevier Ltd. All rights reserved. 1. Introduction A fibre-filled composite consists of polymer matrix and short fibres. During processing such as injection molding, it is melted and becomes fibre suspension in viscoelastic fluid. Since there are no complete analytical solutions for fibre suspension in viscoelastic fluid thus far, research in this area has almost been carried out on experi- mental basis with some measurements of the rheological properties of fibre suspension (e.g. see Ganani and Powell, 1986; Iso et al., 1996a,b; Ramazani et al., 2001; Sepehr et al., 2004, etc.). However, with advancement in numerical methodology/techniques coupled with the vast improvement in computer technology making unit cost of computation becoming more and more affordable, numerical investigation of fibre suspension in viscoelastic flows is becoming a viable alternative approach, supplementing experimental data. In studying fibre suspension, it is convenient to define three regimes of concentrations: dilute, semi-concentrated and concen- trated. The fibres are considered as rigid cylinders of effective length L, diameter d and aspect ratio a r = L/d. In dilute suspension, the vol- ume fraction satisfies the condition of  <d 2 L/V , or a 2 r < 1, where  = nd 2 L/4, n is the number of density. If 1 < a 2 r <a r then the ∗ Corresponding author at: Department of Mechanical Engineering, National Uni- versity of Singapore, Kent Ridge, Singapore 117576, Singapore. Tel.: +65 65162889; fax: +65 67791459. E-mail address: mpekbc@nus.edu.sg (B.C. Khoo). 0009-2509/$ - see front matter © 2008 Elsevier Ltd. All rights reserved. doi:10.1016/j.ces.2008.04.058 suspension is said to be semi-concentrated where each fibre is con- fined in the volume d 2 L<  < dL 2 . The spacing between the fibres is greater than the fibre diameter d but less than the fibre length L. If a r > 1, the suspension is said to be concentrated where the aver- age distance between fibres is less than a fibre diameter d. For these latter two regimes, the fibre interaction affects the motion of the fi- bre. The rheological properties and fibre structures are the results of the fibre orientation in the suspension. The fibre interaction depends not only on the number of fibres per unit volume n but also on the length of the fibres. Therefore, a group of parameters, such as nL 3 and nL 2 d are usually used to characterize fibre suspension. The di- lute regimes is defined as nL 3 >1. If nL 3 > 1 but nL 2 d>1, the suspen- sion is said to be semi-dilute; and nL 2 d> 1 for semi-concentrated. Typically, nL 2 d is taken to be between 1 and 5 in injection molding process of fibre-filled thermoplastics. In simulating the hydrodynamic interactions, one usually as- sumes that the fibre suspension is spatially periodic. When the pe- riodic length approaches to a very large value, the fibre suspension can be considered as the real suspension. In periodic suspension, there is a very large if not an infinite number of particles involved in the far-field hydrodynamic interaction with a generic particle in a reference cell. For periodic suspensions, the boundary integral expression summing over infinite number of particles is not conver- gent and cannot be used in simulating the suspension in practice. As such, O'Brien (1979) suggested an approach using Ewald summa- tion technique (Ewald, 1921) to construct an absolutely convergent