Xi-Jun Fan

A direct simulation of fibre suspensions

Abstract A numerical method to simulate fibre suspensions in shear flow is reported, which takes into account short range interaction via lubrication forces and long range interaction via slender body approximation, together with an appropriate Ewald summation technique. The numerical results are averaged to produce macroscopic properties of the suspension, including the Folgar–Tucker diffusion constant, the structure functions, and the reduced viscosity. In the semi-concentrated to concentrated regime, the fibres no longer follow Jefferys orbits, they align mostly with the shear direction. Numerical data on the diffusivity constant, the structure functions, and the reduced viscosity agree reasonably well with available experimental data.

Direct simulation of flexible fibers

Abstract A numerical simulation of multiple flexible fibers in suspension in Newtonian simple shear flow is presented. The method used is similar to those of previous recent simulation works by Fan et al. [J. Non-Newtonian Fluid Mech. 74 (1998) 113] and Yamane et al. [J. Non-Newtonian Fluid Mech. 54 (1994) 405], however, the method has been modified to allow a small amount of bending and torsion in the fibers. A restoring moment acts to straighten the fibers as they interact in the flow. It is demonstrated that this simulation can be used to extract basic rheological information about the suspension including fiber orientations and suspension viscosity. The viscosity of semi-concentrated to concentrated flexible fiber suspensions are shown to increase by a magnitude of the order 7–10% greater than the equivalent rigid fiber suspension tested. This is in qualitative agreement with previous experimental work by Goto et al. [Rheologica Acta 25 (1986) 119] and Blakeney [J. Colloid Interface Sci. 22 (1966) 324]. The implication is that any constitutive relation involving particulate suspensions described by orientation vectors may quantitatively underestimate suspension viscosity, particularly for fibers of large aspect ratio, or low Young’s modulus, whereby the tendency to flex is greater [Rheologica Acta 25 (1986) 119]. If particulate deformation were accounted for (by whatever means) in the existing constitutive relationship, predictions of bulk suspension parameters such as viscosity should be noticeably improved. A method is developed to modify an existing rigid-fiber viscosity to an equivalent flexible fiber viscosity, hence improving viscosity prediction ability.

Thermoviscoelastic simulation of thermally and pressure-induced stresses in injection moulding for the prediction of shrinkage and warpage for fibre-reinforced thermoplastics

In this paper we present a detailed thermoviscoelastic formulation for the simulation of thermally and pressure induced residual stresses in injection moulded short-fibre-reinforced thermoplastics. The computed residual stresses enable us to predict shrinkage and warpage in the finished products. We also apply an anisotropic version of a rotary diffusion equation to calculate the flow-induced fibre orientation distribution. The predicted fibre orientation state, together with micromechanical theories, allows the incorporation of anisotropy in material properties into the thermoviscoelastic model. Finally we report three numerical examples to indicate the success of the present model.

Molecular dynamics simulation of a liquid in a complex nano channel flow

We report some molecular dynamics simulation results for a complex nano channel flow. In certain flow geometry, some of the flow features cannot be predicted by the Navier–Stokes equations with no-slip boundary conditions. The results show a loss of dynamic similarity for flows with similar geometry and global dimensionless flow parameters. Nano-sized vortex flow can be developed at low Reynolds numbers due to near-wall molecules having large enough momenta, resulting in qualitatively different flow field from that predicted by the Navier–Stokes equations.

Viscosity of curved fibers in suspension

In this paper, the relationship between fiber shape and relative viscosity of a fiber suspension is explored. A numerical simulation has been used to model non-Brownian curved rigid fibers in suspension under the influence of Newtonian shear flow. Curvature in the simulated fibers was taken to represent general deformities of real fibers in suspension. The simulation method was previously used by Joung et al. [J. Non-Newtonian Fluid Mech. 99 (2001) 1] to determine suspension viscosity for flexible fibers in suspension. When compared to the equivalent straight rigid fiber suspension, fiber curvature was found to contribute to a large increase in suspension viscosity. For typical semi-concentrated to concentrated suspensions, curved fibers were observed to produce viscosity increases of the order twice that of straight fiber suspensions. Results indicate that even a small bend in the fibers may cause a large bulk viscosity increase. Suspension viscosity is therefore highly dependent on the quality control measures taken during sample preparation.

Simulation of fibre suspension flows by the Brownian configuration field method

Abstract We describe a robust numerical method for solving general flows of fibre suspensions that couples the CONNFFESSIT idea of Ottinger [H.C. Ottinger, Stochastic Processes in Polymeric Fluids, Springer, Berlin, 1996], and the DAVSS method of Sun et al. [J. Sun, N. Phan-Thien, R.I. Tanner, J. Non-Newtonian Fluid Mech. 65 (1996) 75–91; J. Sun, M.D. Smith, R.C. Armstrong, R.A. Brown, Finite element method for viscoelastic flows based on the discrete adaptive viscoelastic stress splitting and the discontinuous Galerkin method: DAVSS-G/DG, J. Non-Newtonian Fluid Mech., 1998, in press] The method does not require a closure approximation and allows good quality solutions at high volume fractions of fibres to be obtained. The algorithm was tested on the flow past the sphere in a tube problem, which forms the basis of the falling ball viscometry. The numerical results compare well with numerical the results from Phan-Thien and Graham [N. Phan-Thien, A.L. Graham, J. Rheol. Acta 30 (1991) 44–57] and the experimental data of Milliken et al. [W.J. Milliken, M. Gottlieb, A.L. Graham, L.A. Momdy, R.L. Powell, J. Fluid Mech. 202 (1989) 217–232].

Simulation of fibre suspension flow with shear-induced migration

Abstract We report a robust numerical method that allows the simulation of fibre suspensions, together with shear-induced fibre migration, to be carried out at a large volume fraction of the fibres. The method is a combination of the Brownian configuration field (BCF) technique and the adaptive viscosity split stress (AVSS) finite element formulation. The fibre suspension model is adapted from Folgar and Tucker [F.P. Folgar, C.L. Tucker, J. Reinforced Plastics and Composites 3 (1984) 98–119] (for the structure evolution) and Phan-Thien and Graham [N. Phan-Thien, A.L. Graham, Rheol. Acta 30 (1991) 44–57] (for the stress rule), together with a modified version of the shear-induced migration model of Phillips et al. [R.J. Phillips, R.C. Armstrong, R.A. Brown, A.L. Graham, J.R. Abott, Phys. Fluids A4 (1992) 30–40]. The implementation was tested in the circular Couette and the plane Poiseuille flow. The numerical data compared well with the experimental results of Mondy et al. [L.A. Mondy, H. Brenner, S.A. Altobelli, J.R. Abbott, A.L. Graham, J. Rheol. 38 (1994) 444–452].

Viscoelastic Mobility Problem Using A Boundary Element Method

Abstract In this paper, the complete double layer boundary integral equation formulation for Stokes flows is extended to viscoelastic fluids to solve the mobility problem for a particle in an unbounded body of fluid, where the non-linearity is handled by using particular solutions of the Stokes inhomogeneous equation. Some meshless techniques are employed and a point-wise solver is used to solve the viscoelastic constitutive equation, avoiding volume meshing. The method is tested against the numerical solution for a sphere settling in an unbounded Oldroyd-B fluid. Some results on a prolate motion in shear flow for the Oldroyd-B fluid are reported and compared qualitatively with some theoretical and experimental results.

Effective moduli of particulate solids: the completed double layer boundary element method with lubrication approximation

Abstract. This paper reports the numerical results of the effective moduli of a composite consisting of identical rigid spherical inclusions in a homogeneous isotropic elastic medium. The effective moduli are calculated for particles in simple, body-centered, face-centered cubic lattices, and in a random configuration. The numerical technique used is the Completed Double Layer Boundary Element Method (CDLBEM) augmented by a short-range lubrication approximation. The results agree well with previous numerical results obtained by a completely different technique. Agreement with experimental results for particles in random configurations is also observed.

Traction-based Completed Adjoint Double Layer Boundary Element Method in elasticity

This paper is concerned with a traction-based Completed Adjoint Double Layer Boundary Element Method to solve for the surface traction of a system of rigid particles embedded in an elastic matrix. The main feature of the method is a single layer representation of the displacement field, which leads to a system of second-kind integral equations for the traction field, the extreme eigenvalue of which could be deflated, allowing iterative solution strategies to be effectively applied. The method is therefore most suitable for large-scale simulations of particulate solids. The method is benchmarked against some known analytic solutions, including the difficult stress singularity problems at sharp edges. The effectiveness of the method in dealing with a large number of inclusions is also demonstrated with an elongational deformation problem involving up to 25 inclusions.