R. Zheng

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.

On the flow past a sphere in a cylindrical tube: Limiting Weissenberg number

Abstract The paper is concerned with the steady flow generated by a sphere falling along the centreline of a cylindrical tube containing a viscoelastic fluid which is modelled by the Oldroyd-B constitutive relation. By exploiting the similarity solution in the neighbourhood of the centreline of the tube it is found that there is a limiting Weissenberg number above which no steady state axisymmetric solution can exist. The full numerical solution to the problem using a boundary element method is reported and compared with results obtained by other numerical methods. We find an overall agreement between different sets of results pointing to the existence of the limiting Weissenberg number.

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].

The flow past a sphere in a cylindrical tube: effects of intertia, shear-thinning and elasticity

Numerical solutions are presented for the flow past a sphere placed at the centreline of a cylindrical tube for Reynolds numbers ranging from 0 to 150, using a boundary element method. Fluids are modelled by a variety of constitutive equations including the Newtonian, the Carreau and the Phan-Thien-Tanner models. The influence of inertia, shear-thinning and fluid elasticity on the flow field, drag and the pressure drop force-drag ratio is examined. Some results are compared with available experimental data.

A numerical analysis of calendering

Abstract An analysis of calendering of inelastic (power-law) and viscoelastic sheets of finite initial thickness has been carried out using (i) a perturbation method based on lubrication theory; (ii) an approximate treatment including normal stress effects; (ii) a full numerical analysis using the boundary element method. The Phan-Thien-Tanner (PTT) fluid model was used in the viscoelastic analyses. Attention is focused on the separation criterion at the roll exit plane. While it is usual to assume in the inelastic case that separation occurs when the pressure and pressure gradient vanish simultaneously, it is not clear that this is appropriate in the viscoelastic model. The main new results are (a) a method of determining the separation point numerically using the criterion of zero tangential traction; (b) a computation of welling (∼ 5%) after the sheet leaves the nip; (c) a demonstration that the roll force first decreases as Weissenberg number (roll speed) rises, and then increases.

A boundary element simulation of the unsteady motion of a sphere in a cylindrical tube containing a viscoelastic fluid

A boundary element method is used to simulate the unsteady motion of a sphere falling under gravity along the centreline of a cylindrical tube containing a viscoelastic fluid. The fluid is modelled by the upper-convected Maxwell constitutive equation. Results show that the viscoelasticity of the liquid leads to a damped oscillation in sphere velocity about its terminal value. The maximum sphere velocity, which occurs in the first overshoot, is approximately proportional to the square root of the Weissenberg number when the ratio of the sphere radius to the tube radius is sufficiently small. Particular attention is also paid to the wall effects. It is shown that a closer wall reduces the oscillatory amplitude of the sphere velocity but increases its frequency. The results suggest that the falling-ball technique, which is now widely used for viscosity measurement, might also be used for the determination of a relaxation time for a viscoelastic fluid.

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 flow in a curved channel: a similarity solution for the Oldroyd-B fluid

A similarity solution is used to analyse the flow of the Oldroyd fluid B, which includes the Newtonian and Maxwell fluids, in a curved channel modelled by the narrow annular region between two circular concentric cylinders of large radius. The solution is exact, including inertial forces. It is found that the non-Netonian kinematics are very similar to the Newtonian ones, although some stress components can become very large. At high Reynolds number a boundary layer is developed at the inner cylinder. The structure of this boundary layer is asymptotically analysed for the Newtonian fluid. Non-Newtonian stress boundary layers are also developed at the inner cylinder at large Reynolds numbers.

Flow along the centreline behind a sphere in a uniform stream

Abstract The flow along the centreline behind a falling sphere in Maxwell-type fluids is analysed using a finite difference and a path integration scheme. If the flow is unbounded, then there is no limiting Weissenberg number. This conclusion, at least to a Weissenberg number of the order 10, follows from our numerical data; analytical indications are that no limiting values exist at any Weissenberg number, either with the Maxwell, Oldroyd-B, or PTT (Phan-Thien-Tanner) models.

On the non-orthogonal stagnation flow of the Oldroyd-B fluid

This paper is concerned with a non-orthogonal stagnation flow of an Oldroyd-B fluid between two parallel plates. We reduce the problem to a set of ordinary differential equations (ODEs), which is then solved with finite differences using a parameter continuation method. Perturbation analyses are also carried out for small Reynolds numbers and small Weissenberg numbers respectively. The solution of the set of ODEs is discussed. It is known that for a Newtonian fluid, the stagnation point shifts from the potential flow case in the opposite direction of the tangential velocity. The effect of the fluid elasticity is to reduce this shift. It is also shown that the Oldroyd-B model has a limiting Weissenbeg number, depending on the angle of the injected flow.