Numerical modelling of step–strain for stretched filaments

This article addresses the modelling of filament-stretching/step–strain deformation under viscoelastic capillary break-up configurations of the CaBER-type. Start-up, prior to step–strain, is conducted under constant stretch-rate synchronous plate retraction with impulsive sessation of plate motion. The study encompasses variation in material rheology, appealing to Oldroyd, Geisekus and Phan-Thien/Tanner-type models, which display differences in shear and extensional viscosity properties (shear thinning/extension hardening). Two different viscosity ratio settings are considered to reflect high- and low-solvent viscosity constituent components; the former representing typical Boger fluids, the latter high-polymer concentration fluids. We compare and contrast results for three alternative filament aspect ratios at the onset of step–strain. Throughout the step–strain period, we have been able to successfully capture such physical features as drainage to the filament feet, necking at the filament centre, and periods with travelling waves through the axial filament length. In addition, we have identified the suppressive influence that larger capillary forces have upon radial fluctuations, and the minor impact that gravitational forces have upon the ensuing deformation. From this study, estimates for rheometrical data have been derived in terms of characteristic material relaxation time and apparent extensional viscosity. The computational techniques employed include a compressed-mesh (CM) procedure, an Arbitrary Lagrangian–Eulerian scheme (ALE) and a free-surface particle tracking technique. Spatial discretisation of the problem is accomplished through a hybrid finite element/finite volume algorithm implemented in the form of a time-stepping incremental pressure-correction formulation.     

A finite volume approach for unsteady viscoelastic fluid flows

A finite volume, time‐marching for solving time‐dependent viscoelastic flow in two space dimensions for Oldroyd‐B and Phan Thien–Tanner fluids, is presented. A non‐uniform staggered grid system is used. The conservation and constitutive equations are solved using the finite volume method with an upwind scheme for the viscoelastic stresses and an hybrid scheme for the velocities. To calculate the pressure field, the semi‐implicit method for the pressure linked equation revised method is used. The discretized equations are solved sequentially, using the tridiagonal matrix algorithm solver with under‐relaxation. In both, the full approximation storage multigrid algorithm is used to speed up the convergence rate. Simulations of viscoelastic flows in four‐to‐one abrupt plane contraction are carried out. We will study the behaviour at the entrance corner of the four‐to‐one planar abrupt contraction. Using this solver, we show convergence up to a Weissenberg number We of 20 for the Oldroyd‐B model. No limiting Weissenberg number is observed even though a Phan Thien–Tanner model is used. Several numerical results are presented. Smooth and stable solutions are obtained for high Weissenberg number. Copyright © 2002 John Wiley & Sons, Ltd.     

Estimating the viscoelastic moduli of complex fluids using the generalized Stokes–Einstein equation

We obtain the linear viscoelastic shear moduli of complex fluids from the time-dependent mean square displacement, <Δr ²(t)>, of thermally-driven colloidal spheres suspended in the fluid using a generalized Stokes–Einstein (GSE) equation. Different representations of the GSE equation can be used to obtain the viscoelastic spectrum, G˜(s), in the Laplace frequency domain, the complex shear modulus, G ^*(ω), in the Fourier frequency domain, and the stress relaxation modulus, G _r (t), in the time domain. Because trapezoid integration (s domain) or the Fast Fourier Transform (ω domain) of <Δr ²(t)> known only over a finite temporal interval can lead to errors which result in unphysical behavior of the moduli near the frequency extremes, we estimate the transforms algebraically by describing <Δr ²(t)> as a local power law. If the logarithmic slope of <Δr ²(t)> can be accurately determined, these estimates generally perform well at the frequency extremes. Received: 8 September 2000/Accepted: 9 March 2000     

Rectilinear oscillatory viscoelastic flow in rectangular ducts

Rectilinear oscillatory pressure-driven motion of Oldroyd fluids with complex dynamic viscosity within straight rectangular ducts is described. The linear viscoelastic flow solutions are given as functions of a Stokes number, the ratio of cross-section side lenghts and two Oldroyd time constants. Exact analysis and numerical approximate solutions via finite difference methods are shown, as well as suggestions for applying the theory to measure viscoelastic material properties.     

Time periodic electroosmotic flow of the generalized Maxwell fluids between two micro-parallel plates

Analytical solutions are presented using method of separation of variables for the time periodic EOF flow of linear viscoelastic fluids between micro-parallel plates. The linear viscoelastic fluids used here are described by the general Maxwell model. The solution involves analytically solving the linearized Poisson–Boltzmann equation, together with the Cauchy momentum equation and the general Maxwell constitutive equation. By numerical computations, the influences of the electrokinetic width K denoting the characteristic scale of half channel width to Debye length, the periodic EOF electric oscillating Reynolds number Re and normalized relaxation time λ₁ω on velocity profiles and volumetric flow rates are presented. Results show that for prescribed electrokinetic width K, lower oscillating Reynolds number Re and shorter relaxation time λ₁ω reduces the plug-like EOF velocity profile of Newtonian fluids. For given Reynolds number Re and electrokinetic width K, longer relaxation time λ₁ω leads to rapid oscillating EOF velocity profiles with increased amplitude. With the increase of the K, the velocity variations are restricted to a very narrow region close to the EDL for small relaxation time. However, with the increase of the relaxation time, the elasticity of the fluid becomes conspicuous and the velocity variations can be expanded to the whole flow field. As far as volume flow rates are concerned, for given electrodynamic width K, larger oscillating Reynolds number Re results in a smaller volume flow rates. For prescribed oscillating Reynolds number Re, with the changes of relaxation time λ₁ω, volume flow rates will produce some peaks no matter how the electrodynamic width K varies. Moreover, the time periodic evolution of the velocity profiles provides a detail insight of the flow characteristic of this flow configuration.     

An L2‐stable approximation of the Navier–Stokes convection operator for low‐order non‐conforming finite elements

We develop in this paper a discretization for the convection term in variable density unstationary Navier–Stokes equations, which applies to low‐order non‐conforming finite element approximations (the so‐called Crouzeix–Raviart or Rannacher–Turek elements). This discretization is built by a finite volume technique based on a dual mesh. It is shown to enjoy an L² stability property, which may be seen as a discrete counterpart of the kinetic energy conservation identity. In addition, numerical experiments confirm the robustness and the accuracy of this approximation; in particular, in L² norm, second‐order space convergence for the velocity and first‐order space convergence for the pressure are observed. Copyright © 2010 John Wiley & Sons, Ltd.     

Effect of inertia on thermoelastic flow instability

Thermal effects induced by viscous heating cause thermoelastic flow instabilities in curvilinear shear flows of viscoelastic polymer solutions. These instabilities could be tracked experimentally by changing the fluid temperature T₀ to span the parameter space. In this work, the influence of T₀ on the stability boundary of the Taylor–Couette flow of an Oldroyd-B fluid is studied. The upper bound of the stability boundary in the Weissenberg number (We)–Nahme number (Na) space is given by the critical conditions corresponding to the extension of the time-dependent isothermal eigensolution. Initially, as T₀ is increased, the critical Weissenberg number, We_c, associated with this upper branch increases. Increasing T₀ beyond a certain value T^* causes the thermoelastic mode of instability to manifest. This occurs in the limit as We/Pe → 0, where Pe denotes the Péclet number. In this limit, the fluid relaxation time is much smaller than the time scale of thermal diffusion. T₀ = T^* represents a turning point in the We_c–Na_c curve. Consequently, the stability boundary is multi-valued for a wide range of Na values. Since the relaxation time and viscosity of the fluid decrease with increasing T₀, the elasticity number, defined as the ratio of the fluid relaxation time to the time scale of viscous diffusion, also decreases. Hence, O(10) values of the Reynolds number could be realized at the onset of instability if T₀ is sufficiently large. This sets limits for the temperature range that can be used in experiments if inertial effects are to be minimized.     

Ideal elastic fluids of different viscosity and elasticity levels

Four constant viscosity, highly elastic fluids of different viscosity and elasticity levels are presented. The viscosity ranges from 4 × 10^?3 to 5.0 Pa s and the Maxwell relaxation time varies from 0.09 to 4.5 s. The steady and dynamic shear properties are determined. These fluids comply with the requirements of the simple fluid theory except for theG′ andN ₁/2 data where a slight deviation is observed. The results suggest the possibility of preparing a wide range of constant viscosity elastic fluids with specific values of viscosity and relaxation time by manipulating polymer molecular parameters as well as polymer concentration, solvent viscosity and salt addition. The effects of each of these parameters on the rheological behaviour are examined.     

Predictions of linear viscoelastic properties for polydisperse entangled polymers

Different blending laws have been proposed in the literature to describe the polydispersity effect on the rheological behavior of polymer melts. In this paper predictions of linear viscoelastic properties of entangled polydisperse polymers have been derived from the double reptation mixing rule. The results in terms of the relaxation modulus, the zero shear-rate viscosity, η₀, and the steady-state compliance, J _e ⁰, have been obtained using three different relaxation functions for the monodisperse fractions, namely the Tuminello step function, the single exponential function and the BSW function. Both discrete and continuous molecular weight distributions (MWDs) have been investigated. The Generalized Exponential Function (GEX) has been considered in the continuous case. The results showed that, in systems with a large number of components, the predictions of linear viscoelastic properties mainly depend on the double reptation mixing rule assumption, while the choice of the relaxation function is not crucial. In particular, the mathematical simplicity of the Tuminello step relaxation function has allowed analytical computation of the linear viscoelastic properties in closed form. Indeed, the analytical results indicated a dependence of η₀ on the MWD that could be expressed in terms of (M _z/M _w)^0.8, in agreement with experimental results reported in the literature. In the case of J _e ⁰, the analytical model defines a dependence on (M _z/M _w)^5.5, i.e. as expected a strong dependence on the MWD is predicted for the steady-state compliance. Finally, dynamic moduli have been computed from the relaxation modulus and their predictions have been favorably compared with experimental results from the literature. Received: 19 July 1999/Accepted: 24 November 1999     

A new stable space–time formulation for two‐dimensional and three‐dimensional incompressible viscous flow

A space–time finite element method for the incompressible Navier–Stokes equations in a bounded domain in ?^d (with d=2 or 3) is presented. The method is based on the time‐discontinuous Galerkin method with the use of simplex‐type meshes together with the requirement that the space–time finite element discretization for the velocity and the pressure satisfy the inf–sup stability condition of Brezzi and Babu?ka. The finite element discretization for the pressure consists of piecewise linear functions, while piecewise linear functions enriched with a bubble function are used for the velocity. The stability proof and numerical results for some two‐dimensional problems are presented. Copyright © 2001 John Wiley & Sons, Ltd.     

Experimental study on the capillary thinning of entangled polymer solutions

The transient elongation behavior of entangled polymer and wormlike micelles (WLM) solutions has been investigated using capillary breakup extensional rheometry (CaBER). The transient force ratio X = 0.713 reveals the existence of an intermediate Newtonian thinning region for polystyrene and WLM solutions prior to the viscoelastic thinning. The exponential decay of X(t) in the first period of thinning defines an elongational relaxation time λ _x which is equal to elongational relaxation time λ _e obtained from exponential diameter decay D(t) indicating that the initial stress decay is controlled by the same molecular relaxation process as the strain hardening observed in the terminal regime of filament thinning. Deviations in true and apparent elongational viscosity are discussed in terms of X(t). A minimum Trouton ratio is observed which decreases exponentially with increasing polymer concentration leveling off at Tr_min = 3 for the solutions exhibiting intermediate Newtonian thinning and Tr_min ≈ 10 otherwise. The relaxation time ratio λ _e/ λ _s, where λ _s is the terminal shear relaxation time, decreases exponentially with increasing polymer concentration and the data for all investigated solutions collapse onto a master curve irrespective of polymer molecular weight or solvent viscosity when plotted versus the reduced concentration c[ η], with [ η] being the intrinsic viscosity. This confirms the strong effect of the nonlinear deformation in CaBER experiments on entangled polymer solutions as suggested earlier. On the other hand, λ _e ≈λ _s is found for all WLM solutions clearly indicating that these nonlinear deformations do not affect the capillary thinning process of these living polymer systems.     

Elastic and inertial effects in pulsatile flow of polymeric liquids in circular tubes

In the earlier theoretical work of Barnes, Townsend, and Walters on pulsatile flow of viscoelastic fluids, it was found that an Oldroyd model predicted that flow enhancement should decrease with increasing frequency of pulsation, whereas their experimental data showed an increase. Their calculations have been repeated for the Goddard—Miller model using two different expressions for the relaxation modulus; it is found that the Goddard—Miller model leads to the same discrepancy between theory and experiment. It is tentatively concluded that secondary-flow effects are playing an important role. The role of the elastic and inertial effects and the role of the shape of the viscosity vs. shear rate curve are discussed.     

Microsecond relaxation processes in shear and extensional flows of weakly elastic polymer solutions

In this paper, we introduce an experimental protocol to reliably determine extensional relaxation times from capillary thinning experiments of weakly elastic dilute polymer solutions. Relaxation times for polystyrene in diethyl phthalate solutions as low as 80?μ s are reported: the lowest relaxation times in uniaxial extensional flows that have been assessed so far. These data are compared to the linear viscoelastic relaxation times that are obtained from fitting the Zimm spectrum to high frequency oscillatory squeeze flow data measured with a piezo-axial vibrator (PAV). This comparison demonstrates that the extensional relaxation time reduced by the Zimm time, λ _ext/λ _z, is not solely a function of the reduced concentration c/c*, as is commonly stated in the literature: an additional dependence on the molecular weight is observed.     

A least-squares finite element method for time-dependent incompressible flows with thermal convection

The time-dependent Navier–Stokes equations and the energy balance equation for an incompressible, constant property fluid in the Boussinesq approximation are solved by a least-squares finite element method based on a velocity–pressure–vorticity–temperature–heat-flux ( u –P–ω–T– q ) formulation discretized by backward finite differencing in time. The discretization scheme leads to the minimization of the residual in the l²-norm for each time step. Isoparametric bilinear quadrilateral elements and reduced integration are employed. Three examples, thermally driven cavity flow at Rayleigh numbers up to 10⁶, lid-driven cavity flow at Reynolds numbers up to 10⁴ and flow over a square obstacle at Reynolds number 200, are presented to validate the method.     

Elasticity and dynamics of particle gels in non-Newtonian melts

We investigate the relation between the structure and the viscoelastic behavior of a model polymer nanocomposite system based on a mixture of titanium dioxide (TiO₂) nanoparticles and polypropylene. Above a critical volume fraction, Φ _c, the elasticity of the hybrids dramatically increases, and the frequency dependence of the elastic and viscous moduli reflects the superposition of the independent responses of the suspending polymer melt and of an elastic particle network. In addition, the elasticity of the hybrids shows critical behavior around Φ _c. We interpret these observations by hypothesizing the formation of a transient network, which forms due to crowding of particle clusters. Consistent with this interpretation, we find a long-time, Φ-dependent, structural relaxation, which emphasizes the transient character of the structure formed by the particle clusters. For times below this characteristic relaxation time, the elasticity of the network is Φ-independent and reminiscent of glassy behavior, with the elastic modulus, G^′, scaling with frequency, ω, as G^′∼ω ^0.3. We expect that our analysis will be useful for understanding the behavior of other complex fluids where the elasticity of the components could be superimposed.     

Numerical simulations of linear viscoelasticity of monodisperse emulsions of Newtonian drops in a Newtonian fluid from dilute to concentrated regime

The bulk viscoelastic properties of monodisperse emulsions of Newtonian drops in a Newtonian matrix subjected to small amplitude oscillatory shear (SAOS) flow are investigated by means of arbitrary Lagrangian Eulerian finite element method 3D numerical simulations. Volume fractions of the suspended phase from the dilute to the concentrated regime (up to 30 %), and a range of several orders of magnitude of the drops-to-matrix viscosity ratio and of the frequency of the oscillatory flow are examined; the eventual presence of slip between the two fluids is also considered. The computational results are compared with theory, yielding a quantitative agreement with Oldroyd (Proc R Soc Lond A 218:122–132, 1953) predictions in a wide range of values of the considered parameters, even well beyond the dilute regime, and also in the cases with slip.     

Transonic Euler computation in streamfunction co-ordinates

Numerical simulation by a finite element method is used to examine the problem of the rotating flow of a viscoelastic fluid in a cylindrical vessel agitated with a paddle impeller. The mathematical model consists of a viscoelastic constitutive equation of Oldroyd B type coupled to the hydrodynamic equations expressed in a rotating frame. This system is solved by using an unsteady approach for velocity, pressure and stress fields. For Reynolds numbers in the range 0.1–10, viscoelastic effects are taken into account up to a Deborah number De of 1.33 and viscoelasticity and inertia cross-effects are studied. Examining the velocity and stress fields as well as the power consumption, it is found that their evolutions are significantly different for low and moderate inertia. These results confirm the trends of experimental studies and show the specific contribution of elasticity without interference of the pseudoplastic character found in actual fluids.     

Mixed convection flow of second grade fluid along a vertical stretching flat surface with variable surface temperature

In the present study we have explored the effects of thermal buoyancy on flow of a viscoelastic second grade fluid past a vertical, continuous stretching sheet of which the velocity and temperature distributions are assumed to vary according to a power-law form. The governing differential equations are transformed into dimensionless form using appropriate transformations and then solved numerically. The methods here employed are (1) the perturbation method together with the Shanks transformation, (2) the local non-similarity method with second level of truncation and (3) the implicit finite difference method for values of ξ ( = Gr _x /Re _x ², defined as local mixed convection parameter) ranging in [0, 10]. The comparison between the solutions obtained by the aforementioned methods found in excellent agreement. Effects of the elasticity parameter λ on the skin-friction and heat transfer coefficients have been shown graphically for the fluids having the values of the Prandtl number equal to 0.72, 7.03 and 15.0. Effects of the viscoelastic parameter and the mixed convection parameter, ξ, on the temperature and velocity fields have also been studied. We notice that with the increase in visco-elastic parameter λ, velocity decreases whereas temperature increases and that velocity gradient is higher than that of temperature. On leave of absence from the Department of Mathematics, University of Dhaka, Bangladesh.     

Multigrid solution of the Navier–Stokes equations on highly stretched grids

Relaxation-based multigrid solvers for the steady incompressible Navier–Stokes equations are examined to determine their computational speed and robustness. Four relaxation methods were used as smoothers in a common tailored multigrid procedure. The resulting solvers were applied to three two-dimensional flow problems, over a range of Reynolds numbers, on both uniform and highly stretched grids. In all cases the L₂ norm of the velocity changes is reduced to 10^?6 in a few 10's of fine-grid sweeps. The results of the study are used to draw conciusions on the strengths and weaknesses of the individual relaxation methods as well as those of the overall multigrid procedure when used as a solver on highly stretched grids.     

Theoretical prediction of the mass transfer coefficients in bubble columns operating in churn-turbulent flow regime. Study in Newtonian and non-Newtonian fluids under different operation conditions

 A comprehensive experimental study of the volumetric transfer coefficient k _L a with Newtonian and non-Newtonian fluids in bubble columns using CO₂ as gas phase is the objective of this work. The evaluation of the hydrodynamic characteristics of the bubble columns and delineated the different hydrodynamic regimes considering column geometry, gas flow, liquid height and type of fluid (Newtonian and non-Newtonian) suggest a general applicability of the proposed model. An explanation about of the k _L a values in non-Newtonian fluid is offered take into account shear rate, column geometry, viscosity and results reported in the literature previously. Received on 31 July 1999