Numerical Initial Value Problems in Ordinary Differential Equations

We identify the physical mechanism through which newly developed quaternary ammonium salt (QAS) deposit control additives (DCAs) affect the rheological properties of cavitating turbulent flows, resulting in an increase in the volumetric efficiency of clean injectors fuelled with diesel or biodiesel fuels. Quaternary ammonium surfactants with appropriate counterions can be very effective in reducing the turbulent drag in aqueous solutions, however, less is known about the effect of such surfactants in oil-based solvents or in cavitating flow conditions. Small-angle neutron scattering (SANS) investigations show that in traditional DCA fuel compositions only reverse spherical micelles form, whereas reverse cylindrical micelles are detected by blending the fuel with the QAS additive. Moreover, experiments utilising X-ray micro computed tomography (micro-CT) in nozzle replicas, quantify that in cavitation regions the liquid fraction is increased in the presence of the QAS additive. Furthermore, high-flux X-ray phase contrast imaging (XPCI) measurements identify a flow stabilization effect in the region of vortex cavitation by the QAS additive. The effect of the formation of cylindrical micelles is reproduced with computational fluid dynamics (CFD) simulations by including viscoelastic characteristics for the flow. It is demonstrated that viscoelasticity can reduce turbulence and suppress cavitation, and subsequently increase the injector’s volumetric efficiency.

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Turbulence and Cavitation Suppression by Quaternary Ammonium Salt Additives.

The non-Newtonian flow of dilute aqueous polyethylene oxide (PEO) solutions through micro-fabricated planar abrupt contraction-expansions is investigated. The small lengthscales and high deformation rates in the contraction throat lead to significant extensional flow effects even with dilute polymer solutions having time constants on the order of milliseconds. By considering the definition of the elasticity number, El = Wi/Re, we show that the lengthscale of the geometry is key to the generation of strong viscoelastic effects, such that the same flow behaviour cannot be reproduced using the equivalent macro-scale geometry using the same fluid. We observe significant vortex growth upstream of the contraction plane, which is accompanied by an increase of more than 200% in the dimensionless extra pressure drop across the contraction. Streak photography and video-microscopy using epifluorescent particles shows that the flow ultimately becomes unstable and three-dimensional. The moderate Reynolds numbers (0.44 ≤ Re ≤ 64) associated with these high Weissenberg number (0 ≤ Wi ≤ 548) micro-fluidic flows results in the exploration of new regions of the Re-Wi parameter space in which the effects of both elasticity and inertia can be observed. Understanding such interactions will be increasingly important in micro-fluidic applications involving complex fluids and can best be interpreted in terms of the elasticity number, El = Wi/Re, which is independent of the flow kinematics and depends only on the fluid rheology and the characteristic size of the device.

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The inertio-elastic planar entry flow of low-viscosity elastic fluids in micro-fabricated geometries

Particle focusing in planar geometries is essentially required in order to develop cost-effective lab-on-a-chips, such as cell counting and point-of-care (POC) devices. In this study, a novel method for sheathless particle focusing, called “Elasto-Inertial Particle Focusing”, was demonstrated in a straight microchannel. The particles were notably aligned along the centerline of the straight channel under a pressure-driven flow without any additional external force or apparatus after the addition of an elasticity enhancer: PEO (poly(ethylene oxide)) (∼O(100) ppm). As theoretically predicted (elasticity number: El ≈ O(100)), multiple equilibrium positions (centerline and corners) were observed for the viscoelastic flow without inertia, whereas three-dimensional particle focusing only occurred when neither the elasticity nor the inertia was negligible. Therefore, the three-dimensional particle focusing mechanism was attributed to the synergetic combination of the elasticity and the inertia (elasticity number: El ≈ O(1–10)). Furthermore, from the size dependence of the elastic force upon particles, we demonstrated that a mixture of 5.9 and 2.4 µm particles was separated at the exit of the channel in viscoelastic flows. We expect that this method can contribute to develop the miniaturized flow cytometry and microdevices for cell and particle manipulation.

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Sheathless elasto-inertial particle focusing and continuous separation in a straight rectangular microchannel

Microgel pastes and concentrated emulsions are shown to exhibit a generic slip behavior at low stresses when sheared near smooth surfaces. The magnitude of slip depends on the applied stress. Well above the yield stress, slip is negligible compared to the bulk flow. Just above the yield stress, slip becomes significant and the total deformation results from a combination of bulk flow and slip. At and below the yield stress, the bulk flow is negligible and the apparent motion is entirely due to wall slip. By directly imaging the deformation of pastes and from rheological measurements, we show that slip is characterized by universal scaling properties, which depend on solvent viscosity, bulk shear modulus, and particle size. A model based on elastohydrodynamic lubrication between the squeezed particles and the shearing surface explains these properties quantitatively.

https://sor.scitation.org/doi/pdf/10.1122/1.1795171

Slip and flow in pastes of soft particles: Direct observation and rheology

In this paper, a general class of viscoelastic model is used to investigate numerically the pattern and strength of the secondary flows in rectangular pipes as well as the influence of material parameters on them. To solve the coupled governing equation system, an implicit finite volume method based on the SIMPLEST algorithm, which is applicable for both time-dependent and steady-state flow computations, has been developed and extended for viscoelastic flow computations by applying the decoupled techniques. The main feature of the method is to split the solution process into a series of steps in which the continuity of the flow field is enforced by solving a Poisson's equation for the pressure, and at the end of the steps, both the pressure and velocity fields are made to satisfy one and the same momentum equation. For viscoelastic flow computations, artificial diffusion terms are introduced on both sides of the discretized constitutive equations to improve numerical stability. It is found that there are in total two vortices in each quadrant of the pipe at different aspect ratios (from 1 to 16), and at each ratio the pattern of secondary flows takes the same form for different material parameters, but their strength is very sensitive to the viscoelastic material parameters. Numerical results indicate that the presence of secondary flow strongly depends on the primary flow rate and the elasticity of the fluid, namely, the first and the second normal stress differences as well as their functional departure from the constant multiple viscosity.

Numerical study of secondary flows of viscoelastic fluid in straight pipes by an implicit finite volume method

We present in this paper a fully three dimensional (3D) convergent numerical study of planar viscoelastic contraction flows. A 3D finite volume method (FVM) with the primary variable elastic viscous split stress (EVSS) formulation is employed, and a very efficient 3D block solver coupled with block correction is developed to speed up the convergence rate. Full 3D simulations of viscoelastic flows in 4:1 planar abrupt contractions are carried out using experimental conditions. Upstream vortex patterns comparable with the existing flow visualisation observations are captured using the upper convected Maxwell model for a Boger fluid and the Phan-Thien–Tanner model for a shear thinning fluid. Comprehensive comparisons between numerical simulation results and data measured in the dynamic fields in a 4:1 planar abrupt contraction are made, and the results indicate that the experimental measurements can be quantitatively reproduced if the fluid is well characterised by an appropriate viscoelastic model. It is confirmed numerically that the shear thinning of the fluid reduces the intensity of the singularity of viscoelastic flow near the re-entrant corner. With the Oldroyd-B model, by extensive computations on successively refined meshes with the minimum dimensionless size being 0.16–0.014 on the contraction plane in 2D configuration, it is revealed that, although the asymptotic flow behavior near the re-entrant corner and the build-up of the overall pressure and extensional stresses as well as the kinematic behavior along the centreline are insensitive to mesh refinement, completely different vortex activities may be predicted if the mesh is not sufficiently fine. It is verified numerically that, depending on the flow inertia and rheological properties of fluids, both the lip vortex mechanism and the corner vortex mechanism may be responsible for the vortex activities of viscoelastic fluids in 4:1 planar contraction flow, and the elasticity number E and Mach number M of the flow can be used to determine the vortex mechanism approximately. It is clear that the development process of the vortex activities could be underestimated with 2D simplification, and overpredicted with the creeping flow assumption, particularly when Re>0.5. Therefore, in planar contraction flow analyses, numerical artifacts may be produced with a coarse mesh, and 2D flow simulation is only a good approximation to the fully 3D flow if the upstream aspect ratio W/H in the experiment is at least 10.

Three dimensional numerical simulations of viscoelastic flows through planar contractions

Edge fracture is an instability of cone-plate and parallel plate flows of viscoelastic liquids and suspensions, characterised by the formation of a `crack' or indentation at a critical shear rate on the free surface of the liquid. A study is undertaken of the theoretical, experimental and computational aspects of edge fracture. The Tanner-Keentok theory of edge fracture in second-order liquids is re-examined and is approximately extended to cover the Criminale-Ericksen-Filbey (CEF) model. The second-order theory shows that the stress distribution on the semi-circular crack is not constant, requiring an average to be taken of the stress; this affects the proportionality constant, K in the edge fracture equation −N
2c = KΓ/a, where N
2c is the critical second normal stress difference, Γ is the surface tension coefficient and a is the fracture diameter. When the minimum stress is used, K = 2/3 as found by Tanner and Keentok (1983). Consideration is given to the sources of experimental error, including secondary flow and slip (wall effect). The effect of inertia on edge fracture is derived. A video camera was used to record the inception and development of edge fracture in four viscoelastic liquids and two suspensions. The recorded image was then measured to obtain the fracture diameter. The edge fracture phenomenon was examined to find its dependence on the physical dimensions of the flow (i.e. parallel plate gap or cone angle), on the surface tension coefficient, on the critical shear rate and on the critical second normal stress difference. The critical second normal stress difference was found to depend on the surface tension coefficient and the fracture diameter, as shown by the theory of Tanner and Keentok (1983); however, the experimental data were best fitted by the equation −N
2c = 1.095Γ/a. It was found that edge fracture in viscoelastic liquids depends on the Reynolds number, which is in good agreement with the inertial theory of edge fracture. Edge fracture in lubricating grease and toothpaste is broadly consistent with the CEF model of edge fracture. A finite volume method program was used to simulate the flow of a viscoelastic liquid, obeying the modified Phan-Thien-Tanner model, to obtain the velocity and stress distribution in parallel plate flow in three dimensions. Stress concentrations of the second normal stress difference (N
2) were found in the plane of the crack; the velocity distribution shows a secondary flow tending to aid crack formation if N
2 is negative, and a secondary flow tending to suppress crack formation if N
2 is positive.

Edge fracture in cone-plate and parallel plate flows

In this paper, numerical modelling of transient viscoelastic flows in the context of a finite volume formulation is discussed with focus on numerical accuracy, efficiency and stability of the iterative solution algorithms used in solving the coupled system of the discretized governing equations. Generalized three-level schemes are used for temporal discretization, which give at least second-order temporal discretization accuracy for a convection-diffusion equation. However, we will show that numerical stability and efficiency of calculations for the coupled system of the governing equations can be poor if an improper iterative solution algorithm is used. It is found that the explicit solution algorithms in which the parent equations are discretized explicitly in time is not suitable for transient viscoelastic calculations due to the possible spatial oscillations whenever the order of spatial discretization for hyperbolic constitutive equations is higher than the first-order. On the other hand, with the implicit solution algorithms for truly transient flow calculations, only when the indicative errors in the iterative solution process are at a compatible (the same or smaller) order with that embodied in the discretization process, can we take the temporal discretization order to be the accuracy of the final solutions for the coupled system. In this sense, the SIMPLEST and PISO algorithms are two of the most efficient and stable algorithms available. Our analysis has been confirmed by numerical calculations carried out for the start-up and decay of Poiseuille flows between two parallel plates for Newtonian fluid and for viscoelastic fluids with a solvent viscosity described by the Oldroyd-B model. We demonstate that analytical solutions can be reproduced smoothly without limitation on Weissenberg number. Finally, we analyze and numerically demonstrate that with the decoupled method, for accurate simulations of the transient viscoelastic flow with inertia, the governing equations have to be decoupled and solved strictly according to the type of each equation. The stabilizing measures developed for steady state modelling, such as ‘both sides diffusion’ (BSD), elastic viscous stress split (EVSS), and adaptive viscous split stress (AVSS) may lead to change of type in the equation from a purely hyperbolic one to a parabolic one for a purely elastic (Maxwell) fluid, or lead to overdiffusion in transient velocity field calculations for the viscoelastic fluid with a solvent viscosity. Therefore, these kinds of measures should not be taken, and the ‘viscous formulation’ is more proper.

Numerical modelling of transient viscoelastic flows

In drawing microstructured optical fibers (MOFs), the cross-sectional hole structure, including holes' relative size and shape, in a finished fiber drawn down from a preform can be different from that designed in the preform due to combined effects of draw tension and surface tension. As a result, the fiber's optical properties relative to the initial design can be significantly altered. In order to find means of minimizing or exploiting hole deformation so that MOFs with desirable optical functionality can be fabricated, the underlying mechanism of hole deformation is analyzed by numerically investigating the continuous draw process of MOFs of different materials under different drawing conditions. It is found that three dimensionless numbers, i.e., 1) the capillary number (related to material properties), 2) the draw ratio, and 3) the aspect ratio (both related to the drawing conditions), can be used to predict the type of hole deformation. Silica and polymer materials are considered in particular, but the use of these dimensionless numbers allows the analysis to be applied to any other material.

Shicheng Xue

Papers

Numerical study of secondary flows of viscoelastic fluid in straight pipes by an implicit finite volume method

Three dimensional numerical simulations of viscoelastic flows through planar contractions

Edge fracture in cone-plate and parallel plate flows

Numerical modelling of transient viscoelastic flows

Role of material properties and drawing conditions in the fabrication of microstructured optical fibers