Steady and unsteady laminar flows of Newtonian and generalized Newtonian fluids in a planar T‐junction

An investigation of laminar steady and unsteady flows in a two‐dimensional T‐junction was carried out for Newtonian and a non‐Newtonian fluid analogue to blood. The flow conditions considered are of relevance to hemodynamical applications and the localization of coronary diseases, and the main objective was to quantify the accuracy of the predictions and to provide benchmark data that are missing for this prototypical geometry. Under steady flow, calculations were performed for a wide range of Reynolds numbers and extraction flow rate ratios, and accurate data for the recirculation sizes were obtained and are tabulated. The two recirculation zones increased with Reynolds number, but the behaviour was non‐monotonic with the flow rate ratio. For the pulsating flows a periodic instability was found, which manifests itself by the breakdown of the main vortex into two pieces and the subsequent advection of one of them, while the secondary vortex in the main duct was absent for a sixth of the oscillating period. Shear stress maxima were found on the walls opposite the recirculations, where the main fluid streams impinge onto the walls. For the blood analogue fluid, the recirculations were found to be 10% longer but also short lived than the corresponding Newtonian eddies, and the wall shear stresses are also significantly different especially in the branch duct. Copyright © 2007 John Wiley & Sons, Ltd.     

Finite element simulation of dip coating,I: Newtonian fluids

A finite element simulation of the dip coating process based on a discretization of the continuum with discontinuous pressure elements is presented. The algorithm computes the flow field from natural boundary conditions while an extra condition provided by the existence of free surface is employed to displace the meniscus location towards the actual position. The process is iterative and uses a pseudo-time stepping technique coupled to a cubic spline fitting of the free surface. Numerical predictions exhibit good agreement with experimental data for Newtonian fluids in the case of flat plate dip coating as well as in the case of wire dip coating.     

Three-dimensional flow of Newtonian and Boger fluids in square–square contractions

The flow of a Newtonian fluid and a Boger fluid through sudden square–square contractions was investigated experimentally aiming to characterize the flow and provide quantitative data for benchmarking in a complex three-dimensional flow. Visualizations of the flow patterns were undertaken using streak-line photography, detailed velocity field measurements were conducted using particle image velocimetry (PIV) and pressure drop measurements were performed in various geometries with different contraction ratios. For the Newtonian fluid, the experimental results are compared with numerical simulations performed using a finite volume method, and excellent agreement is found for the range of Reynolds number tested (Re₂ ≤ 23). For the viscoelastic case, recirculations are still present upstream of the contraction but we also observe other complex flow patterns that are dependent on contraction ratio (CR) and Deborah number (De₂) for the range of conditions studied: CR = 2.4, 4, 8, 12 and De₂ ≤ 150. For low contraction ratios strong divergent flow is observed upstream of the contraction, whereas for high contraction ratios there is no upstream divergent flow, except in the vicinity of the re-entrant corner where a localized atypical divergent flow is observed. For all contraction ratios studied, at sufficiently high Deborah numbers, strong elastic vortex enhancement upstream of the contraction is observed, which leads to the onset of a periodic complex flow at higher flow rates. The vortices observed under steady flow are not closed, and fluid elasticity was found to modify the flow direction within the recirculations as compared to that found for Newtonian fluids. The entry pressure drop, quantified using a Couette correction, was found to increase with the Deborah number for the higher contraction ratios.     

The behaviour of several stress–velocity–pressure mixed finite elements for Newtonian flows

The Q₂/P₁, P/P₁, P₂/P₀ and Q₁/P₀ velocity–pressure mixed elements are extended to the stress–velocity–pressure formulation, using the same interpolants for stress and velocity, and tested in the 4-to-1 contraction problem for Stokes flow. The comparison shows significant differences among them, which are not present when the velocity–pressure formulation is used. To provide a better understanding of the phenomenon, several variants of the previous elements are introduced, obtained by either changing the pressure space or by enriching the stress space with bubble functions. The formulation exhibits a strong sensitivity to the first alternative, while the second produces only a minor effect. These observations are confirmed by a convergence test effected on a regular problem with the explicit analytical solution. Also, as a result of the whole comparison, the P/P/P₁ element looks promising for three-field calculations.     

Bubble collapse in compressible fluids using a spectral element marker particle method. Part 1. Newtonian fluids

This paper is concerned with the development of a high‐order numerical scheme for the modelling of two‐phase Newtonian flows. The companion paper, herein referred to as Part 2, extends the scheme to two‐phase viscoelastic flows. The particular problem of the collapse of a two‐dimensional bubble in the vicinity of a rigid boundary is considered. The governing equations are discretized using the spectral element method, and the two phases are modelled using a marker particle method. The marker particle scheme is validated using the Zalesak slotted disk rotation test problem. A comprehensive set of results is presented for the problem of bubble collapse near a rigid wall, and qualitative agreement is obtained with other numerical studies and experimental observations. Viscous effects are shown to inhibit bubble collapse and prevent jet formation and are therefore likely to have a mitigating effect on cavitation damage.Copyright © 2011 John Wiley & Sons, Ltd.     

Parallel domain decomposition method for finite element approximation of 3D steady state non‐Newtonian fluids

We introduce a stabilized finite element method for the 3D non‐Newtonian Navier–Stokes equations and a parallel domain decomposition method for solving the sparse system of nonlinear equations arising from the discretization. Non‐Newtonian flow problems are, generally speaking, more challenging than Newtonian flows because the nonlinearities are not only in the convection term but also in the viscosity term, which depends on the shear rate. Many good iterative methods and preconditioning techniques that work well for the Newtonian flows do not work well for the non‐Newtonian flows. We employ a Galerkin/least squares finite element method, with stabilization parameters adjusted to count the non‐Newtonian effect, to discretize the equations, and the resulting highly nonlinear system of equations is solved by a Newton–Krylov–Schwarz algorithm. In this study, we apply the proposed method to some inelastic power‐law fluid flows through the eccentric annuli with inner cylinder rotation and investigate the robustness of the method with respect to some physical parameters, including the power‐law index and the Reynolds number ratios. We then report the superlinear speedup achieved by the domain decomposition algorithm on a computer with up to 512 processors. Copyright © 2015 John Wiley & Sons, Ltd.     

Correlation of data concerning resistance to flow of generalized Newtonian fluids through granular beds

Summary The paper is concerned with the pressure drop during the flow of rheologically complex fluids through granular beds. An approach is proposed which enables the correlation of the data for any generalized Newtonian fluid. As an example, a detailed derivation of a correlation equation is presented for Carreau fluids. The applicability of the derived equation was proved experimentally in the case of flow of molten poly(ethylene terephthalate) through granular beds.

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Zusammenfassung Die Arbeit betrifft den Druckverlust bei der Strömung von Flüssigkeiten mit komplexen rheologischen Eigenschaften durch Kornschüttungen. Es wird ein Verfahren vorgeschlagen, das eine Korrelation der Daten für beliebige verallgemeinerte newtonsche Flüssigkeiten ermöglicht. Als Beispiel wird die Ableitung der Korrelationsgleichung für Carreau-Flüssigkeiten ausführlich dargestellt. Die Anwendbarkeit der abgeleiteten Gleichung wird für die Strömung von geschmolzenem Polyäthylenterephthalat durch Kornschüttungen experimentell bestätigt.

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a _T temperature shift factor - A constant in eq. [1] - b _M molecular weight shift factor - d _p effective particle diameter, m - D capillary diameter, m - f _BK friction factor, defined by eq. [4] - K constant in eq. [22] - K ₀ constant depending on the shape of a conduit cross-section - K ₁ constant in eq. [11] - l bed height, m - l _e average length of the bed channels, m - L length of a conduit, m - M _n number-average molecular weight - M _w weight-average molecular weight - N constant in Carreau model - p pressure drop due to friction, Pa - r _h hydraulic radius, m - T absolute temperature, K - v mean linear velocity, m/s - v _e mean linear velocity in the bed channels, m/s - v ₀ superficial velocity, related to an empty cross-section of the column, m/s - dimensionless factor, defined by eq. [21] - shear rate, s^–1 - nominal shear rate at the wall of a circular pipe, s^–1 - average shear rate at the wall of a noncircular channel, s^–1 - bed porosity - shear dependent viscosity, Pa s - ₀ zero-shear rate viscosity, Pa s - infinite-shear rate viscosity, Pa s - [] intrinsic viscosity - time constant in Carreau model, s - µ Newtonian viscosity, Pa s - fluid density, kg/m³ - _w shear stress at the wall of a circular pipe, Pa - average shear stress at the wall of a noncircular channel, Pa - {Re} general form of Reynolds number in eq. [1] - Re _BK modified Reynolds number for Newtonian fluids, defined by eq. [25] - Re _BK ^* generalized Reynolds number for Carreau fluids, defined by eq. [24] With 5 figures and 2 tables     

Numerical studies on non‐linear free surface flow using generalized Schwarz–Christoffel transformation

This paper deals with a technique to transform a free surface flow problem in the physical domain with an unknown boundary to a standard domain that has a fixed boundary. All the difficulties in the physical domain are reduced to finding an unknown mapping function that can be solved iteratively in a standard domain. A derivation is first presented to express an analytic function in terms of the boundary value of its imaginary part. Using a relationship between boundaries of the standard and the physical domains, a formula for the generalized Schwarz–Christoffel transformation is then developed. Based on the generalized Schwarz–Christoffel integral and the Hilbert transform, a pair of non‐linear boundary integro‐differential equations in an infinite strip is formulated for solving fully non‐linear free surface flow problems. The boundary integral equations are then discretized with quadratic elements in an untruncated standard domain and solved by the Levenberg–Marquardt algorithm. Several examples of supercritical flow past obstructions are provided to demonstrate the flexibility and the accuracy of the proposed numerical scheme. Copyright © 2000 John Wiley & Sons, Ltd.     

广义KdV方程Fourier谱逼近的最优误差估计

A Fourier spectral method for the generalized Korteweg-de Vries equation with periodic boundary conditions is analyzed, and a corresponding optimal error estimate in L^2-norm is obtained. It improves the result presented by Maday and Quarteroni. A modified Fourier pseudospectral method is also presented, with the same convergence properties as the Fourier spectral method.     

Numerical and analytical simulation of peristaltic flows of generalized Oldroyd‐B fluids

In this paper, we study the peristaltic flows of generalized Oldroyd‐B fluids through the gap between concentric uniform tubes under the assumption of large wavelength and low Reynolds number approximations. The inner tube is rigid and the outer tube has a sinusoidal wave travelling down its wall. Homotopy perturbation and variational iteration methods are used for solution of the problem. The obtained solution is then used to discuss various interesting features of peristalsis. The effects of relaxation time, retardation time and radii of the tubes on pressure rise and friction forces (per wavelength on the inner and outer tubes) are discussed with illustrations. It is found that pressure rise diminishes with increase in relaxation time or the ratio of radii of inner and outer tubes. It increases with increasing retardation time. The effects of both time parameters on friction forces have the opposite behavior to that of pressure rise. Copyright © 2010 John Wiley & Sons, Ltd.     

Stability of high‐speed two‐layer film casting of Newtonian fluids

The steady‐state flow and its linear stability are investigated for the isothermal two‐layer film casting process. Newtonian fluids are considered in this study. The continuity of traction is ensured at the interface, and the axial velocity is assumed to be uniform across each film layer separately. The effects of inertia, gravity, fluid parameters and processing conditions on the steady‐state flow and its stability are studied. The results indicate that the fluid properties and the processing conditions have significant influence on the flow. The flow stability is strongly dependent on the layer layout with respect to the take‐up rolling process. The frequency of the (unstable) disturbance is insensitive to flow and processing parameters. Copyright © 2006 John Wiley & Sons, Ltd.     

Approximation of generalized Stokes problems using dual‐mixed finite elements without enrichment

In this work a finite element method for a dual‐mixed approximation of generalized Stokes problems in two or three space dimensions is studied. A variational formulation of the generalized Stokes problems is accomplished through the introduction of the pseudostress and the trace‐free velocity gradient as unknowns, yielding a twofold saddle point problem. The method avoids the explicit computation of the pressure, which can be recovered through a simple post‐processing technique. Compared with an existing approach for the same problem, the method presented here reduces the global number of degrees of freedom by up to one‐third in two space dimensions. The method presented here also represents a connection between existing dual‐mixed and pseudostress methods for Stokes problems. Existence, uniqueness, and error results for the generalized Stokes problems are given, and numerical experiments that illustrate the theoretical results are presented. Copyright © 2010 John Wiley & Sons, Ltd.     

The flow behavior of fiber suspensions in Newtonian fluids and polymer solutions.

This is the second part of a study examining the mechanical properties and capillary flow of fiber suspensions in Newtonian fluids and in polymer solutions. In part I results for the viscous and elastic properties of the fiber suspensions were presented and it was shown that the fiber suspensions exhibited normal stresses in Newtonian as well as in viscoelastic suspending media. It was thus expected that circulating secondary flows would occur near the entrance to a capillary. Four types of fillers (glass, carbon, nylon and vinylon fibers) suspended in glycerin, HEC solutions and Separan solutions were investigated. The entrance flow patterns were visualized and the pressure fluctuations measured. The visualization enabled the eddies occurring in the fiber suspensions in Newtonian fluids to be analysed and classified into two tpyes. The results from the flow visualization were correlated with the pressure fluctuations. Empirical equations for the tube length correction factor due to the elasticity were obtained.     

Analysis of hypersonic flows using finite elements with Taylor–Galerkin scheme

The aim of the present work is to introduce a formulation for the numerical analysis of three‐dimensional thermochemical non‐equilibrium hypersonic flows, using the finite element method and the Taylor–Galerkin scheme and adopting Park's 2‐temperature, 5‐species (N2, O2, NO, N and O) and 17‐reaction model. Examples using Euler and Navier–Stokes equations are included and compared with experimental and numerical works presented by other authors. The results are close to those analysed by other researches and a good computational performance was obtained. Copyright © 2004 John Wiley & Sons, Ltd.     

Numerical simulations of miscible fluids on a rotating Hele–Shaw cell with effects of Coriolis forces

The effects of Coriolis forces to interfacial stabilities on miscible rotating Hele–Shaw flows are investigated by means of highly accurate numerical schemes. Two major influences on the interfaces, i.e. the stabilization of circumferential fingerings and the unstable body distortion of droplets, are observed. These two phenomena can be represented by two characteristic measurements, i.e. interfacial length and radius of gyration, respectively. Without an additional injection, the miscible interface is found to be more stable at higher Coriolis factors, based on both the judgments of growths in interfacial length and the radius of gyration. However, with an additional injection, a visible body distortion with suppressed interfacial fingerings at higher Coriolis factors leads to inconsistent developments in these two characteristic measurements. The stabilized effect of interfacial fingering delays and weakens the growth in interfacial length at higher Coriolis factors, while faster and greater growth in the radius of gyration is observed because of the significant body distortion. As a result, the effects of the Coriolis forces on the miscible interfacial instabilities should be addressed with great caution. Copyright © 2005 John Wiley & Sons, Ltd.     

Numerical analysis of saltwater intrusion using B-spline boundary elements

Continuity of the derivatives of the main variable is an important feature to obtain an accurate representation of moving boundaries with discrete numerical methods, since the value and direction of the velocity are normally used to relocate the nodal points in a time-marching scheme. A recently developed formulation of the boundary element method using cubic B-splines provides up to C² continuity between adjacent elements. This formulation is applied in this work to saltwater intrusion problems in confined, leaky and unconfined aquifers.     

On the stability of multi-layered fluids. Part I: analysis

The linear stability of the Poiseuille flow of multi-layered different fluids, described mathematically by a system of Orr-Somerfeld differential equations, is investigated. A spectral method is used to rewrite this system into a generalized eigenvalue problem, which can be solved with the QZ-algorithm. Special attention is paid to the tractibility of the interfacial conditions of the stability problem. Since we will limit ourselves to a linear stability analysis, the analytical treatment of the interfacial conditions is simplified. Some results related to simple flow configurations are presented. The origin of certain regions of interfacial instability is explained by simple analytical reasoning.     

Numerical modelling of gas–solid fluidized beds using the two‐fluid approach

This paper details an approach to modelling gas–solid fluidized beds using the two‐fluid granular temperature model. Details concerning the difficulties associated with the boundary conditions, particularly for curved boundaries, are described along with a novel means of obtaining the internal stress of the solid‐phase, in part, by solving an implicit equation. This results in a scheme that is stable even when the solid volume fraction is close to maximum packing. A transient, mixed finite element discretization is used to solve the multi‐phase equations with a discontinuous finite element representation of the granular temperature and continuity equations. A new solution method is proposed to solve the coupled momentum and continuity equations based on Arnoldi iteration. Two fluidized beds are modelled, one in the bubbling regime and the other in the slugging regime. These simulations are compared with experiments. Copyright © 2001 John Wiley & Sons, Ltd.     

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.     

Numerical simulations of the steady Navier–Stokes equations using adaptive meshing schemes

In this paper, we consider an adaptive meshing scheme for solution of the steady incompressible Navier–Stokes equations by finite element discretization. The mesh refinement and optimization are performed based on an algorithm that combines the so‐called conforming centroidal Voronoi Delaunay triangulations (CfCVDTs) and residual‐type local a posteriori error estimators. Numerical experiments in the two‐dimensional space for various examples are presented with quadratic finite elements used for the velocity field and linear finite elements for the pressure. The results show that our meshing scheme can equally distribute the errors over all elements in some optimal way and keep the triangles very well shaped as well at all levels of refinement. In addition, the convergence rates achieved are close to the best obtainable. Extension of this approach to three‐dimensional cases is also discussed and the main challenge is the efficient implementation of three‐dimensional CfCVDT generation that is still under development. Copyright © 2007 John Wiley & Sons, Ltd.