Numerical study of particulate suspension flow through wavy‐walled channels

The particulate suspension flow in a channel whose walls describe a travelling wave motion is examined numerically. A perturbation method is employed and the primitive variables are expanded in a series with the wall amplitude as the perturbation parameter. The boundary conditions are applied at the mean surface of the channel and the first‐order perturbation quantities are numerically determined by solving the governing system of ordinary differential equations by shooting technique. The present approach does not impose any restriction on the Reynolds number of the flow and the wave number and frequency of the wavy‐walled channel, although it is limited by the linear analysis. The wall shear stress and the positions of flow separation and reattachment points are computed and the influence of the volume fraction density of the particles is examined. The variations of velocity and pressure of the particulate suspension flow with frequency of excitation are also presented. Copyright © 2005 John Wiley & Sons, Ltd.     

On the validity of the perturbation approach for the flow inside weakly modulated channels

The equations governing the flow of a viscous fluid in a two‐dimensional channel with weakly modulated walls have been solved using a perturbation approach, coupled to a variable‐step finite‐difference scheme. The solution is assumed to be a superposition of a mean and perturbed field. The perturbation results were compared to similar results from a classical finite‐volume approach to quantify the error. The influence of the wall geometry and flow Reynolds number have extensively been investigated. It was found that an explicit relation exists between the critical Reynolds number, at which the wall flow separates, and the dimensionless amplitude and wavelength of the wall modulation. Comparison of the flow shows that the perturbation method requires much less computational effort without sacrificing accuracy. The differences in predicted flow is kept well around the order of the square of the dimensionless amplitude, the order to which the regular perturbation expansion of the flow variables is carried out. Copyright © 2002 John Wiley & Sons, Ltd.     

A low‐dimensional description of transient shear‐thinning free‐surface flow in thin cavities,as applied to injection molding

A spectral methodology is proposed to examine the influence of shear thinning on the transient free‐surface flow inside a three‐dimensional thin cavity. The problem is closely related to the filling stage during the injection molding process. The moving domain is mapped onto a rectangular domain at each time step of the computation. A modified pressure is introduced that is governed by the Laplace's equation. The flow field is expanded in Fourier series along the lateral direction in the mapped domain, and the Galerkin projection is used to derive the equations that govern the expansion coefficients, which are solved using a variable‐step finite‐difference scheme. This approach is valid for simple and complex cavities as illustrated for the cases of a flat plate with variable and constant thickness. It is shown that, even for highly non‐linear shear‐thinning flow, only a few modes are needed for convergence. Shear thinning generally influences the flow behaviour. However, shear thinning may enhance or prohibit the flow, depending whether the flow rate at the entrance of the cavity is fast or slow, respectively. Copyright © 2004 John Wiley & Sons, Ltd.     

Instability of the separated shear layer in flow past a cylinder: Forced excitation

The receptivity of the separated shear layer for Re = 300 flow past a cylinder is investigated by forced excitation via an unsteady inflow. In order to isolate the shear layer instability, a numerical experiment is set up that suppresses the primary wake instability. Computations are carried out for one half of the cylinder, in two dimensions. The flow past half a cylinder with steady inflow is found to be stable for Re = 300. However, an inlet flow with pulsatile perturbations, of amplitude 1% of the mean, results in the excitation of the shear layer mode. The frequency of the perturbation of the inlet flow determines the frequency associated with the shear layer vortices. For a certain range of forced frequencies the recirculation region undergoes a low‐frequency longitudinal contraction and expansion. An attempt is made to relate this instability to a global mode of the wake determined from a linear stability analysis. Interestingly, this phenomenon disappears when the outflow boundary of the computational domain is shifted sufficiently downstream. This study demonstrates the need of carefully investigating the effect of the location of outflow boundaries if the computational results indicate the presence of low‐frequency fluctuations. The effect of Re and amplitude of unsteadiness at the inlet are also presented. All computations have been carried out using a stabilized finite element formulation of the incompressible flow equations. Copyright © 2007 John Wiley & Sons, Ltd.     

Cross-stream forces and velocities of fixed and freely suspended particles in viscoelastic Poiseuille flow: Perturbation and numerical analyses

The cross-stream migration of a circular particles (or infinitely long cylinder) in two dimensional, inertia-less viscoelastic pressure-driven flows is examined through complementary finite element simulations and second-order fluid perturbation analyses for small Deborah number (De), where De is defined as the fluid relaxation time divided by the characteristic flow time. A neutrally buoyant, freely suspended particle migrates toward the center of the channel for all particle sizes and cross-stream positions due to the coupled effects of the linear and quadratic variations of the imposed velocity. A particle that is held at a fixed position, in contrast, experiences a cross-stream force directed toward the wall as a result of the coupled effects of the local shear flow and the flow relative to the particle.     

On the interplay between inertial and viscoelastic effects for the flow in weakly modulated channels

The flow inside a spatially modulated channel is examined for viscoelastic fluids of the Oldroyd‐B type. The lower wall is flat and the upper wall is sinusoidally modulated. The modulation amplitude is assumed to be small. Thus, a regular perturbation expansion of the flow field coupled to a variable‐step finite‐difference scheme is used to solve the problem. Convergence and accuracy assessment against earlier experimental results indicate that there is a significant range of validity of the perturbation approach. The influences of wall geometry, inertia and viscoelasticity on the flow kinematics and stresses are investigated systematically. In particular, the interplay between the flow and fluid parameters effects on the conditions for the onset of backflow, number of vortices, their size and location is revealed. The distance between the flow separation and reattachment locations identifies the vortex size. Non‐monotonic dependence of the vortex size on elasticity is reported. The critical conditions for the onset of negative elasticity effects on vortex size are identified. The critical Reynolds number for the onset of backflow initially decreases then levels off or even increases as elasticity increases. For highly elastic fluid and large enough Reynolds number, more than one vortex appear near the lower wall. Copyright © 2005 John Wiley & Sons, Ltd.     

Time‐dependent finite element simulations of a shear‐thinning viscoelastic fluid with application to blood flow

A new finite element method is developed to simulate time‐dependent viscoelastic shear‐thinning flows characterized by the generalized Oldroyd‐B model. The focus of the algorithm is improved stability through a free‐energy dissipative scheme by using low‐order piecewise‐constant finite element approximations for stress. The algorithm is further modified by incorporating a pressure‐projection method, a DG‐upwinding scheme, a symmetric interior penalty DG method to solve the elliptic pressure‐update equation and a geometric multigrid preconditioner. The improved stability and cost to accuracy is compared when using higher order discontinuous bilinear approximation, where in addition, we consider the influence of a slope limiter for these elements. The algorithm is applied to the 2D start‐up‐driven cavity problem, and the stability of the free energy is illustrated and compared between element choices. An application of the model to modelling blood in small arterioles and channels is considered by simulating pulsatile blood flow through a stenotic arteriole. The individual influences of viscoelasticity and shear‐thinning within the generalized Oldroyd‐B model are investigated by comparing results to the Newtonian, generalized Newtonian and Oldroyd‐B models. Copyright © 2014 John Wiley & Sons, Ltd.     

Simulation of pressure‐tooling wire‐coating flow with Phan‐Thien/Tanner models

Annular pressure‐tooling extrusion is simulated for a low density polymer melt using a Taylor–Petrov–Galerkin finite element scheme. This represents industrial‐scale wire‐coating. Viscoelastic fluids are modeled via three forms of Phan‐Thien/Tanner (PTT) constitutive laws employed for short‐die and full specification pressure‐tooling. Effects of variation in Weissenberg number (We) and polymeric viscosity are investigated. Particular attention is paid to mesh refinement to predict accurate results. The impact of variation in shear‐thinning and strain‐softening properties is considered upon the modelling predictions. For the short‐die flow, the influence of the lack of strain softening is identified. For the full‐die flow and more severe deformation rates, the linear PTT model failed to converge. In contrast, the exponential PTT model is found to be more stable numerically and to adequately reflect the material response. Comparing short‐die and full‐die pressure‐tooling results, shear rates increase 10‐fold, while strain rates increase one hundred times. Copyright © 2002 John Wiley & Sons, Ltd.     

Bounds and perturbation series for incompressible elastic composites with transverse isotropic symmetry

The effective elastic behavior of a transversely isotropic composite made from two incompressible elastic materials is examined. The set of all effective elasticity tensors for transversely isotropic finite rank laminar microstructures is described. The extremal property of this class of microstructures is used to derive a new more precise characterization of the set of effective shear moduli.The perturbation series for the effective elasticity tensor is considered. An explicit formula for the second order perturbation tensor is derived. We describe precisely the set of tensors that correspond to all second order perturbations consistent with transverse isotropy. We apply analytic methods [cf. 27] to show that all second order perturbation tensors are realized by finite rank laminar microstructures.Supported by NSF through Grant DMS-3907658.     

Local stresses and strains in axially cracked cylindrical shells

A perturbation solution is obtained for the local stress-strain fields in an axially cracked cylindrical shell. The tenth-order differential equations are used that take into account the transverse shear deformation. The perturbation of a curvature parameter, λ, is employed, where . The stress intensity factors for finite size cylindrical shells subjected to bending and internal pressure are evaluated. Sufficient accuracy can be obtained without using fine mesh sizes in regions near the crack tip. Also analyzed are the influence of cylinder diameter and shearing stiffness on bulging.     

High‐order numerical simulations of flow‐induced noise

In this paper, the flow/acoustics splitting method for predicting flow‐generated noise is further developed by introducing high‐order finite difference schemes. The splitting method consists of dividing the acoustic problem into a viscous incompressible flow part and an inviscid acoustic part. The incompressible flow equations are solved by a second‐order finite volume code EllipSys2D/3D. The acoustic field is obtained by solving a set of acoustic perturbation equations forced by flow quantities. The incompressible pressure and velocity form the input to the acoustic equations. The present work is an extension of our acoustics solver, with the introduction of high‐order schemes for spatial discretization and a Runge–Kutta scheme for time integration. To achieve low dissipation and dispersion errors, either Dispersion‐Relation‐Preserving (DRP) schemes or optimized compact finite difference schemes are used for the spatial discretizations. Applications and validations of the new acoustics solver are presented for benchmark aeroacoustic problems and for flow over an NACA 0012 airfoil. Copyright © 2010 John Wiley & Sons, Ltd.     

A hybrid immersed boundary and material point method for simulating 3D fluid–structure interaction problems

A numerical method is developed for solving the 3D, unsteady, incompressible Navier–Stokes equations in curvilinear coordinates containing immersed boundaries (IBs) of arbitrary geometrical complexity moving and deforming under forces acting on the body. Since simulations of flow in complex geometries with deformable surfaces require special treatment, the present approach combines a hybrid immersed boundary method (HIBM) for handling complex moving boundaries and a material point method (MPM) for resolving structural stresses and movement. This combined HIBM & MPM approach is presented as an effective approach for solving fluid–structure interaction (FSI) problems. In the HIBM, a curvilinear grid is defined and the variable values at grid points adjacent to a boundary are forced or interpolated to satisfy the boundary conditions. The MPM is used for solving the equations of solid structure and communicates with the fluid through appropriate interface‐boundary conditions. The governing flow equations are discretized on a non‐staggered grid layout using second‐order accurate finite‐difference formulas. The discrete equations are integrated in time via a second‐order accurate dual time stepping, artificial compressibility scheme. Unstructured, triangular meshes are employed to discretize the complex surface of the IBs. The nodes of the surface mesh constitute a set of Lagrangian control points used for tracking the motion of the flexible body. The equations of the solid body are integrated in time via the MPM. At every instant in time, the influence of the body on the flow is accounted for by applying boundary conditions at stationary curvilinear grid nodes located in the exterior but in the immediate vicinity of the body by reconstructing the solution along the local normal to the body surface. The influence of the fluid on the body is defined through pressure and shear stresses acting on the surface of the body. The HIBM & MPM approach is validated for FSI problems by solving for a falling rigid and flexible sphere in a fluid‐filled channel. The behavior of a capsule in a shear flow was also examined. Agreement with the published results is excellent. Copyright © 2007 John Wiley & Sons, Ltd.     

Effects of the Jacobian evaluation on Newton's solution of the Euler equations

Newton's method is developed for solving the 2‐D Euler equations. The Euler equations are discretized using a finite‐volume method with upwind flux splitting schemes. Both analytical and numerical methods are used for Jacobian calculations. Although the numerical method has the advantage of keeping the Jacobian consistent with the numerical residual vector and avoiding extremely complex analytical differentiations, it may have accuracy problems and need longer execution time. In order to improve the accuracy of numerical Jacobians, detailed error analyses are performed. Results show that the finite‐difference perturbation magnitude and computer precision are the most important parameters that affect the accuracy of numerical Jacobians. A method is developed for calculating an optimal perturbation magnitude that can minimize the error in numerical Jacobians. The accuracy of the numerical Jacobians is improved significantly by using the optimal perturbation magnitude. The effects of the accuracy of numerical Jacobians on the convergence of the flow solver are also investigated. In order to reduce the execution time for numerical Jacobian evaluation, flux vectors with perturbed flow variables are calculated only for neighbouring cells. A sparse matrix solver that is based on LU factorization is used. Effects of different flux splitting methods and higher‐order discretizations on the performance of the solver are analysed. Copyright © 2005 John Wiley & Sons, Ltd.     

Numerical study of concentrated fluid–particle suspension flow in a wavy channel

The particle migration effects and fluid–particle interactions occurring in the flow of highly concentrated fluid–particle suspension in a spatially modulated channel have been investigated numerically using a finite volume method. The mathematical model is based on the momentum and continuity equations for the suspension flow and a constitutive equation accounting for the effects of shear‐induced particle migration in concentrated suspensions. The model couples a Newtonian stress/shear rate relationship with a shear‐induced migration model of the suspended particles in which the local effective viscosity is dependent on the local volume fraction of solids. The numerical procedure employs finite volume method and the formulation is based on diffuse‐flux model. Semi‐implicit method for pressure linked equations has been used to solve the resulting governing equations along with appropriate boundary conditions. The numerical results are validated with the analytical expressions for concentrated suspension flow in a plane channel. The results demonstrate strong particle migration towards the centre of the channel and an increasing blunting of velocity profiles with increase in initial particle concentration. In the case of a stenosed channel, the particle concentration is lowest at the site of maximum constriction, whereas a strong accumulation of particles is observed in the recirculation zone downstream of the stenosis. The numerical procedure applied to investigate the effects of concentrated suspension flow in a wavy passage shows that the solid particles migrate from regions of high shear rate to low shear rate with low velocities and this phenomenon is strongly influenced by Reynolds numbers and initial particle concentration. Copyright © 2008 John Wiley & Sons, Ltd.     

The instability mechanism of single and multilayer Newtonian and viscoelastic flows down an inclined plane

The instability mechanism of single and multilayer flow of Newtonian and viscoelastic fluids down an inclined plane has been examined based on a rigorous energy analysis as well as careful examination of the eigenfunctions. These analyses demonstrate that the free surface instability in single and multilayer flows in the limit of longwave disturbances (i.e., the most dangerous disturbances) arise due to the perturbation shear stresses at the free surface. Specifically, for viscoelastic flows, the elastic forces are destabilizing and the main driving force for the instability is the coupling between the base flow and the perturbation velocity and stresses and their gradient at the free surface. For Newtonian flows at finite Re, the driving force for the interfacial instability in the limit of longwaves depends on the placement of the less viscous fluid. If the less viscous fluid is adjacent to the solid surface then the main driving force for the instability is interfacial friction, otherwise the bulk contribution of Reynolds stresses drives the instability. For viscoelastic fluids in the limit of vanishingly small Re, the driving force for the instability is the coupling of the base flow and perturbation velocity and stresses and their gradients across the interface. In the limit of shortwaves the interfacial stability mechanism of flow down inclined plane is the same as plane Poiseuille flows (Ganpule and Khomami 1998, 1999a, b). Received: 20 October 2000/Accepted: 11 January 2001     

Torsional flow stability of highly dilute polymer solutions

We examine the torsional flow stability, to axisymmetric disturbances, of a variety of multimode and non-linear constitutive models in a bounded parallel plate geometry. The analysis is facilitated by the construction of a regular perturbation scheme in the ratio of polymer to total viscosity. As a model for Boger fluids this corresponds to the assumption that the Boger fluid is highly dilute. The consequent simplification provided by the perturbation scheme allows us to examine the effects of a discrete spectrum of relaxation times, shear thinning, second normal stress difference, and finite extensional viscosity on the torsional instability.     

Peristaltic flow of a Jeffrey‐six constant fluid in a uniform inclined tube

In this work, we studied the peristaltic flow of a Jeffrey‐six constant fluid in a uniform tube. The governing equations of the Jeffrey‐six constant fluid were simplified by using the assumptions of long wave length and low Reynolds number approximation. The simplified form of equations were solved using the perturbation, homotopy analysis and finite difference methods. The comparison of the three solutions are shown graphically. The variation of pressure rise and frictional forces with the different parameters were also examined numerically. Results are presented at the end of the article. Copyright © 2011 John Wiley & Sons, Ltd.     

Motion of a nonrectilinear fiber in a viscous fluid flow

The plane problem of dynamic interaction of a laminar viscous fluid flow and an inextensible pliable fiber of finite length is solved using the perturbation method. Two types of rheological twodimensional flows — pure shear and simple shear — are considered. Formulas are obtained for the evolution of the tensile force and the shape of the fiber. Results of asymptotic and numerical calculations are compared.     

Die swell measurements of second‐order fluids: numerical experiments

An analysis of the flow of a second‐order fluid is presented. Reference values for some variables are defined, and with these a non‐dimensional formulation of the governing equations. From this formulation, three dimensionless numbers appear; one is the Reynolds number, and two numbers that are called the first‐ and second‐dimensionless normal stress (NSD) coefficients. The equations of motion are solved by a finite element method using a commercially available program (Fidap), and the steady state converged solution was used to measure the die swell. The factors that influence die swell and that are studied in this work include: the die geometry for circular cross sectional dies, including tubular, converging, diverging, half‐converging/half‐tubular shapes; fluid characteristics such as Reynolds number and first‐ and second‐DNS coefficients (both positive and negative values); and flow rates, as determined by the maximum velocity in a parabolic velocity profile at the entrance to the die. The results suggest that shear and deformation histories of the fluid directly influence not only swell characteristics, but also convergence characteristics of the numerical simulation. © 1999 John Wiley & Sons, Ltd.