Some simple flows of a Burgers’ fluid

The problems dealing with some unidirectional flows of a Burgers’ fluid are investigated. By using the constitutive equation for a Burgers’ fluid in the literature, the governing time-dependent equation is modeled. Based on the flow conditions described, three flow situations are solved and the exact analytic solutions are given using the Fourier transform technique. Finally, the previous solutions, corresponding to second grade fluid, Oldroyd-B fluid, Maxwell fluid and Newtonian fluid, appear as the special cases of the present analysis.     

Axisymmetric non-Newtonian drops treated with a boundary integral method

A boundary integral method for the simulation of the time-dependent deformation of axisymmetric Newtonian or non-Newtonian drops suspended in a Newtonian fluid subjected to an axisymmetric flow field is developed. The boundary integral formulation for Stokes flow is used and the non-Newtonian stress is treated as a source term which yields an extra integral over the domain of the drop. By transforming the integral representation for the velocity to cylindrical coordinates we can reduce the dimension of the computational problem. The integral equation for the velocity remains of the same form as in Cartesian coordinates, and the Green's functions are transformed explicitly to cylindrical coordinates. Besides a numerical validation of the method we present simulation results for a Newtonian drop and a drop consisting of an Oldroyd-B fluid. The results for the Newtonian drop are consistent with results from the literature. The deformation process of the non-Newtonian drop for small capillary numbers appears to be governed by two relaxation times.     

An explicit finite volume method for viscoelastic fluid flows

We discuss a finite volume method for computing solutions of steady incompressible viscoelastic fluid flows. A fourth-order Runge-Kutta method is used in the explicit time-stepping scheme. The computations are carried out mainly on unstructured grids on Newtonian, inelastic and differential-type constitutive equations, which include the Oldroyd-B and the upper-convected Maxwell models. The performance of the scheme on unstructured grids is investigated, with particular reference to the stick-slip problem for the modified upper-convected Maxwell fluid. The results are compared with those obtained by using the finite element method whenever possible.This research was supported by the Australian Research Council (ARC).     

BEM-RBF approach for viscoelastic flow analysis

A new BE-only method is achieved for the numerical solution of viscoelastic flows. A decoupled algorithm is chosen where the fluid is considered as being composed of an artificial Newtonian component and the remaining component which is accordingly defined from the original constitutive equation. As a result the problem is viewed as that of solving for the flow of a Newtonian liquid with the non-linear viscoelastic effects acting as a pseudo-body force. Thus the general solution can be obtained by adding a particular solution (PS) to the homogeneous one. The former is obtained by a BEM for the base Newtonian fluid and the latter is obtained analytically by approximating the pseudo-body force in terms of suitable radial basis functions (RBFs). Embedded in the approximation of the pseudo-body force is the calculation of the polymer stress. This is achieved by solving the constitutive equation using RBF networks (RBFNs). Both the calculations of the PS and the polymer stress are therefore meshless and the resultant BEM-RBF method is a BE-only method. The complete elimination of any structured domain discretisation is demonstrated with a number of flow problems involving the upper convected Maxwell (UCM) and the Oldroyd-B fluids.     

Exact solutions for the unsteady rotational flow of non-Newtonian fluid in an annular pipe

This paper deals with some unsteady transient rotational flows of an Oldroyd-B fluid in an annular pipe. The fractional calculus approach in the constitutive relationship model of an Oldroyd-B fluid is introduced. A generalized Jeffreys model with the fractional calculus is built. Exact solutions for some unsteady rotational flows of an Oldroyd-B fluid in an annular pipe are obtained by using Hankel transform and the theory of Laplace transform for fractional calculus. The well known solutions for a Navier–Stokes fluid, as well as those corresponding to a Maxwell fluid and a second grade one, appear as limiting cases of our solutions.     

Extensional deformation of non-Newtonian liquid bridges

A numerical method for simulating the extensional dynamics of elongating filaments of non-Newtonian fluids in a filament stretching rheometer is presented. The boundary element method, in conjunction with either the Oldroyd-B or the generalized multimode Upper-Convected Maxwell constitutive model, is used to calculate the transient evolution of the liquid interface, the applied force on the stationary end plate and the polymeric stresses. The numerical results are compared to experimental results and are in excellent agreement at low Hencky strains (Newtonian response) but provide less accurate modeling of the stress growth observed in experiments at higher strains. A comparison of different methods for measuring the apparent extensional viscosity from global measurements of the net force and the mid-point radius of the filament is presented. At large strains calculations show that the fluid motion in these devices closely approximates ideal uniaxial elongation.     

Boundary element analysis of planar drop deformation in confined flow. Part II. Viscoelastic fluids

A boundary element formulation is presented for the general two-dimensional simulation of confined two-phase incompressible flow of viscoelastic fluids. A boundary-only formulation is implemented for fluids obeying the Oldroyd-B constitutive equation. Similarly to the formulation in Part I for Newtonian fluids [Khayat et al. Engng Anal Boundary Elements 1997;19:279], the method requires the solution of two simultaneous integral equations on the interface between the two fluids and the confining solid boundary. Although the problem is formulated for any confining geometry, the method is illustrated for a deforming drop as it is driven by the ambient flow inside a convergent channel. The accuracy and convergence of the method are assessed by comparison with the analytical solution for two-phase Taylor–Couette flow, leading to excellent agreement. Further assessment is made by varying the time increment and mesh size of the discretized boundary for a drop deforming in a convergent channel. The influence of fluid elasticity is examined when one or both fluids are viscoelastic.     

Influence of Hall current on the flows of a generalized Oldroyd-B fluid in a porous space

Summary Exact analytical solutions for the magnetohydrodynamic (MHD) flows of an Oldroyd-B fluid through a porous space are developed. The fractional calculus approach is used to describe the constitutive model of a generalized Oldroyd-B fluid. The porous space is taken into account using modified Darcy's law for fractional viscoelastic fluid. The effect of Hall current is taken into account. Some interesting flows induced due to certain special oscillations are given. In each case, the analytical solutions are obtained using Fourier transform for fractional calculus. The present analysis with fractional calculus approach seems a first attempt for the study of MHD viscoelastic flows in a porous medium.     

Hall effects on the unsteady hydromagnetic flows of an Oldroyd-B fluid

The influence of Hall currents and rotation on the oscillatory flows of an infinite plate is investigated. Exact solutions for the two problems are obtained.The fluid considered is a homogeneous Oldroyd-B. During the mathematical analysis it is found that governing differential equation for steady flow in an Oldroyd-B fluid is identical to that of viscous fluid. Further, it is observed that in absence of the strength of transverse magnetic field (B₀) the solution in resonance case does not satisfy the boundary condition at infinity. Physical significance of mathematical results is also discussed.     

On some helical flows of Oldroyd-B fluids

Summary In this study the velocity fields and the associated tangential stresses corresponding to some helical flows of Oldroyd-B fluids between two infinite coaxial circular cylinders and within an infinite circular cylinder are determined in forms of series in terms of Bessel functions. At time t = 0 the fluid is at rest and the motion is produced by the combined action of rotating and sliding cylinders. The solutions that have been obtained satisfy the governing differential equations and all imposed initial and boundary conditions. For λ _r = 0, λ = 0 or λ _r = λ = 0 they reduce to the similar solutions for a Maxwell, second grade or Newtonian fluid, respectively. Finally, for comparison, the velocity profiles corresponding to the four models are plotted for different values of t.     

MHD Flow of an Oldroyd-B fluid between eccentric rotating disks

An exact solution is obtained for the magnetohydrodynamics (MHD) flow of a conducting, incompressible Oldroyd-B fluid between two infinite, parallel, insulated disks rotating about non-coincident axes normal to the disks in the presence of a uniform transverse magnetic field. The effects of the Hartmann number M, the Deborah number D, the Reynolds number R and the elastic number on the velocity field and the force are discussed. It is found that the value of the torque exerted by the fluid on one of the disks is zero.     

Stream spreading in multilayer microfluidic flows of suspensions

The goal of this experimental study is to quantify the spreading of parallel streams with viscosity contrast in multilayer microfluidic flows. Three streams converge into one channel where a test fluid is sheathed between two layers of a Newtonian reference fluid. The test fluids are Newtonian fluids with viscosities ranging from 1.1 to 48.2 cP and suspensions of 10-mum-diameter PMMA particles with particle volume fractions phi = 0.16-0.30. The fluid interface locations are identified through fluorescence microscopy. The steady-state width of the center stream is strongly dependent on the viscosity ratio between the adjacent fluids and exhibits a near power-law relationship. This dependence occurs for both the Newtonian fluids and the suspensions, although the slopes differ. The high-concentration suspension (phi = 0.30) diverges from Newtonian behavior, while the low-concentration suspensions (phi = 0.16, 0.22) closely approximate that of the Newtonian fluids. The observed suspension behavior can be attributed to shear-induced particle migration.     

Hydromagnetic couette flow of an Oldroyd-B fluid in a rotating system

The motion of electrically conducting, Oldroyd-B and incompressible fluid between two infinitely extended non-conducting parallel plates under a uniform transverse magnetic field, fixed relative to the fluid has been considered. The lower plate is at rest and the upper plate is oscillating in its own plane. The governing partial differential equation of this problem, subject to boundary conditions are solved analytically. The expressions for the steady and unsteady velocity fields for the conducting Oldroyd-B fluid are obtained. The graphs are plotted for different values of dimensionless parameters of the problem and the analysis of the results showed that the flow field is appreciably influenced by the applied magnetic field, the rotation and the material parameters of the fluid.     

Simulation of the Phase of Establishment of the Rheometric Flow of a Viscoelastic Fluid

The flow between two plates in the phase of establishment for the case of a viscoelastic fluid in comparison with a Newtonian fluid is investigated numerically and analytically. The quantitative and qualitative features of the flow are analyzed. It is noted that in a number of cases this flow has wave features, and it is determined that the presence of viscoelasticity usually increases the time of establishment of the flow. It is pointed out that some of the revealed features of the flow can be used in experimental investigations of nonNewtonian fluids.     

Lagrangian finite element analysis of Newtonian fluid flows

A fully Lagrangian finite element method for the analysis of Newtonian flows is developed. The approach furnishes, in effect, a Lagrangian implementation of the compressible Navier–Stokes equations. As the flow proceeds, the mesh is maintained undistorted through continuous and adaptive remeshing of the fluid mass. The principal advantage of the present approach lies in the treatment of boundary conditions at material surfaces such as free boundaries, fluid/fluid or fluid/solid interfaces. In contrast to Eulerian approaches, boundary conditions are enforced at material surfaces ab initio and therefore require no special attention. Consistent tangents are obtained for Lagrangian implicit analysis of a Newtonian fluid flow which may exhibit compressibility effects. The accuracy of the approach is assessed by comparison of the solution for a sloshing problem with existing numerical results and its versatility demonstrated through a simulation of wave breaking. The finite element mesh is maintained undistorted throughout the computation by recourse to frequent and adaptive remeshing © 1998 John Wiley & Sons, Ltd.     

Energetic balance for the Rayleigh–Stokes problem of a Maxwell fluid

The energetic balance in the Rayleigh–Stokes problem for a Maxwell fluid is studied for several initial and/or boundary conditions. In the case of the first problem of Stokes, in comparison with the Newtonian fluid, the power of the wall shear stress and the dissipation increase while the boundary layer thickness decreases. A similar result is obtained for a series solution. In the case of the decay of the previous steady motion as well as for the Newtonian fluid, the power of the wall shear stress is null and the boundary layer thickness is the same. Finally, the second problem of Stokes is also considered.     

平行圆盘间电流变液的挤压流研究

采用双粘度本构模型研究了两平行圆盘间电流变液的挤压流特性,壁面边界上采用Navier滑移模型.流场根据其特性被分为两个区域:对称轴附近的牛顿区以及远离对称轴的双粘区;双粘区存在屈服面.本文在牛顿区和双粘区分别求解出其速度场和压力梯度场.壁面上的滑移速度与当地的压力梯度成正比;而压力梯度在牛顿区与r成正比,在双粘区r值较大的地方与r近似成线性关系.通过将压力梯度在双粘区近似为r的线性函数,可积分出流场的压力分布与作用在圆盘上的挤压力.此外,本文还通过计算,考察了速度场的分布特点,分析了滑移系数对速度场、压力梯度场、屈服面位置以及挤压力的影响.     

A simultaneous solution procedure for strong interactions of generalized Newtonian fluids and viscoelastic solids at large strains

The paper presents a methodology for numerical analyses of coupled systems exhibiting strong interactions of viscoelastic solids and generalized Newtonian fluids. In the monolithic approach, velocity variables are used for both solid and fluid, and the entire set of model equations is discretized with stabilized space–time finite elements. A viscoelastic material model for finite deformations, which is based on the concept of internal variables, describes the stress‐deformation behaviour of the solid. In the generalized Newtonian approach for the fluid, the viscosity depends on the shear strain rate, leading to common non‐Newtonian fluid models like the power‐law. The consideration of non‐linear constitutive equations for solid and fluid documents the capability of the monolithic space–time finite element formulation to deal with complex material models. The methodology is applied to fluid‐conveying cantilevered pipes in order to determine the influence of material non‐linearities on stability characteristics of coupled systems. Copyright © 2005 John Wiley & Sons, Ltd.     

Fundamental solutions for micropolar fluids

New fundamental solutions for micropolar fluids are derived in explicit form for two- and three-dimensional steady unbounded Stokes and Oseen flows due to a point force and a point couple, including the two-dimensional micropolar Stokeslet, the two- and three-dimensional micropolar Stokes couplet, the three-dimensional micropolar Oseenlet, and the three-dimensional micropolar Oseen couplet. These fundamental solutions do not exist in Newtonian flow due to the absence of microrotation velocity field. The flow due to these singularities is useful for understanding and studying microscale flows. As an application, the drag coefficients for a solid sphere or a circular cylinder that translates in a low-Reynolds-number micropolar flow are determined and compared with those corresponding to Newtonian flow. The drag coefficients in a micropolar fluid are greater than those in a Newtonian fluid.