Flow Of Shear-thinning Fluids Research Articles

MHD Flow of Shear-Thinning Fluid over a Rotating Disk with Heat Transfer

This paper studied the Magneto hydrodynamic (MHD) flow and heat transfer of an electrically conducting non-Newtonian fluid over a rotating disk in the presence of a uniform magnetic field. The steady, laminar and axial-symmetric flow is driven solely by the rotating disk, and the incompressible fluid obeys the inelastic Ostwald de-Waele power-law model. The governing differential equations were reduced to a set of ordinary differential equations by utilizing the generalized Karman similarity transformation. The nonlinear two-point boundary value problem is solved by multi-shooting method. Numerical results show that the magnetic parameter and the power-law index have significant effects on the swirling flow and heat transfer.

삼각형 단면 덕트 내의 Shear-Thinning 유체에 대한 열전달 촉진에 관한 연구

Abstract The prediction of heat transfer and pressure drops in the exchanger passages is a clue to the problem of heat exchanger design. In order to make such predictions for non-Newtonian fluids, it is necessary to know the relation between the viscous properties of the fluid and the wall shear rate in the duct. This study deals with the limits of validity of the power law equation. The useful methodology of the present research involves a consideration of a more general equation which has power law and Newtonian behavior as asymptotes. It is concluded that use of the power law equation outside of its applicability range can lead to serious errors in predicting the heat transfer and pressure drops. The present computational results of the friction factors times Reynolds number for shear-thinning fluid flows in a triangular duct are compared with previous published results, showing agreement with 0.13 % in Newtonian region and 2.85 % in power law region. These shear-thinning fluid results also showed the 12% increase of convective heat transfer enhancement compared with Newtonian heat transfer.

Subcritical bifurcation of shear-thinning plane Poiseuille flows

AbstractIn a recent article (Nouar, Bottaro & Brancher,J. Fluid. Mech., vol. 592, 2007, pp. 177–194), a linear stability analysis of plane Poiseuille flow of shear-thinning fluids has been performed. The authors concluded that the viscosity stratification delays the transition and that is important to account for the viscosity perturbation. The current paper focuses on the first-principles understanding of the influence of the viscosity stratification and the nonlinear variation of the effective viscosity$\ensuremath{\mu} $with the shear rate$\dot {\gamma } $on the flow stability with respect to a finite-amplitude perturbation. A weakly nonlinear analysis, using the amplitude expansion method, is adopted as a first approach to study nonlinear effects. The bifurcation to two-dimensional travelling waves is studied. For the numerical computations, the shear-thinning behaviour is described by the Carreau model. The rheological parameters are varied in a wide range. The results indicate that (i) the nonlinearity of the viscous terms tends to reduce the viscous dissipation and to accelerate the flow, (ii) the harmonic generated by the nonlinearity$\ensuremath{\mu} (\dot {\gamma } )$is smaller and in opposite phase to that generated by the quadratic nonlinear inertial terms and (iii) with increasing shear-thinning effects, the bifurcation becomes highly subcritical. Consequently, the magnitude of the threshold amplitude of the perturbation, beyond which the flow is nonlinearly unstable, decreases. This result is confirmed by computing higher order-Landau constants.

A Newton–Krylov finite volume algorithm for the power-law non-Newtonian fluid flow using pseudo-compressibility technique

An implicit Newton–Krylov finite volume algorithm has been developed for efficient steady-state computation of the power-law non-Newtonian fluid flows. The pseudo-compressibility technique is used for the coupling of continuity and momentum equations. The spatial discretization is central (second-order) for both convective and diffusive terms and the accuracy of the solution is verified. The nine block diagonal Jacobian matrix (needed for implicit formulation) is computed directly through the flux differentiation. Five-diagonal and three-diagonal block matrices (the simplified versions of the main Jacobian matrix) are used with the ILU(0 & 1) and the Thomas linear solvers for preconditioning, respectively. The performance of the Newton-GMRES solver is examined in detail for different preconditioning strategies. The effects of the power-law behavior index and Re number on the convergence rate are also studied. The performance of the Newton-BiCGSTAB and the Newton-GMRES solvers are compared with each other. The results show, the ILU(1)/Newton-GMRES is the most efficient combination that is robust even in high Reynolds number shear-thinning fluid flow cases.

A perturbation analysis of the flow of a Powell--Eyring fluid between coaxial cylinders

The Powell--Eyring generalised viscous fluid model was developed to model the flow of viscous inelastic shear thinning fluids, such as polymer melts and suspensions of solids in non-Newtonian solvents. We consider the helical flow of a Powell--Eyring fluid between infinitely long coaxial cylinders. Such a flow is of relevance to a number of applications, including rheometry. We show that a perturbation approach to the flow problem yields simple explicit expressions for the fluid velocity field, which may then be applied to obtain approximations to other features of the flow. References C. Chiera, H. J. Connell and J. J. Shepherd, Perturbation analysis of the helical flow of non-Newtonian fluids with application to a recirculating coaxial cylinder rheometer, Mathematical and Computer Modelling, 18, No. 10: 131--140, 1993. B. D. Coleman and W. Noll, Helical flow of General Fluids, J. App. Phys., 10: 1508--1512, 1959. R. I. Tanner, Engineering Rheology, Oxford University Press, Oxford, 1988. M. T. J. Farrugia, A Perturbation Analysis of the Flow of a Powell--Eyring Fluid between Coaxial Cylinders, Honours Thesis, Mathematics Department, RMIT, 1999. R. B. Bird, R. C. Armstrong and D. Hassager, Dynamics of Polymetric liquids, Vol. 1, John Wiley and Sons, 1987.

Drag on a Sphere in Poiseuille Flow of Shear-Thinning Power-Law Fluids

The effect of finite boundaries on the drag experienced by a sphere exposed to the Poiseuille flow of power-law fluids in cylindrical vessels has been investigated numerically. In particular, the momentum equations have been solved over the following ranges of conditions: sphere Reynolds number based on the area average velocity in the pipe, Re 1−100; power law index, n: 0.2−1, and sphere-to-tube diameter ratio, λ: 0−0.5. Due to the obstruction in the path of the fluid caused by the sphere fixed at the axis of the tube and the corresponding changes in the velocity field close to the sphere, there is extra viscous dissipation at the walls and this, in turn, leads to an increase in the drag force acting on the sphere. Conversely, there is an extra pressure drop caused by the fixed sphere for the flow of the fluid in the tube. The effect, however, is more significant at low Reynolds numbers than that at high Reynolds numbers. Similarly, the additional drag due to the confining walls increases with the increasing degree of obstruction, i.e., sphere-to-tube diameter ratio. Overall, all else being equal, the wall effect is seen to be less severe in power-law fluids than that in Newtonian fluids. Furthermore, the confining walls also influence the onset of flow separation and subsequently the size of the recirculation region. The present numerical predictions are consistent with the scant experimental results available in the literature for Newtonian and power-law fluids.

Planar flows of incompressible heat-conducting shear-thinning fluids—Existence analysis

We study the flow of an incompressible homogeneous fluid whose material coefficients depend on the temperature and the shear-rate. For large class of models we establish the existence of a suitable weak solution for two-dimensional flows of fluid in a bounded domain. The proof relies on the reconstruction of the globally integrable pressure, available due to considered Navier’s slip boundary conditions, and on the so-called L ∞-truncation method, used to obtain the strong convergence of the velocity gradient. The important point of the approach consists in the choice of an appropriate form of the balance of energy.

Experimental characterization of the fluid dynamics in an in- vitro system simulating the peristaltic movement of the stomach wall

The difficulty in experimentally characterizing the dynamics of gastric contents during digestion has limited the knowledge required to understand and model the kinetics of food disintegration. The goal of this work was to experimentally characterize, in a closed system, the peristaltic flow of fluids with different rheological properties. An experimental apparatus was developed to reproduce the general characteristics of gastric motility, while supporting the use of a non-intrusive flow measurement technique (Particle Image Velocimetry). The apparatus consisted of a polycarbonate chamber with one neoprene wall that was continuously deformed by vertically moving a stainless steel cylinder at a speed of 3.5mm/s. The fluid dynamics of three Newtonian fluids (water, and water solutions of 10% and 95% corn-syrup) and two shear thinning fluids (water solutions of 0.15% and 0.50% CMC) were investigated. In addition, the effect of the occlusion ratio (30, 45 and 60%) on the fluid dynamics of water within the system was analyzed. Experimental results showed that the loci of maximum velocity were present in the section corresponding to the location of the hump peak. As the viscosity of the Newtonian fluid increased, higher and more localized retropulsive velocities developed. However, these differences were not observed in the dynamics of shear thinning fluid flows. As the viscosity of the fluid increased, a more ordered flow developed, but no differences were found with respect to maximum values of shear rates or vorticity. The differences in the overall flow behaviour of the two CMC solutions were even less noticeable, but they both exhibited lower shear rates and vorticities than the Newtonian fluid. By decreasing the compression ratio of the hump, the maximum water velocity within the system decreased by 45%, while the maximum values of shear and vorticity decreased by 18% and 23%, respectively. This work showed the significant effect of the rheological properties of the flow on the fluid-dynamics within a closed system due to the peristaltic deformation of its wall. From a boarder perspective, this work demonstrated the relevance of acquiring a good characterization of the rheology of gastric contents to fully characterize the fluid- mechanical forces involved in food digestion.

Effect of the contraction ratio upon viscoelastic fluid flow in three-dimensional square–square contractions

In this work we investigate the laminar flow through square–square sudden contractions with various contraction ratios ( CR=2.4, 4, 8 and 12), using a Newtonian fluid and a shear-thinning viscoelastic fluid. Visualizations of the flow patterns were carried out using streak line photography and detailed velocity field measurements were performed using particle image velocimetry. The experimental results are compared with numerical predictions obtained using a finite-volume method. For the Newtonian fluid, a corner vortex is found upstream of the contraction and increasing flow inertia leads to a reduction of the vortex size. Good agreement is observed between experiments and numerical simulations. For the shear-thinning fluid flow a corner vortex is also observed upstream of the contraction independently of the contraction ratio. Increasing the elasticity of the flow, while still maintaining low inertia flow conditions, leads to a strong increase of the vortex size, until an elastic instability sets in and the flow becomes time-dependent at De≈200, 300, 70 and 450 for CR=2.4, 4, 8 and 12, respectively. At low contraction ratios, viscoelasticity brings out an anomalous divergent flow upstream of the contraction. For both fluids studied the flow presents a complex three-dimensional helical vortex structure which is well predicted by numerical simulations. However, for the viscoelastic fluid flow the maximum Deborah number achieved in the numerical simulations is about one order of magnitude lower than the critical Deborah number for the onset of the elastic instability found in the experiments.

Relationship between induced fluid structure and boundary slip in nanoscale polymer films

The molecular mechanism of slip at the interface between polymer melts and weakly attractive smooth surfaces is investigated using molecular dynamics simulations. In agreement with our previous studies on slip flow of shear-thinning fluids, it is shown that the slip length passes through a local minimum at low shear rates and then increases rapidly at higher shear rates. We found that at sufficiently high shear rates, the slip flow over atomically flat crystalline surfaces is anisotropic. It is demonstrated numerically that the friction coefficient at the liquid-solid interface (the ratio of viscosity and slip length) undergoes a transition from a constant value to the power-law decay as a function of the slip velocity. The characteristic velocity of the transition correlates well with the diffusion velocity of fluid monomers in the first fluid layer near the solid wall at equilibrium. We also show that in the linear regime, the friction coefficient is well described by a function of a single variable, which is a product of the magnitude of surface-induced peak in the structure factor and the contact density of the adjacent fluid layer. The universal relationship between the friction coefficient and induced fluid structure holds for a number of material parameters of the interface: fluid density, chain length, wall-fluid interaction energy, wall density, lattice type and orientation, thermal or solid walls.

PIV Study of Laminar Wall Jets of Non-Newtonian Fluids

Three-dimensional laminar wall jet flows of shear-thinning non-Newtonian fluids have been studied using a particle image velocimetry technique. The non-Newtonian fluids were prepared from xanthan gum solutions of various concentrations. The velocity measurements were performed in various streamwise-transverse and streamwise-spanwise planes at various inlet Reynolds numbers. From these measurements, the maximum velocity decay, jet half-widths, and velocity profiles were obtained to study the effects of Reynolds number and fluid type on the characteristics of the wall jet flows. It was observed that the maximum velocity decay and jet half-widths depend on inlet Reynolds number and fluid but the similarity velocities profiles are independent of both Reynolds number and specific fluid type.

Petrov–Galerkin computation of nonlinear waves in pipe flow of shear-thinning fluids: First theoretical evidences for a delayed transition

A pseudospectral Petrov–Galerkin code is developed in order to compute nonlinear traveling waves in pipe flow of shear-thinning fluids. The framework is continuum mechanics and the rheological model used is the purely viscous Carreau model. The code is validated, and a study of its convergence properties is made. It is shown that exponential convergence is obtained, despite the highly-nonlinear nature of the viscous diffusion terms. Physical computations show that, as compared with the case of a constant-viscosity fluid, i.e., a Newtonian fluid, in the case of shear-thinning fluids the critical Reynolds number of the saddle-node bifurcation where the waves with an azimuthal wavenumber m 0 = 3 appear increases significantly when the non-Newtonian effects come into play.

Stability of core-annular flow of power-law fluids in the presence of interfacial surfactant

The shear-thinning influence on the core-annular flow stability of two immiscible power-law fluids is considered by making a linear stability analysis. The flow is driven by an axial pressure gradient in a straight pipe with the interface between the two fluids occupied by an insoluble surfactant. Given the basic flow for this core-annular arrangement, the analytical solution is obtained with respect to the power-law fluid model. The linearized equations for the evolution of infinitesimal disturbances are derived and the stability problem is formulated as a generalized matrix eigenvalue problem, which is solved by using the software package Matlab based on the QZ algorithm. The shear-thinning property is found to have marked influence on the power-law fluid core-annular flow stability, which is reflected in various aspects. First, the capillary instability is magnified by the shear-thinning property, which may lead to an essential difference between power-law and Newtonian fluid flows. Especially when the interface is close to the pipe wall, the power-law fluid flow may be unstable while the Newtonian fluid flow is stable. Second, under disturbances to the interface a velocity discontinuity at the interface appears which is destabilizing to the flow. The magnitude of this velocity discontinuity is affected by the power-law index and the flow stability is influenced correspondingly. Besides, the shear-thinning property may induce new stability modes which do not appear in the Newtonian fluid flow. The flow stability shows much dependence on the interface location, the role of which was neglected in most previous studies. The shear-thinning fluid flow is more unstable to long wave disturbances when the interface is close to the pipe wall, while the Newtonian fluid flow is more unstable when the interface is close to the pipe centerline. But this trend is changed by the addition of interfacial surfactant, for which the power-law fluid flow is more stable no matter where the interface is located.

Turbulent flow of shear-thinning liquids in stirred tanks—The effects of Reynolds number and flow index

LDA measurements are reported on the turbulent velocity fields in vessels agitated by a Rushton turbine and containing Newtonian as well as non-Newtonian, shear-thinning fluids. Ten different liquids were investigated, with flow indices varying from 1.00 down to 0.56. Experiments were performed in three vessel sizes, viz. 28.6, 44.1, and 62.7 cm in diameter, at various impeller speeds. The main issue of the paper is the question whether or not, and if so to what extent, turbulent flow of shear-thinning fluids differs from that of Newtonian liquids, or – in other words – whether and when turbulent flow of shear-thinning liquids exhibits Reynolds number similarity. The experimental data presented comprise profiles of the mean velocity components and the rms fluctuating velocity components as a function of the radial position in the tank at the height of the impeller disc as well as similar profiles in the impeller outflow near to the impeller tip. The effects – if any – of both Reynolds number and flow index on these profiles are assessed. Fit equations are presented for the various profiles in the various liquids. These fit equations are claimed to be valid throughout the ranges of Reynolds numbers and flow indices covered by the experiments presented. The idea is that these fit equations may be used to validate Computational Fluid Dynamics (CFD) simulations of the Reynolds Averaged Navier–Stokes (RANS) type for agitated shear-thinning liquids, even for liquids and conditions not investigated in this study as long as falling within the Reynolds number and flow index ranges investigated.

Stability of plane Couette–Poiseuille flow of shear-thinning fluid

A linear stability analysis of the combined plane Couette and Poiseuille flow of shear-thinning fluid is investigated. The rheological behavior of the fluid is described using the Carreau model. The linearized stability equations and their boundary conditions result in an eigenvalue problem that is solved numerically using a Chebyshev collocation method. A parametric study is performed in order to assess the roles of viscosity stratification and the Couette component. First of all, it is shown that for shear-thinning fluid, the critical Reynolds number for a two-dimensional perturbation is less than for a three dimensional. Therefore, it is sufficient to deal only with a modified Orr–Sommerfeld equation for the normal velocity component. The influence of the velocity of the moving wall on the critical conditions is qualitatively similar to that for a Newtonian fluid. Concerning the effect of the shear thinning, the computational results indicate that this behavior leads to a decrease in the phase velocity of the traveling waves and an increase in stability, when an appropriate viscosity is used in the definition of the Reynolds number. Using a long-wave version of the Orr–Sommerfeld equation, the cutoff velocity is derived. The mechanisms responsible for the changes in the flow stability are discussed in terms of the location of the critical layers, Reynolds stress distribution, and the exchange of energy between the base flow and the disturbance.

Stability analysis of stratified coating flow of shear-thinning fluids

This study is a linear stability analysis of a two-layer non-Newtonian film flow driven by gravity down an inclined plane. The fluids rheology is described by the Carreau law. The effects of density stratification are studied in the case of a less viscous layer adjacent to the wall, which is favourable to interfacial instabilities. We highlight that shear-thinning can have stabilizing as well as destabilizing effects according to the wavelength and the position of the Carreau fluid layers.

Symmetry breaking phenomena of purely viscous shear-thinning fluid flow in a locally constricted channel

The goal of a present study is to investigate the effects of generalized Newtonian fluids on the threshold of the transition from flow symmetry to its asymmetry for the flow through a locally constricted channel. We consider purely viscous shear-thinning fluid and compare it with the Newtonian fluid. Fluid flow is studied numerically by solving the two dimensional momentum equations along with the continuity equation and the Carreau-Yasuda rheological model. We report systematic results in a range of generalized Reynolds number 150≤Re≤250 with a focus on its critical value. Results indicate that the shear-thinning viscous behaviour decreases the onset of bifurcation phenomena and the critical value of Reynolds number. Last but not least, a systematic grid refinement analysis and numerical accuracy study is performed and present numerical results may be treated as the benchmark.

Viscoelastic flow in a 3D square/square contraction: Visualizations and simulations

The inertialess three-dimensional (3D) flow of viscoelastic shear-thinning fluids in a 4:1 sudden square-square contraction was investigated experimentally and numerically and compared with the flow of inelastic fluids. Whereas for a Newtonian fluid the vortex length remains unchanged at low Reynolds numbers, with the non-Newtonian fluid there is a large increase in vortex length with fluid elasticity leading to unstable periodic flow at higher flow rates. In the steady flow regime the vortices are 3D and fluid particles enter the vortex at the middle plane, rotate towards its eye, drift sideways to the corner-plane vortex, rotate to its periphery, and exit to the downstream duct. Such dynamic process is reverse of that observed and predicted with Newtonian fluids. Numerical predictions using a multimode Phan-Thien–Tanner viscoelastic model are found to match the visualizations accurately and in particular are able to replicate the observed flow reversal. The effect of fluid rheology on flow reversal, vor...

Galerkin Least-Squares Multifield Approximations for Flows of Inelastic Non-Newtonian Fluids

The aim of this work is to investigate a Galerkin least-squares (GLS) multifield formulation for inelastic non-Newtonian fluid flows. We present the mechanical modeling of isochoric flows combining mass and momentum balance laws in continuum mechanics with an inelastic constitutive equation for the stress tensor. For the latter, we use the generalized Newtonian liquid model, which may predict either shear-thinning or shear-thickening. We employ a finite element formulation stabilized via a GLS scheme in three primal variables: extra stress, velocity, and pressure. This formulation keeps the inertial terms and has the capability of predicting viscosity dependency on the strain rate. The GLS method circumvents the compatibility conditions that arise in mixed formulations between the approximation functions of pressure and velocity and, in the multifield case, of extra stress and velocity. The GLS terms are added elementwise, as functions of the grid Reynolds number, so as to add artificial diffusivity selectively to diffusion and advection dominant flow regions—an important feature in the case of variable viscosity fluids. We present numerical results for the lid-driven cavity flow of shear-thinning and shear-thickening fluids, using the power-law viscosity function for Reynolds numbers between 50 and 500 and power-law exponents from 0.25 to 1.5. We also present results concerning flows of shear-thinning Carreau fluids through abrupt planar and axisymmetric contractions. We study ranges of Carreau numbers from 1 to 100, Reynolds numbers from 1 to 100, and power-law exponents equal to 0.1 and 0.5. Besides accounting for inertia effects in the flow, the GLS method captures some interesting features of shear-thinning flows, such as the reduction of the fluid stresses, the flattening of the velocity profile in the contraction plane, and the separation of the boundary layer downstream the contraction.

Galerkin least-squares solutions for purely viscous flows of shear-thinning fluids and regularized yield stress fluids

This paper aims to present Galerkin Least-Squares approximations for flows of Bingham plastic fluids. These fluids are modeled using the Generalized Newtonian Liquid (GNL) constitutive equation. Their viscoplastic behavior is predicted by the viscosity function, which employs the Papanastasiou's regularization in order to predict a highly viscous behavior when the applied stress lies under the material's yield stress. The mechanical modeling for this type of flow is based on the conservation equations of mass and momentum, coupled to the GNL constitutive equation for the extra-stress tensor. The finite element methodology concerned herein, the well-known Galerkin Least-Squares (GLS) method, overcomes the two greatest Galerkin shortcomings for mixed problems. There is no need to satisfy Babuška-Brezzi condition for velocity and pressure subspaces, and spurious numerical oscillations, due to the asymmetric nature of advective operator, are eliminated. Some numerical simulations are presented: first, the lid-driven cavity flow of shear-thinning and shear-thickening fluids, for the purpose of code validation; second, the flow of shear-thinning fluids with no yield stress limit, and finally, Bingham plastic creeping flows through 2:1 planar and axisymmetric expansions, for Bingham numbers between 0.2 and 133. The numerical results illustrate the arising of two distinct unyielded regions: one near the expansion corner and another along the flow centerline. For those regions, velocity and pressure fields are investigated for the various Bingham numbers tested.