Dispersion of linear gravity waves on a viscoelastic fluid in an horizontal canal

A characteristic equation is derived that describes the spatial decay of linear surface gravity waves on Maxwell fluids. Except at small frequencies, the derived dispersion relation is different from the temporal decay dispersion relation which is normally studied within fluid mechanics. The implications for waves on viscous Newtonian fluids using the two different dispersion relations is briefly discussed. The wave number is measured experimentally as function of the frequency in a horizontal canal. Seven Newtonian fluids and four viscoelastic liquids with constant viscosity have been used in the experiments. The spatial decay theory for Newtonian fluids fits well to the experimental data. The model and experiments are used to determine limits for the Maxwell fluid time numbers for the four viscoelastic liquids. As a result of low viscosity it was not possible within this study to obtain these time numbers from oscillatory experiments. Therefore, a comparison of surface gravity wave experiments with theory is applicable as a method to evaluate memory times of low viscosity viscoelastic fluids.     

Effects of shear-thinning/thickening nature on the stability of Couette-like flow with uniform crossflow

The linear stability of wall-injected pressure- driven Couette-like flow in power-law fluids is studied. Previous study on this kind of flow for Newtonian fluids by Nicoud and Angilella [Phys. Rev. E 56, 3000 （1997）] was extended to power-law fluids to understand the effects of shear-thinning/thickening nature on the flow stability. A related expression between the critical crossflow Reynolds number for Newtonian fluids and that for power-law fluids is obtained as the streamwise Reynolds number is large enough based on numerical computations, and verified theoretically in the case of a limiting condition with the power-law index.     

Front‐tracking by the level‐set and the volume penalization methods in a two‐phase microfluidic network

Two‐phase immiscible fluids in a two‐dimensional micro‐channels network are considered. The incompressible Stokes equations are used to describe the Newtonian fluid flow, while the Oldroyd‐B rheological model is used to capture the viscoelastic behavior. In order to perform numerical simulations in a complex geometry like a micro‐channels network, the volume penalization method is implemented. To follow the interface between the two fluids, the level‐set method is used, and the dynamics of the contact line is modeled by Cox law. Numerical results show the ability of the method to simulate two‐phase flows and to follow properly the contact line between the two immiscible fluids. Finally, simulations with realistic parameters are performed to show the difference when a Newtonian fluid is pushed by a viscoelastic fluid instead of a Newtonian one. Copyright © 2015 John Wiley & Sons, Ltd.     

Mixed convection boundary layer flow of non-Newtonian fluids along vertical wavy plates

Mixed convection boundary layer flows of non-Newtonian fluids over the wavy surfaces are studied by the coordinate transformation and the cubic spline collocation numerical method. The effects of the wavy geometry, the buoyancy parameter and the generalized Prandtl number for pseudoplastic fluids, Newtonian fluids and dilatant fluids on the skin-friction coefficient, local and mean Nusselt numbers have been graphically studied. Results show that both higher generalized Prandtl numbers and buoyancy parameters are seen to enhance the influence of wavy surfaces on the local Nusselt number, irrespective of whether the fluids are Newtonian fluids or non-Newtonian fluids. Moreover, the irregular surfaces have higher total heat flux than that of corresponding flats plate for any fluid.     

Laminar,transitional and turbulent annular flow of drag-reducing polymer solutions

Mean and rms axial velocity-profile data obtained using laser Doppler anemometry are presented together with pressure-drop data for the flow through a concentric annulus (radius ratio κ = 0.506) of a Newtonian (a glycerine–water mixture) and non-Newtonian fluids—a semi-rigid shear-thinning polymer (a xanthan gum) and a polymer known to exhibit a yield stress (carbopol). A wider range of Reynolds numbers for the transitional flow regime is observed for the more shear-thinning fluids. In marked contrast to the Newtonian fluid, the higher shear stress on the inner wall compared to the outer wall does not lead to earlier transition for the non-Newtonian fluids where more turbulent activity is observed in the outer wall region. The mean axial velocity profiles show a slight shift (～5%) of the location of the maximum velocity towards the outer pipe wall within the transitional regime only for the Newtonian fluid.     

Newtonian stratified flow through an abrupt expansion

The present paper is devoted to the numerical simulations of stratified generalized Newtonian flow. The main feature of the algorithm is to include the coordinates of the interface as primitive unknowns of the flow problem together with the nodal velocity components and pressures calculated on a deformable finite element mesh. Newton-Raphson's iterative method is used for solving the non-linear problem. Special attention is given to the location of the intersection between the interface and a solid boundary, when it is unknown at the outset. The method is applied to the creeping stratified flow of two Newtonian fluids through a one-to-four abrupt expansion; the results remain valid for the contraction problem. It is found that, under appropriate conditions, the small Newtonian corner vortex can increase by a large factor, both in size and intensity; the degrees of freedom of the simulation are the ratios of flow-rate and of viscosity of the fluids.     

Three‐dimensional boundary element analysis of drop deformation in confined flow for Newtonian and viscoelastic systems

An adaptive (Lagrangian) boundary element approach is proposed for the general three‐dimensional drop deformation in confined flow. The adaptive method is stable as it includes remeshing capabilities of the deforming interface between drop and suspending fluid, and thus can handle large deformations. Both drop and surrounding fluid are viscous incompressible and can be Newtonian or viscoelastic. A boundary‐only formulation is implemented for fluids obeying the linear Jeffrey's constitutive equation. Similarly to the formulation for two‐dimensional Newtonian fluids (Khayat RE, Luciani A, Utracki LA. Boundary element analysis of planar drop deformation in confined flow. Part I. Newtonian fluids. Engineering Analysis of 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 cylindrical tube. The accuracy of the method is assessed by comparison with the analytical solution for two‐phase radial spherical flow, leading to good agreement. The influence of mesh refinement is examined for a drop in simple shear flow. Copyright © 2000 John Wiley & Sons, Ltd.     

A projection scheme for Navier-Stokes with variable viscosity and natural boundary condition

Combiningthe Navier-Stokes systems with Neumann (or natural) boundary condition to characterize a fluid flow is frequent. The popular projection (or pressure correction) methods inspired by Chorin and Temam are not well adapted to such boundary condition, which translate in loss of accuracy. If some alternative projection methods have been proposed to reduce the accuracy loss due to the Neumann condition in case of Newtonian fluids, little has been proposed for generalized Newtonian fluids. In this work, we propose two methods derived from the incremental pressure correction projection that can be used for fluids with inhomogeneous or variable viscosity with natural boundary condition. Both time and space accuracy of the methods will be illustrated using a manufactured solution.     

MHD free-convective flow of micropolar and Newtonian fluids through porous medium in a vertical channel

We have studied the fully-developed free-convective flow of an electrically conducting fluid in a vertical channel occupied by porous medium under the influence of transverse magnetic field. The internal prefecture of the channel is divided into two regions; one region filled with micropolar fluid and the other region with a Newtonian fluid or both the regions filled by Newtonian fluids. Analytical solutions of the governing equations of fluid flow are found to be in excellent agreement with analytical prediction. Analytical results for the details of the velocity, micro-rotation velocity and temperature fields are shown through graphs for various values of physical parameters. It is noticed that Newtonian fluids prop up the linear velocity of the fluid in contrast to micropolar fluid. Also the skin friction coefficient at both the walls is derived and its numerical values are offered through tables.     

Influence of the magnetic field on merging flow of the Powell-Eyring fluids: an exact solution

In this article, the merging flow of the stagnation point and the stretching (or shrinking) flows for the Powell-Eyring fluid (one of the non-Newtonian fluids) in the presence of magnetic field is formulated and analyzed mathematically. An analytical solution was developed on the basis of the homotopy analysis method. The effect of the Hartmann number on fluid-velocity and skin-friction is examined. It is observed that the intensive magnetic field reduces the growth of the reverse/secondary flow which is generated after the mixing of the stagnation-point and shrinking-sheet flows. The magnetic force dominates on the viscous force for stretching as well as for shrinking flows. Furthermore, the magnetic force intensifies the axial velocity of the fluids (the Newtonian as well as the Powell-Eyring fluids) but it decays the transverse-velocity of the fluids. Present results are validated with the existing results for the Newtonian fluids and found comparable with negligible errors.     

A simple mixture theory for ν Newtonian and generalized Newtonian constituents

This work presents the development of mathematical models based on conservation laws for a saturated mixture of ν homogeneous, isotropic, and incompressible constituents for isothermal flows. The constituents and the mixture are assumed to be Newtonian or generalized Newtonian fluids. Power law and Carreau–Yasuda models are considered for generalized Newtonian shear thinning fluids. The mathematical model is derived for a ν constituent mixture with volume fractions ${\phi_\alpha}$ using principles of continuum mechanics: conservation of mass, balance of momenta, first and second laws of thermodynamics, and principles of mixture theory yielding continuity equations, momentum equations, energy equation, and constitutive theories for mechanical pressures and deviatoric Cauchy stress tensors in terms of the dependent variables related to the constituents. It is shown that for Newtonian fluids with constant transport properties, the mathematical models for constituents are decoupled. In this case, one could use individual constituent models to obtain constituent deformation fields, and then use mixture theory to obtain the deformation field for the mixture. In the case of generalized Newtonian fluids, the dependence of viscosities on deformation field does not permit decoupling. Numerical studies are also presented to demonstrate this aspect. Using fully developed flow of Newtonian and generalized Newtonian fluids between parallel plates as a model problem, it is shown that partial pressures p _α of the constituents must be expressed in terms of the mixture pressure p. In this work, we propose ${p_\alpha=\phi_\alpha p}$ and ${\sum_\alpha^\nu p_\alpha = p}$ which implies ${\sum_\alpha^\nu \phi_\alpha = 1}$ which obviously holds. This rule for partial pressure is shown to be valid for a mixture of Newtonian and generalized Newtonian constituents yielding Newtonian and generalized Newtonian mixture. Modifications of the currently used constitutive theories for deviatoric Cauchy stress tensor are proposed. These modifications are demonstrated to be essential in order for the mixture theory for ν constituents to yield a valid mathematical model when the constituents are the same. Dimensionless form of the mathematical models is derived and used to present numerical studies for boundary value problems using finite element processes based on a residual functional, that is, least squares finite element processes in which local approximations are considered in ${H^{k,p}\left(\bar{\Omega}^e\right)}$ scalar product spaces. Fully developed flow between parallel plates and 1:2 asymmetric backward facing step is used as model problems for a mixture of two constituents.     

Mass transfer from small capillaries with wall resistance in the laminar flow regime

The Graetz problem in heat transfer is extended to the analysis of mass transfer in circular ducts for the cases where wall resistance is included and where non-Newtonian fluids that obey Casson's equation are considered. The eigenvalues and fluid bulk coefficients are presented for the fluid between the extremes of Newtonian and slug flow. It is found that for fluids which are only slightly non-Newtonian, such as blood, which is closely approximated by Casson's equation, the mass transfer rate can be predicted by Newtonian fluid analysis without appreciable error. Some experimental results give support to the theory.     

New model for single spherical particle settling velocity in power law (visco-inelastic) fluids

Particle settling in a non-Newtonian power law fluid is of interest to many industrial applications, including chemical, food, pharmaceutical, and petroleum industry. Conventionally, the Newtonian model for the drag coefficient prediction is extended to non-Newtonian fluids. The approach of merely replacing a viscosity term in Newtonian correlation with a power law apparent viscosity is reported to be inadequate.     

Turbulent flow of Newtonian and shear-thinning liquids through a sudden axisymmetric expansion

 Measurements are reported for the turbulent flow through a sudden expansion of a moderately elastic shear-thinning liquid and also for two Newtonian liquids. The differences in the mean velocity fields for the two fluid types are relatively small, including the length of the recirculation region which is essentially unaffected by the fluid rheology. Although turbulent kinetic energy levels for the non-Newtonian fluids are always lower than for the Newtonian fluids, no significant difference is found in the relative contributions to the turbulent kinetic energy of the axial, radial and tangential normal stresses. Since the vorticity thicknesses are much the same for all flows, viscoelasticity appears to be responsible for the reduced levels of turbulent kinetic energy for the non-Newtonian fluids. Received: 6 November 1998/Accepted: 27 January 1999     

Breakup of Newtonian and non-Newtonian fluids in air jets

The breakup of droplets of non-Newtonian fluids has been investigated by high speed photography and impaction following preliminary results of Newtonian fluids, which confirmed the suitability of the measurement techniques. Single droplets with diameters from 2.4 to 3.3 mm, were arranged to fall under gravity into a jet of air with velocities up to 36o m/s. The droplets of Newtonian fluids, water and Diesel oil, were atomised in the expected manner within three main regimes characterised by the Weber number of the droplet and air jet conditions, while similar droplets of non-Newtonian fluids were found not to atomise but to develop under shear and stretching into ligaments of fluid separated from a local region of their surface; these ligaments were elongated until breakup occurred, though not into small droplets as with the Newtonian fluids. Some of the non-Newtonian fluids (TEP with 7.5% and lo% K125, with and without water) were found not to break up at the maximum speed of the tests and they will be re-examined at higher jet velocities. Increase in the concentration of K125 in TEP resulted in higher critical speed for a given droplet diameter.The authors would like to thank Prof. J. H. Whitelaw of Imperial College for many discussions and useful suggestions during the course of this work, and Dr. G. Cambray of CBDE for his valuable administrative support     

A pressure-smoothing scheme for incompressible flow problems

A pressure-smoothing scheme for Stokes and Navier–Stokes flows of Newtonian fluids and for Stokes flow of Maxwell fluids is described. The stress deviator obtained from the calculated velocity field is substituted into the governing equilibrium equation. The resulting equation is then solved to obtain a new, smoothed pressure by a least square finite element method.     

On co-rotational and other rate type constitutive equations

We derive expressions for the dilatational properties of suspensions of gas bubbles in incompressible fluids, using a cell model for the suspension. A cell, consisting of a gas bubble centered in a spherical shell of incompressible fluid, is subjected to a purely dilatational boundary motion and the resulting stress at the cell boundary is obtained. The same dilatational boundary motion is prescribed at the boundary of an “equivalent” cell composed of a one-phase, uniformly compressible fluid with unknown dilatational properties. By specifying that the stress at the boundary of the one-phase cell is equal to the stress at the boundary of the two-phase suspension cell, we obtain expressions for the unknown dilatational properties as a function of observable properties of the suspension. The dilatational viscosity of a suspension with a Newtonian continuous phase and the analogous properties for suspensions with non-Newtonian continuous phases are obtained as functions of the boundary motion, volume fraction of gas, and properties of the incompressible continuous phase. Results are presented for continuous phases which are Newtonian fluids, second-order fluids, and Goddard—Miller model fluids.     

Drag of a sphere in dilute polymer solutions in high Reynolds number range

We have measured by means of four ultrasonic transducers the fall velocity of a sphere at high Reynolds number range in dilute polyacrylamide solutions which have viscoelastic effects. The polymer solutions were 5, 20 and 50ppm in the concentration. Basset-Bousinessq-Oseen equation for the falling sphere was analyzed numerically on Newtonian fluids in order to compare with the fall velocity of a sphere in the polymer solutions, and the experimental data of the fall velocity in tap water is in agreement with the range of no effect of the test tank wall. In polymer solutions, it was shown that the fall velocity is larger than that in Newtonian fluids within the critical Reynolds number range such that the drag reduction occurs and is smaller than that of Newtonian fluids over the range. The experimental data for the drag reduction ratio of polymer solutions is arranged by Weissenberg number calculating the experimental data of the first normal stress differences. It was shown that the maximum drag reduction ratio in the polymer solutions lies in the range of We=3∼10. Received: 15 October 1997 Accepted: 12 May 1998     

Variational principles and entropy production in creeping flow of non-newtonian fluids

The extension of Helmholtz—Korteweg's theorem to purely viscous non-Newtonian fluids yields a variational principle for steady creeping flow for a rather wide class of fluids. Only for a restricted subclass (Newtonian and power-law fluids) does the quantity to be minimized coincide with the total rate of entropy production. It is argued that the Helmholtz—Korteweg theorem is a purely mechanical result which has no significant thermodynamic content.     

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.