THE SLOW MOTION OF A SPHERICAL PARTICLE IN A CARREAU FLUID

The drag on a spherical particle is studied for two limiting cases, namely for the rigid sphere and for the bubble. An approximate solution is found for creeping flow around a particle suspended in a shear-thinning fluid. The three parameter Carreau model is used to represent the suspending liquid. The drag force on the particle for both cases is calculated by a perturbation method around the Newtonian solution in the limit of small Carreau number. The resulting expressions are found to be dependent on the Carreau number and on the power-law index.     

Electroviscous effects in a Carreau liquid flowing through a cylindrical microfluidic contraction

Electroviscous effects in steady, pressure-driven flow of a Carreau shear-thinning liquid in a cylindrical microfluidic 4:1:4 contraction–expansion at low Reynolds number are investigated numerically by solving the equations governing the flow, the electric field, and ion transport, using a finite volume method. The channel wall is considered to have a uniform surface charge density and the liquid is assumed to be a symmetric 1:1 electrolyte solution. Predictions are presented for a range of values of the shear-thinning parameters in the Carreau model for various surface charge densities and Debye lengths. The apparent/physical viscosity ratio is shown to increase as the degree of shear-thinning increases. Thus the electroviscous effect is stronger in shear-thinning liquids than it is when the liquid is Newtonian, a result previously obtained for uniform pipe flow of power-law liquids. The trend holds true regardless of the choice of surface charge density or Debye length, although the magnitude of the trend decreases as the surface charge density and/or the Debye length is reduced. Comparison between uniform pipe flow of a Carreau liquid and the corresponding power-law liquid that approximates it at large shear rates shows that the apparent/physical viscosity ratios for the two models are almost identical. A previous prediction that a near-wall region of reduced velocity can occur for pipe flow of a shear-thinning power-law liquid when EDLs are overlapping and surface charge density is elevated is confirmed for a Carreau liquid.     

Drag on a sphere in a spherical dispersion containing Carreau fluid

The drag on a rigid sphere in a spherical dispersion containing Carreau fluid is investigated theoretically based on a free surface cell model for Reynolds number in the range [0.1,100], Carreau number in the range [0,10], the power-law index in the range [0.3,1], and the void fraction in the range [0.271,0.999]. The influences of the particle concentration, the nature of the Carreau fluid, and Reynolds number, on the drag coefficient are examined. We show that the drag coefficient declines with the decreasing particle concentration, and the reversal of the flow field in the rear region of a sphere is enhanced by the shear-thinning nature of the fluid. An empirical relation, which correlates the drag coefficient with the void fraction (= 1 − particle concentration), the nature of the Carreau fluid, and Reynolds number, is proposed.     

泡沫薄膜液在直管内的流变学特性

针对泡沫液在多孔介质内的流动特点,对泡沫薄膜液在直管内的流变学特性进行了实验研究。按单相幂律流体假设,获得二氧化碳及氮气泡沫薄膜液的表观黏度数据。结果表明：泡沫薄膜液具有较大表观黏度并呈现剪切变稀的非牛顿流体特性。由于水溶性影响,二氧化碳泡沫液的表观黏度要小于氮气泡沫液。利用量纲分析法确定量纲1参数,对泡沫薄膜液的流变学特性进行了量纲1分析并给出基于两相流动分析的阻力模型。     

Drag on two coaxial rigid spheres moving along the axis of a cylinder filled with Carreau fluid

The boundary effect on the drag on two identical, rigid spheres moving along the axis of a long cylinder filled with a Carreau fluid for Reynolds number ranges from 0.1 to 40 is investigated. The influences of the key parameters of the problem under consideration, including the separation distance between two spheres, the relaxation time constant and the power-law index of a Carreau fluid, the Reynolds number, and the ratios (radius of sphere/radius of cylinder), on the drag acting on two spheres are investigated. We show that the boundary effect for the present case is more significant than that for the corresponding Newtonian fluid. The presence of the cylinder has the effect of enhancing the convective motion in the rear part of a sphere, thereby forming wakes and a reverse flow field, and this phenomenon is enhanced by the shear-thinning nature of a fluid. If the boundary effect is insignificant, the shear-thinning nature of a fluid has the effect of reducing the deviation of the ln(drag coefficient)-ln(Reynolds number) curve from a Stokes'-law-like relation. On the other hand, if it is significant, this deviation has a local minimum as the shear-thinning nature of a fluid varies.     

A flow model for non-newtonian liquids inside a slot die

A one-dimensional flow model for non-Newtonian liquids inside a dual-cavity slot die is presented. The model is capable of analyzing slot dies of any cavity shape, cavity taper, slot-length variations, and slot-gap variations. The proposed model incorporates a truncated-power-law model for the viscosity of non-Newtonian liquids. According to flow models with power-law approximation for liquid viscosity, the distribution of non-Newtonian liquid through a slot die depends on the slot Reynolds number only. With our model, we find that the zero shear viscosity and the relaxation time of a non-Newtonian liquid have large effects on its distribution. For non-Newtonian liquids which are expected to experience shear-thinning over portion of a slot die, it is concluded that a flow model with a truncated-power-law approximation for liquid viscosity be used to predict the liquid distribution from the die.     

A SIMPLE THEORY FOR THE ENTRAINED FILM THICKNESS DURING MENISCUS COATING

An analysis is presented of the fluid dynamics of meniscus coating, in which a horizontal substrate contacts the top surface of a liquid film flowing down an inclined plane, forming a meniscus. The substrate is then moved and a film is entrained on it. The analysis shows the dependence of the entrained film thickness on the substrate translation rate, suspension viscosity, geometry and falling film flowrate. An idealized geometry is assumed which retains the physics of the coating process but which allows a leading-order analytic solution by requiring continuity of the meniscus curvature across three regions: an entrained film flow region near the substrate, static meniscus region and a falling film region down the inclined plane. It is shown that, except for a small term associated with the falling film flow, the film thickness equation is identical to the leading order result for the familiar dip-coating problem. Results of the model agree very well with optical film thickness measurements for films deposited from a Newtonian suspension, and departures from the theory due to non-Newtonian rheology are discussed     

A macroscopic model for shear-thinning flow in packed beds based on network modeling

The flow of non-Newtonian fluids in packed beds and other porous media is important in several applications such as polymer processing, filtration, and enhanced oil recovery. Expressions for flowrate versus pressure gradient are desirable for a-priori prediction and for substitution into continuum models. In this work, physically representative network models are used to model the flow of shear-thinning fluids, including power-law and Ellis fluids. The networks are used to investigate the effects of fluid rheology and bed morphology on flow.A simple macroscopic model is developed for the flow of power-law and Ellis fluids in packed beds using results from the network model. The model has the same general functionality as those developed using the popular bundle-of-tubes approach. The constant β, which appears in these models, is often directly derived from the tortuosity and a simple representation of the porous media. It is shown here that this can lead to incorrect and ambiguous values of the constant. Furthermore, the constant is a weak function of the shear-thinning index, indicating that no single bundle-of-tubes could ever properly model flow for a wide variety of shear-thinning fluids.The macroscopic model is compared to experimental data for shear-thinning fluids available in the literature. The model fits the data well when β is treated as an experimental parameter. The best-fit values of β vary, which is expected because even the constant C in the Blake-Kozeny equation varies depending on the source consulted. Additionally, physical effects, such as adsorption and filtration, as well as rheological effects such as viscoelasticity may affect the value of β. We believe that in the absence of these effects, β equals approximately 1.46 for packed beds of uniform spheres at relatively moderate values of the shear-thinning index (>0.3).     

Transient response of slot coating flows of shear-thinning fluids to periodic disturbances

Slot coating is a popular coating method, in which the film thickness is precisely controlled by adjusting the flow rate and production speed. When the coating flow undergoes small-scale disturbances generated by rotating elements such as motors, pumps, or uneven rolls, the downstream meniscus fluctuates, which causes film thickness variation in the flow direction. Although most coating liquids including polymeric and particulate solutions exhibit a shear-thinning rheological property, their effect on transient coating flow behaviors is not deeply understood. Here, the effect of shear-thinning property on film thickness variation under different disturbances is investigated using a computer-aided analysis of transient slot coating flow. In this study, the Carreau model is used to describe the shear-thinning property, and four different disturbances are considered.     

Deep channel operating characteristics of a single screw extruder: Finite element predictions and experimental results for isothermal non-Newtonian flow

Finite element solutions are presented for developed, isothermal, power-law flow in an extruder channel, based on a variational formulation in helical co-ordinates. The success of the method in handling non-Newtonian flow in a deep, highly-curved channel is tested by comparison with experimental results. Operating characteristics were determired under virtually isothermal conditions for a screw with depth to diameter ratio 0.25 pumping an aqueous glucose solution containing high molecular weight hydroxyethyl cellulose. Shear stress and first normal stress differences of this solution were determined in a rheogoniometer. It was both highly shear-thinning and elastic. Computed and experimental screw operating characteristics agree well, except near closed discharge, where effects due to viscous heating and elasticity are tentatively suggested.     

The effect of mesh refinement on the computation of entrance effects for shear-thinning fluids

The effect of mesh refinement on the finite element computation of velocity and pressure is assessed for the creeping flow of a power law fluid and a Carreau fluid through an axisymmetric sudden contraction. The emphasis is placed on the determination of the entrance pressure correction n_ent. The augmented Lagrangian formalism is used and the discretization of the variational problem is based on the Crouzeix-Raviart triangular element. For power-law fluids, it is found that fine meshes yield values of n_ent which are significantly lower than those obtained in previous numerical studies, especially for high shear-thinning fluids. A similar trend is observed in the Carreau model.     

Numerical study of flow at a liquid-porous medium interface

The problem of the two-dimensional creeping flow of a viscous incompressible liquid in a flat channel partially filled with a model fibrous porous medium that is represented by a regular system of square cylinders (prisms) located across the flow is solved. Two types of flows are considered: a shear flow due to the motion of the upper wall of the channel and a gradient flow due to the presence of a pressure drop along the channel. The hydrodynamic microscopic fields of velocity are numerically found. The macroparameters such as the rate of filtration, the permeability of a system of cylinders, the flow rate of the liquid through the channel, tangential stresses on the upper wall of the channel and the porous boundary, and a slip coefficient, the use of which in the Saffman slip boundary condition makes it possible to considerably simplify the solution of the original problem, are obtained as a result of averaging.     

Drag on two co-axial rigid spheres moving normal to a plane: Newtonian and Carreau fluids

The boundary effect and the presence of a nearby entity on the drag of a rigid entity is investigated by considering the movement of two identical, rigid, coaxial spheres normal to a plane in both a Newtonian and a Carreau fluid at a low to medium large Reynolds number. The parameters key to the phenomenon under consideration, including the nature of the fluid, the separation distance between two spheres, the distance between the near sphere and the plane, and the Reynolds number, on the drag coefficient are discussed. We show that the influence of a boundary on the drag coefficient is more important than that of the nature of a fluid and that of the separation distance between two spheres. The variation of the drag coefficient as a function of Reynolds number for a Carreau fluid is similar to that for a Newtonian fluid. Due to the shear-thinning nature of the former the drag coefficient in the former is smaller than that in the latter. The influence of the index parameter of a Carreau fluid becomes appreciable only if the Carreau number is sufficiently large. Correlations between the drag coefficient and the key parameters of a system are developed for the case when the Reynolds number is smaller than l.     

Determining the velocity of a non-newtonian medium flowing past spherical particles

The general hydrodynamic approach is used to formulate and solve the problem of determining the velocity field and pressure of a non-Newtonian medium flowing past a spherical particle. The rheological properties of the non-Newtonian medium are described in terms of a two-parameter power-law model. Expressions are obtained for the velocity of a sphere (in particular, a bubble) relative to the flow of the non-Newtonian liquid in different hydrodynamic pattern. The effects of rheological parameters on the flow velocity and on the limiting sizes of bubbles what retain spherical form, are analyzed.     

Instabilities in a liquid film flow over an inclined heated porous substrate

Stability of a thin viscous Newtonian fluid draining down a uniformly heated porous inclined plane is examined. The long-wave linear stability analysis is performed within the generic Orr-Sommerfeld framework both theoretically and numerically. An evolution equation for the local film thickness for two-dimensional disturbances is derived to analyze the effect of long-wave instabilities. The parameters governing the film flow system and the porous substrate strongly influence the wave forms and their amplitudes and hence the stability of the fluid. The long-time wave forms are either time-independent wave forms that propagate or time-dependent modes that oscillate slightly in the amplitude. The role of permeability and Marangoni number is to increase the amplitude of the disturbance leading to the destabilization state of the film flow system. The permeability of the porous medium promotes the oscillatory behavior.     

Filtration of non-Newtonian fluids

Equations of motion characterizing the flow of incompressible, time-independent non-Newtonian fluids, exhibiting the anomalous surface effect, through compressible porous media are developed. Approximations are made to arrive at workable filtration equations for slurries of non-Newtonian (power-law) fluids. The constant-pressure and constant-rate filtration relationships developed are verified experimentally using slurries of calcium carbonate in water and dilute CMC solutions. The anomalous surface effect is found to exist in the filtration of the non-Newtonian fluids. The specific cake resistance in the case of the non-Newtonian sludge and the ratio of the effective slip velocity to the pore velocity are found to be functions of both the CMC concentration and the pressure drop across the filter bed.     

Non-Newtonian power-law flow across a confined triangular bluff body in a channel

Wall effects on the flow of incompressible non-Newtonian power-law fluids across an equilateral triangular cylinder confined in a horizontal plane channel have been investigated for the range of conditions: Reynolds number, Re=1–40, power-law index, n=0.4–1.8 (covering shear-thinning, Newtonian and shear-thickening behaviors) and blockage ratio=0.125–0.5. Extensive numerical results on flow pattern, wake/recirculation length, individual and overall drag coefficients, variation of pressure coefficient on the surface of the triangular cylinder and so forth are reported to elucidate the combined effect of power-law index, blockage ratio and Reynolds number. The size of vortices decreases with an increase in the value of the blockage ratio and/or power-law index. For a fixed value of the Reynolds number, individual and overall drags decrease with decrease in power-law index and/or blockage ratio in steady confined flow regime. Simple correlations of wake length and drag are also obtained for the range of settings considered.     

Non-darcy forced convective heat transfer in a channel embedded in a non-newtonian inelastic fluid-saturated porous medium

The present analysis investigates non-Darcy forced convective heat transfer in a channel confined by two parallel walls subjected to uniform heat flux in a highly porous medium saturated with a non-Newtonian power-law fluid. Extensive numerical integrations have been carried out utilizing the Brinkman-Forchheimer extension of the Darcy model in order to study the effects of pseudoplasticity, and Brinkman and Forchheimer terms on the heat transfer characteristics.     

BLASIUS AND SAKIADIS FLOW WITH UNIFORM BLOWING OR SUCTION IN NON-NEWTONIAN POWER-LAW FLUIDS

The Blasius and Sakiadis flows of a non-Newtonian power-law fluid are considered. The plate is porous and fluid can be either injected or sucked through it. The boundary layer equations are transformed into a nondimensional form and are solved with a finite difference method. For the case of uniform suction, new results have been found, although this problem has been investigated in the past. Among them are analytical solutions for dilatant fluids of the Blasius flow and analytical solutions of the Sakiadis flow for all values of the power-law index. For the case of uniform injection, the characteristics of the flow until a separation state are investigated and discussed.     

SLOW NON-NEWTONIAN FLOW PAST AN ASSEMBLAGE OF RIGID SPHERES

Previous work on slow flow of non-Newtonian fluids past particles assemblages has been reviewed. Using a combination of Happel's free surface model and variational principles, bounds on the drag have been obtained for the creeping flow of a Carreau Model fluid past an assemblage of rigid spheres. The bounds are related to friction factor for flow through fixed beds of spherical particles. Numerical results covering a wide range of model parameters and bed voidages are presented.

Theoretical predictions are validated by comparing with experimental results reported in the literature that involve viscoelastic fluids. Arithmetic averages of the two bounds compare well for 182 data points with an average error of 12%. It is demonstrated that the present analysis, though based on a purely viscous model, can predict creeping flow behaviour in rigid particles assemblage for both inelastic and viscoelastic fluids.