Millimeter wave rheometry: theory and experiment

A novel millimeter wave (MMW) rheometry is developed to determine the viscosity of fluids based on an unsteady film flow on an inclined plane. The method measures fringes due to MMW interference between the front and back surfaces of a fluid flowing across the field of view of a ceramic wave guide coupled to a MMW receiver operating at 137 GHz. With knowledge of the dielectric constant, the interference fringe spacing is used to calculate the thickness of the fluid layer. This thickness is then transformed into the viscosity by means of a simple hydrodynamic theory. Our results show that the MMW rheometry can practically distinguish between the 30, 100, and 200 Pa·s silicone oils. The geometry of the method allows for potential industrial applications such as measuring viscosity of the flowing slag down the walls of coal gasifiers. The MMW rheometry with simple modifications can be easily extended to measure important non-Newtonian fluid characteristics such as yield stress.     

Moving wedge and flat plate in a power-law fluid

The steady boundary-layer flow of a non-Newtonian fluid, represented by a power-law model, over a moving wedge in a moving fluid is studied in this paper. The transformed boundary-layer equation is solved numerically for some values of the involved parameters. The effects of these parameters on the skin friction coefficient are analyzed and discussed. It is found that multiple solutions exist when the wedge and the fluid move in the opposite directions, near the region of separation. It is also found that the drag force is reduced for dilatant fluids compared to pseudo-plastic fluids.     

An extended study of the hydrodynamics of gravity-driven film flow of power-law fluids

A new similarity transformation has been devised for extensive studies of accelerating non-Newtonian film flow. The partial differential equations governing the hydrodynamics of the flow of a power-law fluid down along an inclined plane surface are transformed into a set of two ordinary differential equations by means of the dimensionless velocity component approach. Although the analysis is applicable for any angle of inclination (0<π/2), the resulting one-parameter problem involves only the power-law index n. Nevertheless, physically essential quantities, like the velocity components and the skin-friction coefficient, do depend on and relevant relationships are deduced between the vertical and inclined cases. Accurate numerical similarity solutions are provided for n in the range from 0.1 to 2.0. The present method enables solutions to be obtained also for highly pseudo-plastic films, i.e. for n below 0.5. The mass flow rate entrained into the momentum boundary layer from the inviscid freestream is expressed in terms of a dimensionless mass flux parameter Φ, which depends on the dimensionless boundary layer thickness and the velocity components at the edge of the viscous boundary layer. Φ, which is thus an integral part of the similarity solution, turns out to decrease monotonically with n. This parameter is of particular relevance in the determination of the streamwise position at which the entire freestream has been entrained and viscous stresses prevail all the way to the free surface of the film. A short-cut method to facilitate rapid and yet accurate estimates of the mass flux parameter is developed to this end.     

Comparative study of forced convection of a power-law fluid in a channel with a built-in square cylinder

A detailed comparison between the lattice Boltzmann method and the finite element method is presented for an incompressible steady laminar flow and heat transfer of a power-law fluid past a square cylinder between two parallel plates. Computations are performed for three different blockage ratios (ratios of the square side length to the channel width) and different values of the power-law index n covering both pseudo-plastic fluids (n < 1) and dilatant fluids (n > 1). The methodology is validated against the exact solution. The local and averaged Nusselt numbers are also presented. The results show that the relatively simple lattice Boltzmann method is a good alternative to the finite element method for analyzing non-Newtonian fluids.     

A remark on similarity analysis of boundary layer equations of a class of non-Newtonian fluids

A similarity analysis of three-dimensional boundary layer equations of a class of non-Newtonian fluid in which the stress, an arbitrary function of rates of strain, is studied. It is shown that under any group of transformation, for an arbitrary stress function, not all non-Newtonian fluids possess a similarity solution for the flow past a wedge inclined at arbitrary angle except Ostwald-de-Waele power-law fluid. Further it is observed, for non-Newtonian fluids of any model only $90°$ of wedge flow leads to similarity solutions. Our results contain a correction to some flaws in Pakdemirli׳s [14] (1994) paper on similarity analysis of boundary layer equations of a class of non-Newtonian fluids.     

Instability and waves during generalized Newtonian fluid film flow down a vertical wall

The problem of linear stability of a non-Newtonian fluid film flowing down a vertical plane under the action of gravity is considered. The linear stability of steady-state flow with a plane free boundary and the nonlinear waves that arise if this flow is unstable are investigated. The results obtained for two rheological models, the power-law and Eyring fluids, are compared.     

Asymptotic Nusselt numbers for dissipative non-Newtonian flow through ducts

General expressions for fully developed temperature profiles and Nusselt numbers are obtained for heat transfer to non-Newtonian fluid flow between parallel plates and through circular pipes subjected to a uniform wall heat flux. The effect of viscous dissipation is taken into account since it may often be significant in the flow of non-Newtonian fluids. Asymptotic Nusselt numbers for three widely used models, i.e. the power law fluid, the Bingham plastic, and the Ellis fluid are obtained as specific results.     

The Falkner–Skan Wedge Flows of Power-Law Fluids Embedded in a Porous Medium

The non-Darcy flow characteristics of power-law non-Newtonian fluids past a wedge embedded in a porous medium have been studied. The governing equations are converted to a system of first-order ordinary differential equations by means of a local similarity transformation and have been solved numerically, for a number of parameter combinations of wedge angle parameter m, power-law index of the non-Newtonian fluids n, first-order resistance A and second-order resistance B, using a fourth-order Runge–Kutta integration scheme with the Newton–Raphson shooting method. Velocity and shear stress at the body surface are presented for a range of the above parameters. These results are also compared with the corresponding flow problems for a Newtonian fluid. Numerical results show that for the case of the constant wedge angle and material parameter A, the local skin friction coefficient is lower for a dilatant fluid as compared with the pseudo-plastic or Newtonian fluids.     

A consistent thin-layer theory for Bingham plastics

Thin-layer theory is developed for Bingham plastic fluids; the specific case of a fluid flowing down an inclined plane is considered. In contrast to previous work it is indicated how the Bingham model leads directly to a self-consistent thin-layer theory; this does not rely upon adopting a bi-viscous approximation. The theory describes the fluid in terms of regions of fully plastic flow bounded by a `fake' yield surface. Above this fake yield surface are `pseudo-plugs' – regions in which the leading-order equations predict a plug, but which are seen to be weakly yielded at higher order.     

Effects of inertia and surface tension on a power-law fluid flowing down a wavy incline

We consider a thin film of a power-law liquid flowing down an inclined wall with sinusoidal topography. Based on the von Kármán–Pohlhausen method an integral boundary-layer model for the film thickness and the flow rate is derived. This allows us to study the influence of the non-Newtonian properties on the steady free surface deformation. For weakly undulated walls we solve the governing equation analytically by a perturbation approach and find a resonant interaction of the free surface with the wavy bottom. Furthermore, the analytical approximation is validated by numerical simulations. Increasing the steepness of the wall reveals that nonlinear effects like the resonance of higher harmonics grow in importance. We find that shear-thickening flows lead to a decrease while shear thinning flows lead to an amplification of the steady free surface. A linear stability analysis of the steady state shows that the bottom undulation has in most cases a stabilizing influence on the free surface. Shear thickening fluids enhance this effect. The open questions which occurred in the linear analysis are then clarified by a nonlinear stability analysis. Finally, we show the important role of capillarity and discuss its influence on the steady solution and on the stability.     

热变形求解及其对高速点接触弹流润滑影响研究

提出了一种固体表面热变形求解新方法(ITD),由此研究了热变形对高速点接触弹流润滑行为的影响. 为此,基于计入流体惯性项的Reynolds方程获得了油膜压力,采用追赶法对润滑剂和接触固体的温度进行了求解,进而研究了不同工况下有无热变形的高速点接触非牛顿热弹流润滑性能. 采用有限元法和离散累加法对ITD法进行了验证,通过中心膜厚试验验证了考虑热变形的正确性. 结果表明:ITD法可准确快速地计算表面热变形;考虑热变形后,油膜厚度降低且向油膜出口倾斜,考虑热变形后的中心膜厚更接近试验结果.      

A correlation for yield stress fluid flow through packed beds

Relatively few correlations are available for non-Newtonian fluid flows through packed beds, even though such fluids are frequently used in industry. In this paper, a correlation is presented for yield stress fluid flow through packed beds. The correlation is developed by introducing the yield stress model in place of the Newtonian model used in deriving Erguns equation. The resulting model has three parameters that are functions of the geometry and roughness of the particle surfaces. Two of the parameters can be deduced in the limit as the yield stress becomes negligible and the model reduces to Erguns equation for Newtonian fluids. The third model parameter is determined from experimental data. The correlation relates a defined friction factor to the dimensionless Reynolds and Hedstrom numbers and can be used to predict pressure drop for flow of a yield stress fluid through a packed bed of spherical particles. Conditions for flow or no-flow are also determined in the correlation. Comparison of model calculations, between a Newtonian and a yield stress fluid for flow penetration into a packed bed of spheres, shows the yield stress fluid initially performs similar to the Newtonian fluid, at large Reynolds numbers. At lower Reynolds numbers the yield stress effect becomes important and the flow rate significantly decreases when compared to the Newtonian fluid.     

Comparison of different theoretical approaches to experiments on film flow down an inclined wavy channel

We study the flow of a liquid down an inclined channel with a sinusoidal bottom profile. We show how wavy bottom variations, which are long compared with the film thickness or the amplitude, modify the flow with respect to that down a flat inclined channel. We consider different perturbation analyses. Their results are compared with experimental data on the velocity profiles and on the film thickness. We discuss the effect of waviness, inclination angle, film thickness, and Reynolds number.     

Evaluation of unsteadiness in effervescent sprays by analysis of droplet arrival statistics - The influence of fluids properties and atomizer internal design

The ideal spray theory of Edwards and Marx was used to investigate the dependence of effervescent spray unsteadiness on fluid properties and atomizer internal design. Results demonstrate that fluid properties and internal design of atomizer directly affect the two-phase flow pattern inside the atomizer which consequently affects the spray unsteadiness of the atomizer. Water sprays are more unsteady when the air to liquid ratio (ALR) increases, whereas, more unsteady is observed for using glycerol/water mixture (high-viscosity Newtonian fluid) or glycerol/water/xanthan (non-Newtonian fluid) mixture as ALR reduces. In addition, sprays using low-viscosity or strong non-Newtonian fluids usually are more unsteady, regardless of ALR.A short mixing chamber results in less unsteady for water but has no effect on spray unsteadiness for high-viscosity or non-Newtonian fluids at ALR of 0.15. Otherwise, the influence of mixing chamber distance on the spray quality is weak at ALR of 0.15. Large diameter of inclined aeration holes shows the low spray unsteadiness and good spray quality for water but causes more unsteady for glycerol/water/xanthan mixture at ALR of 0.15. Furthermore, the diameter of the inclined aeration holes has little influence on spray unsteadiness for glycerol/water mixture. Spray unsteadiness and quality are not affected by the angle of aeration holes for three fluids at ALR of 0.15.     

Peristaltic motion of a non-Newtonian fluid

Summary To understand theoretically the flow properties of physiological fluids, we have considered as a model the peristaltic motion of a power law fluid in a tube, with a sinusoidal wave of small amplitude travelling down its wall. The solution for the stream function is obtained as a power series in terms of the amplitude of the wave. The stream function and the velocity components are evaluated by solving numerically two point boundary value problems with a singular point at the origin. The influence of the applied pressure gradient along with non-Newtonian parameters on the streamlines and velocity profiles are discussed in detail.     

Yield stress of structured fluids measured by squeeze flow

Various structured fluids were placed between the parallel circular plates of a squeeze-flow rheometer and squeezed by a force F until the fluid thickness h was stationary. Fluid thickness down to a few microns could be measured. Most fluids showed two kinds of dependence of f on h according to an experimentally-determined thickness h ^*. If h > h ^* then F varied in proportion to h ⁻¹ as predicted by Scott (1931) for a fluid with a shear yield stress τ₀. The magnitude of τ₀ from squeeze-flow data in this region was compared with the yield stress measured by the vane method. For some fluids τ₀ measured by squeeze flow was less than the vane yield stress, suggesting that the yield stress of fluid in contact with the plates was less than the bulk yield stress. If h < h ^* then F varied approximately as h ^−5/2 and the squeeze-flow data in this region analysed with Scott's relationship gave a yield stress which increased as the fluid thickness decreased. This previously unreported effect may result from unconnected regions of large yield stress in the fluid of size similar to h ^* which are not sensed by the vane and which become effective in squeeze flow only when h < h ^*. Received: 13 December 1999/Accepted: 4 January 2000     

Multilayer film casting of modified Giesekus fluids Part 2. Linear stability analysis

A linear stability analysis of the multilayer film casting of polymeric fluids has been conducted. A modified Giesekus model was used to characterize the rheological behaviors of the fluids. The critical draw ratio at the onset of draw resonance was found to depend on the elongational and shear viscosities of the fluids. Extensional-thickening has a stabilizing effect, whereas shear-thinning and extensional-thinning have destabilizing effects. The critical draw ratios for bilayer films of various thickness fractions are bounded by those for single layer films of the two fluids. When the two fluids have a comparable elongational viscosity, the critical draw ratio at a given Deborah number varies linearly with thickness fraction. When one fluid has a much larger elongational viscosity, it dominates the flow and the critical draw ratios at most thickness fractions remain close to its critical draw ratio as a single layer film. When the dominating fluid exhibits extensional-thickening, a film with a certain thickness fraction has more than one critical draw ratio at a given Deborah number and may not exhibit draw resonance within some range of the Deborah number.     

Pore-scale modeling of viscoelastic flow in porous media using a Bautista–Manero fluid

This article examines the extensional flow and viscosity and the converging–diverging geometry as the basis of the peculiar viscoelastic behavior in porous media. The modified Bautista–Manero model, which successfully describes elasticity, thixotropic time dependency and shear-thinning, was used for modeling the flow of viscoelastic materials which also show thixotropic attributes. An algorithm, originally proposed by Philippe Tardy, that employs this model to simulate steady-state time-dependent flow was implemented in a non-Newtonian flow simulation code using pore-scale modeling. The simulation results using two topologically-complex networks confirmed the importance of the extensional flow and converging–diverging geometry on the behavior of non-Newtonian fluids in porous media. The analysis also identified a number of correct trends (qualitative and quantitative) and revealed the effect of various fluid and flow parameters on the flow process. The impact of some numerical parameters was also assessed and verified.     

基于舌/上颚微间隙下流体流动行为研究

为探讨口腔环境下流体的流动行为,采用数值方法与流变试验深入研究舌/上颚微间隙下流体流量的影响因素. 建立舌/上颚微间隙的简化模型及Reynolds方程,通过数值方法获取微间隙下流量变化;在DHR-2流变仪上研究非牛顿流体的黏度与剪切率的变化,探讨牛顿流体和非牛顿流体的流量影响. 结果表明:牛顿流体流量平方的倒数同载荷和黏度比值和时间均呈线性函数关系;所制备的非牛顿流体近似为幂律流体,其黏度随脂肪含量的增加而增大,而非牛顿流体流量率先高于后低于等效牛顿流体,其研究结果将为特定人群功能产品的研发提供技术支持.      

Simulation of non-Newtonian ink transfer between two separating plates for gravure-offset printing

The inks used in gravure-offset printing are non-Newtonian fluids with higher viscosities and lower surface tensions compared to Newtonian fluids. This paper examines the transfer of a non-Newtonian ink between two parallel plates when the top plate is moved upward with a constant velocity while the bottom plate is held fixed. Numerical simulations were carried out using the Carreau model to explore the behavior of a non-Newtonian ink in gravure-offset printing. The volume of fluid (VOF) model was adopted to demonstrate the stretching and break-up behaviors of the ink. The results indicate that the ink transfer ratio is greatly influenced by the contact angle, especially the contact angle at the upper plate (α). For lower values of α, oscillatory or unstable behavior of the position of minimum thickness of the ink between the two parallel plates during the stretching period is observed. This oscillation gradually diminishes as the contact angle at the upper plate is increased. Moreover, the number of satellite droplets increases as the velocity of the upper plate is increased. The surface tension of the conductive ink shows a positive impact on the ink transfer ratio to the upper plate. Indeed, the velocity of the upper plate has a significant influence on the ink transfer in gravure-offset printing when the Capillary number (Ca) is greater than 1 and the surface tension dominates over the ink transfer process when Ca is less than 1.