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.     

Influence of the capillarity on a creeping film flow down an inclined plane with an edge

Summary In creeping flows of thin films, the capillarity can play a dominant role. In this paper, the creeping film flow down an inclined plane with an edge is considered. The influence of the capillarity on the velocity and the film surface is studied analytically, numerically and experimentally. Received 12 April 1999; accepted for publication 9 May 1999     

Planar dynamics of inclined curved flexible riser carrying slug liquid–gas flows

Flexible risers transporting hydrocarbon liquid–gas flows may be subject to internal dynamic fluctuations of multiphase densities, velocities and pressure changes. Previous studies have mostly focused on single-phase flows in oscillating pipes or multiphase flows in static pipes whereas understanding of multiphase flow effects on oscillating pipes with variable curvatures is still lacking. The present study aims to numerically investigate fundamental planar dynamics of a long flexible catenary riser carrying slug liquid–gas flows and to analyse the mechanical effects of slug flow characteristics including the slug unit length, translational velocity and fluctuation frequencies leading to resonances. A two-dimensional continuum model, describing the coupled horizontal and vertical motions of an inclined flexible/extensible curved riser subject to the space–time varying fluid weights, flow centrifugal momenta and Coriolis effects, is presented. Steady slug flows are considered and modelled by accounting for the mass–momentum balances of liquid–gas phases within an idealized slug unit cell comprising the slug liquid (containing small gas bubbles) and elongated gas bubble (interfacing with the liquid film) parts. A nonlinear hydrodynamic film profile is described, depending on the pipe diameter, inclination, liquid–gas phase properties, superficial velocities and empirical correlations. These enable the approximation of phase fractions, local velocities and pressure variations which are employed as the time-varying, distributed parameters leading to the slug flow-induced vibration (SIV) of catenary riser. Several key SIV features are numerically investigated, highlighting the slug flow-induced transient drifts due to the travelling masses, amplified mean displacements due to the combined slug weights and flow momenta, extensibility or tension changes due to a reconfiguration of pipe equilibrium, oscillation amplitudes and resonant frequencies. Single- and multi-modal patterns of riser dynamic profiles are determined, enabling the evaluation of associated bending/axial stresses. Parametric studies reveal the individual effect of the slug unit length and the translational velocity on SIV response regardless of the slug characteristic frequency being a function of these two parameters. This key observation is practically useful for the identification of critical maximum response.     

Three-dimensional CFD simulation of bubble–melt two-phase flow with air injecting and melt stirring

This paper reports on progress in developing CFD simulations of gas bubble–metallic melt turbulent flows induced by a pitched-blade impeller with an inclined shaft. Foaming process of aluminum foams, in which air is injected into molten aluminum composites and the melt is mechanical stirred by the impeller, has been investigated. A two-fluid model, incorporated with the Multiple Reference Frames (MRF) method is used to predict the three-dimensional gas–liquid flow in the foaming tank, in which a stirring shaft is positioned inclined into the melt. Locally average bubble size is also predicted by additively solving a transport equation for the bubble number density function, which accounts for effects of bubble breakup and coalescence phenomena. The computed bubble sizes are compared with experimental data from our water model measurement and reasonable agreements are obtained. Further, simulated results show that the volume averaged total and local gas fractions are generally increased with rising impeller speed and gas flow rate. The local averaged bubble size increases with increasing gas flow rate and orifice diameter and decreasing liquid viscosity, and decreases also with rising rotation speed of the impeller.     

The influence of the drag force due to the interstitial gas on granular flows down a chute

Fully-developed steady flow of granular material down an inclined chute has been a subject of much research interest, but the effect of the interstitial gas has usually been ignored. In this paper, new expressions for the drag force and energy dissipation caused by the interstitial gas (ignoring the turbulent fluctuations of the gas phase) are derived and used to modify the governing equations derived from the kinetic theory approach for granular–gas mixture flows, where particles are relatively massive so that velocity fluctuations are caused by collisions rather than the gas flow. This new model is applied to fully-developed, steady mixture flows down an inclined chute and the results are compared with other simulations. Our results show that the effect of the interstitial gas plays a significant role in modifying the characteristics of fully developed flow. Although the effect of the interstitial gas is less pronounced for large particles than small ones, the flowfields with large particles are still very different from granular flows which do not incorporate any interactions with the interstitial gas.     

涂布流动的非线性阈值问题

本文研究了沿斜面流动薄层液体的非线性稳定性,即涂布流动的非线性稳定性问题。我们将周恒对平面Poiseuille流提出的弱非线性理论应用于涂布流动。文中对自由表面的世界条件提出了一个合理的简化方法,对亚临界时不同Reynolds数及扰动频率,求出了有限扰动的阈值。     

Viscous liquid film flow down an inclined corrugated surface. Calculation of the flow stability to arbitrary perturbations using an integral method

Viscous liquid film flow along an inclined corrugated (sinusoidal) surface has been studied. Calculations were performed using an integral model. The stability of nonlinear steady-state flows to arbitrary perturbations was examined using the Floquet theory. It has been shown that for each type of corrugation there is a critical Reynolds number for which unstable perturbations occur. It has been found that this value greatly depends on the physical properties of the liquid and geometric parameters of the flow. In particular, in the case of film flow down a smooth wall, the critical waveformation parameter depends only on the angle of inclination of the flow surface. The values of the corrugation parameters (amplitude and period) were obtained for which the film flow down a wavy wall is stable to arbitrary perturbations up to moderate Reynolds numbers. Such parameter values exist for all investigated angles of inclination of the flow surface.     

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.     

Exact solutions of core-annular laminar inclined flows

An exact solution for laminar two-phase eccentric core-annular flows (CAF) in inclined pipes is derived. This solution complements the exact solutions that were obtained for inclined stratified flows with curved interfaces as to provide a set of solutions for two-phase laminar separated flows. A unified set of three dimensionless parameters for separated flows is defined and used to explore the effects of the system parameters and separated flow configurations on the velocity profiles and the resulting holdup, pressure gradient and pumping power requirement in horizontal and inclined concurrent and countercurrent flows. It is shown that similarly to stratified flows, also in CAF multiple solutions for the holdup and the associated flow characteristics can be obtained in inclined flows. The boundaries of the multiple solution regions are mapped and the effect of the core eccentricity and system parameters boundaries are demonstrated and discussed.The benefits of adding a lubricating phase for transportation of a viscous fluid in inclined CAFs is investigated. An adverse effect of the upward pipe inclination on the power savings in all of the separate flow configurations is demonstrated. Independently of the density of the lubricant, namely, whether it is lighter or heavier than the viscous fluid, the effect of hydrostatic pressure gradient always hinders the possibility of reducing the pumping requirement for transporting the viscous phase. However, surprisingly, a heavier lubricant is preferable form the view point of power saving. The implications of turbulent flow of the lubricating phase and the susceptibility to Ledinegg instability on the potential power savings are also considered and discussed. The application of the model for the analysis of experimental data of the holdup and pressure drop obtained in horizontal and inclined CAF is also demonstrated.     

A two-parameter model of wave regimes for viscous liquid film flows

Wave regimes of viscous liquid film flows are considered when the viscosity coefficients vary in a wide range. An approximate model system of differential equations with two external governing parameters for the film layer thickness and the local flow rate is derived. The viscous dissipation of a film layer is taken into account in this system more accurately than in the well-known one-parameter Shkadov model. New properties of linear and nonlinear waves caused by the hydrodynamic instability of high-viscous liquid flows under gravity and surface tension are found.     

Liquid Filtration in an Unbounded Water‐Bearing Stratum with an Inclined Confining Bed

PolubarinovaKochina, Numerov, and other authors paid much attention to filtration problems of a heavy incompressible liquid in inclined waterbearing strata. In this work, therefore, classical schemes of liquid filtration on inclined confining beds are considered along with the general problem of filtration for arbitrary polygonal impermeable walls of a waterbearing stratum. In doing so, we also consider direct problems of physical and geometrical parameters of filtration flows.     

Instability of a liquid layer under periodic influence: Falling film in an alternating electric field

The linear analysis of stability of a plane-parallel time-periodic flow is carried out. The numerical method which makes it possible to reduce the spectral problem for the time-dependent Orr–Sommerfeld equation to an algebraic eigenvalue problem is used. The film of viscous conducting liquid which flows down a vertical wall in the normal electric field is considered and parametric resonances are revealed.     

The flow of non-Newtonian fluids down inclines

Equipment units (e.g. tray column heaters) in many chemical engineering and mineral processing industries involve the flow of non-Newtonian fluids down inclined plates. When designing these equipment units the non-Newtonian fluid flows often are not fully understood and so the designs are not properly optimised. In this study the flow down a series of inclined plates was experimentally and numerically investigated to better understand the flow for various fluids and to validate a computational fluid dynamics (CFD) model.In the experimental rig there were a series of consecutive plates inclined at 45°. An optically clear polymer solution was used to simulate a yield pseudo-plastic material and allowed flow visualisation to be undertaken of the flow. The fluid film thickness was observed to decrease down the consecutive plates. Experiments were also carried out using a yield pseudo-plastic mineral slurry and the results were found to be qualitatively similar.An analytical model was developed to calculate the fluid film layer thickness on the first plate and a CFD model was used to compute the flow down a series of flat plates. The CFD model employed a homogeneous multiphase model and surface-sharpening algorithm. The CFD model accurately predicted the fluid film thicknesses and flow patterns. The validated CFD model can now be used with confidence as a design tool.     

The instability mechanism of single and multilayer Newtonian and viscoelastic flows down an inclined plane

The instability mechanism of single and multilayer flow of Newtonian and viscoelastic fluids down an inclined plane has been examined based on a rigorous energy analysis as well as careful examination of the eigenfunctions. These analyses demonstrate that the free surface instability in single and multilayer flows in the limit of longwave disturbances (i.e., the most dangerous disturbances) arise due to the perturbation shear stresses at the free surface. Specifically, for viscoelastic flows, the elastic forces are destabilizing and the main driving force for the instability is the coupling between the base flow and the perturbation velocity and stresses and their gradient at the free surface. For Newtonian flows at finite Re, the driving force for the interfacial instability in the limit of longwaves depends on the placement of the less viscous fluid. If the less viscous fluid is adjacent to the solid surface then the main driving force for the instability is interfacial friction, otherwise the bulk contribution of Reynolds stresses drives the instability. For viscoelastic fluids in the limit of vanishingly small Re, the driving force for the instability is the coupling of the base flow and perturbation velocity and stresses and their gradients across the interface. In the limit of shortwaves the interfacial stability mechanism of flow down inclined plane is the same as plane Poiseuille flows (Ganpule and Khomami 1998, 1999a, b). Received: 20 October 2000/Accepted: 11 January 2001     

Lattice Boltzmann modeling of shallow water flows over discontinuous beds

Numerical modeling of shallow water flows over discontinuous beds is presented. The flows are described with the shallow water equations and the equations are solved using the lattice Boltzmann method (LBM) with single relaxation time (Bhatnagar–Gross–Krook‐LBM (BGK‐LBM)) and the multiple relaxation time (MRT‐LBM). The weighted centered scheme for force term together with the bed height for a bed slope is described to improve simulation of flows over discontinuous bed. Furthermore, the resistance stress is added to include the local head loss caused by flow over a step. Four test cases, one‐dimensional tidal over regular bed and steps, dam‐break flows, and two‐dimensional shallow water flow over a square block, are considered to verify the present method. Agreements between predictions and analytical solutions are satisfactory. Furthermore, the performance and CPU cost time of BGK‐LBM and MRT‐LBM are compared and studied. The results have shown that the lattice Boltzmann method is simple and accurate for simulating shallow water flows over discontinuous beds. This demonstrates the capability and applicability of the lattice Boltzmann method in modeling shallow water flows on bed topography with a discontinuity in practical hydraulic engineering. Copyright © 2014 John Wiley & Sons, Ltd.     

基于应力波动的修正非局部流变模型

基于Pouliquen 提出的非局部流变模型,考虑颗粒流中某个位置重新排列引起的自激发过程,详细分析颗粒介质中应力波动幅值的概率密度分布形式以及剪切速率与体积分数的耦合作用,提出一种修正的非局部流变模型. 采用此修正非局部流变模型对斜面剪切颗粒流的流动特性进行了预测,颗粒流动的临界厚度、平均流动速度及剪切速率廓线的预测结果与实验结果定量吻合. 此修正模型的提出为复杂的密集颗粒流的描述和表征提供了一种新的研究思路.      

Stability of the convective flow in an inclined layer with heat release at its center

The flow of a viscous incompressible fluid in a plane infinite inclined layer in the presence of internal heat sources concentrated on its axis is considered. The stability of the plane-parallel motion is investigated, the neutral curves are plotted, and the stability regions are determined. The results are compared with the case of uniform distribution of the heat sources. Supercritical fluid flows are calculated numerically.     

Long waves on a viscoelastic film flow down a wavy incline

Long waves on a viscoelastic film flow down a wavy inclined plane is investigated. The analysis is performed to see how long non-linear waves on viscoelastic film down an uneven inclined wall are deformed due to the non-uniformity of the basic flow. The results are then compared with those corresponding to Newtonian film down a wavy inclined wall as well as viscoelastic film down a plane inclined wall.     

An inverse boundary value problem for Maxwell's equations

We consider steady, two-dimensional motions of an incompressible, Newtonian fluid flowing under gravity down an inclined channel. If the bottom of the channel is flat, the flow is the classical Poiseuille-Nusselt flow and the free surface is then a plane parallel to the bottom. Motivated by the recent experimental and numerical studies of Pritchard, Scott & Tavener, we look at bottom configurations which possess some localized, non-uniform structure. We present an existence theory for steady, highly viscous flow over such configurations. An important consequence of our theory is that the steady flows whose existence is established decay exponentially rapidly to the unperturbed Poiseuille-Nusselt flow away from the local variation in the channel bottom profile.