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.     

A study of thin water film flow down an inclined plate without and with countercurrent air flow

Characteristics of thin water film flow down an inclined plane surface without as well as with superposed countercurrent air flow were studied experimentally. Three different angles of inclination of the channel with horizontal were investigated. At each inclination angle, five film Reynolds numbers were studied. Experiments were performed beginning at zero air flow and then increasing the air flow rate in steps until water entrainment occurred. Visual observation of the film surface was carried out as were film thickness measurements by means of capacitance probes. Results presented include mean film thickness and distribution, frequency spectra and propagation velocities of interfacial disturbances, and the incipience condition for entrainment of water from the film into the air stream.     

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.     

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.     

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.     

Effect of permeability on the instability of a non-Newtonian film down a porous inclined plane

A thin film of a power–law fluid flowing down a porous inclined plane is considered. It is assumed that the flow through the porous medium is governed by the modified Darcy’s law together with Beavers–Joseph boundary condition for a general power–law fluid. Under the assumption of small permeability relative to the thickness of the overlying fluid layer, the flow is decoupled from the filtration flow through the porous medium and a slip condition at the bottom is used to incorporate the effects of the permeability of the porous substrate. Applying the long-wave theory, a nonlinear evolution equation for the thickness of the film is obtained. A linear stability analysis of the base flow is performed and the critical condition for the onset of instability is obtained. The results show that the substrate porosity in general destabilizes the film flow system and the shear-thinning rheology enhances this destabilizing effect. A weakly nonlinear stability analysis reveals the existence of supercritical stable and subcritical unstable regions in the wave number versus Reynolds number parameter space. The numerical solution of the nonlinear evolution equation in a periodic domain shows that the fully developed nonlinear solutions are either time-dependent modes that oscillate slightly in the amplitude or time independent stable two-dimensional nonlinear waves with large amplitude referred to as ‘permanent waves’. The results show that the shape and the amplitude of the nonlinear waves are strongly influenced by the permeability of the porous medium and the shear-thinning rheology.     

Heat transfer in droplet-laden turbulent channel flow with phase transition in the presence of a thin film of water

We present results of a numerical study of turbulent droplet-laden channel flow with phase transition. Previous studies of the same system did not take into account the presence of gravity. Here, we do so introducing a thin film of water at the bottom wall and permitting droplets to fall into and merge with it. We treat the carrier phase with the Eulerian approach. Each droplet is considered separately in the Lagrangian formulation, adopting the point–particle approximation. We maintain the film thickness constant by draining water from the bottom wall to compensate for (a) the droplets that fall onto the film and (b) evaporation/condensation. We also maintain on average the total mass of water in the channel by inserting new droplets at the top wall to compensate for the water that has been drained from the bottom wall. We analyze the behavior of the statistically averaged gas and droplet quantities focusing on the heat exchange between the two phases. We increase (a) the initial droplet diameter keeping the same initial droplet volume fraction and (b) the initial number of droplets in the channel keeping their diameter the same. In both parameter studies we find that droplets grow less than in the reference case. In case (a) this is explained by the larger velocity with which they travel to the bottom wall and in case (b) by the lower rate of condensation of vapor due to the presence of neighboring droplets.     

Effect of bottom topography on the flow of a non-Newtonian liquid film down an inclined plane

The problem of flow of a nonlinear viscous liquid film down an inclined surface with local microtopography is considered. Numerical and approximate analytic solutions are obtained for steady flows of power-law liquid films down inclined surfaces with topography. Steps, hills, and periodic structures are considered as local topography. Basic properties of flows are found.     

Multi-layer film flow down an inclined plane: experimental investigation

We report the results from an experimental study of the flow of a film down an inclined plane where the film itself is comprised of up to three layers of different liquids. By measuring the total film thickness for a broad range of parameters including flow rates and liquid physical properties, we provide a thorough and systematic test of the single-layer approximation for multi-layer films for Reynolds numbers \(Re = \rho Q/\mu \approx 0.03 - 60\) . In addition, we also measure the change in film thickness of individual layers as a function of flow rates for a variety of experimental configurations. With the aid of high-speed particle tracking, we derive the velocity fields and free-surface velocities to compare to the single-layer approximation. Furthermore, we provide experimental evidence of small capillary ridge formations close to the point where two layers merge and compare our experimental parameter range for the occurrence of this phenomenon to those previously reported.     

Macro-to-microchannel transition in two-phase flow: Part 1 - Two-phase flow patterns and film thickness measurements

The classification of macroscale, mesoscale and microscale channels with respect to two-phase processes is still an open question. The main objective of this study focuses on investigating the macro-to-microscale transition during flow boiling in small scale channels of three different sizes with three different refrigerants over a range of saturation conditions to investigate the effects of channel confinement on two-phase flow patterns and liquid film stratification in a single circular horizontal channel (Part 2 covers the flow boiling heat transfer and critical heat flux). This paper presents the experimental two-phase flow pattern transition data together with a top/bottom liquid film thickness comparison for refrigerants R134a, R236fa and R245fa during flow boiling in small channels of 1.03, 2.20 and 3.04 mm diameter. Based on this work, an improved flow pattern map has been proposed by determining the flow patterns transitions existing under different conditions including the transition to macroscale slug/plug flow at a confinement number of Co ≈ 0.3-0.4. From the top/bottom liquid film thickness comparison results, it was observed that the gravity forces are fully suppressed and overcome by the surface tension and shear forces when the confinement number approaches 1, Co ≈ 1. Thus, as a new approximate rule, the lower threshold of macroscale flow is Co = 0.3-0.4 while the upper threshold of symmetric microscale flow is Co ≈ 1 with a transition (or mesoscale) region in-between.     

Long interfacial waves inside an inclined permeable channel

The time evolution of superposed layers of fluid flowing down inside an inclined permeable channel is investigated. Using the Kármán-Pohlhausen approximation, the problem is reduced to the study of the evolution equation for the liquid–liquid interface of the liquids film derived through a long-wave approximation. A linear stability analysis of the base flow is performed. The solutions and stability of the non-linear stationary long waves are investigated. A special form of the stationary long waves (say Shkadov waves) is introduced.     

Determination of yield stress fluid behaviour from inclined plane test

The aim of this paper is to determine precisely under which conditions an inclined plane can be used as a rheometer, which could represent a practical and rapid technique for various types of industrial or natural viscoplastic coarse suspensions. We first examine its efficiency and relevancy for determining fluid yield stress in a straight way by measuring the deepest fluid layer able to stay on the inclined plane. We have made experiments with different materials (clay-water suspensions) whose yield stress ranged from 35 to 90 Pa, using 1 m long open rectangular channels with a slope ranging from 10 to 30° and a width ranging from 5 to 25 cm. Our procedure involved measuring the final fluid depth far from edges a long time after the end of the slow gravity-induced emptying of a dam placed upstream. The fluid yield stress was also estimated independently by fitting a Herschel-Bulkley model to simple shear rheometry data obtained within a relatively wide shear rate range. A good agreement between inclined rectangular channel tests and independent usual rheometrical tests is obtained even for aspect ratios (flow depth to channel width ratio) as large as 1 when one assumes that, when the fluid has stopped, the side and bottom wall shear stresses are equal to the fluid yield stress. These results prove the efficiency of the inclined plane test for determining yield stress when appropriate experimental precautions are taken for both tests. In addition we examine the possibility of determining the simple shear flow curve of a mud suspension from fluid depth, velocity and discharge measurements of different steady flows in a wide open channel (8 m long; 60 cm wide) equipped with a recirculating system. The results obtained from inclined plane tests are in good agreement with independent rheometrical data (with torsional geometries). However it is technically difficult to cover a wide shear rate range from the inclined plane technique since this requires a rather wide channel flow rate range.     

Turbulence modification in vertical upward annular flow passing through a throat section

Experimental studies on the turbulence modification in annular two-phase flow passing through a throat section were carried out. The turbulence modification in multi-phase flow due to the interactions between two-phases is one of the most interesting scientific issues and has attracted research attention. In this study, the gas-phase turbulence modification in annular flow due to the gas–liquid phase interaction is experimentally investigated. The annular flow passing through a throat section is under the transient state due to the changing cross sectional area of the channel and resultantly the superficial velocities of both phases are changed compared with a fully developed flow in a straight pipe. The measurements for the gas-phase turbulence were precisely performed by using a constant temperature hot-wire anemometer, and made clear the turbulence structure such as velocity profiles, fluctuation velocity profiles. The behavior of the interfacial waves in the liquid film flow such as the ripple or disturbance waves was also observed. The measurements for the liquid film thickness by the electrode needle method were also performed to measure the base film thickness, mean film thickness, maximum film thickness and wave height of the ripple or the disturbance waves.     

Study of the thermocapillary effect on a wavy falling film using a fiber optical thickness probe

The wave flow of a water film down a vertical plate with a 150×150 mm heater has been experimentally studied. The effect of the heat flux on the film flow leads to the formation of periodically flowing rivulets and thin film between them due to the action of thermocapillary forces in spanwise direction. The local film thickness between rivulets is measured by means of a noncontact fiber optical probe. As the heat flux grows, the average film thickness continuously decreases but upon reaching about 50% of the initial thickness, the film spontaneously breaks down. It is found that the decrease of the wave amplitude between rivulets is caused by the reduction of the local Reynolds number and is in a qualitative agreement with the laws of the hydrodynamics for the isothermal case. That is, no appreciable effect of streamwise thermocapillary forces on the wave amplitudes is detected. The experimental results are in good agreement with recently published data obtained by the capacitance method.     

Drag reduction and improvement of material transport in creeping films

It is widely accepted that for bodies in turbulent flows a reduction of skin friction can be reached if the surface of the body is provided with small ridges aligned in the local flow direction. This surprising and counterintuitive phenomenon is called the shark-skin effect, motivated from the dermal surface morphology of sharks. In the present article we examine the possibility of resistance reduction due to a rippled surface topography in Stokes flow. We especially analyse the influence of wall riblets perpendicular to the flow direction on the mean transport velocity in gravity-driven creeping film flows following the idea that eddies generated in the valleys of the riblets act like fluid roller bearings and hence may reduce drag. Using a theoretical treatment of the Stokes equations with complex function theory, parameter studies with varying flow rate, bottom amplitude and bottom shape are presented. For the given bottom shapes the maximum enhancement of transport velocity is found by optimising the film thickness.     

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     

Stability of conducting liquid flowing down an inclined plane at moderate Reynolds number in the presence of constant electromagnetic field

The stability of a conducting viscous film flowing down an inclined plane at moderate Reynolds number in the presence of electromagnetic field is investigated under induction-free approximation. Using momentum integral method a non-linear evolution equation for the development of the free surface is derived. The linear stability analysis of the evolution equation shows that the magnetic field stabilizes the flow whereas the electric field stabilizes or destabilizes the flow depending on its orientation with the flow. The weakly non-linear study reveals that both the supercritical stability and subcritical instability are possible for this type of thin film flow. The influence of magnetic field on the different zones is very significant, while the impact of electric field is very feeble in comparison.     

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.     

Stability of flow of a thin layer of viscous magnetic liquid

The laminar flow of a thin layer of heavy viscous magnetic liquid down an inclined wall is examined. The stability and control of the flow of an ordinary liquid are affected only by alteration of the angle of inclination of the solid wall and the velocity of the adjacent gas flow. When magnetic liquids are used [1, 2], an effective method of flow control may be control of the magnetic field. By using magnetic fields of various configurations it is possible to control the flow of a thin film of viscous liquid, modify the stability of laminar film flow, and change the shape of the free surface of the laminarly flowing thin film, a factor which plays a role in mass transfer, whose rate depends on the phase contact surface area. The magnetic field significantly affects the shape of the free surface of a magnetic liquid [3, 4]. In this paper the velocity profile of a layer of viscous magnetic liquid adjoining a gas flow and flowing down an inclined solid wall in a uniform magnetic field is found. It is shown that the flow can be controlled by the magnetic field. The problem of stability of the flow is solved in a linear formulation in which perturbations of the magnetic field are taken into account. The stability condition is found. The flow stability is affected by the nonuniform nature of the field and also by its direction.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 5, pp. 59–65, September–October, 1977.     

Long waves between a subsonic gas flow-film interface flowing down an inclined plate

The present work deals with temporal stability properties of a falling liquid film down an inclined plane in the presence of a parallel subsonic gas flow. The waves are described by evolution equation previously derived as a generalization of the model for the Newtonian liquid. We confirm linear stability results of the basic flow using the Orr–Sommerfeld analysis to that obtained by long wave approximation analysis. The non-linear stability criteria of the model are discussed analytically and stability branches are obtained. Finally, the solitary wave solutions at the liquid–gas interface are discussed, using specially envelope transform and direct ansatz approach to Ginzburg–Landau equation. The influence of different parameters governing the flow on the stability behavior of the system is discussed in detail.