The effect of an electrostatic field on inviscid liquid flow down an inclined plane

For a study of nonlinear wave motion under an electrostatic field, the Korteweg-de Vries (KdV) equation for inviscid liquid film flowing under gravity down an inclined plane has been derived, and the effect of the electric field on the stability of a solitary wave as a solution of the KdV equation is examined. Under a constant electrostatic potential the stability of the wave is not affected. However, with a slowly varying potential it becomes unstable.     

Hydraulic-jump behavior of a thin film flowing down an inclined plane under an electrostatic field

The hydraulic-jump phenomenon of a thin fluid layer flowing down an inclined plane under an electrostatic field is explored by using a global bifurcation theory. First, the existence of hydraulic-hump wave has been found from heteroclinic trajectories of an associated ordinary differential equation. Then, the jump behavior has been characterized by introducing an intensity function on the variations of Reynolds number and surfave-wave speed. Finally, we have investigated the nonlinear stability of traveling shock waves triggered from a hydraulic jump by integrating the initial-value problem directly. At a given wave speed there exists a certain value of Reynolds number beyond which a time-dependent buckling of the free surface appears. Like the other wave motions such as periodic and pulse-like solitary waves, the hydraulic-jump waves are also found to become more unstable as the electrostatic field is getting stronger.     

Characteristics of solitary waves on a running film down an inclined plane under an electrostatic field

For the study on the nonlinear dynamics of thin-film flow running down an inclined plane under the effect of an electrostatic field, the mechanism of solitary waves has been examined by using a global bifurcation theory. First, the existence of solitary waves has been chased by using an orbit homoclinic to a fixed point of saddle-focus type in a linearized third-order ordinary differential equation which resulted from the evolution equation in a steady moving frame. Then, the trajectories with several kinds of solitary waves have also been searched numerically for the nonlinear system. In addition, the behavior of these waves has been directly confirmed by integrating the initial-value problem. The slightly perturbed waves at the inception eventually evolve downstream into almost permanent pulse-like solitary waves through the processes of coalescence and repulsion of the triggered subharmonics. In the global aspects the flow system at a given Reynolds number becomes more unstable and chaotic than when there is no electrostatic force applied.     

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.     

Long-wave instabilities of film flow under an electrostatic field

The free-surface behavior of a viscous liquid layer flowing down an inclined plane by gravity and interacting with an overlying uniform electrostatic field is examined in the limit of long-wave approximation. Both linear and nonlinear stability analyses are performed to address two-dimensional surface-wave evolution initiating from a flat interface. The growth of a periodic disturbance is first investigated for a linear analysis, and then to examine the nonlinear surface-wave instabilities the evolution equation for film height is solved numerically by a Fourier-spectral method. For small evolution time the calculated nonlinear modes of instability are consistent with the results obtained from the linear theory. The effect of an electrostatic field increases the wavenumbers showing a maximum linear growth rate as well as a cutoff. A significant phenomenon as Reynolds number is increasing is the appearance of the catastrophic surface waves in the long run whenever any initial wavenumber making a traveling wave linearly unstable is employed into the initial simple-harmonic disturbance.     

Hydrodynamic behavior of an electrified thin film flowing down an inclined plane at high Reynolds numbers

To answer the questions on the dynamics of thin liquid flow down an inclined plane at high Reynolds numbers subjected to a uniform normal electrostatic field, we have derived evolution equations describing the free-surface behavior by using the von Kármán-Pohlhausen approximation. The integration of the evolution equations is numerically performed to address two-dimensional finite-amplitude surface-wave propagation modes. The growth of a periodic disturbance is first examined to compare with the results linear-stability theory, and then to investigate the nonlinear surface-wave behavior the evolution equations are solved numerically by a Fourier-spectral method. For small evolution time the computed nonlinear modes of instability are well consistent with the results from the linear theory. The effect of an electrostatic field makes the flow system significantly unstable.     

The flow of a thin conducting film over a spinning disc in the presence of an electric field

We consider the flow of a thin liquid film over a spinning disc in the presence of an electric field. This is imposed by applying a potential between the disc and an electrode above the film. The integral method and lubrication theory are used to derive a coupled set of evolution equations for the film thickness, radial and azimuthal flow rates and the surface charge density. These equations are parameterized by a modified Weber number, a modified Masuda number, dimensionless conductivities and a dimensionless electrode separation. We focus on the limit of large liquid conductivity and explore the influence of the applied electric field on the film dynamics via numerical simulations. The effects of spatially and temporally varying electric fields and electrode geometry on the formation of waves are also examined. Our results indicate that increasing the intensity of the electric field or decreasing the electrode separation exerts a destabilizing effect, leading to the formation of interfacial waves of larger amplitude. Spatial and temporal variations of the electric field also lead to complex film dynamics which, beyond a certain threshold of the relevant parameter values, give rise to sustained formation of waves over a large fraction of the spinning disc. These results suggest that electric fields have the potential to enhance the degree of wave-induced process intensification.     

Effects of an electrostatic field in pneumatic conveying of granular materials through inclined and vertical pipes

The pneumatic transport of granular materials through an inclined and vertical pipe in the presence of an electrostatic field was studied numerically using the discrete element method (DEM) coupled with computational fluid dynamics (CFD) and a simple electrostatic field model. The simulation outputs corresponded well with previously reported experimental observations and measurements carried out using electrical capacitance tomography and high-speed camera techniques in the present study. The eroding dunes and annular flow regimes, observed experimentally by previous research workers in inclined and vertical pneumatic conveying, respectively, were reproduced computationally by incorporating a simplified electrostatic field model into the CFD-DEM method. The flow behaviours of solid particles in these regimes obtained from the simulations were validated quantitatively by experimental observations and measurements. In the presence of a mild electrostatic field, reversed flow of particles was seen in a dense region close to the bottom wall of the inclined conveying pipe and forward flow in a more dilute region in the space above. At sufficiently high field strengths, complete backflow of solids in the inclined pipe may be observed and a higher inlet gas velocity would be required to sustain a net positive flow along the pipe. However, this may be at the expense of a larger pressure drop over the entire conveying line. In addition, the time required for a steady state to be attained whereby the solids flow rate remains substantially constant with respect to time was also dependent on the amount of electrostatic effects present within the system. The transient period was observed to be longer when the electrostatic field strength was higher. Finally, a flow map or phase diagram was proposed in the present study as a useful reference for designers of inclined pneumatic conveying systems and a means for a better understanding of such systems.     

Accelerating laminar liquid film along an inclined wall

The paper presents an analytical study which describes the laminar accelerating flow of a thin film along an inclined wall. When interfacial shear is taken into consideration it is found that the non-dimensional film thickness depends on two parameters (Fr/Re) and ρu-ratio, the effect of the latter being more pronounced for larger values of the former. The results are compared with those of earlier workers.     

The effect of an electrostatic field on the rupture of a thin viscous static film by van der Waals attractions

This research examines rupture phenomena of a horizontal static thin viscous layer on a solid plate under an electrostatic field generating from a charged foil above the film. The dynamics of the electrified liquid film is formulated to derive a long-wave evolution equation of local film thickness. It determines two-dimensional nonlinear behavior of the film subject to surface tension, viscous, electrically induced forces, and van der Waals attractions. Linear stability analysis is used to obtain the maximum growth rate of a periodic disturbance and its corresponding wavenumber. To see the development of film rupture the strongly nonlinear partial differential equation is numerically solved for the unlimited or limited foil length as part of an initial-value problem with spatially periodic boundary conditions. The stronger electric forces make the thin layer more unstable and speed up its rupture.     

An experimental study of the flow of dry powders over inclined surfaces

The behavior of dry particulate solids during unconfirmed flow over inclined surfaces has been investigated. The motion of individual particles is found to depend strongly on the nature of the surface over which they flow. For smooth surfaces, flow occurs primarily by sliding at the surface, and little or no shear is introduced into the stream. In the case of highly roughened surfaces consisting of, for example, a stationary layer of the same particles, there appears to be no slip at the surface, and flow occurs entirely by shear within the flowing stream. Surfaces of intermediate roughness lead to flow in which both slip at the surface and shear within the bed contribute significantly.Velocity profiles have been measured experimentally under a variety of conditions, and the effects of such variables as roughness and inclination of the surface, depth of the flowing stream and particle size have been evaluated quantitatively. Empirical relationships have been obtained which describe the flow behavior in all cases studied.     

A theoretical model of a free flow solution over an inclined long flat plate solar still

This analysis is used to investigate a free flow inclined flat plate solar still.To study the effect of significant parameters on a laminar falling liquid solution over an inclined long flat plate solar still, a mathematical two dimensional flow analysis is carried out based on continuity, momentum and energy equations for liquid and vapor phases together with the interface. As far as the liquid film is concerned, the velocity, film thickness, pressure, temperature and the hourly evaporated water volume profiles are found. It is shown that the significant parameters which affect the solar still are the initial film thickness (i.e, the initial mass flow rate), the plate inclination, the still length and the absorber heat flux (i.e., the solar radiation reaching the plate). It is also shown that the variation in liquid film thickness down the still has a significant effect on the evaporation rate (i.e., the condensed water) and should not be neglected particularly in the case of a long inclined solar still.     

温差引起管子在斜面上的蠕动现象

在斜面上布置管路时,虽然设计上满足了管子自重下滑力与其摩擦力的平衡,但工程实际中管子仍会下滑从而造成变形破坏。详细分析了破坏的原因,认为是由于温差引起管子在斜面上蠕动而造成的。     

Inertial two- and three-dimensional thin film flow over topography

The effect of inertia on gravity-driven thin film free-surface flow over substrates containing topography is considered. Flow is modelled using a depth-averaged form of the governing Navier–Stokes equations and the discrete analogue of the coupled equation set solved accurately using an efficient full approximation storage (FAS) and full multigrid (FMG) technique. The free-surface disturbance induced by topographic features is illustrated by considering examples of gravity-driven flow over and around peak, trench and occlusion topography. Results are presented which demonstrate how increasing Reynolds number can significantly enhance the magnitude of free-surface disturbances, a feature which may have important consequences for the wide range of coating process that aim to maximise free-surface planarity.     

The flow of solid particles in an aerated inclined channel

The dynamic behavior of the flow of solid particles in an inclined open channel with air flow introduced through a porous base plate was examined. The solid particle velocity distributions were measured in detail using an optical probe.Five types of flow patterns were found depending on the air velocity and the slope of the channel. They were: sliding flow, immature sliding flow, splashing flow, bubbling flow, and gliding flow. The first two could be observed when the air velocity was less than the minimum fluidization velocity, U_mf, for the particles. The last two were observed when it became higher than U_mf. Splashing flow occurred when the slope of the channel was steep.     

Thin film flow over structured packings at moderate Reynolds numbers

A model describing the dynamic evolution of waves on laminar falling wavy films at low to moderate Reynolds numbers (Re<30) over corrugated surfaces is presented. This model is based on the integral balance method, which is used to simplify the analysis by assuming a parabolic velocity profile within the bulk of the film. The predictions of this model are compared to those obtained from a computational fluid dynamics (CFD) code for which no assumptions regarding the film velocity profile are made. The film evolution profiles, obtained via numerical solution of the model equations, are used to calculate the ratio of film interfacial area to that of the substrate. The model predicts suppression of wave growth on a corrugated surface. The model is also used to predict the effect of packing geometry on the flow characteristics, which may improve packing design. Solutions obtained for Re∼200 using the CFD code demonstrate the degree of fluid accumulation within the ‘valleys’ of the structured substrate wherein re-circulation occurs.     

Exact solution for diffusion to flow down an incline

The concentration of solute in a film flowing down an inclined plane has been obtained by solving the complete diffusion equation, including the diffusion effect in the direction of flow. The formulation leads to the necessity of solving non-orthogonal characteristic value problem, treatment of which is, however, quite straightforward. Though the motivation for the study was the diffusion of oxygen to plasma or blood films, solutions are presented for the complete range of fluid properties, due to the great significance of this mode of flow for wider application in chemical engineering.     

Numerical investigation of monodisperse granular flow through an inclined rotating chute

A discrete element model of spherical glass particles flowing down a rotating chute is validated against high quality experimental data. The simulations are performed in a corotating frame of reference, taking into account Coriolis and centrifugal forces. In view of future extensions aimed at segregation studies of polydisperse granular flows, several validation steps are required. In particular, the influence of the interstitial gas, a sensitivity study of the collision parameters, and the effect of system rotation on particle flow is investigated. Shirsath et al. have provided the benchmark laboratory measurements of bed height and surface velocities of monodisperse granular flow down an inclined rotating chute. With a proper choice of the friction coefficients, the simulations show very good agreement with our experimental results. The effect of interstitial gas on the flow behavior is found to be relatively small for 3‐mm granular particles. © 2014 American Institute of Chemical Engineers AIChE J, 60: 3424–3441, 2014