Wave regimes on power-law fluid film flowing down a porous plane

Non-linear waves on the surface of a falling film of power-law fluid on a vertical porous plane are investigated. The waves are described by evolution equations generalising equations previously derived in the case of solid plane. It is shown that the slip condition on the interface between pure liquid and the porous substrate drastically changes structure of the steady waves travelling in the film.     

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.     

Three-dimensional surface characteristics of a falling liquid film

Optical methods are described for examining the three-dimensional character of waves on a falling liquid film. This involved monitoring the motion of the local film surface normal through the use of laser beam refraction. The wavy motion was found to be primarily of a two-dimensional nature only for Re (equal to 4Q/v) less than 1500.

Surface characteristics were examined for Reynolds numbers from 217 to 4030 and for different distances along the direction of flow.     

Evaporation and flow dynamics of thin, shear-driven liquid films in microgap channels

Thin and ultra-thin shear-driven liquid films in a narrow channel are a promising candidate for the thermal management of advanced semiconductor devices in earth and space applications. Such flows experience complex, and as yet poorly understood, two-phase flow phenomena requiring significant advances in fundamental research before they could be broadly applied. This paper focuses on the results obtained in experiments with locally heated shear-driven liquid films in a flat mini-channel. A detailed map of the flow sub-regimes in a shear-driven liquid film flow of water and FC-72 have been obtained for a 2 mm channel operating at room temperature. While the water film can be smooth under certain liquid/gas flow rates, the surface of an intensively evaporating film of FC-72 is always distorted by a pattern of waves and structures. It was found, that when heated the shear-driven liquid films are less likely to rupture than gravity-driven liquid films. For shear-driven water films the critical heat flux was found of up to 10 times higher than that for a falling film, which makes shear-driven films (annular or stratified two-phase flows) more suitable for cooling applications than falling liquid films.     

Experimental investigation into three-dimensional wavy liquid films under the influence of electrostatic forces

Three-dimensional interfacial waves that develop on the free surface of falling liquid films are known to intensify heat and mass transfer. In this context, the present paper studies the effect of electrostatic forces applied to a falling film of dielectric liquid on its three-dimensional nonlinear wave dynamics. Therefore, measurements of the local film thickness using a confocal chromatic imaging method were taken, and the complex wave topology was characterized through photography. The experiments show a complex interaction between the electric field and the hydrodynamics of the falling film, whereby electrostatic forces were found to both increase and decrease wave peak height in different regions of the wave. Additionally, an electrically induced breakup of the three-dimensional wave fronts, which leads to a locally doubled frequency in streamwise direction, is found. The ability to influence the wave topology demonstrated here opens the possibility to optimize heat transfer processes in falling liquid films.     

Wave formation on vertical falling liquid films

The method of integral relations is used to derive a nonlinear “two-wave” structure equation for long waves on the surface of vertical falling liquid films. This equation is valid in a wide range of Reynolds numbers and reduces to the known equations for high and low Re. Theoretical data for the fastest growing waves are compared with the experimental results on velocities, wave numbers and growth rates of the waves in the inception region. The validity of theoretical assumptions is also confirmed by the direct measurements of the instantaneous velocity profiles in a wave liquid film.     

Viscous flowing film instability down an inclined plane in the presence of constant electromagnetic field

The present work deals with temporal stability properties of a falling liquid film down an inclined plane in the presence of constant electromagnetic field. Using the Kármán approximation, the problem is reduced to the study of the evolution equation for the free surface of the liquid film derived through a long-wave approximation. A linear stability analysis of the base flow is performed. Also, the solutions of stationary waves and Shkadov waves are introduced and discussed analytically by analyzing the linearized instability of the fixed points and Hopf bifurcation.     

Waves on the surface of a falling power-law fluid film

Waves that occur at the surface of a falling film of thin power-law fluid on a vertical plane are investigated. Using the method of integral relations an evolution equation is derived for two types of waves equation which are possible under long wave approximation. This equation reveals the presence of both kinematic and dynamic wave processes which may either act together or singularly dominate the wave field depending on the order of different parameters. It is shown that, at a small flow rate, kinematic waves dominate the flow field and the energy is acquired from the mean flow during the interaction of the waves, while, for high flow rate, inertial waves dominate and the energy comes from the kinematic waves. It is also found that this exchange of energy between kinematic and inertial waves strongly depends on the power-law index n. Linear stability analysis predicts the contribution of different terms in the wave mechanism. Further, it is found that the surface tension plays a double role: for a kinematic wave process, it exerts dissipative effects so that a finite amplitude case may be established, but for a dynamic wave process it yields dispersion. Further, it is shown that the non-Newtonian character n plays a vital role in controlling the role of the term that contains surface tension in the above processes.     

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.     

Local film thickness and temperature distribution measurement in wavy liquid films with a laser-induced luminescence technique

Heat transfer in falling liquid film systems is enhanced by waviness. Comprehension of the underlying kinetic phenomena requires experimental data of the temperature field with high spatiotemporal resolution. Therefore a non-invasive measuring method based on luminescence indicators is developed. It is used to determine the temperature distribution and the local film thickness simultaneously. Results are presented for the temperature distribution measurement in a laminar-wavy water film with a liquid side Reynolds number of 126 flowing down a heated plane with an inclination angle of 2° at two positions in flow direction. The measured temperature distributions are used to calculate the local heat transfer coefficient for solitary waves at two positions in flow direction.     

Electroviscous effects in steady fully developed flow of a power-law liquid through a cylindrical microchannel

Electroviscous effects in steady, fully developed, pressure-driven flow of power-law liquids through a uniform cylindrical microchannel have been investigated numerically by solving the Poisson–Boltzmann and the momentum equations using a finite difference method. The pipe wall is considered to have uniform surface charge density and the liquid is assumed to be a symmetric 1:1 electrolyte solution. Electroviscous resistance reduces the velocity adjacent to the wall, relative to the velocity on the axis. The effect is shown to be greater when the liquid is shear-thinning, and less when it is shear-thickening, than it is for Newtonian flow. For overlapping electrical double layers and elevated surface charge density, the electroviscous reduction in the near-wall velocity can form an almost stationary (zero shear) layer there when the liquid is shear-thinning. In that case, the liquid behaves approximately as if it is flowing through a channel of reduced diameter. The induced axial electrical field shows only a weak dependence on the power-law index with the dependence being greatest for shear-thinning liquids. This field exhibits a local maximum as surface charge density increases from zero, even though the corresponding electrokinetic resistance increases monotonically. The magnitude of the electroviscous effect on the apparent viscosity, as measured by the ratio of the apparent and physical consistency indices, decreases monotonically as the power-law index increases. Thus, overall, the electroviscous effect is stronger in shear-thinning, and weaker in shear-thickening liquids, than it is when the liquid is Newtonian.     

Longitudinal flow characteristics of vertically falling liquid films without concurrent gas flow

Flow characteristics of liquid films vertically falling along the outer wall of a circular tube without concurrent gas flow are experimentally studied, and attention is given to the longitudinally developing liquid film flow in the flow direction. Flow measurements are carried out by the methods of needle contact and electric capacity, and the obtained data are statistically processed.There exists a definite difference in flow characteristics such as wave motion patterns, film thicknesses, critical Reynolds number, and so on, depending strongly on the longitudinal distance in the flow direction as well as the liquid film Reynolds number. Measured probability distributions of interfacial waves can be well expressed by the functions of probability distribution statistically well-known as normal, logarithmic normal and gamma distributions. In terms of these functions, interfacial wave patterns are definitely classified over the whole experimental flow regime. As a rule, interfacial wave motion proceeds vigorously with increases of the longitudinal distance and Reynolds number; however, there exists a flow condition that wave fluctuation never grows up but declines regardless of an increase of Reynolds number.     

Rheological properties of some biphenyl liquid crystals

Rheological measurements have been carried out using a rotational viscometer with a system of coaxial cylinders on four liquid crystalline substances from the group of cyanobiphenyls. On the basis of results of these investigations it was found that in the investigated range of shear rates the nematic phases exhibited Newtonian flow behaviour, while the smectic phases exhibited non-Newtonian behaviour. For shear rates up to ca. 1000 s^–1 the dependence of the shear stress on shear rate is well described by a power-law model.     

Experimental and numerical study on the falling film flow characteristics outside circular tube applied in floating liquefied natural gas (FLNG) under offshore conditions

Spiral wound heat exchanger (SWHE) relying on falling film evaporation and boiling is often used for FLNG. The performance of SWHE can be impacted strongly by the motion of the FLNG caused by the wave and typhoon. The falling film characteristics of SWHE outside circular tube are studied experimentally and numerically by a visualization experimental device based on the high-speed camera and a numerical model based on the dynamic grid. The results show that the wave crest of the liquid film moves to the titled side under offshore conditions. The evolution process of falling film flow pattern outside circular tube with the tilt angle of 9° can be divided into four stages: droplet formation and migration, liquid column formation and migration, liquid column coalescence, liquid sheet formation. A correlation permitting the prediction of the falling film flow pattern outside circular tube and the other one permitting the prediction of the average film thickness of circular tube are developed respectively based on the experimental and numerical data.     

Deformation and breakup of a non-Newtonian slender drop in an extensional flow

The deformation and breakup of a non-Newtonian slender drop in a Newtonian liquid in a simple extensional and creeping flow has been theoretically studied. The power-law was chosen for the fluid inside the drop, and the deformation of the drop is described by a single ordinary differential equation, which was numerically solved. Asymptotic analytical expressions for the local radius were derived near the center and close to the end of the drop. The results for the shape of the drop and the breakup criterion are presented as a function of the capillary number, the viscosity ratio and type of non-Newtonian fluid inside the drop. An approximate analytical solution is also suggested which is in good agreement with the numerical results.     

Effects of shear-thinning/thickening nature on the stability of Couette-like flow with uniform crossflow

The linear stability of wall-injected pressure- driven Couette-like flow in power-law fluids is studied. Previous study on this kind of flow for Newtonian fluids by Nicoud and Angilella [Phys. Rev. E 56, 3000 （1997）] was extended to power-law fluids to understand the effects of shear-thinning/thickening nature on the flow stability. A related expression between the critical crossflow Reynolds number for Newtonian fluids and that for power-law fluids is obtained as the streamwise Reynolds number is large enough based on numerical computations, and verified theoretically in the case of a limiting condition with the power-law index.     

Heat transfer in non-Newtonian falling liquid film on a horizontal circular cylinder

 This study aims to investigate numerically the laminar flow and heat transfer in a pseudoplastic non-Newtonian falling liquid film on a horizontal cylinder for the constant heat flux and isothermal boundary conditions. The inertia terms are taken into account. An implicit finite difference method is carried out to solve the governing boundary layer equations. The effects of operational parameters on the hydrodynamic and heat transfer characteristics are examined and discussed in detail. The results presented show that the local and average Nusselt numbers varies significantly as a function of the concentration of aqueous carboxymethylcellulose (CMC) solutions and the cylinder diameter. Higher concentration of aqueous CMC solutions generate larger heat transfer coefficients. Finally, a comparison with the experimental and numerical results available in the literature for Newtonian fluids shows clearly that the present analysis is reasonably accurate. Received on 29 March 2001 / Published online: 29 November 2001     

Pure steam condensation model with laminar film in a vertical tube

A new physical model for calculating the liquid film thickness and condensation heat transfer coefficient in a vertical condenser tube is proposed by considering the effects of gravity, liquid viscosity, and vapor flow in the core region of the flow. To estimate the velocity profile in the liquid film, the liquid film was assumed to be in Couette flow forced by the interfacial velocity at the liquid–vapor interface. For simplifying the calculation procedures, the interfacial velocity was estimated by introducing an empirical power-law velocity profile. The resulting film thickness and heat transfer coefficient from the model were compared with the experimental data and the results obtained from the other condensation models. The results demonstrated that the proposed model described the liquid film thinning effect by the vapor shear flow and predicted the condensation heat transfer coefficient from experiments reasonably well.     

A variational approach to non-steady non-newtonian flow in a circular pipe

The transient response of a non-Newtonian power-law fluid to several assumed forms of pressure pulse in a circular tube is analysed by the semi-direct variational method of Kanntovorich. Velocity profiles are shown for several power-law indices, and by comparing the results for the Newtonian case with the exact solution given by Szymanski, it is observed that the results are good to 5%. More accurate solutions have been found for the case involving Newtonian fluid flow. New results are reported concerning the effect of a triangular pressure pulse on the development and transient response of the flow field of a non-Newtonian fluid.     

Gravity-driven laminar film flow of power-law fluids along vertical walls

A theoretical analysis is presented which brings steady laminar film flow of power-law fluids within the framework of classical boundary layer theory. The upper part of the film, which consists of a developing viscous boundary layer and an external inviscid freestream, is treated separately from the viscous dominated part of the flow, thereby taking advantage of the distinguishing features of each flow region. It is demonstrated that the film boundary layer developing along a vertical wall can be described by a generalized Falkner-Skan type equation originally developed for wedge flow. An exact similarity solution for the velocity field in the film boundary layer is thus made available.Downstream of the boundary layer flow regime the fluid flow is completely dominated by the action of viscous shear, and fairly accurate solutions are obtained by the Von Karman integral method approach. A new form of the velocity profile is assumed, which reduces to the exact analytic solution for the fully-developed film. By matching the downstream integral method solution to the upstream generalized Falkner-Skan similarity solution, accurate estimates for the hydrodynamic entrance length are obtained. It is also shown that the flow development in the upstream region predicted by the approximate integral method closely corresponds to the exact similarity solution for that flow regime. An analytical solution of the resulting integral equation for the Newtonian case is compared with previously published results.