Gravity-driven Film Flow Research Articles

Magnetic resonance velocimetry of thin falling films

Falling film microreactors are a type of liquid–gas contactor based on a gravity-driven film flow. These devices achieve high interfacial areas per unit volume compared to other contactors. Nevertheless, they are limited by the liquid-side mass transfer. To overcome this issue, the operating conditions must be chosen in a way to enhance gas absorption and ensure efficient transfer within the film. The present study looks into the use of magnetic resonance velocimetry to study the velocity field inside of falling films at the microscale. Two parameters were considered, the channel geometry, and the volumetric flow rate. The aim was not only to determine the conditions that optimize the interfacial area, but also to showcase the capabilities and advantages of magnetic resonance velocimetry for this type of flow configuration. In addition, a straightforward mathematical model, based on the Navier–Stokes equations, is used to predict the thickness of the film, since previous approaches, mainly those of Kapitza and Nusselt, are not valid for flows at the microscale, where surface tension phenomena predominate. It was shown that the round channel yields by far the highest surface-to-volume ratio, twice as high as for the rectangular channel for the same volumetric flow rate. In addition, film breakup was not observed in the round channel even for volumetric flow rates of down to 0.27mLmin−1.

Gravity-driven thin film flow of a third-grade fluid on a movable inclined plane with slip boundary condition

The prime aim and essence of this study are to present a closed-form solution of the unidirectional velocity field for thin film flow generated by a third-grade liquid moving over a fixed or movable inclined plane in the presence of partial slip boundary condition. Consideration of partial slip makes this problem different from other published research works. Here lies the novelty of this study. The nondimensional, unidirectional velocity profiles rely on the material parameter of third-grade fluid, slip parameter and initial velocity of the movable plane. Study discloses that the fluid’s velocity decelerates with raising the material property of grade-3 fluid and it accelerates with the slip parameter and initial velocity of the inclined plane. The outcomes of this study are deliberated physically at length.

Surface turbulence in film flow over periodic ripples

Turbulence in gravity-driven film flow is usually discussed in terms of three-dimensional solitary-wave pulses as they are frequently observed in flows along smooth walls. Here, we show that free-surface turbulence arises in films along rippled substrates, as they are commonly employed in process engineering applications, already at rather low Reynolds numbers from the irregular break-up of the solitary-wave fronts. Short waves in the capillary regime replace the broken solitary waves beyond a certain Reynolds number. The crossover coincides with the occurrence of steady three-dimensional surface patterns upstream, which suppress travelling waves. The waves show power spectral densities with power-law exponents typical for weak capillary-wave turbulence. With increasing Reynolds number, the steepness of the power law declines to lower levels.

Evaluation of the variation of the GPR frequency spectra created by the activities of earthworms

The analysis and utilization of frequency spectrum in emission signals from ground penetrating radar (GPR) are essential for investigating soil characteristics and enhancing urban management. This research focuses on assessing the effectiveness of earthworm activities in improving rainwater management in regions with clay soil, where water infiltration is challenging. Specifically, we examined the impact of two earthworm species, Lumbricus terrestris and Aporrectodea caliginosa, as well as a combination of both, on the permeability of clay loam soil. To evaluate earthworm activities, a gravity-driven film flow setup with nine tanks was employed. Furthermore, we suggest utilizing the Discrete Fourier transform to analyse the frequency spectra of different data types (traces, transects, and the average of tanks) and comprehend their influence on earthworm behavior. To investigate the relationship between fluctuations in infiltration velocity and the frequency spectrum signal over time and activity position, a correlation analysis was performed using linear regression models. The goodness of fit was assessed based on the coefficient of determination (R2). For the earthworm species L1, L2, and L12, the R2 values were determined as 0.8858, 0.847, and 0.9493, respectively. These results demonstrate that the frequency domain was derived from variations in ground penetrating radar signals, and a significant correlation exists between infiltration velocity and spectra influenced by earthworm activity.

Interplay of fluid rheology and micro-patterning toward modulating draining characteristics on an inclined substrate

We investigate a gravity-driven thin film flow of a non-Newtonian liquid over an inclined micro-patterned surface. We demonstrate the effect of micro-patterning on the film draining rate and the velocity profile by varying the relative slit width (Tr) and the length of the periodic irregularities (L). We unveil the interplay of the substrate structure and the fluid rheology by modeling the non-Newtonian thin film using the Carreau model, and the rheology of the film is varied for different values of power index n. Through numerical simulations, we delineate the effects of inertia, viscous, and capillary forces on the physics of thin film flow. We report a significant augmentation of flow velocity for both shear-thinning and shear-thickening fluids as a result of substrate micro-patterning, with the relative slit width playing a dominant role while the length of the periodic irregularities has only a minor influence on drainage characteristics. However, when the sole effect of fluid rheology is considered, flow velocity enhances for pseudoplastic fluid and decreases for dilatant fluid in comparison to Newtonian fluid. We examine the combined effect of rheology and substrate topography, revealing the dominating influence of micro-patterning at high slit-widths, while the fluid rheology has a greater role to play at lower slit-widths. We also demonstrate that the susceptibility of flow physics on varying rheology or topography is greatest for low viscosity liquids. Finally, we mark different regimes where the augmentation of average velocity and surface velocity are individually achieved. Hence, we propose a suitable combination of substrate structure and fluid rheology to engineer a flow characteristic. Based on the suitability for various applications, we provide the key to simultaneously optimizing the fluid rheology and substrate micro-patterning for precise engineering and controlling the draining characteristics of a thin film.

Hydrodynamic and thermal model for gravity-driven smooth laminar film flow undergoing flash evaporation cooling: Case study and correlation development

Hydrodynamic and thermal analyses have been carried out for gravity-driven smooth laminar film flow, undergoing flash evaporation at the free surface. A classical one-dimensional semi-analytical approach has been adopted to address a unique problem where hydrodynamic and thermal boundary layers (TBLs) approach from opposite directions and eventually intersect each other. This occurs due to the rapid evaporation cooling at the film-free surface exposed to the low-pressure ambiance, leading to the growth of a TBL from the free surface. In contrast, the hydrodynamic boundary layer (HBL) grows from the solid wall over which the film flow occurs. The intersections between the TBL and HBL edges, HBL edge and the free surface, and TBL edge and the wall, in conjunction with the attainment of a fully developed hydrodynamic condition, result in the division of the overall film domain into three distinct hydrodynamic and five distinct thermal sub-zones requiring zone-specific formulations. The model is successfully validated for hydrodynamic formulations with the existing experimental data. However, the lack of available experimental studies limits the validation of the proposed thermal model. Correlations for relevant thermal and hydrodynamic parameters, such as local Nusselt number, local free surface temperature, local bulk mean temperature, and local film thickness, are developed based on the model predictions. The proposed model and the correlations derived from its predictions are anticipated to serve as crucial benchmarks for optimizing the design of thermal management and desalination systems that are fundamentally driven by the film evaporation process.

A new elastic instability in gravity-driven viscoelastic film flow

We examine the linear stability of the gravity-driven flow of a viscoelastic fluid film down an inclined plane. The viscoelastic fluid is modeled using the Oldroyd-B constitutive equation and, therefore, exhibits a constant shear viscosity and a positive first normal stress difference in viscometric shearing flows; the latter class of flows includes the aforesaid film-flow configuration. We show that the film-flow configuration is susceptible to two distinct purely elastic instabilities in the inertialess limit. The first instability owes its origin entirely to the existence of a free surface and has been examined earlier [Shaqfeh et al., “The stability of gravity driven viscoelastic film-flow at low to moderate Reynolds number,” J. Non-Newtonian Fluid Mech. 31, 87–113 (1989)]. The second one is the analog of the centermode instability recently discovered in plane Poiseuille flow [Khalid et al., “Continuous pathway between the elasto-inertial and elastic turbulent states in viscoelastic channel flow,” Phys. Rev. Lett. 127, 134502 (2021)] and owes its origin to the base-state shear; it is an example of a purely elastic instability of shearing flows with rectilinear streamlines. One may draw an analogy of the aforesaid pair of unstable elastic modes with the inertial free-surface and shear-driven instabilities known for the analogous flow configuration of a Newtonian fluid. While surface tension has the expected stabilizing effect on the Newtonian and elastic free-surface modes, its effect on the corresponding shear modes is, surprisingly, more complicated. For both the Newtonian shear mode and the elastic centermode, surface tension plays a dual role, with there being parameter regimes where it acts as a stabilizing and destabilizing influence. While the Newtonian shear mode remains unstable in the limit of vanishing surface tension, the elastic centermode becomes unstable only when the appropriate non-dimensional surface tension parameter exceeds a threshold. In the limit of surface tension being infinitely dominant, the free-surface boundary conditions for the film-flow configuration reduce to the centerline symmetry conditions satisfied by the elastic centermode in plane Poiseuille flow. As a result, the regime of instability of the film-flow centermode becomes identical to that of the original channel-flow centermode. At intermediate values of the surface tension parameter, however, there exist regimes where the film-flow centermode is unstable even when its channel-flow counterpart is stable. We end with a discussion of the added role of inertia on the aforementioned elastic instabilities.

Experimental study of dripping, jetting and drop-off from thin film flows on inclined fibers

Gravity driven film flows on vertical fibers are known to exhibit a variety of flow dynamics including the formation of droplet trains induced by the hydrodynamic (Kapitza) and Plateau–Rayleigh instability mechanisms. Through an experimental study, it is shown how inclination of the fiber from the vertical influences these dynamics. The formation of waves, regime transitions from dripping to jetting regimes, as well as the onset of drop-off in the form of droplet detachment from the fiber are illustrated and described in dependence of the fiber inclination angle and the liquid mass flow rate. Additionally, the influence of fiber diameter and nozzle geometry on regime transitions and the onset of drop-off from the substrate are examined. It is shown that the onset of drop-off is strongly related to the transition from a regime characterized by a regular wave pattern to a regime characterized by an irregular wave pattern. It is also demonstrated that this regime transition depends not only on flow rate and fiber geometry, but also strongly on the inclination angle. Interestingly, a stabilizing effect of increasing the fiber inclination is detected for constant fiber geometry and film flow rate.

Steady three-dimensional patterns in gravity-driven film flow down an inclined sinusoidal bottom contour

We experimentally studied gravity-driven film flow in an inclined corrugated channel. Beyond a critical Reynolds number, three-dimensional patterns appear. We identified two different types of patterns: a synchronous and a checkerboard one. While the synchronous pattern appears at all inclination angles studied, we observed the checkerboard one only at higher inclination angles and Reynolds numbers. The patterns suppress traveling waves and stabilize the steady flow. We characterize the patterns and their generation and provide a flow-regime map.

Gravity-driven film flow inside an inclined corrugated pipe: An experimental investigation of corrugation shape and tip width

Fluorescence images were acquired in gravity-driven film flow through inclined corrugated pipes representing a range of corrugation shapes and tip widths. The film flow developed an identifiable statically deformed free surface with a wavelength similar to the substrate for most cases of corrugation shape and tip width. The amplitude and phase shift of the statically deformed free surface, as well as the steady-state film thickness, varied more with tip width than with corrugation shape. Transient fluctuations in the free surface elevation were examined for evidence of periodic traveling waves. In general, the film flow produced transient free surface fluctuations, and in many cases, periodic traveling waves with parameters that varied similarly with corrugation shape as with tip width. For flow conditions that produced positive phase shift, low amplitude, or minimal curvature of the statically deformed free surface, transient and periodic behavior were suppressed, supporting previous findings on the importance of the shape and position of the statically deformed free surface. An increase in corrugation tip width also reduced the transient and periodic response. These two findings implicate flow dynamics in the substrate trough as a leading factor in the development of transient and periodic behavior. Steady-state response and the existence of time-dependent behavior are influenced more by tip width than corrugation shape, in agreement with two-dimensional film flow over topography; however, transient fluctuation and periodic traveling wave parameters are similarly influenced by corrugation shape and tip width, which contrasts two-dimensional findings.

Modelling falling film flow: an adjustable formulation

A new two-equation model for gravity-driven liquid film flow based on the long-wave expansion has been derived. The novelty of the model consists in using a base velocity profile combining parabolic (Ruyer-Quil & Manneville, Eur. Phys. J. B, vol. 15, issue 2, 2000, pp. 357–369) and ellipse (Usha et al., Phys. Fluids, vol. 32, issue 1, 2020, 013603) profile functions in the wall-normal coordinate. The dependence on a free parameter $A$ related to the eccentricity of an ellipse serves as an adjustable parameter. The resulting models are consistent at $O(\varepsilon )$ for inertia terms and at $O(\varepsilon ^2)$ for viscous diffusion effects, and predict accurately the primary instability. Appropriate tuning of the adjustable parameter helps to recover accurate predictions for the asymptotic wave celerity of nonlinear solitary waves. Further, the model is shown to capture the closed separation vortices that can form underneath the troughs of precursory capillary ripples.

The Mechanism of Flow Patterns and Rivulet Instability in Gravity-Driven Film Flow on a Porous Wall with Uniform Heating

Properties of porous mediums have significant impacts on the spreading pattern of falling-film along a vertical heated wall. In this paper, we investigate the combined effect of porosity and uniform heating on the flow instability of a falling liquid film. Based on the film thickness equation derived by the long wave theory, linear stability analysis and numerical simulations are given to verify the influences of various dimensionless parameters, and the physical mechanism for the flow instability is explained. With the uniform heating, it is shown that the increasing Marangoni number and Biot number both enhance the rivulet instability because the Marangoni force becomes larger with bigger values of the two numbers. For porous properties, the existence of Darcy number causes the contact line to move faster and results in a destabilizing effect, while a bigger Beavers–Joseph coefficient causes the contact line to move slower and plays a stabilizing role. Increment of porous thickness and the thermal conductivity ratio slightly enhances or impedes the flow instability, respectively, and neither of the two parameters influences the moving speed of the contact lines.

Viscosity and effusion rate identification from free surface data

Based on new explicit expressions for sensitivity coefficients, this study proposes a new method to solve two inverse problems in time-dependent three-dimensional gravity-driven thin film flow down an inclined plane governed by free-surface lubrication approximation. The flow is developed by time-dependent injection through a circular hole in the plane. The first inverse problem, a parameter estimation problem, consists in simultaneously estimating the temperature-dependent viscosity and a constant injection. The second inverse problem, a function estimation problem, involves the determination of the time-dependent injection. It is shown here that using the derived sensitivity coefficients, a parameter estimation approach can still be used for the second inverse problem, efficiently and accurately, and without the need to solve the adjoint and sensitivity problems as well as additional direct problem solutions.

Gravity-driven film flow down a uniformly heated smoothly corrugated rigid substrate

Gravity induced film flow over a rigid smoothly corrugated substrate heated uniformly from below, is explored. This is achieved by reducing the governing equations of motion and energy conservation to a manageable form within the mathematical framework of the well-known long-wave approximation; leading to an asymptotic model of reduced dimensionality. A key feature of the approach and to solving the problem of interest, is proof that the leading approximation of the temperature field inside the film must be nonlinear to accurately resolve the thermodynamics beyond the trivial case of ‘a flat film flowing down a planar uniformly heated incline.’ Superior predictions are obtained compared with earlier work and reinforced via a series of corresponding solutions to the full governing equations using a purpose written finite element analogue, enabling comparisons to be made between free-surface disturbance and temperature predictions, as well as the streamline pattern and temperature contours inside the film. In particular, the free-surface temperature is captured extremely well at moderate Prandtl numbers. The stability characteristics of the problem are examined using Floquet theory, with the interaction between the substrate topography and thermo-capillary instability modes investigated as a set of neutral stability curves. Although there are no relevant experimental data currently available for the heated film problem, recent existing predictions and experimental data concerning the behaviour of corresponding isothermal flow cases are taken as a reference point from which to explore the effect of both heating and cooling.

The effect of substrate amplitude and wavelength on gravity-driven film flow inside an inclined corrugated pipe

An experimental investigation was conducted to determine the effect of substrate amplitude and wavelength on gravity-driven film flow inside an inclined corrugated pipe. Nine different geometries were examined, with substrate amplitude and wavelength varied independently. A statically deformed free surface occurred for all conditions. The amplitude of the statically deformed free surface depended on substrate amplitude and wavelength, with phase shift unaffected by changes in substrate geometry for many conditions under investigation. Fluctuations in free surface elevation were enhanced at low substrate amplitude and intermediate substrate wavelength. Notable reductions in transient free surface behavior were observed for conditions that resulted in a positive phase shift. Transient free surface behavior developed into periodic traveling waves without applied external forcing. Frequency selection for traveling waves was strong, and traveling waves were detected for a majority of the conditions examined. The frequency, phase velocity, and wavelength of the traveling waves showed a potential dependence on substrate geometry; however, there were ranges of substrate amplitude and wavelength for which traveling wave characteristics remained unaffected by changes in substrate geometry. An examination of the amplitude of the statically deformed free surface and transient free surface fluctuations revealed that waviness is a potentially suitable method for combining the effects of substrate amplitude and wavelength on film flow in corrugated pipes. The comparison of amplitudes highlighted a possible link between the statically deformed free surface and the emergence of transient behavior and traveling waves. Length scales proposed in our original work showed promise for characterizing some results.

Role of suction pressure in the stability of a gravity-driven thermoviscous liquid film flow down the interior surface of a cylinder.

This study aims to analyze the stability of a gravity-driven thin film flow in the heated/cooled interior surface of a vertical hollow cylinder. The model development involves simplifying the flow and energy equations using the usual thin-film approximation, where the average film thickness is considered to be much smaller than the radius of cylinder. A dispersion relation is then derived to study the temporal stability of the system in order to quantify the effect of various non-dimensional parameters present in the model, such as the thermoviscous number, Marangoni number, Biot number, and Bond number. Another non-dimensional parameter is introduced by considering an opposing suction pressure in the annulus region. The thermocapillary stress and the thermoviscous effect are shown to strongly affect the temporal stability of the flow. It is shown that although the suction pressure affects the velocity profile of the flow, it does not affect the temporal stability results. The suction pressure is then shown to have some effect on the spatiotemporal stability. Critical condition is presented for the transition between absolutely and convectively unstable systems, and parameter regimes are presented to quantify the effect of the above-mentioned parameters.

Irreversibility analysis in micropolar fluid film along an incline porous substrate with slip effects

The current article examined the semi-analytical solution for the rate of entropy generation in a steady, gravity-driven thin film flow of a micropolar fluid descending over a heated inclined substrate with slip constraints. A comprehensive semi-analytical approach (differential transformation method (DTM)) was instrumental in obtaining the solution of the nonlinear governing model. The characteristics of velocity, angular velocity, heat balance, entropy generation, skin-friction, wall couple stress, Nusselt number and Bejan number have been deliberated under the influence of involved flow controlling physical parameters. The outcomes of the present study shows that an incline in values of Microrotation parameter declines the velocity gradient and inclines thermal, dimensionless Bejan number and entropy generation profiles.

Stability of liquid film flow laden with the soluble surfactant sodium dodecyl sulphate: predictions versus experimental data

Gravity-driven liquid film flows laden with a soluble surfactant are considered, and aqueous solutions of sodium dodecyl sulphate (SDS) are taken as a case-study. Literature measurements of the critical Reynolds number for the onset of instability are set in perspective with predictions of linear stability theory. The theory is based on a Frumkin model of adsorption equilibrium, modified by the inclusion of finite compressibility of the adsorbed monolayer. Quantitative comparison between data and theory is first attempted in the limit of infinite wavelength. Though wave characteristics are satisfactorily predicted, the theoretical critical Reynolds number is an order of magnitude below measurements. This discrepancy is understood in terms of the large difference between momentum and mass diffusivities and indicates that the assumption of infinite wavelength is far more restrictive for the mass transfer than for the flow problem. Finite-wavelength effects are taken into account by numerical solution of the Orr–Sommerfeld eigenvalue problem, leading to predictions of maximum stabilization in good agreement with the measurements. Introduction of realistic values of monolayer compressibility improves further the agreement at high surfactant loadings. Finally, a strong stabilizing effect of salinity is confirmed.

Mixed convection in gravity-driven thin nano-liquid film flow with homogeneous–heterogeneous reactions

A modified nanofluid model for homogeneous–heterogeneous reactions in a gravity-driven liquid film is proposed based on the assumptions of the homogeneous reaction governed by isothermal cubic autocatalytic kinetics and the heterogeneous reaction given by first-order kinetics. The Buongiorno model is introduced to describe behaviors of nanofluids with the Boussinesq approximation being used for simplification of the buoyancy term. Multiple solutions are captured, whose stabilities are checked and discussed based on the theory of Sturm–Liouville. The influence of various physical parameters on important physical quantities is presented for different cases, including buoyancy assisting, no buoyancy, and buoyancy opposing. Different from previous studies on chemical reactions in which reactants are presumed in nanometer scale, we assume that the chemical species are of regular size, which react with each other in a nanofluid. This configuration makes our model physically more realistic than previous ones.

Stability of the channel flow—new phenomena in an old problem

In this article, we show an unexpected effect of side walls on the stability of gravity-driven viscous film flows over flat substrates in an open channel. Until now, it was assumed that side walls have a stabilizing effect on the flow, but the qualitative structure of the neutral curve remains the same. We show a fragmentation of the stability chart that was only known from undulated substrates. The cause of the fragmentation is a distinct damping of waves with a certain frequency that is independent of the underlying instability of the two-dimensional flow. This paper provides a detailed parameter study that identifies the channel width as the decisive parameter for the damping frequency. In addition, the damping is not an effect of flat side walls exclusively, since corrugated side walls have qualitatively the same impact on the stability chart. Furthermore, we demonstrate that the curvature of the emerging wave’s crest line has a non-monotonous dependency on the frequency and shows a distinct maximum. The frequency of this maximum shows the same behavior as the damping frequency when the channel width is varied. We therefore assume a relation between the wave’s curvature and its growth rate, both massively affected by the side walls.