Steven J. Weinstein

The Effect of Applied Pressure on the Shape of a Two-Dimensional Liquid Curtain Falling under the Influence of Gravity

The shape of a two-dimensional liquid curtain issuing from a slot and falling under the influence of gravity is predicted theoretically and verified experimentally for cases where a pressure is applied to the curtain. A set of approximate equations is derived which governs the location of the curtain for a liquid having surface tension σ, density ρ, volumetric flow per unit width Q , and local free-fall velocity V . These equations possess a singularity at the point where the local Weber number, We = ρ QV /2σ, is equal to 1. Despite the fact that previous work on the stability of two-dimensional curtains shows that curtains having locations where We It is found that the singularity can be eliminated from the governing equations if the curtain assumes a definite direction as it leaves the slot. By contrast, if the curtain leaves the slot such that We > 1, there is no such restriction, and experimentally it is found that the curtain leaves parallel to the slot walls. The theoretical predictions of the curtain shapes are in agreement with those measured experimentally for all Weber numbers investigated.

Time-dependent equations governing the shape of a two-dimensional liquid curtain, Part 1: Theory

Approximate equations have been derived that govern the time-dependent response of a two-dimensional liquid curtain falling under the influence of gravity and subjected to ambient pressure disturbances. Starting with the assumptions of potential flow and constant surface tension, and using the approximation that the curtain is long and thin, a steady-state base flow is first determined. In agreement with previous literature results, the analysis reveals that the curtain flow is essentially in free fall, where the velocity profile is only slightly curved across the curtain thickness. Then, by assuming that the disturbances to the curtain are small, the time-dependent equations are linearized about the approximated base flow. The approximate nature of the base flow necessitates a careful ordering of terms to assure that the linearization is valid. Two equations governing the curtain shape are derived: the first governs the deflection of the curtain centerline, and the second governs the thickness variations. Previous literature results regarding wave propagation and steady curtain deflections can be predicted via the derived equations. It is also found that to lowest order, pressure disturbances induce a deflection of the curtain centerline while preserving the local thickness associated with the undisturbed curtain.

Long‐wavelength instabilities in three‐layer flow down an incline

This paper reports a long‐wavelength instability which has not previously been identified for three‐layer free‐surface flow down an inclined plane. The instability is identified in the zeroth‐order asymptotic solution in wave number, indicating that neither inertial nor finite wavelength effects are necessary to induce instabilities in three‐layer systems. Various neutral stability boundaries are presented which demonstrate the effect of viscosity stratification, density stratification, and layer thicknesses. It is found that destabilization occurs in cases where the middle‐layer viscosity (for equal densities in each layer) or density (for equal viscosities in each layer) is smaller than those of the adjacent layers. The regions of instability afford a smooth transition between neutrally stable regions of the parameter space where the in‐phase and out‐of‐phase characteristics of the interfaces differ.

Flow Field Analysis in Expanding Healthy and Emphysematous Alveolar Models Using Particle Image Velocimetry

Particulates that deposit in the acinus region of the lung have the potential to migrate through the alveolar wall and into the blood stream. However, the fluid mechanics governing particle transport to the alveolar wall are not well understood. Many physiological conditions are suspected to influence particle deposition including morphometry of the acinus, expansion and contraction of the alveolar walls, lung heterogeneities, and breathing patterns. Some studies suggest that the recirculation zones trap aerosol particles and enhance particle deposition by increasing their residence time in the region. However, particle trapping could also hinder aerosol particle deposition by moving the aerosol particle further from the wall. Studies that suggest such flow behavior have not been completed on realistic, nonsymmetric, three-dimensional, expanding alveolated geometry using realistic breathing curves. Furthermore, little attention has been paid to emphysemic geometries and how pathophysiological alterations effect deposition. In this study, fluid flow was examined in three-dimensional, expanding, healthy, and emphysemic alveolar sac model geometries using particle image velocimetry under realistic breathing conditions. Penetration depth of the tidal air was determined from the experimental fluid pathlines. Aerosol particle deposition was estimated by simple superposition of Brownian diffusion and sedimentation on the convected particle displacement for particles diameters of 100-750 nm. This study (1) confirmed that recirculation does not exist in the most distal alveolar regions of the lung under normal breathing conditions, (2) concluded that air entering the alveolar sac is convected closer to the alveolar wall in healthy compared with emphysematous lungs, and (3) demonstrated that particle deposition is smaller in emphysematous compared with healthy lungs.

TIME-DEPENDENT EQUATIONS GOVERNING THE SHAPE OF A TWO-DIMENSIONAL LIQUID CURTAIN, PART 2 : EXPERIMENT

In Part I of this paper, two governing equations have been derived that describe the shape of a falling liquid sheet (a curtain) subjected to ambient pressure disturbances. These equations are termed varicose and sinuous. The varicose equation governs thickness variations in the curtain, for which the two air–liquid interfaces move exactly out of phase. The sinuous equation governs the deflection of the curtain centreline, i.e., the two air–liquid interfaces move in phase such that the local thickness of the liquid is preserved. To the order of the approximations used, the theory presented in Part 1 indicates that pressure disturbances invoke a sinuous curtain deflection with no varicose contribution. In Part 2 of this paper, the sinuous equation is verified by means of a localised pressure disturbance induced by an electrostatic field. After initiation, the propagation of this disturbance is followed and the shape of the air liquid interface is measured using a laser reflection technique. Both the genera...

Large growth rate instabilities in three-layer flow down an incline in the limit of zero Reynolds number

In this paper, we examine the effect of viscosity stratification on wave propagation in three-layer flow down an inclined plane at vanishingly small Reynolds number and at finite wavelengths, for cases of negligible liquid–liquid interfacial tensions. We have found that the long-wavelength interface mode inertialess instability of Weinstein and Kurz [Phys. Fluids A 3, 2680 (1991)] persists into the finite wavelength domain in the form of nearly complex conjugate wave speed pairs; in certain limits, the interface modes are precisely complex conjugates. As in the case of Weinstein and Kurz, the physical configuration necessary to achieve inertialess instability is a low viscosity and thin internal layer with respect to the other layers in the film. The largest growth rate of the inertialess instability is found at finite wavelengths on the order of the total thickness of the film, and is orders of magnitude larger than the maximum growth rates identified by Lowenhurz and Lawrence [Phys. Fluids A 1, 1686 (1989)] for two-layer flows. We have also found an additional configuration exhibiting extremely large growth rates, also characterized by nearly complex conjugate behavior, that is not accessible via a long or short wavelength asymptotic limit; these three-layer structures have thin, high viscosity internal layers. The characteristic wavelengths associated with the largest growth rates are on the order of ten times smaller than those for the low viscosity internal layer cases. The influence of the deformable free surface on the growth rates of these interface modes is studied and found to be significant.

A theoretical study of two-phase flow through a narrow gap with a moving contact line : viscous fingering in a Hele-Shaw cell

The problem of viscous fingering in a Hele-Shaw cell with moving contact lines is considered. In contrast to the usual situation where the displaced fluid coats the solid surface in the form of thin films, here, both the displacing and the displaced fluids make direct contact with the solid. The principal differences between these two situations are in the ranges of attainable values of the gapwise component of the interfacial curvature (the component due to the bending of the fluid interface across the small gap of the Hele-Shaw cell), and in the introduction of two additional parameters for the case with moving contact lines. These parameters are the receding contact angle, and the sensivity of the dynamic angle to the speed of the contact line. Our objective is the prediction of the shape and widths of the fingers in the limit of small capillary number, U μ/σ. Here, U denotes the finger speed, μ denotes the dynamic viscosity of the more viscous displaced fluid, and σ denotes the surface tension of the fluid interface. As might be expected, there are similarities and differences between the two problems. Despite the fact that different equations arise, we find that they can be analysed using the techniques introduced by McLean & Saffman and Vanden-Broeck for the thin-film case. The nature of the multiplicity of solutions also appears to be similar for the two problems. Our results indicate that when contact lines are present, the finger shapes are sensitive to the value of the contact angle only in the vicinity of its nose, reminiscent of experiments where bubbles or wires are placed at the nose of viscous fingers when thin films are present. On the other hand, in the present problem at least two distinct velocity scales emerge with well-defined asymptotic limits, each of these two cases being distinguished by the relative importance played by the two components of the curvature of the fluid interface. It is found that the widths of fingers can be significantly smaller than half the width of the cell.

Low-Reynolds-number instabilities in three-layer flow down an inclined wall

The finite wavelength instability of viscosity-stratified three-layer flow down an inclined wall is examined for small but finite Reynolds numbers. It has previously been demonstrated using linear theory that three-layer zero-Reynolds-number instabilities can have growth rates that are orders of magnitude larger than those that arise in twolayer structures. Although the layer configurations yielding large growth instabilities have been well characterized, the physical origin of the three-layer inertialess instability remains unclear. Using analytic, numerical and experimental techniques, we investigate the origin and evolution of these instabilities. Results from an energy equation derived from linear theory reveal that interfacial shear and Reynolds stresses contribute to the energy growth of the instability at finite Reynolds numbers, and that this remains true in the limit of zero Reynolds number. This is thus a rare example that demonstrates how the Reynolds stress can play an important role in flow instability, even when the Reynolds number is vanishingly small. Numerical solutions of the Navier–Stokes equations are used to simulate the nonlinear evolution of the interfacial deformation, and for small amplitudes the predicted wave shapes are in excellent agreement with those obtained from linear theory. Further comparisons between simulated interfacial deformations and linear theory reveal that the linear evolution equations are surprisingly accurate even when the interfaces are highly deformed and nonlinear effects are important. Experimental results obtained using aqueous gelatin systems exhibit large wave growth and are in agreement with both the theoretical predictions of small-amplitude behaviour and the nonlinear simulations of the large-amplitude behaviour. Quantitative agreement is confounded owing to water diffusion driven by differences in gelatin concentration between the layers in experiments. However, the qualitative agreement is sufficient to confirm that the correct mechanism for the experimental instability has been determined.

Dip coating on a planar non-vertical substrate in the limit of negligible surface tension

Abstract The problem of dip coating of a planar non-vertical substrate is considered for negligible surface tension effects. As in the problem of vertical withdrawal (Cerro and Scriven, J. Fluid Mech. 208 (1980) 40), a singularity arises in the approximate steady-state equation governing the shape of the air–liquid interface; the ultimate thickness of the entrained film on the substrate follows directly from the elimination of this singularity. The removable singularity corresponds to a critical point that divides regions where waves propagate upstream and downstream (subcritical flow) and waves only propagate downstream (supercritical flow). These waves move at the speed of characteristics of the linearized time-dependent hyperbolic equation governing the film flow. The range of flows and film thicknesses that may be achieved by dip coating in an inclined substrate configuration are examined; these film thicknesses are self-metered, in that it they are determined by the substrate speed, substrate angle, and fluid properties. We also examine the parameter space to find flow conditions where the final uniform thickness of the film is not determined by a critical point. In these cases, the film thickness is set by some other means, such as in premetered die coating where both the flow rate and substrate speed are imposed. Examination of the asymptotic behavior of the film downstream shows that some of the possible uniform film thicknesses are not attainable. While there may be two possible uniform film thickness solutions for an imposed volumetric flow rate, there can be at most one supercritical and one subcritical flow solution.

Thin-Film Flow at Moderate Reynolds Number

Viscous, laminar, gravitationally-driven flow of a thin film over a round-crested weir is analyzed for moderate Reynolds numbers. A previous analysis of this flow utilized a momentum integral approach with a semiparabolic velocity profile to obtain an equation for the film thickness (Ruschak, K. J., and Weinstein, S. J.). In this work, a viscous boundary layer is introduced in the manner of Haugen. As in the previous analysis of Ruschak and Weinstein, the approximate equations have a critical point that provides an internal boundary condition for a bounded solution