Mechanism of the long-wave inertialess instability of a two-layer film flow

2008 ◽  
Vol 608 ◽  
pp. 379-391 ◽  
Author(s):  
PENG GAO ◽  
XI-YUN LU
Keyword(s):  
Free Surface ◽  
Film Flow ◽  
Underlying Mechanism ◽  
Continuity Condition ◽  
Layer Film ◽  
Long Wave ◽  
Pressure Driven Flow ◽  
Intuitive Interpretation ◽  
Additional Phase Shift ◽  
Additional Phase

This paper provides an intuitive interpretation of the long-wave inertialess instability of a two-layer film flow. The underlying mechanism is elucidated by inspecting the longitudinal perturbation velocity associated with the surface and interfacial deflections. The velocity is expressed by the composition of three parts, related to the shear stress at the free surface, the continuity condition at the interface, and the pressure disturbance induced by gravity. The effect of each velocity component on the evolutions of the surface and the interface is examined in detail. Specifically, the growth of the free surface is caused by the continuity-induced first-order velocity disturbance associated with an additional phase shift between the surface and interfacial waves, while the growth of the interface is due to the pressure-driven flow. The proposed mechanism gives an alternatively reliable prediction of the wave velocity and growth rate.

Effect of surfactants on the long-wave stability of two-layer oscillatory film flow

2021 ◽  
Vol 928 ◽  
Author(s):  
Cheng-Cheng Wang ◽  
Haibo Huang ◽  
Peng Gao ◽  
Xi-Yun Lu
Keyword(s):  
Film Flow ◽  
Viscosity Ratio ◽  
Density Ratio ◽  
Thickness Ratio ◽  
High Frequency Oscillation ◽  
Key Factors ◽  
Layer Film ◽  
Wave Disturbances ◽  
Long Wave ◽  
The Stability

The stability of the two-layer film flow driven by an oscillatory plate under long-wave disturbances is studied. The influence of key factors, such as thickness ratio ( $n$ ), viscosity ratio ( $m$ ), density ratio ( $r$ ), oscillatory frequency ( $\beta$ ) and insoluble surfactants on the stability behaviours is studied systematically. Four special Floquet patterns are identified, and the corresponding growth rates are obtained by solving the eigenvalue problem of the fourth-order matrix. A small viscosity ratio ( $m\le 1$ ) may stabilize the flow but it depends on the thickness ratio. If the viscosity ratio is not very small ( $m>0.1$ ), in the $(\beta ,n)$ -plane, stable and unstable curved stripes appear alternately. In other words, under the circumstances, if the two-layer film flow is unstable, slightly adjusting the thickness of the upper film may make it stable. In particular, if the upper film is thin enough, even under high-frequency oscillation, the flow is always stable. The influence of density ratio is similar, i.e. there are curved stable and unstable stripes in the $(\beta ,r)$ -planes. Surface surfactants generally stabilize the flow of the two-layer oscillatory membrane, while interfacial surfactants may stabilize or destabilize the flow but the effect is mild. It is also found that gravity can generally stabilize the flow because it narrows the bandwidth of unstable frequencies.

Object Wave Determination by Single-Sideband Methods

1973 ◽  
Vol 31 ◽  
pp. 266-267
Author(s):  
Kenneth H. Downing ◽  
Benjamin M. Siegel
Keyword(s):  
Phase Shift ◽  
Carbon Films ◽  
Object Wave ◽  
Electron Wave ◽  
Phase Object ◽  
Heavy Atoms ◽  
Single Sideband ◽  
Thin Carbon ◽  
Additional Phase Shift ◽  
Additional Phase

Under the “weak phase object” approximation, the component of the electron wave scattered by an object is phase shifted by π/2 with respect to the unscattered component. This phase shift has been confirmed for thin carbon films by many experiments dealing with image contrast and the contrast transfer theory. There is also an additional phase shift which is a function of the atomic number of the scattering atom. This shift is negligible for light atoms such as carbon, but becomes significant for heavy atoms as used for stains for biological specimens. The light elements are imaged as phase objects, while those atoms scattering with a larger phase shift may be imaged as amplitude objects. There is a great deal of interest in determining the complete object wave, i.e., both the phase and amplitude components of the electron wave leaving the object.

Characteristics of an Electrostatic Phase Plate

1972 ◽  
Vol 30 ◽  
pp. 618-619
Author(s):  
William Krakow ◽  
Benjamin Siegel
Keyword(s):  
Phase Contrast ◽  
Objective Lens ◽  
Phase Plate ◽  
Net Charge ◽  
Phase Plates ◽  
Beam Currents ◽  
Contrast Image ◽  
Bright Phase ◽  
Additional Phase Shift ◽  
Additional Phase

Unwin has used a metallized non-conducting thread in the back focal plane of the objective lens that stops out a portion of the unscattered beam, takes on a localized positive charge and thus produces an additional phase shift to give a different transfer function of the lens. Under the particular conditions Unwin used, the phase contrast image was shifted to bright phase contrast for optimum focus.We have investigated the characteristics of this type of electrostatic phase plate, both analytically and experimentally, as functions of the magnitude of charge and defocus. Phase plates have been constructed by using Wollaston wire to mount 0.25μ diameter platinum wires across apertures ranging from 50 to 200μ diameter and vapor depositing SiO and gold on the mounted wires to give them the desired charging characteristics. The net charge was varied by adjusting only the bias on the Wehnelt shield of the gun, and hence the beam currents and effective size of the source.

STABILITY ANALYSIS OF THE THIN POWER LAW LIQUID FILM FLOWING DOWN ON A VERTICAL CYLINDER

2006 ◽  
Vol 30 (1) ◽  
pp. 81-96 ◽  
Author(s):  
Po-Jen Cheng ◽  
Kuo-Chi Liu
Keyword(s):  
Liquid Film ◽  
Power Law ◽  
Film Flow ◽  
Vertical Cylinder ◽  
Flow Index ◽  
Free Film ◽  
Long Wave ◽  
Lateral Curvature ◽  
The Stability ◽  
Film Interface

The paper investigates the stability theory of a thin power law liquid film flowing down along the outside surface of a vertical cylinder. The long-wave perturbation method is employed to solve for generalized linear kinematic equations with free film interface. The normal mode approach is used to compute the stability solution for the film flow. The degree of instability in the film flow is further intensified by the lateral curvature of cylinder. This is somewhat different from that of the planar flow. The analysis results also indicate that by increasing the flow index and increasing the radius of the cylinder the film flow can become relatively more stable as traveling down along the vertical cylinder.

scholarly journals Effect of imposed shear on the dynamics of a contaminated two-layer film flow down a slippery incline

2020 ◽  
Vol 32 (10) ◽  
pp. 102113
Author(s):  
Muhammad Sani ◽  
Siluvai Antony Selvan ◽  
Sukhendu Ghosh ◽  
Harekrushna Behera
Keyword(s):  
Film Flow ◽  
Layer Film

Stability of a Shear-Thinning Film on a Porous Substrate

2010 ◽  
Author(s):  
R. Usha ◽  
S. Millet ◽  
H. BenHadid ◽  
F. Rousset
Keyword(s):  
Thin Film ◽  
Free Surface ◽  
Shear Rate ◽  
Film Flow ◽  
Shear Thinning ◽  
Free Surface Flows ◽  
Significant Feature ◽  
Porous Substrate ◽  
Carreau Model ◽  
Porous Substrates

A significant feature of gravity-driven film flows of Newtonian and rheologically complex fluids down an inclined/vertical substrate is the instability of the free surface which manifests as surface waves having wavelengths much larger than the film thickness. There are a number of applications which can be modeled as thin film flow systems on porous substrates. Pascal [1] investigated the stability of a falling power-law film on an inclined porous substrate. This model for the fluid predicts a viscosity that goes to infinity as the shear rate approaches zero. There is a need to employ a more appropriate model to examine the effects of non-Newtonian rheology on the dynamics and stability of thin film free surface flows down inclined or vertical rigid/porous substrates. The four-parameter Carreau model predicts a viscosity that remains finite as the shear rate approaches zero and is given by η−η∞η0−η∞=[1+(δγ)˙2]n−12.(1) Weinstein [2] and Rousset et al. [3] have considered the Carreau model and have examined the temporal stability of a film flow down an impermeable rigid inclined substrate. The authors show that a shear-thinning Carreau fluid film on a rigid impermeable substrate is more unstable than a Newtonian film. This calls for an analysis that includes both the effects of Carreau non-Newtonian rheology and bottom permeability and the present study reports such an investigation of a Carreau non-Newtonian film on a porous inclined substrate.

Thin-Film Flow at Moderate Reynolds Number

2000 ◽  
Vol 122 (4) ◽  
pp. 774-778 ◽  
Author(s):  
Kenneth J. Ruschak ◽  
Steven J. Weinstein
Keyword(s):  
Thin Film ◽  
Boundary Layer ◽  
Free Surface ◽  
Film Flow ◽  
Previous Analysis ◽  
Vertical Wall ◽  
Integral Approach ◽  
Internal Boundary ◽  
Thin Film Flow ◽  
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., 1999, “Viscous Thin-Film Flow Over a Round-Crested Weir,” ASME J. Fluids Eng., 121, pp. 673–677). In this work, a viscous boundary layer is introduced in the manner of Haugen (Haugen, R., 1968, “Laminar Flow Around a Vertical Wall,” ASME J. Appl. Mech. 35, pp. 631–633). 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. The complication of a boundary layer is found to have little effect on the thickness profile while introducing a weak singularity at its beginning. The thickness of the boundary layer grows rapidly, and there is little cumulative effect of the increased wall friction. Regardless of whether a boundary layer is incorporated, the approximate free-surface profiles are close to profiles from finite-element solutions of the Navier-Stokes equation. Similar results are obtained for the related problem of developing flow on a vertical wall (Cerro, R. L., and Whitaker, S., 1971, “Entrance Region Flows With a Free Surface: the Falling Liquid Film,” Chem. Eng. Sci., 26, pp. 785–798). Less accurate results are obtained for decelerating flow on a horizontal wall (Watson, E. J., 1964, “The Radial Spread of a Liquid Jet Over a Horizontal Plane,” J. Fluid Mech. 20, pp. 481–499) where the flow is not gravitationally driven. [S0098-2202(00)01904-0]

Mechanics of a Free-Surface Liquid Film Flow

1987 ◽  
Vol 54 (4) ◽  
pp. 951-954 ◽  
Author(s):  
Cyrus K. Aidun
Keyword(s):  
Differential Equation ◽  
Free Surface ◽  
Film Flow ◽  
Navier Stokes ◽  
Experimental Measurements ◽  
Navier Stokes Equation ◽  
Higher Order Terms ◽  
Liquid Film Flow ◽  
Surface Liquid ◽  
Similar Equation

The mechanics of a free surface viscous liquid curtain flowing steadily between two vertical guide wires under the influence of gravity is investigated. The Navier-Stokes equation is integrated over the film thickness and an integro-differential equation is derived for the average film velocity. An approximate nonlinear differential equation, attributed to G. I. Taylor, is obtained by neglecting the higher order terms. An analytical solution is obtained for a similar equation which neglects the surface tension effects and the results are compared with the experimental measurements of Brown (1961).

Sign in / Sign up

Export Citation Format

Share Document