Developing Film Flow on an Inclined Plane With a Critical Point

2001 ◽  
Vol 123 (3) ◽  
pp. 698-702 ◽  
Author(s):  
Kenneth J. Ruschak ◽  
Steven J. Weinstein ◽  
Kam Ng
Keyword(s):  
Velocity Profile ◽  
Critical Point ◽  
Film Flow ◽  
Boundary Layer Equation ◽  
Critical Conditions ◽  
Integral Approach ◽  
Inclined Plane ◽  
Navier Stokes Equation ◽  
Moderate Reynolds Number ◽  
Momentum Integral

Viscous, laminar, gravitationally-driven flow of a thin film on an inclined plane is analyzed for moderate Reynolds number under critical conditions. A previous analysis of film flow utilized a momentum integral approach with a semiparabolic velocity profile to obtain an ordinary differential equation for the film thickness for flow over a round-crested weir, and the singularity associated with the critical point for a subcritical-to-supercritical transition was removable. For developing flow on a plane with a supercritical-to-subcritical transition, however, the same approach leads to a nonremovable singularity. To eliminate the singularity, the film equations are modified for a velocity profile of changing shape. The resulting predictions compare favorably with those from the two-dimensional boundary-layer equation obtained by finite differences and with those from the Navier-Stokes equation obtained by finite elements.

Laminar, Gravitationally Driven Flow of a Thin Film on a Curved Wall

2003 ◽  
Vol 125 (1) ◽  
pp. 10-17 ◽  
Author(s):  
Kenneth J. Ruschak ◽  
Steven J. Weinstein
Keyword(s):  
Thin Film ◽  
Differential Equations ◽  
Velocity Profile ◽  
Film Thickness ◽  
Standing Wave ◽  
Flow Direction ◽  
Critical Conditions ◽  
Navier Stokes ◽  
Navier Stokes Equation ◽  
Moderate Reynolds Number

Gravitationally driven flow of a thin film down an arbitrarily curved wall is analyzed for moderate Reynolds number by generalizing equations previously developed for flow on a planar wall. In the analysis, the ratio of the characteristic film thickness to the characteristic dimension of the wall is presumed small, and terms estimated to be first order in this parameter are retained. Partial differential equations are reduced to ordinary differential equations by the method of von Ka´rma´n and Pohlhausen; namely, an expression for the velocity profile is assumed, and the equation for conservation of linear momentum is averaged across the film. The assumed velocity profile changes shape in the flow direction because a self-similar profile, one of fixed shape but variable magnitude, leads to an equation that typically fails under critical conditions. The resulting equations for film thickness routinely accommodate subcritical-to-supercritical transitions and supercritical-to-subcritical transitions as classified by the underlying wave propagation. The more severe supercritical-to-subcritical transition is manifested by a standing wave where the film noticeably thickens; this standing wave is a simple analogue of a hydraulic jump. Predictions of the film-thickness profile and variations in the velocity profile compare favorably with those from the Navier-Stokes equation obtained by the finite element method.

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]

On Laminar Thin-Film Flow Along a Vertical Wall

1984 ◽  
Vol 51 (3) ◽  
pp. 691-692 ◽  
Author(s):  
T. R. Roy
Keyword(s):  
Thin Film ◽  
Boundary Layer ◽  
Velocity Profile ◽  
Film Flow ◽  
Vertical Wall ◽  
Thin Film Flow ◽  
Runge Kutta Method ◽  
Parabolic Velocity Profile ◽  
Cubic Polynomial ◽  
Momentum Integral

An accelerating laminar thin-film flow along a vertical wall is investigated in this paper. Using a cubic polynomial for the velocity profile inside the boundary layer the momentum integral equation is solved by a Runge-Kutta method to determine the boundary layer thickness. The corresponding film-thickness is then calculated for the entrance region. These results are compared with the existing results obtained by using a parabolic velocity profile.

Instability of a liquid film flow over a vibrating inclined plane

1995 ◽  
Vol 294 ◽  
pp. 391-407 ◽  
Author(s):  
David R. Woods ◽  
S. P. Lin
Keyword(s):  
Gravity Waves ◽  
Film Flow ◽  
Shear Waves ◽  
Faraday Waves ◽  
Critical Amplitude ◽  
Inclined Plane ◽  
Flow Parameters ◽  
Chebychev Polynomials ◽  
Liquid Film Flow ◽  
Faraday Wave

The problem of the onset of instability in a liquid layer flowing down a vibrating inclined plane is formulated. For the solution of the problem, the Fourier components of the disturbance are expanded in Chebychev polynomials with time-dependent coefficients. The reduced system of ordinary differential equations is analysed with the aid of Floquet theory. The interaction of the long gravity waves, the relatively short shear waves and the parametrically resonated Faraday waves occurring in the film flow is studied. Numerical results show that the long gravity waves can be significantly suppressed, but cannot be completely eliminated by use of the externally imposed oscillation on the incline. At small angles of inclination, the short shear waves may be exploited to enhance the Faraday waves. For a given set of relevant flow parameters, there exists a critical amplitude of the plane vibration below which the Faraday wave cannot be generated. At a given amplitude above this critical one, there also exists a cutoff wavenumber above which the Faraday wave cannot be excited. In general the critical amplitude increases, but the cutoff wavenumber decreases, with increasing viscosity. The cutoff wavenumber also decreases with increasing surface tension. The application of the theory to a novel method of film atomization is discussed.

scholarly journals On the Kármán momentum-integral approach and the Pohlhausen paradox

2020 ◽  
Vol 32 (12) ◽  
pp. 123605
Author(s):  
Joseph Majdalani ◽  
Li-Jun Xuan
Keyword(s):  
Integral Approach ◽  
Momentum Integral

scholarly journals Criticality of plasma membrane lipids reflects activation state of macrophage cells

2020 ◽  
Vol 17 (163) ◽  
pp. 20190803 ◽  
Author(s):  
Eugenia Cammarota ◽  
Chiara Soriani ◽  
Raphaelle Taub ◽  
Fiona Morgan ◽  
Jiro Sakai ◽  
...  
Keyword(s):  
Plasma Membrane ◽  
Critical Point ◽  
Membrane Lipids ◽  
Membrane Receptors ◽  
Interleukin 4 ◽  
Critical Conditions ◽  
Membrane Composition ◽  
Macrophage Cells ◽  
Lipid Species ◽  
Anti Inflammatory

Signalling is of particular importance in immune cells, and upstream in the signalling pathway many membrane receptors are functional only as complexes, co-locating with particular lipid species. Work over the last 15 years has shown that plasma membrane lipid composition is close to a critical point of phase separation, with evidence that cells adapt their composition in ways that alter the proximity to this thermodynamic point. Macrophage cells are a key component of the innate immune system, are responsive to infections and regulate the local state of inflammation. We investigate changes in the plasma membrane’s proximity to the critical point as a response to stimulation by various pro- and anti-inflammatory agents. Pro-inflammatory (interferon γ , Kdo 2-Lipid A, lipopolysaccharide) perturbations induce an increase in the transition temperature of giant plasma membrane vesicles; anti-inflammatory interleukin 4 has the opposite effect. These changes recapitulate complex plasma membrane composition changes, and are consistent with lipid criticality playing a master regulatory role: being closer to critical conditions increases membrane protein activity.

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