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  • 1
    Electronic Resource
    Electronic Resource
    Large growth rate instabilities in three-layer flow down an incline in the limit of zero Reynolds number (1999)
    [S.l.] : American Institute of Physics (AIP)
    Physics of Fluids 11 (1999), S. 3270-3282 
    Details
    ISSN: 1089-7666
    Source: AIP Digital Archive
    Topics: Physics
    Notes: 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. © 1999 American Institute of Physics.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1063/1.870187
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  • 2
    Electronic Resource
    Electronic Resource
    Time-dependent equations governing the shape of a three-dimensional liquid curtain (1998)
    [S.l.] : American Institute of Physics (AIP)
    Physics of Fluids 10 (1998), S. 1815-1818 
    Details
    ISSN: 1089-7666
    Source: AIP Digital Archive
    Topics: Physics
    Notes: In a previous paper, Weinstein et al. derive time-dependent equations that govern the response of a planar liquid curtain falling under the influence of gravity and subjected to ambient pressure disturbances. In the previous study, disturbances to the curtain are assumed to be two-dimensional, and thus, the curtain response is independent of widthwise location in the curtain. In this paper, we generalize the previous equations to incorporate the widthwise dimension. The validity of these equations is demonstrated by their ability to predict standing wave shapes in agreement with those studied by Lin and Roberts. © 1998 American Institute of Physics.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1063/1.869730
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  • 3
    Electronic Resource
    Electronic Resource
    Long-wavelength instabilities in three-layer flow down an incline (1991)
    New York, NY : American Institute of Physics (AIP)
    Physics of Fluids 3 (1991), S. 2680-2687 
    Details
    ISSN: 1089-7666
    Source: AIP Digital Archive
    Topics: Physics
    Notes: 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.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1063/1.858158
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  • 4
    Electronic Resource
    Electronic Resource
    Time-dependent equations governing the shape of a two-dimensional liquid curtain, Part 1: Theory (1997)
    [S.l.] : American Institute of Physics (AIP)
    Physics of Fluids 9 (1997), S. 3625-3636 
    Details
    ISSN: 1089-7666
    Source: AIP Digital Archive
    Topics: Physics
    Notes: 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. © 1997 American Institute of Physics.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1063/1.869500
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  • 5
    Electronic Resource
    Electronic Resource
    COATING FLOWS (2004)
    Palo Alto, Calif. : Annual Reviews
    Annual Review of Fluid Mechanics 36 (2004), S. 29-53 
    Details
    ISSN: 0066-4189
    Source: Annual Reviews Electronic Back Volume Collection 1932-2001ff
    Topics: Mechanical Engineering, Materials Science, Production Engineering, Mining and Metallurgy, Traffic Engineering, Precision Mechanics , Physics
    Notes: Coating is the process of applying thin liquid layers to a substrate, often a moving web. Complex coating processes can be approached through examination of their fluid mechanical components. The flow elements reviewed in this article include the boundary layer along a moving wall, the dynamic wetting line, withdrawal from a pool, flow metered by a narrow channel, die flow, flow on an incline, the freely falling liquid curtain, premetered coating with a small gap, and flow after coating. Although some flow elements are well studied and understood, others require additional investigation. Genuinely predictive modeling of complex coating processes is not yet possible and coating practice remains largely empirical. Nonetheless, coating science is sufficiently advanced that physical insights and mathematical models greatly benefit design and practice.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1146/annurev.fluid.36.050802.122049
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  • 6
    Electronic Resource
    Electronic Resource
    Wave propagation in the flow of shear-thinning fluids down an incline (1990)
    Hoboken, NJ : Wiley-Blackwell
    AIChE Journal 36 (1990), S. 1873-1889 
    Details
    ISSN: 0001-1541
    Keywords: Chemistry ; Chemical Engineering
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Chemistry and Pharmacology , Process Engineering, Biotechnology, Nutrition Technology
    Notes: The effect of shear-thinning rheology on the spatial growth of waves in multilayered flow down an inclined plane is modeled, utilizing the Carreau viscosity constitutive equation. It is shown that waves associated with the free surface propagate as if they were in a Newtonian system, where the viscosity is some average of the varying viscosities in the shear-thinning layer. This averaging is due to the global effects of shear thinning, such as changes in velocity profile and film thicknesses. In contrast, waves that are associated with the interfaces between adjacent fluid layers are largely affected by the local interfacial viscosities; wave propagation is not governed by some average Newtonian viscosity across the layer. It is found that shear-thinning rheology can either increase or decrease the growth of waves associated with a fluid-fluid interface, compared with a purely Newtonian case.
    Additional Material: 12 Ill.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1002/aic.690361211
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  • 7
    Electronic Resource
    Electronic Resource
    The propagation of surface waves in flow down an oscillating inclined plane (1993)
    Hoboken, NJ : Wiley-Blackwell
    AIChE Journal 39 (1993), S. 1113-1123 
    Details
    ISSN: 0001-1541
    Keywords: Chemistry ; Chemical Engineering
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Chemistry and Pharmacology , Process Engineering, Biotechnology, Nutrition Technology
    Notes: The propagation of surface waves is investigated theoretically and experimentally for the case of a single layer of viscous liquid flowing down an inclined plane, where the plane is oscillating in the flow direction. This work focuses on the linearized wavemaker problem, where the oscillations create waves which are small perturbations from the undisturbed flow. Downstream from the entrance region to the incline where the fluid is introduced, the undisturbed interface is parallel to the incline surface, and theory predicts that oscillations do not interact with waves that travel along the free surface. These waves grow as if there were no oscillation at all, and their propagation is governed by a dispersion relation between frequency, wavelength, and wave growth for single layer flow down a nonoscillating inclined plane. The entrance region to the incline is therefore responsible for exciting the various wave frequencies which are observed down the incline, as well as the initial amplitude of these waves. Experiments performed verify that waves propagate as predicted. Theory indicates that these conclusions are valid when the oscillations are perpendicular to the incline, as well as for the case of multiple stacked layers.
    Additional Material: 5 Ill.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1002/aic.690390704
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  • 8
    Electronic Resource
    Electronic Resource
    Asymptotic analysis of die flow for shear-thinning fluids (1996)
    Hoboken, NJ : Wiley-Blackwell
    AIChE Journal 42 (1996), S. 1501-1513 
    Details
    ISSN: 0001-1541
    Keywords: Chemistry ; Chemical Engineering
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Chemistry and Pharmacology , Process Engineering, Biotechnology, Nutrition Technology
    Notes: The flow in a single-cavity extrusion die is considered by using 1-D equations. Since the fluid flow in a well-performing die does not deviate much from the case of perfect widthwise distribution, linearization of 1-D governing equations is often justified. It is demonstrated how linearization can lead to significant simplifications in the analysis of die flow. A self-consistent asymptotic expansion is identified to accomplish this linearization; the expansion can be generalized to obtain higher-order corrections if necessary. In contrast to previous analyses, which have generally been numerical, analytic solutions are obtained to predict widthwise flow nonuniformities in a die having a specific cavity area variation and in which cavity inertial effects are important; these solutions are valid for power-law and Newtonian fluids. A novel truncated power-law approach to modeling the flow of shear-thinning fluids in a die is proposed that combines, by using analytical criteria, the solutions for purely Newtonian or power-law flow in the cavity and slot. The validity of this approach is demonstrated through comparison with numerical results of a 1-D die model for a Carreau constitutive equation.
    Additional Material: 12 Ill.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1002/aic.690420602
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  • 9
    Electronic Resource
    Electronic Resource
    One-dimensional equations governing single-cavity die design (1996)
    Hoboken, NJ : Wiley-Blackwell
    AIChE Journal 42 (1996), S. 2401-2414 
    Details
    ISSN: 0001-1541
    Keywords: Chemistry ; Chemical Engineering
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Chemistry and Pharmacology , Process Engineering, Biotechnology, Nutrition Technology
    Notes: The 1-D approximate equations governing flow in a single-cavity die have been examined in detail. Although the 1-D momentum equation in the slot and mass conservation equation used in previous studies are generally the same, various forms of the 1-D momentum equation governing flow in the cavity have been utilized. A rigorous accounting of the origin of the 1-D cavity equation is given, via asymptotic techniques, integral equations, and the use of approximate velocity fields. Additionally, these techniques are employed to place the slot flow and mass conservation equations, utilized in previous studies, on a firmer theoretical ground. The derived cavity equation and differential equation system is found to be identical to that of Leonard (1985).
    Additional Material: 5 Ill.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1002/aic.690420902
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