Search

This paper reports a breakdown in linear stability theory under conditions of neutral stability that is deduced by an examination of exponential modes of the form |$h\approx{{e}^{i(kx-\omega t)}}$|⁠ , where |$h$| is a response to a disturbance, |$k$| is a real wavenumber and |$\omega (k)$| is a wavelength-dependent complex frequency. In a previous paper, King et al. (2016, Stability of algebraically unstable dispersive flows. Phys. Rev. Fluids , 1, 073604) demonstrates that when Im |$[\omega (k)]=0$| for all |$k$|⁠ , it is possible for a system response to grow or damp algebraically as |$h\approx{{t}^{s}}$| where |$s$| is a fractional power. The growth is deduced through an asymptotic analysis of the Fourier integral that inherently invokes the superposition of an infinite number of modes. In this paper, the more typical case associated with the transition from stability to instability is examined in which Im |$[\omega (k)]=0$| for a single mode (i.e. for one value of |$k$|⁠) at neutral stability. Two partial differential equation systems are examined, one that has been constructed to elucidate key features of the stability threshold, and a second that models the well-studied problem of rectilinear Newtonian flow down an inclined plane. In both cases, algebraic growth/decay is deduced at the neutral stability boundary, and the propagation features of the responses are examined. [ABSTRACT FROM AUTHOR]

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. [DOI: 10.1115/1.1385516]

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]

A coating die forms liquid layers of uniform thickness for application to a substrate. In a dual-cavity coating die an outer cavity and slot improves flow distribution from an inner cavity and slot. A model for axial flow in the outer cavity must consider the ever-present cross flow. A 1-D equation for the pressure gradient for a power-law liquid is obtained as a small departure from a uniform flow distribution and no axial flow. The equation contains a shape factor dependent on cavity shape, Reynolds number, and power-law index. The shape factor for five triangular cavity shapes is obtained by finite-element analysis and correlated for application to die design up to the onset of flow recirculation which arises at the junction of the cavity and outer slot. The performance of the combined cavity and slot is considered and the most effective design determined. © 2017 American Institute of Chemical Engineers AIChE J, 64: 708-716, 2018 [ABSTRACT FROM AUTHOR]