LINEAR HYDRODYNAMIC STABILITY THEORY FOR THIN FILM FLOW DOWN CONES AND CYLINDERS

Zollars, Richard L.

Graduate Thesis Or Dissertation

LINEAR HYDRODYNAMIC STABILITY THEORY FOR THIN FILM FLOW DOWN CONES AND CYLINDERS Public Deposited

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Citeable URL: https://scholar.colorado.edu/concern/graduate_thesis_or_dissertations/kk91fn442

Abstract

The stability of thin film flows has been studied for some time because of the profound effect that surface ripples can have on heat and mass transfer coefficients in certain types of chemical processing equipment such as packed beds, trickling filters, wettedwall columns, and vertical condensers. However, as a simplification the basic flow for all classical stapility studies has been the flow down a flat plate, i.e. a parallel flow.

Since the flows encountered practically have nonparallel basic flows, this study considered the effect of the nonparallel nature of the basic flow on the stability of thin film flow down a cone. The resulting basic flow equations and also the linearized stability equations are partial differential equations, in contrast to the classical stability studies where these equations are ordinary differential equations. An analytical solution to the linear stability problem for a nonparallel film flow was obtained using a regular perturbation expansion in a small parameter which is a measure of the characteristic length for diffusion of vorticity in the cross-stream direction to the characteristic length in the streamwise direction.

One method commonly used to analyze the stability of nonparallel flows is to assume the flow behaves locally as a parallel flow and analyze the stability at that point classically. This approximation, called the quasi-parallel flow assumption, can be applied to thin film flow down a cone. In order to use the quasiparallel f~ow assumption for flow down a cone, an analytical solution for thin film flow down a cylinder had to be obtained.

When compared with the results predicte~ using the quasiparallel flow assumption, the nonparallel flow solution indicates that the nonparallel nature of the basic flow acts to stabilize the flow. These results appear to be compatible with existing nonparallel stability studies for shear flows. Furthermore, the nonparallel flow exhibits two unstable regimes; a relatively unstable regime and an absolutely unstable regime. While thin film flow down a cone may exhibit regions of instability, the flow is stabilized by increasing distance from the apex ·of the cone. Thus thin film flow down a cone is globally asymptotically stable but not uniformly asymptotically stable. This appears to be the first report of a hydrodynamic flow exhibiting this behavior.

The solution of the nonparallel linear stability problem for flow down a cone also predicts that the wave length and phase velocity of disturbances decrease with increasing distance from the apex. The flow is shown to be stabilized by the lateral curvature effects of the basic flow but destabilized by the lateral curvature effects of the perturbed flow.

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