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On the Navier-Stokes flow down an inclined plane

This paper concerns the two-dimensional motion of a viscous liquid down a perturbed inclined plane under the influence of gravity, and the main goal is the prediction of the surface height as the fluid flows over the perturbations. The specific perturbations chosen for the present study were two humps stretching laterally across an otherwise uniform plane, with the flow being confined in the lateral direction by the walls of a channel. Theoretical predictions of the flow have been obtained by finite-element approximations to the Navier-Stokes equations and also by a variety of lubrication approximations. The predictions from the various models are compared with experimental measurements of the free-surface profiles. The principal aim of this study is the establishment and assessment of certain numerical and asymptotic models for the description of a class of free-surface flows, exemplified by the particular case of flow over a perturbed inclined plane. The laboratory experiments were made over a range of flow rates such that the Reynolds number, based on the volume flux per unit width and the kinematical viscosity of the fluid, ranged between 0.369 and 36.6. It was found that, at the smaller Reynolds numbers, a standard lubrication approximation provided a very good representation of the experimental measurements but, as the flow rate was increased, the standard model did not capture several important features of the flow. On the other hand, a lubrication approximation allowing for surface tension and inertial effects expanded the range of applicability of the basic theory by almost an order of magnitude, up to Reynolds numbers approaching 10. At larger flow rates, numerical solutions to the full equations of motion provided a description of the experimental results to within about 4% , up to a Reynolds number of 25, beyond which we were unable to obtain numerical solutions. It is not known why numerical solutions were not possible at larger flow rates, but it is possible that there is a bifurcation of the Navier-Stokes equations to a branch of unsteady motions near a Reynolds number of 25.

http://rsta.royalsocietypublishing.org/content/340/1656/1.full.pdf

Numerical and asymptotic methods for certain viscous free-surface flows

The objective of this paper is to study weakly nonlinear waves on a viscous magnetic fluid flow down an inclined plane under the influence of gravity and a constant magnetic field parallel to the plane. An approximate evolution equation is derived and a critical Reynolds number for the stability of three‐dimensional long waves obtained. It is found that the component of the applied magnetic field aligned with the flow has a stabilizing effect but that the transverse component, however, has a destabilizing effect on the long waves. Stability regions of two‐dimensional long waves at the critical Reynolds number and numerical results for the solution of a nonlinear ill‐posed problem are presented. Several progressive‐wave solutions are also given.

Surface waves on viscous magnetic fluid flow down an inclined plane

We prove a continuous dependence theorem for weak solutions of equations governing a fluid–structure interaction problem in two spatial dimensions. The proof is based on a priori estimates which, in particular, convey uniqueness of weak solutions. The estimates are obtained using Eulerian coordinates, without remapping the problem into a fixed domain.

Continuous Dependence on Initial Data in Fluid–Structure Motions

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On stability of liquid flow down an inclined plane

An asymptotic method is developed for the linearized Navier-Stokes equations governing the time-dependent motion of a viscous fluid flow down an inclined plane. A diffusion equation for the first order approximation of the fluid surface elevation in a perturbation scheme is derived and a critical Reynolds number is defined based upon the well-posedness of the equation. Under a set of sufficient conditions it is shown that the solution of the diffusion equation is a uniform asymptotic approximation to the generalized solution of the full equations for all time by means of various $L_2 $ and pointwise estimates.

S. M. Shih

Papers

Uniform Asymptotic Approximation for Viscous Fluid Flow Down an Inclined Plane

Uniform asymptotic approximation for viscous fluid flow down an inclined plane