Skip to main content Accessibility help
×
Home
Hostname: page-component-7ccbd9845f-wr4x4 Total loading time: 0.297 Render date: 2023-01-30T01:33:31.456Z Has data issue: true Feature Flags: { "useRatesEcommerce": false } hasContentIssue true

Wave formation in laminar flow down an inclined plane

Published online by Cambridge University Press:  28 March 2006

T. Brooke Benjamin
Affiliation:
Department of Engineering, University of Cambridge

Abstract

This paper deals theoretically with a problem of hydrodynamic stability characterized by small values of the Reynolds number R. The primary flow whose stability is examined consists of a uniform laminar stream of viscous liquid running down an inclined plane under the action of gravity, being bounded on one side by a free surface influenced by surface tension. The problem thus has a direct bearing on the properties of thin liquid films such as have important uses in chemical engineering.

Numerous experiments in the past have shown that in flow down a wall the stream is noticeably agitated by waves except when R is quite small; on a vertical water film, for instance, waves may be observed until R is reduced to some value rather less than 10. The present treatment is accordingly based on methods of approximation suited to fairly low values of R, and thereby avoids the severe mathematical difficulties usual in stability problems at high R. The formulation of the problem resembles that given by Yih (1954); but the method of solution differs from his, and the respective results are in conflict. In particular, there is dis-agreement over the matter of the stability of a strictly vertical stream at very small R. In contrast with the previous conclusions, it is shown here that the flow is always unstable: that is, a class of undamped waves exists for all finite values of R. However, the rates of amplification of unstable waves are shown to become very small when R is made fairly small, and their wavelengths to become very large; this provides a satisfactory explanation for the apparent absence of waves in some experimental observations, and also for the wide scatter among existing estimates of the ‘quasi-critical’ value of R below which waves are undetectable. In view of the controversial nature of these results, emphasis is given to various points of agreement between the present work and the established theory of roll waves; the latter theory gives a clear picture of the physical mechanism of wave formation on gravitational flows, and in its light the results obtained here appear entirely reasonable.

The conditions governing neutral stability are worked out to the third order in a parameter which is shown to be small; but a less accurate approximation is then justified as an adequate basis for an easily workable theory providing a ready check with experiment, This theory is used to predict the value of R at which observable waves should first develop on a vertical water film, and also the length and velocity of the waves. These three predictions are compared with the experimental results found by Binnie (1957), and are substantially confirmed.

Type
Research Article
Copyright
© 1957 Cambridge University Press

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Bellman, R. & Pennington, R. H. 1954 Quart. Appl. Math. 12, 151.
Benjamin, T. B. & Ursell, F. 1954 Proc. Roy. Soc. A, 225, 505.
Binnie, A. M. 1957 J. Fluid Mech. 2, 551.
Dressler, R. F. 1952 Gravity Waves, Nat. Bur. Stan., Wash., Circular no. 521, p. 237.Google Scholar
Dukler, A. E. & Bergelin, O. P. 1952 Chem. Engng Progres 48, 557.
Friedman, S. J. & Miller, C. O. 1941 Industr. Engng Chem. 33, 885.
Grimley, S. S. 1945 Trans. Inst. Chem. Engrs 23, 228.
Jeffreys, H. 1925 Phil. Mag. (6), 49, 793.
Kapitza, P. L. 1948 J. Exp. Theor. Phys., U.S.S.R., 18, 3.
Kelvin, Lord 1887 Mathematical and Physical Papers, Vol. 4, 321. Cambridge University Press.
Kirkbride, C. G. 1934 Industr. Engng Chem. 26, 425. Trans. Amer. Inst. Chem. Engrs 30, 170.Google Scholar
Lamb, H. 1932 Hydrodynamics, 6th Ed. Cambridge University Press.
Lighthill, M. J. & Whitham, G. B. 1955 Proc. Roy. Soc. A, 229, 281.
Lin, C. C. 1955 The Theory of Hydrodynamic Stability. Cambridge University Press.
Nusselt, W. 1916 Z. ver. Dtsch. Ing. 60, 541.
Rayleigh, Lord 1894 The Theory of Sound, 2nd Ed., Vol. 1. London: Macmillan.
Squire, H. B. 1933 Proc. Roy. Soc. A, 142, 621.
Taylor, G. I. 1950 Proc. Roy. Soc. A, 201, 192.
Yih, C.-S. 1954 Proc. 2nd U.S. Congr. Appl. Mech., Amer. Soc. Mech. Engrs, 623.
724
Cited by

Save article to Kindle

To save this article to your Kindle, first ensure coreplatform@cambridge.org is added to your Approved Personal Document E-mail List under your Personal Document Settings on the Manage Your Content and Devices page of your Amazon account. Then enter the ‘name’ part of your Kindle email address below. Find out more about saving to your Kindle.

Note you can select to save to either the @free.kindle.com or @kindle.com variations. ‘@free.kindle.com’ emails are free but can only be saved to your device when it is connected to wi-fi. ‘@kindle.com’ emails can be delivered even when you are not connected to wi-fi, but note that service fees apply.

Find out more about the Kindle Personal Document Service.

Wave formation in laminar flow down an inclined plane
Available formats
×

Save article to Dropbox

To save this article to your Dropbox account, please select one or more formats and confirm that you agree to abide by our usage policies. If this is the first time you used this feature, you will be asked to authorise Cambridge Core to connect with your Dropbox account. Find out more about saving content to Dropbox.

Wave formation in laminar flow down an inclined plane
Available formats
×

Save article to Google Drive

To save this article to your Google Drive account, please select one or more formats and confirm that you agree to abide by our usage policies. If this is the first time you used this feature, you will be asked to authorise Cambridge Core to connect with your Google Drive account. Find out more about saving content to Google Drive.

Wave formation in laminar flow down an inclined plane
Available formats
×
×

Reply to: Submit a response

Please enter your response.

Your details

Please enter a valid email address.

Conflicting interests

Do you have any conflicting interests? *