Longitudinal flow of a lenticular liquid filament down an inclined plane

Allen, Robert F.

doi:10.1063/1.1694713

Cited by 51 publications

(29 citation statements)

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“…Duffy and Moffatt [2] used the lubrication approximation employed by Allen and Biggin [3] to obtain analytically the leading-order solution in the special case when the cross-sectional profile of the rivulet in the direction transverse to the flow is thin. Duffy and Moffatt [2] calculated the shape of the rivulet (and, in particular, its width and maximum height) as a function of α, the angle of inclination of the substrate to the horizontal, for 0 ≤ α ≤ π. Duffy and Moffatt [2] also interpreted their results as describing the locally unidirectional flow down a locally planar substrate whose local slope α varies slowly in the flow-wise direction and, in particular, used them to describe the flow in the azimuthal direction round a large horizontal circular cylinder.…”

Section: Introductionmentioning

confidence: 99%

Untitled

Wilson

Duffy

2002

Journal of Engineering Mathematics

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This version is available at https://strathprints.strath.ac.uk/2089/ Strathprints is designed to allow users to access the research output of the University of Strathclyde. Unless otherwise explicitly stated on the manuscript, Copyright © and Moral Rights for the papers on this site are retained by the individual authors and/or other copyright owners. Please check the manuscript for details of any other licences that may have been applied. You may not engage in further distribution of the material for any profitmaking activities or any commercial gain. You may freely distribute both the url (https://strathprints.strath.ac.uk/) and the content of this paper for research or private study, educational, or not-for-profit purposes without prior permission or charge.Any correspondence concerning this service should be sent to the Strathprints administrator: strathprints@strath.ac.ukThe Strathprints institutional repository (https://strathprints.strath.ac.uk) is a digital archive of University of Strathclyde research outputs. It has been developed to disseminate open access research outputs, expose data about those outputs, and enable the management and persistent access to Strathclyde's intellectual output.On the gravity-driven draining of a rivulet of fluid with temperature-dependent viscosity down a uniformly heated or cooled substrate AbstractWe use the lubrication approximation to investigate the unsteady gravity-driven draining of a thin rivulet of Newtonian fluid with temperature-dependent viscosity down a substrate that is either uniformly hotter or uniformly colder than the surrounding atmosphere. First we derive the general nonlinear evolution equation for a thin film of fluid with an arbitrary dependence of viscosity on temperature. Then we show that at leading order in the limit of small Biot number the rivulet is isothermal, as expected, but that at leading order in the limit of large Biot number (in which the rivulet is not isothermal) the governing equation can, rather unexpectedly, always be reduced to that in the isothermal case with a suitable rescaling. These results are then used to give a complete description of steady flow of a slender rivulet in the limit of large Biot number in two situations in which the corresponding isothermal problem has previously been solved analytically, namely non-uniform flow down an inclined plane, and locally unidirectional flow down a slowly varying substrate. In particular, we find that if a suitably defined integral measure of the fluidity of the film is a decreasing function of the temperature of the atmosphere (as it is for all three specific viscosity models we consider) then decreasing the temperature of the atmosphere always has the effect of making the rivulet wider and deeper.2

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Section: Introductionmentioning

confidence: 99%

Untitled

Wilson

Duffy

2002

Journal of Engineering Mathematics

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“…Hence the magnetic pressure vanishes at both the solid plate and the free surface. The latter fact means that the location of the free surface is exactly the same as that in a nonmagnetic case [9] where is the cross-sectional area. This means that in dimensional terms the total traction exerted by the plate on the fluid is independent of the induction of the magnetic field, and is the same as that in a non-magnetic case.…”

Section: Bo= A~pgitmentioning

confidence: 59%

“…Suppose that the location of the free surface is given by the equation z = h(y). At the free surface the jump in pressure is balanced by the surface tension [9], [10]. This gives:…”

Section: Formulationmentioning

confidence: 99%

Fully developed magnetohydrodynamic flow in a rivulet.

Reed¹,

Molokov²

2000

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“…In particular, Towell & Rothfeld (1966) and Allen & Biggin (1974) studied the steady unidirectional flow of a uniform rivulet (i.e. a rivulet with constant crosssectional profile) of Newtonian fluid down an inclined plane, and subsequently Duffy & Moffatt (1995) obtained the solution for a thin rivulet and interpreted their results as describing the locally unidirectional flow of a thin rivulet down a slowly varying substrate, and, in particular, as describing the flow in the azimuthal direction around a large horizontal cylinder.…”

Section: Introductionmentioning

confidence: 99%

Three-dimensional coating and rimming flow: a ring of fluid on a rotating horizontal cylinder

2013

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This version is available at https://strathprints.strath.ac.uk/42349/ Strathprints is designed to allow users to access the research output of the University of Strathclyde. Unless otherwise explicitly stated on the manuscript, Copyright © and Moral Rights for the papers on this site are retained by the individual authors and/or other copyright owners. Please check the manuscript for details of any other licences that may have been applied. You may not engage in further distribution of the material for any profitmaking activities or any commercial gain. You may freely distribute both the url (https://strathprints.strath.ac.uk/) and the content of this paper for research or private study, educational, or not-for-profit purposes without prior permission or charge.Any correspondence concerning this service should be sent to the Strathprints administrator: strathprints@strath.ac.ukThe Strathprints institutional repository (https://strathprints.strath.ac.uk) is a digital archive of University of Strathclyde research outputs. It has been developed to disseminate open access research outputs, expose data about those outputs, and enable the management and persistent access to Strathclyde's intellectual output.Under consideration for publication in J. Fluid Mech. The steady three-dimensional flow of a thin, slowly varying ring of Newtonian fluid on either the outside or the inside of a uniformly rotating large horizontal cylinder is investigated. Specifically, we study "full-ring" solutions, corresponding to a ring of continuous, finite and non-zero thickness that extends all the way around the cylinder. In particular, it is found that there is a critical solution corresponding to either a critical load above which no full-ring solution exists (if the rotation speed is prescribed) or a critical rotation speed below which no full-ring solution exists (if the load is prescribed). We describe the behaviour of the critical solution and, in particular, show that the critical flux, the critical load, the critical semi-width and the critical ring profile are all increasing functions of the rotation speed. In the limit of small rotation speed, the critical flux is small and the critical ring is narrow and thin, leading to a small critical load. In the limit of large rotation speed, the critical flux is large and the critical ring is wide on the upper half of the cylinder and thick on the lower half of the cylinder, leading to a large critical load. We also describe the behaviour of the non-critical full-ring solution, and, in particular, show that the semi-width and the ring profile are increasing functions of the load but, in general, non-monotonic functions of the rotation speed. In the limit of large rotation speed, the ring approaches a limiting non-uniform shape, whereas in the limit of small load, the ring is narrow and thin with a uniform parabolic profile. Finally, we show that, while for most values of the rotation speed and the load the azimuthal velocity is in the same direction as the rotation of the cylinder, there ...

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Longitudinal flow of a lenticular liquid filament down an inclined plane

Cited by 51 publications

References 1 publication

Untitled

Untitled

Fully developed magnetohydrodynamic flow in a rivulet.

Three-dimensional coating and rimming flow: a ring of fluid on a rotating horizontal cylinder

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