Similarity Solutions for Unsteady Gravity-Driven Slender Rivulets

Yatim, Yazariah Mohd; Duffy, Brian; Wilson, S. K.; Hunt, Roland

doi:10.1093/qjmam/hbr014

Cited by 16 publications

(17 citation statements)

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“…However, exactly the same kind of solution for a steadily translating dry patch on a substrate moving parallel to itself at constant velocity U 0 i upwards or downwards (that is, with U 0 < 0 or U 0 > 0) may be obtained simply by means of a shift in the value of c by U 0 , that is, with c replaced by c − U 0 in the above discussion. In particular, a solution (19) representing an unsteady flow with a dry patch translating steadily with velocity ci on a stationary substrate also provides a solution for a steady flow (with a stationary dry patch) on a substrate moving with velocity −ci provided simply that x − ct is replaced by x in η.…”

Section: Discussionmentioning

confidence: 99%

“…Lastly, it is worth commenting on the question of travelling-wave similarity solutions of (13) of the form (19) for the rather different physical context of flow of a steadily translating symmetric slender rivulet occupying |y| ≤ a(x, t) on an inclined planar substrate, driven by gravity, a constant shear stress on its free surface and/or steady motion of the substrate.…”

Section: Discussionmentioning

confidence: 99%

“…To understand the nature of the dry-patch solutions (19) we investigate the properties of solutions of (21), (23) and (24); in section III we derive analytical results that provide valuable information about the motion of the dry patch, and which guide the numerical study described in subsection IV A.…”

Section: B a Travelling-wave Similarity Solutionmentioning

confidence: 99%

“…Equation (58) has a travelling-wave similarity solution of the form (19) with S ∞ = +1, again representing a steadily translating dry patch. The resulting ordinary differential equation for 33 F (η), replacing (21), is 28…”

Section: Purely Surface-shear-stress-driven Flowmentioning

confidence: 99%

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Travelling-wave similarity solutions for a steadily translating slender dry patch in a thin fluid film

2013

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A novel family of three-dimensional travelling-wave similarity solutions describing a steadily translating slender dry patch in an infinitely wide thin fluid film on an inclined planar substrate is obtained, the flow being driven by gravity and/or a prescribed constant shear stress on the free surface of the film. For both driving mechanisms, the dry patch has a parabolic shape (which may be concave up or concave down the substrate), and the film thickness increases monotonically away from the contact lines to its uniform far-field value. The two most practically important cases of purely gravity-driven flow and of purely surface-shear-stress-driven flow are analysed separately.

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Section: Discussionmentioning

confidence: 99%

Section: Discussionmentioning

confidence: 99%

Section: B a Travelling-wave Similarity Solutionmentioning

confidence: 99%

Section: Purely Surface-shear-stress-driven Flowmentioning

confidence: 99%

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Travelling-wave similarity solutions for a steadily translating slender dry patch in a thin fluid film

2013

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“…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. There has been considerable work on many different aspects of rivulet flow; in the absence of a recent review article, the work by Schmuki & Laso (1990), Duffy & Moffatt (1997), Wilson & Duffy (1998), Roy & Schwartz (1999), Wilson et al (2002a), Kim et al (2004), Perazzo & Gratton (2004), Saber & El-Genk (2004), Wilson & Duffy (2005), Le Grand-Piteira et al (2006), Alekseenko et al (2008), Sullivan et al (2008), Benilov (2009), Diez et al (2009), Tanasijczuk et al (2010), Daerr et al (2011), Wilson et al (2011) and Yatim et al (2011) (and the references therein) provide a representative selection of the recent literature.…”

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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Note on semi‐analytical nonstationary solution for the rivulet flows of non‐Newtonian fluids

Ershkov

Leshchenko

2022

Math Methods in App Sciences

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We have presented in this analytical research the revisiting of approach for mathematical modeling the rivulet flows on inclined surface in terms of viscous‐plastic theory of two‐dimensional movements in a frame of (x, y)‐plane in Cartesian coordinates. A semi‐analytical solution has been obtained here for nonstationary creeping approximation of plane‐parallel flow slowly moving on inclined surface. The main motivation of the current development is that each additional enhancement of knowledge in regard to this phenomenon should obviously improve the accuracy of calculations for possible applications in technological or engineering areas of fluid dynamics researches. Even in such simple formulation, equations of motion that govern the dynamics of rivulet flows of non‐Newtonian fluids are hard to be solved analytically. Nevertheless, we have succeeded in obtaining analytical expressions for the components of velocity in {Ox, Oy}‐directions of motion for slowly moving the advancing front of rivulet flow (where Ox‐axis coincides to the initial main direction of slowly moving rivulet flow). Restrictions to the form of flow (stemming from the continuity equation) have been used accordingly.

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Similarity Solutions for Unsteady Gravity-Driven Slender Rivulets

Cited by 16 publications

References 0 publications

Travelling-wave similarity solutions for a steadily translating slender dry patch in a thin fluid film

Travelling-wave similarity solutions for a steadily translating slender dry patch in a thin fluid film

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

Note on semi‐analytical nonstationary solution for the rivulet flows of non‐Newtonian fluids

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