Shock waves in Stokes flows down an inclined plate

Benilov, E. S.; Lapin, Vladimir

doi:10.1103/physreve.83.066321

Cited by 6 publications

(22 citation statements)

References 16 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…A different situation has been reported for bores with an order-one slope of the free surface (but still small Re): if α is sufficiently large and/or h + /h − is sufficiently small, all bores overturn and no steady solution exists (Benilov & Lapin 2011, 2015Benilov 2014). A similar conclusion has been drawn for flows on the inside of a horizontal rotating cylinder (Benilov, Lapin & O'Brien 2012).…”

Section: Introductionsupporting

confidence: 51%

“…Note, however, even though the computed shapes of near-limiting solutions cannot be trusted, the (intuitively correct) tendency of bore steepening for steeper inclines should still be. (iii) Evidently, the solutions presented in figure 3 are monotonic (as well as all examples computed for, but not included in, the paper) -whereas bores computed by Bertozzi et al (2001) and Benilov & Lapin (2011) have oscillatory structure.…”

Section: The Resultsmentioning

confidence: 78%

See 1 more Smart Citation

Hydraulic jumps in a shallow flow down a slightly inclined substrate

Benilov

2015

J. Fluid Mech.

View full text Add to dashboard Cite

This work examines free-surface flows down an inclined substrate. The slope of the free surface and that of the substrate are both assumed small, whereas the Reynolds number Re remains unrestricted. A set of asymptotic equations is derived, which includes the lubrication and shallow-water approximations as limiting cases (as Re → 0 and Re → ∞, respectively). The set is used to examine hydraulic jumps (bores) in a two-dimensional flow down an inclined substrate. An existence criterion for steadily propagating bores is obtained for the (η, s) parameter space, where η is the bore's downstream-to-upstream depth ratio, and s is a non-dimensional parameter characterising the substrate's slope. The criterion reflects two different mechanisms restricting bores. If s is sufficiently large, a 'corner' develops at the foot of the bore's front -which, physically, causes overturning. If, in turn, η is sufficiently small (i.e. the bore's relative amplitude is sufficiently large), the non-existence of bores is caused by a stagnation point emerging in the flow.

show abstract

Section: Introductionsupporting

confidence: 51%

Section: The Resultsmentioning

confidence: 78%

Hydraulic jumps in a shallow flow down a slightly inclined substrate

Benilov

2015

J. Fluid Mech.

View full text Add to dashboard Cite

show abstract

“…Thus, mathematically, the inner (smoothed-shock) solution examined in this article is a particular case (for Q = 2 3 ) of the more general problem of films on an inclined plate examined in [18]. 2 Physically, however, this particular case is important (as it describes rimming flows) and, thus, deserves to be explored in detail.…”

Section: Applicability Of the Results Obtainedmentioning

confidence: 99%

“…The full Eq. 20 is not valid either, as a simple estimate shows that the applicability condition (18) does not hold near the shock. Thus, the LA [which was used to derive (20)] is invalid.…”

Section: The Shock Regionmentioning

confidence: 88%

See 1 more Smart Citation

On rimming flows with shocks

2011

Self Cite

View full text Add to dashboard Cite

We examine rimming flows, i.e. flows of a liquid film on the inside of a horizontal rotating cylinder. So far this problem has mostly been explored using the so-called lubrication approximation (LA). It was shown that, if the volume of the liquid in the cylinder exceeds a certain threshold, then a shock similar to a tidal bore appears in the lower half of the cylinder on its rising side. The position of the shock can be characterized by the polar angle θ s , with a value between θ s = −90 • (the bottom of the cylinder) and θ s = 0 • (the horizontal direction). In this study, we examine rimming flows without the LA, by solving numerically the exact Stokes equations. It is shown that a steady solution describing a (smoothed) shock exists only if −60 • θ s < 0 • . Shocks with lower locations overturn, so no steady solution exists. It is also shown that smoothed-shock solutions have an oscillating structure upstream from the shock. If, however, capillary effects are taken into account, the range of θ s where solutions overturn contracts, and if surface tension is sufficiently strong, solutions exist for all values of θ s .

show abstract

Lubrication theory for free-surface flows with finite slopes and fluxes

Qin

Gao

2022

Acta Mech. Sin.

View full text Add to dashboard Cite

Shock waves in Stokes flows down an inclined plate

Cited by 6 publications

References 16 publications

Hydraulic jumps in a shallow flow down a slightly inclined substrate

Hydraulic jumps in a shallow flow down a slightly inclined substrate

On rimming flows with shocks

Lubrication theory for free-surface flows with finite slopes and fluxes

Contact Info

Product

Resources

About