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. 2020 Jun 26;378(2174):20190515.
doi: 10.1098/rsta.2019.0515. Epub 2020 Jun 8.

Shallow free-surface Stokes flow around a corner

Edward M Hinton  1 , Andrew J Hogg  2 , Herbert E Huppert  3
Affiliations

Shallow free-surface Stokes flow around a corner

Edward M Hinton et al. Philos Trans A Math Phys Eng Sci. .

Abstract

The steady lateral spreading of a free-surface viscous flow down an inclined plane around a vertex from which the channel width increases linearly with downstream distance is investigated analytically, numerically and experimentally. From the vertex the channel wall opens by an angle α to the downslope direction and the viscous fluid spreads laterally along it before detaching. The motion is modelled using lubrication theory and the distance at which the flow detaches is computed as a function of α using analytical and numerical methods. Far downslope after detachment, it is shown that the motion is accurately modelled in terms of a similarity solution. Moreover, the detachment point is well approximated by a simple expression for a broad range of opening angles. The results are corroborated through a series of laboratory experiments and the implication for the design of barriers to divert lava flows are discussed. This article is part of the theme issue 'Stokes at 200 (Part 1)'.

Keywords: gravity currents; lava flows; viscous flows.

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Conflict of interest statement

The authors declare that they have no competing interests.

Figures

Figure 1.
Figure 1.
Schematic for gravitationally driven viscous flow down an inclined plane at an angle β to the horizontal. The channel expands with opening angle α to the downslope direction, which we take to be the X axis, while the Y axis is in the cross-slope direction. (Online version in colour.)
Figure 2.
Figure 2.
Contour plots of the dimensionless thickness of the steady flow, h(x, y) in a channel that expands at x = 0 for two values of the opening angle, α: (a) α = π/4 and (b) α = π/2. The results are obtained numerically, as described in §3. Contours are plotted at h = 0.01, 0.1, 0.2, 0.3, 0.4, 0.5, 0.6, 0.7, 0.8, 0.9. (Online version in colour.)
Figure 3.
Figure 3.
(a) Steady flow depth along three cross-sections (x = 1, 3, 5) in the case of a channel opening angle of α = π/2 as a function of the similarity variable, y/x1/2. The similarity solution (obtained in §4) provides good agreement with the numerical result for x ≥ 5 for this value of α. There is slight divergence for very small h because the extra source term added for the numerical method has an influence here. (b) Steady flow depth along the line y = 0− as a function of streamwise distance for four values of the opening angle, α. The upstream flow depth is unity. The downstream flow depth is given by the similarity solution, which is a constant, h(x, 0) = χ(η = 0) = 0.812. The numerically computed solutions adjust to the similarity value over relatively short lengths. The results are obtained through numerical integration of the governing partial differential equation (2.3), with the exception of the case α = π, for which there is an analytic solution (see §5). (Online version in colour.)
Figure 4.
Figure 4.
Contours of the steady flow depth for the case α = π. Contours of the numerical result (continuous coloured lines) are plotted at h = 0.01, 0.1, 0.2, 0.3, 0.4, 0.5, 0.6, 0.7, 0.8, 0.9 The thick black line represents the wall. Contours of the exact solution (5.4), plotted as dashed lines for h = 0.01, h = 0.4 and h = 0.7 show excellent agreement. (Online version in colour.)
Figure 5.
Figure 5.
The position of the contact line as a function of downstream distance, x, obtained from our numerical method for three values of α. The translated similarity solution (which is also the exact solution for α = π) provides a much better prediction for the contact line than the untranslated similarity solution. The agreement is good for all α but better for wider opening angles. (Online version in colour.)
Figure 6.
Figure 6.
The distance along the wall, d, at which detachment occurs as a function of the channel opening angle, α. The prediction from our numerical simulations is plotted as a continuous black line and the red pluses indicate the experimental results (table 1). The prediction from the similarity solution (equation (6.2)) is plotted as a blue dashed line and we also include its behaviour as α → 0, which is d = (η0/α)2 (a yellow dot-dashed line). (Online version in colour.)
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