Viscous flow and collapse of macroscopic cavities in a granular material in terms of a Darcylet

Abstract

<jats:p>Viscous flow through a granular material that has a macroscopic cavity is obtained on the basis of the Stokes and the generalized Darcy’s equation. A new singularity termed ‘Darcylet’, a fundamental flow in the granular material that acts like a point force with negative direction, is proposed, which is applied to elucidate the interaction of two cavities. Depending on the configuration of the latter, the volume flux into the cavity increases, which enhances the local stresses on the boundary, causes the collapse of cavities and triggers landslides.</jats:p> <jats:p>This article is part of the theme issue ‘Stokes at 200 (Part 1)’.</jats:p>

Viscous flow around three-dimensional macroscopic cavities in a granular material

“…The interaction of two cavities, in which the asymptotic effect of the other cavity is taken into account, was analysed (see Sano 2020), which also reproduced the experimental findings. As a result of interaction, the critical value U c , where the collapse of cavities results, may be lowered, and the spread of the fluidized region may be enlarged, so that the network formation of the water channel and the onset of a landslide arising from heavy rainfalls may be predicted (see Kaneko & Sano 2005;Sano & Nagata 2006;Sano 2011Sano , 2020, which will be discussed in § 6. We found that the concentration of the flow and the stresses are more pronounced in the 3-D configuration, which is often the case in real situations.…”

“…What is the effect in the interacting cavities? To see this, the velocity at the centre of 2-D cavities positioned in tandem was reproduced (Sano 2020), which is given explicitly by…”

“…We show the variation of Q Relative for the case of two equally sized spherical cavities positioned in tandem (figure 10). The corresponding results for the 2-D case, as in Sano (2020), Kaneko & Sano (2003), are also shown. It is seen that the influence of the second cavity is felt when the centre-to-centre distance is within approximately seven times the radius of the cavity in a 3-D case with ζ 0 = 100.…”

“…These 3-D results were partially confirmed by Sano et al (2013), who measured the velocity field inside a hemispherical cavity by LDV. If the far upstream velocity U ∞ exceeds a certain critical value U c , the grains at the cavity boundary can no longer sustain the structure, so that the grains of the upstream-side boundary are removed and are carried downstream (see Kaneko & Sano 2005;Sano & Nagata 2006;Sano 2011Sano , 2020. Repetition of these processes create a fluidized region toward the upstream direction, which will be discussed in § 6.…”

Viscous flow around three-dimensional macroscopic cavities in a granular material: asymptotic theory for two sufficiently distant spherical cavities of arbitrary configuration

“…The variations increase as much as as the distance becomes small, but decrease to a few per cent as becomes as large as 5. They also pointed out the possibility of the collapse of the cavities and fluidisation, which are caused by the local increase of the stress on the cavity boundaries due to the increase of the velocity (Kaneko & Sano 2005; Sano & Kaneko 2005; Sano & Nagata 2006 a , b ; Sano 2011, 2020).…”