scholarly journals Ultra-sharp pinnacles sculpted by natural convective dissolution

2020 ◽  
Vol 117 (38) ◽  
pp. 23339-23344
Author(s):  
Jinzi Mac Huang ◽  
Joshua Tong ◽  
Michael Shelley ◽  
Leif Ristroph

The evolution of landscapes, landforms, and other natural structures involves highly interactive physical and chemical processes that often lead to intriguing shapes and recurring motifs. Particularly intricate and fine-scale features characterize the so-called karst morphologies formed by mineral dissolution into water. An archetypal form is the tall, slender, and sharply tipped karst pinnacle or rock spire that appears in multitudes in striking landforms called stone forests, but whose formative mechanisms remain unclear due to complex, fluctuating, and incompletely understood developmental conditions. Here, we demonstrate that exceedingly sharp spires also form under the far-simpler conditions of a solid dissolving into a surrounding liquid. Laboratory experiments on solidified sugars in water show that needlelike pinnacles, as well as bed-of-nails-like arrays of pinnacles, emerge robustly from the dissolution of solids with smooth initial shapes. Although the liquid is initially quiescent and no external flow is imposed, persistent flows are generated along the solid boundary as dense, solute-laden fluid descends under gravity. We use these observations to motivate a mathematical model that links such boundary-layer flows to the shape evolution of the solid. Dissolution induces these natural convective flows that, in turn, enhance dissolution rates, and simulations show that this feedback drives the shape toward a finite-time singularity or blow-up of apex curvature that is cut off once the pinnacle tip reaches microscales. This autogenic mechanism produces ultra-fine structures as an attracting state or natural consequence of the coupled processes at work in the closed solid-fluid system.

2020 ◽  
Vol 2020 ◽  
pp. 1-8
Author(s):  
Linrui Li ◽  
Shu Wang

In this paper, we study the finite-time singularity formation on the coupled Burgers–Constantin–Lax–Majda system with the nonlocal term, which is one nonlinear nonlocal system of combining Burgers equations with Constantin–Lax–Majda equations. We discuss whether the finite-time blow-up singularity mechanism of the system depends upon the domination between the CLM type’s vortex-stretching term and the Burgers type’s convection term in some sense. We give two kinds of different finite-time blow-up results and prove the local smooth solution of the nonlocal system blows up in finite time for two classes of large initial data.


Author(s):  
D. W. Beard ◽  
K. Walters

AbstractThe Prandtl boundary-layer theory is extended for an idealized elastico-viscous liquid. The boundary-layer equations are solved numerically for the case of two-dimensional flow near a stagnation point. It is shown that the main effect of elasticity is to increase the velocity in the boundary layer and also to increase the stress on the solid boundary.


2020 ◽  
Vol 17 (6) ◽  
pp. 1507-1533 ◽  
Author(s):  
Roger D. Finlay ◽  
Shahid Mahmood ◽  
Nicholas Rosenstock ◽  
Emile B. Bolou-Bi ◽  
Stephan J. Köhler ◽  
...  

Abstract. Plant nutrients can be recycled through microbial decomposition of organic matter but replacement of base cations and phosphorus, lost through harvesting of biomass/biofuels or leaching, requires de novo supply of fresh nutrients released through weathering of soil parent material (minerals and rocks). Weathering involves physical and chemical processes that are modified by biological activity of plants, microorganisms and animals. This article reviews recent progress made in understanding biological processes contributing to weathering. A perspective of increasing spatial scale is adopted, examining the consequences of biological activity for weathering from nanoscale interactions, through in vitro and in planta microcosm and mesocosm studies, to field experiments, and finally ecosystem and global level effects. The topics discussed include the physical alteration of minerals and mineral surfaces; the composition, amounts, chemical properties, and effects of plant and microbial secretions; and the role of carbon flow (including stabilisation and sequestration of C in organic and inorganic forms). Although the predominant focus is on the effects of fungi in forest ecosystems, the properties of biofilms, including bacterial interactions, are also discussed. The implications of these biological processes for modelling are discussed, and we attempt to identify some key questions and knowledge gaps, as well as experimental approaches and areas of research in which future studies are likely to yield useful results. A particular focus of this article is to improve the representation of the ways in which biological processes complement physical and chemical processes that mobilise mineral elements, making them available for plant uptake. This is necessary to produce better estimates of weathering that are required for sustainable management of forests in a post-fossil-fuel economy. While there are abundant examples of nanometre- and micrometre-scale physical interactions between microorganisms and different minerals, opinion appears to be divided with respect to the quantitative significance of these observations for overall weathering. Numerous in vitro experiments and microcosm studies involving plants and their associated microorganisms suggest that the allocation of plant-derived carbon, mineral dissolution and plant nutrient status are tightly coupled, but there is still disagreement about the extent to which these processes contribute to field-scale observations. Apart from providing dynamically responsive pathways for the allocation of plant-derived carbon to power dissolution of minerals, mycorrhizal mycelia provide conduits for the long-distance transportation of weathering products back to plants that are also quantitatively significant sinks for released nutrients. These mycelial pathways bridge heterogeneous substrates, reducing the influence of local variation in C:N ratios. The production of polysaccharide matrices by biofilms of interacting bacteria and/or fungi at interfaces with mineral surfaces and roots influences patterns of production of antibiotics and quorum sensing molecules, with concomitant effects on microbial community structure, and the qualitative and quantitative composition of mineral-solubilising compounds and weathering products. Patterns of carbon allocation and nutrient mobilisation from both organic and inorganic substrates have been studied at larger spatial and temporal scales, including both ecosystem and global levels, and there is a generally wider degree of acceptance of the “systemic” effects of microorganisms on patterns of nutrient mobilisation. Theories about the evolutionary development of weathering processes have been advanced but there is still a lack of information connecting processes at different spatial scales. Detailed studies of the liquid chemistry of local weathering sites at the micrometre scale, together with upscaling to soil-scale dissolution rates, are advocated, as well as new approaches involving stable isotopes.


2015 ◽  
Vol 31 (4) ◽  
pp. 457-465
Author(s):  
C. Fang

AbstractA zero-order turbulence closure model of a dry granular dense flow is proposed, with the boundary considered an energy source and sink of the turbulent kinetic energy of the grains. Muller-Liu entropy principle is carried out to derive the equilibrium closure relations, with their dynamic responses postulated from the experimental calibrations. A gravity-driven flow with incompressible grains down an inclined moving plane is studied to investigate the influence of velocity slip near solid boundary on the turbulent features of the flow. While the calculated mean porosity and velocity correspond to the experimental outcomes, increasing velocity slip on the boundary tends to enhance the turbulent dissipation nearby. The distribution of the turbulent dissipation shows a similarity with that of conventional Newtonian fluids in turbulent boundary layer flows. Boundary as an energy sink is more apparent in the zero-order model.


Author(s):  
Beomjun Choi ◽  
Panagiota Daskalopoulos ◽  
John King

AbstractThis work concerns with the existence and detailed asymptotic analysis of type II singularities for solutions to complete non-compact conformally flat Yamabe flow with cylindrical behavior at infinity. We provide the specific blow-up rate of the maximum curvature and show that the solution converges, after blowing-up around the curvature maximum points, to a rotationally symmetric steady soliton. It is the first time that the steady soliton is shown to be a finite time singularity model of the Yamabe flow.


2021 ◽  
Vol 18 (2) ◽  
pp. 85-89
Author(s):  
Brajesh K. Dwivedi

The physical and chemical characteristics of spring and well water samples were studied for two years to assess the origin of groundwater and determine the factors driving the geochemical composition. The ionic speciation and mineral dissolution/precipitation were calculated. Water wells, characterising groundwater circulation at shallow depths are moderate to high mineralised waters of Na-HCO3 type. In contrast to the shallow environment, the CO2-rich, deeper water is of the Ca-HCO3-SO4 type and undergoes significant changes in the baseline chemistry along flow lines with increasing residence time. The main factors controlling the groundwater composition and its seasonal variations are the geology, because of the presence of carbonate formations, the elevation and the rate of karst development. In both groups, the carbonate chemistry was a diagnostic approach. The super-saturation with respect to calcite indicates CO2 degassing, occurring either inside the aquifer in open conduits or at the outlet in reservoirs. Interaction between groundwater and surrounding rocks is believed to be the main process responsible for the observed chemical characteristics of groundwater in the study area. Mathematical equations were also derived involving the hydro geological variables for better prediction of the aquifer.


KURVATEK ◽  
2019 ◽  
Vol 4 (2) ◽  
pp. 111-116
Author(s):  
Sri Ning Peni ◽  
. . T. Listyani R.A.

 Groundwater in the Banjararum area is found in various rocks that build  the West Progo Hills, including the Nanggulan and the Andesite Breccia Formations. Groundwater can be found in dug wells and springs. Field surveys show that the groundwater is generally clear, tasteless and odorless, except for the well at Ngipikrejo. Chemical test results from four samples showed that groundwater in limestone has Ca, Mg - HCO3 type, whereas in volcanic rocks of type Ca - HCO3 and Ca, Na - HCO3. Groundwater chemistry type is generated from the soil, mineral dissolution in sedimentary rocks and igneous rocks as well as the atmosphere through rainwater. The considerable content of NO3 in the Ngipikrejo area indicates pollution from human activities, plants or animal dung. In general, groundwater in the study area is suitable for use as clean water or drinking water, characterized by a measurable pH in the field of 6.6 - 7.2 and TDS 54 - 155 ppm. Nevertheless, laboratory test results show that groundwater samples are water-filled, with a hardness of 135,866 - 778,048 (moderately hard - very hard), so special treatment is required before drinking.


2000 ◽  
Vol 409 ◽  
pp. 51-68 ◽  
Author(s):  
H. K. MOFFATT

The interaction of two propagating vortex pairs is considered, each pair being initially aligned along the positive principal axis of strain associated with the other. As a preliminary, the action of accelerating strain on a Burgers vortex is considered and the conditions for a finite-time singularity (or ‘blow-up’) are determined. The asymptotic high Reynolds number behaviour of such a vortex under non-axisymmetric strain, and the corresponding behaviour of a vortex pair, are described. This leads naturally to consideration of the interaction of the two vortex pairs, and identifies a mechanism by which blow-up may occur through self-similar evolution in an interaction zone where scale decreases in proportion to (t* − t)1/2, where t* is the singularity time. The relevance of Leray scaling in this interaction zone is discussed.


2018 ◽  
Vol 13 (1) ◽  
pp. 7 ◽  
Author(s):  
Kolade M. Owolabi

In this paper, we consider a numerical approach for fourth-order time fractional partial differential equation. This equation is obtained from the classical reaction-diffusion equation by replacing the first-order time derivative with the Atangana-Baleanu fractional derivative in Riemann-Liouville sense with the Mittag-Leffler law kernel, and the first, second, and fourth order space derivatives with the fourth-order central difference schemes. We also suggest the Fourier spectral method as an alternate approach to finite difference. We employ Plais Fourier method to study the question of finite-time singularity formation in the one-dimensional problem on a periodic domain. Our bifurcation analysis result shows the relationship between the blow-up and stability of the steady periodic solutions. Numerical experiments are given to validate the effectiveness of the proposed methods.


Axioms ◽  
2019 ◽  
Vol 8 (1) ◽  
pp. 7
Author(s):  
Marisa Fernández ◽  
Victor Manero ◽  
Jonatan Sánchez

We consider the Laplacian flow of locally conformal calibrated G 2 -structures as a natural extension to these structures of the well-known Laplacian flow of calibrated G 2 -structures. We study the Laplacian flow for two explicit examples of locally conformal calibrated G 2 manifolds and, in both cases, we obtain a flow of locally conformal calibrated G 2 -structures, which are ancient solutions, that is they are defined on a time interval of the form ( − ∞ , T ) , where T > 0 is a real number. Moreover, for each of these examples, we prove that the underlying metrics g ( t ) of the solution converge smoothly, up to pull-back by time-dependent diffeomorphisms, to a flat metric as t goes to − ∞ , and they blow-up at a finite-time singularity.


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