Theoretical Analysis of Mechanical Instability in Geological System
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Abstract
Geological systems show buckling instabilities in layers of high viscosity hosted in a matrix of lower viscosity. Such instabilities occur on a wide spectrum of scales, ranging from a nanometre to hundreds of kilometres. This phenomenon dates back to the finding of Euler, who considered a thin rod under compressive stresses along its axis, and showed that, above a critical stress the rod deforms with a wavelength twice as long as the rod. Later theories, developed by Biot and Ramberg, describe the wavelength of buckling instabilities as a function of layer thickness and the layer to matrix viscosity ratio. Their theories are applicable to large scale systems where the viscous forces far dominate over other forces. However, the dynamic becomes completely different in small scale systems where other forces, such as surface tension can be competing, and thereby influence the buckling instabilities. One of the objectives of my present work is to investigate the phenomenon of buckling instabilities accounting the effects of interfacial tension between the layer and the embedding matrix. Many geological settings involve rapid migration of melts and different types of lowviscosity fluids, such as hydrothermal fluids and volatilerich magma through narrow passages. They develop enormous pressure to the wall, resulting in intense deformation, often leading to initiation of wave instabilities. Another direction of this study aims to recognize the dynamic and rheological conditions in which flowinduced instabilities can develop in geological conditions.
Abbreviations
FII  Flow Induced Instabilities 
PDMS  Polydimethylsiloxane 
INTRODUCTION
Background/Rationale
In geology the flow induced instabilty is significant. Many continental back arcs have thin lithosphere. The flow induced instabilty is an integral and important part of shear localization. In the ductile field, instability of flow (in time) is necessarily associated with localization (in space), leading to shear zones at various scales, from slip lines and slip bands due to the glide of dislocations in glide planes of single crystals, to shear bands in polycrystals deformed in plane strain. In Structural Geology and metallurgy flow induced instability or nonuniform instability is rule rather than the exception. Differential stress exerts both static and dynamic effects on rockmass permeability, modulating fluid flow in the Earth’s crust. In general, granitic magmas ascend via propagating fractures, as dykes, in extremely short time periods. In these phenomena flow induced instability plays an important role. A competent layer with a nonlinear rheology can, under extension, exhibit pinch‐and‐swell instabilities. Such instabilities can explain small‐scale regular deformations of rock. These pinchandswells are defined as Boudin. The name has been used by geologists to describe deformed rock layers with pinches and swells. These are some of the significance of the Flow Induced Instability(FII).The buckling theory states that when an elastic plate is subjected to a horizontal force, the plate can buckle, if the applied force is sufficiently large. Fold trains in mountain belts are believed to result from the warping of strata under horizontal compression. In FII, the viscosity plays an important role. The different viscosity among the competent layers show deformities resulting in instability. This is how the FII differs from other instabilities. To recognize the dynamic and rheological aspects in which FII can develop, we need to understand the basic concepts of fluid motion along with their rheological equations.
Basic Concepts of Fluid Motion
Rheological Equations of Fluid Motion:
Newtonian Fluids: Most common fluids i.e. air, water and gasoline are Newtonian fluids under normal conditions. Mathematically for Newtonian fluid we can write:
σ_{xy }∝ $\frac{du}{dy}$
If one consider the deformation of two different Newtonian fluids, say Glycerin and water, one recognizes that they will deform at different rates under the action of same applied stress. Glycerin exhibits much more resistance to deformation than water. Thus we say it is more viscous. The constant of proportionality is called, ‘µ’. Thus,
σ_{xy }= µ$\frac{du}{dy}$_{}
NonNewtonian Fluids: A nonNewtonian fluid is a fluid that does not follow Newton’s law of viscosity. Most commonly, the viscosity (the gradual deformation by shear or tensile stresses) of nonNewtonian fluids is dependent on shear rate or shear rate history. In a nonNewtonian fluid, the relation between the shear stress and the shear rate is different. The fluid can even exhibit timedependent viscosity. Therefore, a constant coefficient of viscosity cannot be defined.
σ_{xy} = k$(\frac{du}{dy})^n$
NavierStokes Equation Of Fluid Motion : In Physics the Navier–Stokes equation, named after ClaudieLouis Navier and George Gabriel Stokes, describe the motion of vicous fluid substances. It describes the state of a liquid. The NavierStokes equation can be viewed as an application of Newton's second law, F=ma, which states that force is the product of the mass of an object times its acceleration.
The NavierStokes Equation is:
Dv_{i }/Dt = 1/ρ(δp/δx_{i})+1/3v(δ/δx_{i})(δv_{j}/δx_{j})+vΔ^{2}v_{i}+X_{i}
Where v_{i} is velocity field, ρ is the density of the fluid, p is te applied pressure, v is the kinematic viscosity (v=η/ρ) and X_{i} is the body force term.
Statement of the Problems
Any material that flows in response to an applied stress is a fluid. Although solids acquire a finite deformation or strain upon being stressed, fluids deform continuously under the action of applied forces. On many scales crustal rocks appear to have been folded. Folding can be attributed to the fluid behavior of these rocks. A fluid instability can also explain the formation of dykes and its propagation. Other than structural geology, in metallurgy too flow induced instability or nonuniform instability is rule rather than the exception. The flow induced instabilty is an integral and important part of shear localization. The theories proposed by Biot and Ramberg are applicable to large scale systems where the viscous forces far dominate over other forces. However, the dynamic becomes completely different in small scale systems where other forces, such surface tension can be competing, and thereby influence the instabilities. The fundamental approach of my project is to observe the influence of surface tension. The experiments performed focused to establish the importance of surface tension during FII. During FII it may be assumed that in small scale systems the surface tension may be the driving force of wall deformation along the channel flow.
Objectives of the Research
One of the primary objectives of my present work under the summer fellowship programme intends to investigate the phenomenon of buckling instabilities accounting the effects of interfacial tension between the layer and the embedding matrix. Many geological settings involve rapid migration of melts and different types of lowviscosity fluids, such as hydrothermal fluids and volatilerich magma through narrow passages. They develop enormous pressure to the wall, resulting in intense deformation, often leading to initiation of sinusoidal wave instabilities. Another direction of this study aims to recognize the dynamic and rheological conditions in which flowinduced instabilities can develop in geological conditions. The theories proposed by Biot and Ramberg are applicable to large scale systems where the viscous forces far dominate over other forces. However, the dynamic becomes completely different in small scale systems where other forces, such surface tension can be competing, and thereby influence the instabilities.
Scope
On many scales crustal rocks appear to have been folded. Folding can be attributed to the fluid behavior of these rocks. A fluid instability can also explain the formation of dykes and its propagation. Other than structural geology, in metallurgy too flow induced instability or nonuniform instability is rule rather than the exception.
LITERATURE REVIEW
The concept of instability is quite old. Geological systems show buckling instabilities in layers of high viscosity hosted in a matrix of lower viscosity. Such instabilities occur on a wide spectrum of scales, ranging from a nanometre to hundreds of kilometres scale. This concept dates back to the finding of Euler, who considered a thin rod under compressive stresses along its axis, and showed that, above a critical stress the rod deforms with a wavelength twice as long as the rod. Later theories, developed by Biot and Ramberg, describe the wavelength of buckling instabilities as a function of layer thickness and the layer to matrix viscosity ratio. Their theories are applicable to large scale systems where the viscous forces far dominate over other forces. However, the dynamic becomes completely different in small scale systems where other forces, such as surface tension can be competing, and thereby influence the buckling instabilities. But when we consider instabilities or FII from the literature point of view in gelogical aspect the resource is quite limited. The experimental setup which I came across various papers was quite satisfying as well as it instigated me to perform the experiments on the intricacies of FII on different scales.
METHODOLOGY
Concepts
The basic concept of instability is such that when a high viscous layer hosted in a matrix of lower viscosity the more viscous layer show buckling instabilities. In the experiments perfomed we took two different materials: 1. PDMS(polydimethylsiloxane) and Glycerin or Engine Oil. PDMS is a silicon based organic ploymer with extraordinary rheological properties. The other element glycerin is a dense substance having density of 1.25g/cc i.e. heavier than water. Both the materials have different viscosity. PDMS is more viscous than Glycerin. Thus the experiment will show the effects of interfacial tension between the layer and the embedding matrix.
Methods
The experiment performed in the following steps:
1. The experiments was performed in a long transparent cylinder.
2. The cylinder was completely filled with PDMS.
3. The cylinder was kept overnight to allow the PDMS to settle.
4. A long hollow glass tube was inserted inside the PDMS to initiate the channel formation.
5. After the PDMS is settled a continuous flow of glycerin or engine oil is administered through the formed channel.
6. The administered glycerin will flow through the channel and due to the difference in viscosity the PDMS wall will be deformed.
7. Continuous pictures are taken in order to analyse the deformation.
RESULTS AND DISCUSSION
The experimental setup showed a particular result regarding instability and deformation. In the experiment was setted in a cylinder and a channel was made with the help of a hollow glass rod. Through the channel a continuous flow of glycerin or engine oil was maintained which gave rise to instability i.e., deformation along the channel. The PDMS wall collapsed and showed instability due to the difference in viscosity among the experinental materials.
The experimental apparatus available brought some limitations along with it thus failed to show the sinusoidal wavy nature but clearly emphasized the instability i.e., deformation along the PDMS channel due to diferrence in viscosity.
CONCLUSION
The experiments perfomed showed different results. Two competent layers were included in the experiment. The high viscous layer embedded in a low viscous matrix showed buckling instability. The PDMS wall deformed in a sinusoidal wave. This show the influence of viscosity and surface tension. The instabilities or the deformation thus found is the result of Viscosity and Surface Tension. This phenomena takes place during dyke propagation as far as geological aspect is concerned. The importance of instability in geological scenario is immense.
REFERENCES
RESEARCH PAPERS
1.The importance of interfacial instability for viscous folding in mechanically heterogeneous layers EVANGELOS MOULAS & STEFAN M. SCHMALHOLZ
2.Challenging dyke ascent models using novel laboratory experiments: Implications for reinterpreting evidence ofmagma ascent and volcanism J.L. Kavanagh et al published in Journal of Volcanolohgy and Geothermal Research(Elseiver)
3.A constant influx model for dike propagation: Implications for magma reservoir dynamicsPaola Traversa et al published in Journal of Geophysical Research Atmosphere
4.The buckling and stretching of a viscida  J. D. BUCKMASTER published in Journal Of Fluid Mechanics
6.Numerical modelling of singlelayer folding: clarification of an issue regarding the possible effect of computer codes and the influence of initial irregularities.Y. Zhanga, N.S. Mancktelow, B.E. Hobbs, A. Ord, H.B. Muhlhaus published in Journal Of Structural Geology
7.Folded microthreads: Role of viscosity and interfacial tension Thomas Cubaud, Bibin M. Jose, and Samira Darvishi published in "Physics Of Fluid"
BOOK
1.Geodynamics by Donald L. Turcotte and Gerald Schubert
WEBSITES
1.https://www.wikipedia.org/ 2.https://www.google.com/ 3.https://scholar.google.co.in/
APPLICATIONS USED
1.MS Paint
ACKNOWLEDGEMENTS
I thank Dr. Subhadip Bhadra for his guidance and support during the application process. I thank Prof. Nibir Mandal for his constant guidance, care,encouragement and support during the project. Being my first summer project, it was a wonderful experience to learn a few experimental techniques in Structural Geology and Geodynamics under his esteemed guidance. This project wouldn't have been possible without his continuous support and overwhelming insight. My special thanks to Dr. Puspendu Saha for his tremendous help during the experiments, without him it wouldn't be possible. I thank Nandan Roy, Joyjeet Sen, Dip Ghosh, Nilkamal Barai, Manaska Mukhopadhyay for their valuable suggestion during my project and Arnab Roy for his help in taking the photographs during the experiments. My parents who let me pursue my own interests and were always there whenever I needed them. A simple “thanks” wouldn’t be sufficient. Finally, I am indebted to the Indian Academies of Sciences for selecting me for the Summer Research Fellowship  2019.
Source

Fig 1: http://www.engineeringarchives.com
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