Consequences of viscous anisotropy in a deforming, two-phase aggregate. Part 1. Governing equations and linearized analysis

2013 ◽  
Vol 734 ◽  
pp. 424-455 ◽  
Author(s):  
Yasuko Takei ◽  
Richard F. Katz
Keyword(s):  
Shear Stress ◽  
Numerical Solutions ◽  
Shear Plane ◽  
High Porosity ◽  
Governing Equations ◽  
Two Phase ◽  
The Matrix ◽  
Continuum Scale ◽  
Torsional Flow ◽  
Matrix Shear

AbstractIn partially molten regions of Earth, rock and magma coexist as a two-phase aggregate in which the solid grains of rock form a viscously deformable framework or matrix. Liquid magma resides within the permeable network of pores between grains. Deviatoric stress causes the distribution of contact area between solid grains to become anisotropic; in turn, this causes anisotropy of the matrix viscosity at the continuum scale. In this two-paper set, we predict the consequences of viscous anisotropy for flow of two-phase aggregates in three configurations: simple shear, Poiseuille, and torsional flow. Part 1 presents the governing equations and an analysis of their linearized form. Part 2 (Katz & Takei, J. Fluid Mech., vol. 734, 2013, pp. 456–485) presents numerical solutions of the full, nonlinear model. In our theory, the anisotropic viscosity tensor couples shear and volumetric components of the matrix stress/strain rate. This coupling, acting over a gradient in shear stress, causes segregation of liquid and solid. Liquid typically migrates toward higher shear stress, but under specific conditions, the opposite can occur. Furthermore, it is known that in a two-phase aggregate with a porosity-weakening viscosity, matrix shear causes porosity perturbations to grow into a banded or sheeted structure. We show that viscous anisotropy reduces the angle between these emergent high-porosity features and the shear plane. Laboratory experiments produce similar, high-porosity features. We hypothesize that the low angle of porosity bands in such experiments is the result of viscous anisotropy. We therefore predict that experiments incorporating a gradient in shear stress will develop sample-wide liquid–solid segregation due to viscous anisotropy.

Consequences of viscous anisotropy in a deforming, two-phase aggregate. Part 2. Numerical solutions of the full equations

2013 ◽  
Vol 734 ◽  
pp. 456-485 ◽  
Author(s):  
Richard F. Katz ◽  
Yasuko Takei
Keyword(s):  
Simple Shear ◽  
Poiseuille Flow ◽  
Numerical Solutions ◽  
High Porosity ◽  
Two Phase ◽  
Band Formation ◽  
Porosity Evolution ◽  
The Matrix ◽  
Continuum Scale ◽  
Torsional Flow

AbstractIn partially molten regions of Earth, rock and magma coexist as a two-phase aggregate in which the solid grains of rock form a viscously deformable framework or matrix. Liquid magma resides within the permeable network of pores between grains. Deviatoric stress causes the distribution of contact area between solid grains to become anisotropic; this, in turn, causes anisotropy of the matrix viscosity at the continuum scale. In the second of a two-paper set, we use numerical methods to solve the full, nonlinear, time-dependent equations governing this system. We consider porosity evolution in simple shear, Poiseuille and torsional flow. Under viscous anisotropy, there are two modes of porosity evolution: base-state segregation, which modifies the domain-scale porosity distribution, and growth of porosity perturbations into melt-rich bands. Simulation results with fixed anisotropy confirm and extend the linearized analysis of Part 1 (Takei & Katz, J. Fluid Mech., vol. 734, 2013, pp. 424–455). Most importantly, numerical solutions capture the interaction of the two modes: under Poiseuille flow, base-state segregation enhances band formation; under torsional flow, bands are suppressed. Simulations also show that low band angle is maintained by nonlinear processes such as reconnection of high-porosity segments and by back-rotation of the compacted regions between bands. Simulations with dynamic anisotropy modify these results, further lowering the average band angle. The effective viscosity of each flow is controlled by base-state segregation; it does not evolve under simple shear, decreases in Poiseuille flow and increases in torsion. We propose a reinterpretation of experimental results in terms of the consequences of viscous anisotropy.

Numerical Solutions of Natural Convection Flow of a Dusty Nanofluid About a Vertical Wavy Truncated Cone

2016 ◽  
Vol 139 (2) ◽  
Author(s):  
Sadia Siddiqa ◽  
Naheed Begum ◽  
M. A. Hossain ◽  
Rama Subba Reddy Gorla
Keyword(s):  
Natural Convection ◽  
Mass Transport ◽  
Numerical Solutions ◽  
Convection Flow ◽  
Governing Equations ◽  
Natural Convection Flow ◽  
Two Phase ◽  
Heat And Mass Transport ◽  
Implicit Finite Difference ◽  
Control Skin

This paper reports the numerical results for the natural convection flow of a two-phase dusty nanofluid along a vertical wavy frustum of a cone. The general governing equations are transformed into parabolic partial differential equations, which are then solved numerically with the help of implicit finite difference method. Comprehensive flow formations of carrier and dusty phases are given with the aim to predict the behavior of heat and mass transport across the heated wavy frustum of a cone. The effectiveness of utilizing the nanofluids to control skin friction and heat and mass transport is analyzed. The results clearly show that the shape of the waviness changes when nanofluid is considered. It is shown that the modified diffusivity ratio parameter, NA, extensively promotes rate of mass transfer near the vicinity of the cone, whereas heat transfer rate reduces.

scholarly journals On the consistency of two-phase local/nonlocal piezoelectric integral model

2021 ◽  
Vol 42 (11) ◽  
pp. 1581-1598
Author(s):  
Yanming Ren ◽  
Hai Qing
Keyword(s):  
Differential Equations ◽  
Boundary Conditions ◽  
Numerical Solutions ◽  
Differential Quadrature Method ◽  
Integral Model ◽  
Governing Equations ◽  
Two Phase ◽  
Differential Model ◽  
Piezoelectric Beam ◽  
General Strain

AbstractIn this paper, we propose general strain- and stress-driven two-phase local/nonlocal piezoelectric integral models, which can distinguish the difference of nonlocal effects on the elastic and piezoelectric behaviors of nanostructures. The nonlocal piezoelectric model is transformed from integral to an equivalent differential form with four constitutive boundary conditions due to the difficulty in solving intergro-differential equations directly. The nonlocal piezoelectric integral models are used to model the static bending of the Euler-Bernoulli piezoelectric beam on the assumption that the nonlocal elastic and piezoelectric parameters are coincident with each other. The governing differential equations as well as constitutive and standard boundary conditions are deduced. It is found that purely strain- and stress-driven nonlocal piezoelectric integral models are ill-posed, because the total number of differential orders for governing equations is less than that of boundary conditions. Meanwhile, the traditional nonlocal piezoelectric differential model would lead to inconsistent bending response for Euler-Bernoulli piezoelectric beam under different boundary and loading conditions. Several nominal variables are introduced to normalize the governing equations and boundary conditions, and the general differential quadrature method (GDQM) is used to obtain the numerical solutions. The results from current models are validated against results in the literature. It is clearly established that a consistent softening and toughening effects can be obtained for static bending of the Euler-Bernoulli beam based on the general strain- and stress-driven local/nonlocal piezoelectric integral models, respectively.

scholarly journals Gradient discretization of two-phase poro-mechanical models with discontinuous pressures at matrix fracture interfaces

2021 ◽  
Author(s):  
Francesco Bonaldi ◽  
Konstantin Brenner ◽  
Jérôme Droniou ◽  
Roland Masson ◽  
Antoine Pasteau ◽  
...  
Keyword(s):  
Convergence Analysis ◽  
Numerical Solutions ◽  
Coupled Model ◽  
Constitutive Law ◽  
Finite Volume Scheme ◽  
Two Phase ◽  
Flux Approximation ◽  
Matrix Fracture ◽  
The Matrix ◽  
Discontinuous Pressure

We consider a two-phase Darcy flow in a fractured and deformable porous medium for which the fractures are described as a network of planar surfaces leading to so-called hybrid-dimensional models. The fractures are assumed open and filled by the fluids and small deformations with a linear elastic constitutive law are considered in the matrix. As opposed to \cite{bonaldi:hal-02549111}, the phase pressures are not assumed continuous at matrix fracture interfaces, which raises new challenges in the convergence analysis related to the additional interfacial equations and unknowns for the flow. As shown in \cite{BHMS2018,gem.aghili}, unlike single phase flow, discontinuous pressure models for two-phase flows provide a better accuracy than continuous pressure models even for highly permeable fractures. This is due to the fact that fractures fully filled by one phase can act as barriers for the other phase, resulting in a pressure discontinuity at the matrix fracture interface.    The model is discretized using the gradient discretization method \cite{gdm}, which covers a large class of conforming and non conforming schemes. This framework allows for a generic convergence analysis of the coupled model using a combination of discrete functional tools. In this work, the gradient discretization of \cite{bonaldi:hal-02549111} is extended to the discontinuous pressure model and the convergence to a weak solution is proved.  Numerical solutions provided by the continuous and discontinuous pressure models are compared on gas injection and suction test cases using a Two-Point Flux Approximation (TPFA) finite volume scheme for the flows and $\P_2$ finite elements for the mechanics.

Numerical Investigation of Oil Slug Trapped in Capillary Tube With a Level Set Method

2013 ◽  
Author(s):  
Liming Dai ◽  
Xiaojie Wang
Keyword(s):  
Level Set Method ◽  
Level Set ◽  
Annular Flow ◽  
High Efficiency ◽  
Numerical Solutions ◽  
Capillary Tube ◽  
Water Film ◽  
Governing Equations ◽  
Two Phase

An investigation of the motion of oil drops trapped in an axisymmetric capillary tube saturated with water is carried out. The governing equations for core-annular flow are derived in detail. Numerical solutions for capturing the evolution of interface are developed by making use of a level set method. A second-order projection method is applied to obtain the velocity field of the incompressible two-phase flow. The results show the high efficiency of the method and the key role of water film in improving the mobilization of oil slugs.

scholarly journals Consequences of viscous anisotropy in a deforming, two-phase aggregate. Why is porosity-band angle lowered by viscous anisotropy?

2015 ◽  
Vol 784 ◽  
pp. 199-224 ◽  
Author(s):  
Yasuko Takei ◽  
Richard F. Katz
Keyword(s):  
Cross Section ◽  
Laboratory Experiments ◽  
A Priori ◽  
Shear Plane ◽  
High Porosity ◽  
Anisotropic Model ◽  
Maximum Compressive Stress ◽  
Two Phase ◽  
The Difference ◽  
Do So

In laboratory experiments that impose shear deformation on partially molten aggregates of initially uniform porosity, melt segregates into high-porosity sheets (bands in cross-section). The bands emerge at $15^{\circ }$–$20^{\circ }$ to the shear plane. A model of viscous anisotropy can explain these low angles whereas previous simpler models have failed to do so. The anisotropic model is complex, however, and the reason that it produces low-angle bands has not been understood. Here we show that there are two mechanisms: (i) suppression of the well-known tensile instability, and (ii) creation of a new shear-driven instability. We elucidate these mechanisms using linearised stability analysis in a coordinate system that is aligned with the perturbations. We consider the general case of anisotropy that varies dynamically with deviatoric stress, but approach it by first considering uniform anisotropy that is imposed a priori and showing the difference between static and dynamic cases. We extend the model of viscous anisotropy to include a strengthening in the direction of maximum compressive stress. Our results support the hypothesis that viscous anisotropy is the cause of low band angles in experiments.

The transition between solitary wave and diapir emergence from a high porosity disturbance in two-phase flow

2020 ◽  
Author(s):  
Janik Dohmen ◽  
Harro Schmeling
Keyword(s):  
Solitary Wave ◽  
Constant Velocity ◽  
Strong Influence ◽  
Wave Dispersion ◽  
High Porosity ◽  
Initial Radius ◽  
Two Phase ◽  
The Earth ◽  
Porosity Wave ◽  
The Matrix

<p>Many processes in the earth involve the melting of rocks and the percolation of the produced melt through the residuum. These processes have been extensively studied but there is still much left what is not completely understood. In this work we focus on the emergence of solitary porosity waves, which can emerge from disturbances in regions where melt is allowed to percolate relatively to the matrix. These waves are regions of higher melt fractions that ascend with a constant velocity while not changing their shape during this ascending process. The size of these waves depends on the compaction length, which depends on just poorly known parameters such as the permeability and the viscosity of the matrix. As they can vary over several orders of magnitudes it might have a strong influence on porosity waves and their emergence from local disturbances with higher porosities than the background.</p><p>In this work we start with a 2D Gaussian-bell shaped disturbance with a certain porosity amplitude and vary the initial radius which is non-dimensionized by the characteristic compaction length. For some cases this disturbance results in an ascending solitary wave and for others it rises upwards as a diapir. For a few cases a kind of fingering can be observed which  looks like a small emerging porosity wave which is just slightly faster than the following melt of the initial larger disturbance. This leads to a melt ascent with a strongly focused front.</p><p>Comparison of porosity wave dispersion curves with analytical ascent rates of a Stokes sphere helps explaining this transition of diapirs to solitary waves via a melt ascent with a strongly focused front.</p>

The Representation of Regenerator Fluid Carryover by Bypass Flows

1984 ◽  
Vol 106 (1) ◽  
pp. 216-220 ◽  
Author(s):  
P. J. Banks
Keyword(s):  
Numerical Solutions ◽  
Fluid Flows ◽  
Porous Matrix ◽  
Heat Transfer Process ◽  
The Other ◽  
Sensible Heat ◽  
Governing Equations ◽  
Fluid Stream ◽  
The Matrix ◽  
Two Fluid

A regenerator transfers sensible heat between two fluid streams by means of a porous matrix through which the streams are passed alternately. The fluid contained in the matrix passages is carried over from one fluid stream to the other, and contributes to the heat transfer process. It has been suggested that this contribution may be predicted by treating the fluid carryover as fluid flows bypassing the matrix. The validity of this representation is explained, and its accuracy is explored in a case for which numerical solutions of the governing equations are available. A published analysis of the representation is discussed and completed.

scholarly journals The effect of effective rock viscosity on 2D magmatic porosity waves

2019 ◽  
Author(s):  
Janik Dohmen ◽  
Harro Schmeling ◽  
Jan Philipp Kruse
Keyword(s):  
Bulk Viscosity ◽  
Numerical Solutions ◽  
Two Phase Flow ◽  
Fluid Phase ◽  
Analytic Solutions ◽  
Phase Flow ◽  
Two Phase ◽  
Source Regions ◽  
The Matrix ◽  
Melt Segregation

Abstract. In source regions of magmatic systems the temperature is above solidus and melt ascent is assumed to occur predominantly by two-phase flow which includes a fluid phase (melt) and a porous deformable matrix. Since McKenzie (1984) introduced his equations for two-phase flow, numerous solutions have been studied one of which predicts the emergence of solitary porosity waves. By now most analytical and numerical solutions for these waves used strongly simplified models for the shear- and bulk viscosity of the matrix, significantly overestimating the viscosity or completely neglecting the porosity-dependence of the bulk viscosity. Schmeling et al. (2012) suggested viscosity laws in which the viscosity decreases very rapidly for small melt fractions. They are incorporated into a 2D finite difference mantle convection code with two-phase flow (FDCON) to study the ascent of solitary porosity waves. The models show that, starting with a Gaussian shaped wave, they rapidly evolve into a solitary wave with similar shape and a certain amplitude. Despite the strongly weaker rheologies compared to previous viscosity laws the effect on dispersion curves and wave shape are only moderate as long as the background porosity is fairly small. The models are still in good agreement with semi-analytic solutions which neglect the shear stress term in the melt segregation equation. However, for higher background porosities and wave amplitudes associated with a viscosity decrease of 50% or more, the phase velocity and the width of the waves are significantly decreased. Our models show that melt ascent by solitary waves is still a viable mechanism even for more realistic matrix viscosities.

Modelling of Extrusion Process for Aluminium A356 Alloy

2014 ◽  
Vol 217-218 ◽  
pp. 188-194 ◽  
Author(s):  
Sudip Simlandi ◽  
Nilkanta Barman ◽  
Himadri Chattaopadhyay
Keyword(s):  
Shear Stress ◽  
Solid State ◽  
Numerical Solutions ◽  
Energy Requirement ◽  
A356 Alloy ◽  
Extrusion Process ◽  
Solid Alloy ◽  
Governing Equations ◽  
Semisolid Alloy ◽  
Semi Solid

In the present work, a model is developed to study extrusion process of A356 alloy in semi-solid state. The distinct rheology of the semisolid alloy reduces energy necessity during extrusion process. Accordingly, a proper rheological model of the alloy is considered in the model towards a detailed study of the process. A combination of analytical and numerical solutions is considered for solving the governing equations. The work finally predicts distribution of velocity and shear stress of the alloy under shear in the considered domain. It also predicts the energy requirement during the extrusion process. It is demonstrated that for semisolid extrusion, reasonably less energy is required as compared to a conventional extrusion process Keywords: Extrusion, semi-solid alloy, apparent viscosity, extrusion power

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