1,721,172 research outputs found
Insights from a pseudospectral study of a potentially singular solution of the three-dimensional axisymmetric incompressible Euler equation
Full text linkWe develop a Fourier-Chebyshev pseudospectral direct numerical simulation
(DNS) to examine a potentially singular solution of the radially bounded,
three-dimensional (3D), axisymmetric Euler equations [G. Luo and T.Y. Hou,
Proc. Natl. Acad. Sci. USA, 111.36 (2014)]. We demonstrate that: (a) the time
of singularity is preceded, in any spectrally truncated DNS, by the formation
of oscillatory structures called tygers, first investigated in the
one-dimensional (1D) Burgers and two-dimensional (2D) Euler equations; (b) the
analyticity-strip method can be generalized to obtain an estimate for the
(potential) singularity time.Comment: 17 pages. 13 figure
Cahn-Hilliard-Navier-Stokes Investigations of Binary-Fluid Turbulence and Droplet Dynamics
Full text linkThe study of finite-sized, deformable droplets adverted by turbulent flows is an active area of research. It spans many streams of sciences and engineering, which include chemical engineering, fluid mechanics, statistical physics, nonlinear dynamics, and also biology. Advances in experimental techniques and high-performance computing have made it possible to investigate the properties of turbulent fluids laden with droplets. The main focus of this thesis is to study the statistical properties of the dynamics of such finite-size droplets in turbulent flows by using direct numerical simulations (DNSs). The most important feature of the model we use is that the droplets have a back-reaction on the advecting fluid: the turbulent fluid affects the droplets and they, in turn, affect the turbulence of the fluid. Our study uncovers (a) statistical properties that characterize the spatiotemporal evolution of droplets in turbulent flows, which are statistically homogeneous and isotropic, and (b) the modification of the statistical properties of this turbulence by the droplets.
This thesis is divided into seven Chapters. Chapter 1 contains an introduction to the background material that is required for this thesis, especially the details about the equations we use; it also contains an outline of the problems we study in subsequent Chapters. Chapter 2 contains our study of “Droplets in Statistically Homogeneous Turbulence: From Many Droplets to a few Droplets”. Chapter 3 is devoted to our study of “Coalescence of Two Droplets”. Chapter 4 deals with “Binary-Fluid Turbulence: Signatures of Multifractal Droplet Dynamics and Dissipation Reduction”. Chapter 5 deals with “A BKM-type theorem and associated computations of solutions of the three-dimensional Cahn-Hilliard-Navier-Stokes equations”. Chapter 6 is devoted to our study of “Turbulence-induced Suppression of Phase Separation in Binary-Fluid Mixtures”. Chapter 7 is devoted to our study of “Antibubbles: Insights from the Cahn-Hilliard-Navier-Stokes Equations”
Spiral-Wave Dynamics in Ionically Realistic Mathematical Models for Human Ventricular Tissue
Full text linkThere is a growing consensus that life-threatening cardiac arrhythmias like ven- tricular tachycardia (VT) or ventricular fibrillation (VF) arise because of the formation of spiral waves of electrical activation in cardiac tissue; unbroken spiral waves are associated with VT and broken ones with VF. Several experimental studies have shown that in homogeneities in cardiac tissue can have dramatic effects on such spiral waves.
In this thesis we focus on spiral-wave dynamics in mathematical models of human ventricular tissue which contain (a) conduction in homogeneities, (b) ionic in- homogeneities, (c) fibroblasts, (d) Purkinje fibers. We also study the effect of a periodic deformation of the simulation domain on spiral wave-dynamics. Chapter 2 contains our study of “Spiral-Wave Dynamics and Its Control in the Presence of In homogeneities in Two Mathematical Models for Human Cardiac Tissue”; this chapter follows closely parts of a paper we have published [1]. Chapter 3 contains our study of “Spiral-wave dynamics in a Mathematical Model of Human Ventricular Tissue with Myocytes and Fibroblasts”; this chapter follows closely a paper that we have submitted for publication. Chapter 4 contains our study of “Spiral-wave Dynamics in Ionically Realistic Mathematical Models for Human Ventricular Tis- sue: The Effects of Periodic Deformation”; this chapter follows closely a paper that we have submitted for publication. Chapter 5 contains our study of “Spiral-wave dynamics in a Mathematical Model of Human Ventricular Tissue with Myocytes and Purkinje fibers”; this chapter follows closely a paper that we will submit for publication soon.
In chapter 2, we study systematically the AP morphology in a state-of-the-art mathematical model of human ventricular tissue due to ten-Tusscher, Noble, Noble, and Panfilov (the TNNP04 model); we also look at the contribution of individual ionic currents to the AP by partially or completely blocking ion channels associated with the ionic currents. We then carry out systematic studies of plane- wave and circular-wave dynamics in the TNNP04 model for cardiac tissue model. We present a detailed and systematic study of spiral-wave turbulence and spa- tiotemporal chaos in two mathematical models for human cardiac tissue due to (a) ten-Tusscher and Panfilov (the TP06 model) and (b) ten-Tusscher, Noble, Noble, and Panfilov (the TNNP04 model). In particular, we use extensive numerical simulations to elucidate the interaction of spiral waves in these models with conduction and ionic in homogeneities. Our central qualitative result is that, in all these models, the dynamics of such spiral waves depends very sensitively on such in homogeneities. A major goal here is to develop low amplitude defibrillation schemes for the elimination of VT and VF, especially in the presence of in homogeneities that occur commonly in cardiac tissue. Therefore, we study a control scheme that has been suggested for the control of spiral turbulence, via low-amplitude current pulses, in such mathematical models for cardiac tissue; our investigations here are designed to examine the efficacy of such control scheme in the presence of in homogeneities in biophysical realistic models. We find that a scheme that uses control pulses on a spatially extended mesh is more successful in the elimination of spiral turbulence than other control schemes. We discuss the theoretical and experimental implications of our study that have a direct bearing on defibrillation, the control of life-threatening cardiac arrhythmias such as ventricular fibrillation.
In chapter 3, we study the role of cardiac fibroblasts in ventricular tissue; we use the TNNP04 model for the myocyte cell, and the fibroblasts are modelled as passive cells. Cardiac fibroblasts, when coupled functionally with myocytes, can modulate their electrophysiological properties at both cellular and tissue levels. Therefore, it is important to study the effects of such fibroblasts when they are coupled with myocytes. Chapter 3 contains our detailed and systematic study of spiral-wave dynamics in the presence of fibroblasts in both homogeneous and inhomogeneous domains of the TNNP04 model for cardiac tissue. We carry out extensive numerical studies of such modulation of electrophysiological properties in mathematical models for (a) single myocyte fibroblast (MF) units and (b) two-dimensional (2D) arrays of such units; our models build on earlier ones and allow for no, one-way, or two-way MF couplings. Our studies of MF units elucidate the dependence of the action-potential (AP) morphology on parameters such as Ef , the fibroblast resting membrane potential, the fibroblast conductance Gf , and the MF gap-junctional coupling Ggap. Furthermore, we find that our MF composite can show autorhythmic and oscillatory behaviors in addition to an excitable response. Our 2D studies use
(a) both homogeneous and inhomogeneous distributions of fibroblasts, (b) various ranges for parameters such as Ggap, Gf , and Ef , and (c) intercellular couplings that can be no, one-way, and two-way connections of fibroblasts with myocytes. We show, in particular, that the plane-wave conduction velocity CV decreases as a function of Ggap, for no and one-way couplings; however, for two-sided coupling, CV decreases initially and then increases as a function of Ggap, and, eventually, we observe that conduction failure occurs for low values of Ggap. In our homogeneous studies, we find that the rotation speed and stability of a spiral wave can be controlled either by controlling Ggap or Ef . Our studies with fibroblast inhomogeneities show that a spiral wave can get anchored to a local fibroblast inhomogeneity. We also study the efficacy of a low-amplitude control scheme, which has been suggested for the control of spiral-wave turbulence in mathematical models for cardiac tissue, in our MF model both with and without heterogeneities.
In chapter 4, we carry out a detailed, systematic study of spiral-wave dynamics in the presence of periodic deformation (PD) in two state-of-the-art mathematical models of human ventricular tissue, namely, the TNNP04 model and the TP06 model. To the best of our knowledge, our work is the first, systematic study of the dynamics of spiral waves of electrical activation and their transitions, in the presence of PD, in such biophysically realistic mathematical models of cardiac tissue. In our studies, we use three types of initial conditions whose time evolutions lead to the following states in the absence of PD: (a) a single rotating spiral (RS),
(b) a spiral-turbulence (ST) state, with a single meandering spiral, and (c) an ST state with multiple broken spirals for both these models. We then show that the imposition of PD in these three cases leads to a rich variety of spatiotemporal pat- terns in the transmembrane potential including states with (a) an RS state with n-cycle temporal evolution (here n is a positive integer), (b) rotating-spiral states with quasiperiodic (QP) temporal evolution, (c) a state with a single meandering spiral MS, which displays spatiotemporal chaos, (d) an ST state, with multiple bro- ken spirals, and (e) a quiescent state in which all spirals are absorbed (SA). For all three initial conditions, precisely which one of the states is obtained depends on the amplitudes and the frequencies of the PD in the x and y directions. We also suggest specific experiments that can test the results of our simulations. We also study, in the presence of PD, the efficacy of a low-amplitude control scheme that has been suggested, hitherto only without PD, for the control of spiral-wave turbulence, via low-amplitude current pulses applied on a square mesh, in mathematical models for cardiac tissue. We also develop line-mesh and rectangular-mesh variants of this control scheme. We find that square- and line-mesh-based, low-amplitude control schemes suppress spiral-wave turbulence in both the TP06 and TNNP04 models in the absence of PD; however, we show that the line-based scheme works with PD only if the PD is applied along one spatial direction. We then demonstrate that a minor modification of our line-based control scheme can suppress spiral-wave turbulence: in particular, we introduce a rectangular-mesh-based control scheme, in which we add a few control lines perpendicular to the parallel lines of the line- based control scheme; this rectangular-mesh scheme is a significant improvement over the square-mesh scheme because it uses fewer control lines than the one based on a square mesh.
In chapter 5, we have carried out detailed numerical studies of (a) a single unit of an endocardial cell and Purkinje cell (EP) composite and (b) a two-dimensional bilayer, which contains such EP composites at each site. We have considered bio- physically realistic ionic models for human endocardial cells (Ecells) and Purkinje cells (Pcells) to model EP composites. Our study has been designed to elucidate the sensitive dependence, on parameters and initial conditions, of (a) the dynamics of EP composites and (b) the spatiotemporal evolution of spiral waves of electrical activation in EP-bilayer domains. We examine this dependence on myocyte parameters by using the three different parameter sets P1, P2, and P3; to elucidate the initial-condition dependence we vary the time at which we apply the S2 pulse in our S1-S2 protocol; we also investigate the dependence of the spatiotemporal dynamics of our system on the EP coupling Dgap, and on the number of Purkinje- ventricular junctions (PVJs), which are measured here by the ratio R, the ratio of the total number of sites to the number of PVJs in our simulation domain.
Our studies on EP composites show that the frequency of autorhythmic activity of a P cell depends on the diffusive gap-junctional conductance Dgap. We perform a set of simulations to understand the source-sink relation between the E and P cells in an EP composite; such a source-sink relation is an important determinant of wave dynamics at the tissue level. Furthermore, we have studied the restitution properties of an isolated E cell and a composite EP unit to uncover this effect on wave dynamics in 2D, bilayers of EP composites.
Autorhythmicity is an important property of Purkinje cell; it helps to carry electrical signals rapidly from bundle of His to the endocardium. Our investigation of an EP composite shows that the cycle length (CL) of autorhythmic activity decreases, compared to that of an uncoupled Purkinje cell. Furthermore, we find that the APD increases for an EP composite, compared to that of an uncoupled P cell. In our second set of simulations for an EP-composite unit, we have obtained the AP behaviors and the amount of flux that flows from the E to the P cell during the course of the AP. The direction of flow of this flux is an important quantity that identifies which one of these cells act as a source or a sink in this EP composite. We have found that the P cell in an EP composite acts as a stimulation-current source for the E cell in the depolarization phase of the AP, when the stimulus is applied to both cells or to the P cell only. However, the P cell behaves both as a source and a sink when the stimulus is applied to the E cell only. In our third set of simulations for an EP composite unit, we have calculated the restitution of the APD; this plays an important role in deciding the stability of spiral waves in mathematical models for cardiac tissue. Our simulation shows that, for the EP composite with high coupling (Dgap = Dmm~10), the APDR slope decreases, relative to its value for an isolated E cell, for parameter sets P1 and P2, and first increases (for 50 ≤ DI ≤ 100 ms) and then decreases for the parameter set P3 ; however, for low coupling (Dgap = Dmm~100), the variation of the AP D as function of DI, for an EP composite, shows biphasic behavior for all these three parameter sets. We found that the above dynamics in EP cable type domains, with EP composites, depends sensitively on R.
We hope our in silico studies of spiral-wave dynamics in a variety of state-of-the- art ionic models for ventricular tissue will stimulate more experimental studies that examine such dynamics
Particles and Fields in Partial-differential-equation Models for fluid and Superfluid Turbulence
No full textWe have shown that uniformly rotating vortex-containing gravitationally-bound
solutions of the GPPE can be generated by starting the evolution from initial data
obtained by integrating to convergence the (imaginary-time) ARGLPE (6.7). We
have built on this GPPE and introduced a minimal model, with a single, angular,
dynamical variable for a solid crust coupled with a rotating GPPE star. We
have demonstrated that this model exhibits stick-slip dynamics, whose statistical
properties we have characterized by computing the event-size and event-duration
CPDFs Q( Jc=Jc0) and Q(ted
0), which show power-law forms, and the waitingtime
CPDF Q(twt
0), which exhibits an exponential tail. These SOC-type desiderata
are in consonance with measurements on a class of pulsars [22].
We plan to study pulsar-glitch models that are more realistic than our minimal
model. Examples include models with (a) a solid crust with 6 degrees of freedom, 3
rotational and 3 translational, instead of only one angle of rotation, or (b) a super6.4.
Conclusions 129
conducting component with magnetic flux tubes. We expect that such generalizations
of our minimal model should help us to undertand all the types of statistical
properties that are displayed by pulsar glitches in different pulsar
Particles and Fields in Superfluid Turbulence : Numerical and Theoretical Studies
Full text linkIn this thesis we study a variety of problems in superfluid turbulence, princi-pally in two dimensions. A summary of the main results of our studies is given below; we indicate the Chapters in which we present these.
In Chapter 1, we provide an overview of several problems in superfluid turbulence with special emphasis on background material for the problems we study in this thesis. In particular, we give: (a) a brief introduction of fluid turbulence; (b) an overview of superfluidity and the phenomenological two-fluid model; (c) a brief overview of experiments on superfluid turbulence; (d) an introductory accounts of the phenomenological models used in the study of superfluid turbulence. We end with a summary of the problems we study in subsequent Chapters of this thesis.
In Chapter 2, we present a systematic, direct numerical simulation of the two-dimensional, Fourier-truncated, Gross-Pitaevskii equation to study the turbulent evolutions of its solutions for a variety of initial conditions and a wide range of parameters. We find that the time evolution of this system can be classified into four regimes with qualitatively different statistical properties. First, there are transients that depend on the initial conditions. In the second regime, power- law scaling regions, in the energy and the occupation-number spectra, appear and start to develop; the exponents of these power laws and the extents of the scaling regions change with time and depend on the initial condition. In the third regime, the spectra drop rapidly for modes with wave numbers k > kc and partial thermalization takes place for modes with k < kc ; the self-truncation wave number kc(t) depends on the initial conditions and it grows either as a power of t or as log t. Finally, in the fourth regime, complete thermalization is achieved and, if we account for finite-size effects carefully, correlation functions and spectra are consistent with their nontrivial Berezinskii-Kosterlitz-Thouless forms. Our work is a natural generalization of recent studies of thermalization in the Euler and other hydrodynamical equations; it combines ideas from fluid dynamics and turbulence, on the one hand, and equilibrium and nonequilibrium statistical mechanics on the other.
In Chapter 3, we present the first calculation of the mutual-friction coefficients α and α (which are parameters in the Hall-Vinen-Bekharevich-Khalatnikov two-fluid model that we study in chapter 5) as a function of temperature in a homogeneous Bose gas in two-dimensions by using the Galerkin-truncated Gross-Pitaevskii equation, with very special initial conditions, which we obtain by using the advective, real, Ginzburg-Landau equation (ARGLE) and an equilibration procedure that uses a stochastic Ginzburg-Landau equation (SGLE). We also calculate the normal-fluid density as a function of temperature.
In Chapter 4, we elucidate the interplay of particles and fields in superfluids, in both simple and turbulent flows. We carry out extensive direct numerical simulations (DNSs) of this interplay for the two-dimensional (2D) Gross-Pitaevskii (GP) equation. We obtain the following results: (1) the motion of a particle can be chaotic even if the superfluid shows no sign of turbulence; (2) vortex motion depends sensitively on particle charateristics; (3) there is an effective, superfluid-mediated, attractive interaction between particles; (4) we introduce a short-range repulsion between particles, with range rSR, and study two- and many-particle collisions; in the case of two-particle, head-on collisions, we find that, at low values of rSR, the particle collisions are inelastic with coefficient of restitution e = 0; and, as we in-crease rSR, e becomes nonzero at a critical point, and finally attains values close to 1; (5) assemblies of particles and vortices show rich, turbulent, spatio-temporal evolution.
In Chapter 5, we present results from our direct numerical simulations (DNSs) of the Hall-Vinen-Bekharevich-Khalatnikov (HVBK) two-fluid model in two dimensions. We have designed these DNSs to study the statistical properties of inverse and forward cascades in the HVBK model. We obtain several interesting results that have not been anticipated hitherto: (1) Both normal-fluid and superfluid energy spectra, En(k) and Es(k), respectively, show inverse- and forward-cascade regimes; the former is characterized by a power law Es(k) En(k) kα whose exponent is consistent with α 5/3. (2) The forward-cascade power law depends on (a) the friction coefficient, as in 2D fluid turbulence, and, in addition, on (b) the coefficient B of mutual friction, which couples normal and superfluid compo-nents. (3) As B increases, the normal and superfluid velocities, un and us, re-spectively, get locked to each other, and, therefore, Es(k) En(k), especially in the inverse-cascade regime. (4) We quantify this locking tendency by calculating the probability distribution functions (PDFs) P(cos(θ)) and P(γ), where the angle θ ≡ (un • us)/( |un||us|) and the amplitude ratio γ = |un|/|us |; the former has a peak at cos(θ) = 1; and the latter exhibits a peak at γ = 1 and power-law tails on both sides of this peak. (4) This locking increases as we increase B, but the power-law exponents for the tails of P(γ) are universal, in so far as they do not depend on B, ρn/ρ, and the details of the energy-injection method. (5) We characterize the energy and enstrophy cascades by computing the energy and enstrophy fluxes and the mutual-friction transfer functions for all wave-number scales k.
In Chapter 6, we examine the multiscaling of structure functions in three-dimensional superfluid turbulence by using a shell-model for the three-dimensional HVBK equations. Our HVBK shell model is based on the GOY shell model. In particular, we examine the dependence of multiscaling on the normal-fluid fraction and the mutual-friction coefficients.
We hope our in silico studies of 2D and 3D superfluid turbulence will stimulate new experimental, numerical, and theoretical studies
Numerical Studies of Problems in Turbulence : 1) Fluid Films with Polymer Additives; 2) Fluid Films with Inertial and Elliptical Particles; 3) Scaled Vorticity Moments in Three- and Two-dimensional Turbulence
Full text linkIn this thesis we study a variety of problems in fluid turbulence, principally in two dimensions. A summary of the main results of our studies is given below; we indicate the Chapters in which we present these.
In Chapter 1, we provide an overview of several problems in turbulence with special emphasis on background material for the problems we study in this thesis. In particular, we give (a) natural and laboratory examples of fluid turbulence, (b) and introductory accounts of the equations of hydrodynamics, without and with polymer additives, Eulerian and Lagrangian frameworks, and the equations of motion of inertial particles in fluid flows. We end with a summary of the problems we study in subsequent Chapters of this thesis.
In Chapter 2, we carry out the most extensive and high-resolution direct numerical simulation, attempted so far, of homogeneous, isotropic turbulence in two-dimensional fluid films with air-drag-induced friction and with polymer additives. Our study reveals that the polymers (a) reduce the total fluid energy, enstrophy, and palinstrophy, (b) modify the fluid energy spectrum both in inverse- and forward-cascade regimes, (c) reduce small-scale intermittency, (d) suppress regions of large vorticity and strain rate, and (e) stretch in strain-dominated regions. We compare our results with earlier experimental studies and propose new experiments.
In Chapter 3, we perform a direct numerical simulation (DNS) of the forced, incompressible two-dimensional Navier-Stokes equation coupled with the FENE-P equations for the polymer- conformation tensor. The forcing is such that, without polymers and at low Reynolds numbers Re, the lm attains a steady state that is a square lattice of vortices and anti-vortices. We nd that, as we increase the Weissenberg number (Wi), this lattice undergoes a series of nonequilibrium phase transitions, first to spatially distorted, but temporally steady, crystals and then to a sequence of crystals that oscillate in time, periodically, at low Wi, and quasiperiodically, for slightly larger Wi. Finally, the system becomes disordered and displays spatiotepmoral chaos and elastic turbulence. We then obtain the nonequilibrium phase diagram for this system, in the Wi − Re plane, and show that (a) the boundary between the crystalline and turbulent phases has a complicated, fractal-type character and (b) the Okubo-Weiss parameter provides us with a natural measure for characterizing the phases and transitions in this diagram.
In Chapter 4, our study is devoted to heavy, inertial particles in two-dimensional (2D) tur- bulent, but statistically steady, flows that are homogeneous and isotropic. The inertial particles are distributed uniformly in our simulation domain when St = 0; they start to cluster as St increases; this clustering tendency reaches a maximum at St 1 and decreases thereafter. We then obtain PDFs of and show that their left tails, which come from extensional regions, do not depend sensitively on St; in contrast, their right tails, from the vortical regions of the flow, are consistent with the exponential form ∼ exp ‰− + Ž; and we nd that the scale + decreases with St until St _0:1 and then saturates at a value _0:75. Our persistence-type studies yield the following results, when we consider forcing that leads to an energy spectrum that is dominated by a forward-cascade regime: In strain-dominated or extensional regions of the flow, wend that the cumulative PDF of the persistence time decays exponentially; this decay yields a time scale T−, which increases rapidly with St, at low values of St, but more slowly after St _0:75. By contrast, in vortical regions of the flow, this cumulative PDF displays a tail that has power-law and exponential parts; the power-law part yields the persistence exponent _ and the exponential tail gives a time scale T−; _ increases with St, whereas T− decreases with St; _ and T− reach saturation values as St increases. From the cumulative PDF of the particle mean-square displacement r2, we obtain the time scale Ttrans at which there is a crossover from ballistic to diffusive behavior; we _nd that Ttrans increases with St. The PDFs of v2, the square of the particle velocity, and v2 ejected, the square of the velocity of a particle just as it is ejected from a region with _ > 0 (vortical region) to one that has _ < 0 (extensional region), do not show a significant dependence on St; the tails of these PDFs are characterized by power-law decays with exponents _1 and _5~3, respectively. Our next set of results deal with statistical properties of special combinations of the acceleration a =dv~dt and the velocity v. For instance, the curvature of the trajectory is _ =aÙ~v2, where the subscript Ù denotes the component perpendicular to the particle trajectory; we obtain PDFs of _ and _nd there from that particles in regions of elongational flow have, on average, trajectories with a lower curvature than particles in vortical regions; this . We also determine how the number of number of points NI , at which a ×v changes sign along a particle trajectory, as time increases; we _nd that the increase of NI with time and decrease as St increases. Our ninth set of results show that the characteristic decay time T_ for decreases with St.
In Chapter 5, we study the statistical properties of orientation and rotation dynamics of elliptical tracer particles in two-dimensional, homogeneous and isotropic turbulence by direct numerical simulations. We consider both the cases in which the turbulent flow is generated by forcing at large and intermediate length scales. We show that the two cases are qualitatively different. For the large-scale forcing, the spatial distribution of particle orientations forms large- scale structures, which are absent for the intermediate-scale forcing. The alignment with the local directions of the flow is much weaker in the latter case than in the former. For the intermediate- scale forcing, the statistics of rotation rates depends weakly on the Reynolds number and on the aspect ratio of particles. In contrast with what is observed in three-dimensional turbulence, in two dimensions the mean-square rotation rate decreases as the aspect ratio increases.
In Chapter 6, we study the issue of intermittency in numerical solutions of the 3D Navier-Stokes equations on a periodic box [0; L]3. This is addressed through four sets of numerical simulations that calculate a new set of variables defined by Dm(t) = where All four simulations unexpectedly show that the Dm are ordered for m =1 ….,9 such that Dm+1 <Dm. Moreover, the Dm squeeze together such that Dm+1/Dm 1 as m increases. The values of D1 lie far above the values of the rest of the Dm, giving rise to a suggestion that a depletion of nonlinearity is occuring which could be the cause of Navier{Stokes regularity. The first simulation, by R. Kerr, is of very anisotropic decaying turbulence ; the second and third, which have been carried out by me, are of decaying isotropic turbulence from random initial conditions and forced isotropic turbulence at fixed Grashof number, respectively ; the fourth, by D. Donzis, is of very-high-Reynolds-number forced, stationary, isotropic turbulence at resolutions up to 40963 collocation points. For the sake of completeness and for a comparison of the data from all these four simulations, all the results are presented; however, in the Sections that deal with the simulations, I indicate who carried out the calculations reported there. I also present an extension of this work to two-dimensional fluid turbulence; this has not been submitted for publication so far.
We hope our in silico studies of 2D and 3D turbulence will stimulate new experimental, numerical, and theoretical studies
Scaling And Universality In Driven Systems : The Sandpile Model And The GOY Shell Model Of Turbulence
Full text linkSpiral- And Scroll- Wave Dynamics In Ironically And Anatomically Realistic Mathematical Models For Mammalian Ventricular Tissue
Full text linkCardiac arrhythmias, such as ventricular tachycardia (VT) and ventricular fibrillation (VF), are among the leading causes of death in the industrialized world. There is growing consensus that these arrhythmias are associated with the formation of spiral and scroll waves of electrical activation in mammalian cardiac tissue; whereas single spiral and scroll waves are believed to be associated with VT, their turbulent analogs are associated with VF. Thus, the study of these waves is an important biophysical problem in-so-far-as to develop an understanding of the electrophysiological basis of VT and VF.
In this thesis, we provide a brief overview of recent numerical studies of spiral- and scroll-wave dynamics in mathematical models of mammalian cardiac tissue. In addition to giving a description of how spiral and scroll waves can be initiated in such models, how they evolve, how they interact with conduction and ionic inhomogeneities, how their dynamics is influenced by the size and geometry of the heart, we also discuss how active Purkinje networks and passive fibroblast clusters modify the electrical activity of cardiomyocytes, and the relevance of such studies to defibrillation.
In Chapter 2 we present a systematic study of the combined effects of muscle-fiber rotation and inhomogeneities on scroll-wave dynamics in the TNNP (ten Tusscher Noble Noble Panfilov) model for human cardiac tissue. In particular, we use the three-dimensional (3D) TNNP model with fiber rotation and consider both conduction and ionic inhomogeneities. We find that, in addition to displaying a sensitive dependence on the positions, sizes, and types of inhomogeneities, scroll-wave dynamics also depends delicately upon the degree of fiber rotation. We find that the tendency of scroll waves to anchor to cylindrical conduction inhomogeneities increases with the radius of the inho-mogeneity. Furthermore, the filament of the scroll wave can exhibit drift or meandering, transmural bending, twisting, and break-up. If the scroll-wave filament exhibits weak meandering, then there is a fine balance between the anchoring of this wave at the inho-mogeneity and a disruption of wave-pinning by fiber rotation. If this filament displays strong meandering, then again the anchoring is suppressed by fiber rotation; also, the scroll wave can be eliminated from most of the layers only to be regenerated by a seed wave. Ionic inhomogeneities can also lead to an anchoring of the scroll wave; scroll waves can now enter the region inside an ionic inhomogeneity and can display a coexistence of spatiotemporal chaos and quasi-periodic behavior in different parts of the simulation domain. We discuss the experimental implications of our study.
In Chapter 3 we present a comprehensive numerical study of plane and scroll waves of electrical activation in two state-of-the-art ionic models for rabbit and pig cardiac tissue. We use anatomically realistic, 3D simulation domains, account for muscle-fiber rotation, and show how to include conduction and ionic inhomogeneities in these models; we consider both localized and randomly distributed inhomogeneities. Our study allows us to compare scroll-wave dynamics, with and without inhomogeneities, in these rabbit-and pig-heart models at a level that has not been attempted hitherto. We begin with a comparison of single-cell action potentials (APs) and ionic currents in the Bers-Puglisi (BP) and modified-Luo-Rudy I (mLRI) models for rabbit- and pig-myocytes, respec-tively. We then show how, for plane-wave propagation in rabbit- and pig-heart models, the conduction velocity CV and wavelength λ depend on the distance of the plane of measurement from the plane containing the heart apex. Without inhomogeneities, and in the parameter r´egime in which these models display scroll waves, the rabbit-heart model supports a single scroll wave, which rotates periodically, whereas the pig-heart model supports two scroll waves, which rotate periodically, but with a slight difference in phase; this is partly because the rabbit-heart model is smaller in size, than the pig-heart one. With randomly-distributed inhomogeneities, we find that the rabbit-heart model loses its ability to support electrical activity, even at inhomogeneity concentra-tions as low as 5%. In the pig-heart model, we obtain rich, scroll-wave dynamics in the presence of localized or distributed inhomogeneities, both of conduction and ionic types; often, but not always, scroll waves get anchored to localized inhomogeneities; and distributed inhomogeneities can lead to scroll-wave break up.
In Chapter 4, we present a comprehensive numerical study of spiral-and scroll-wave dynamics in a state-of-the-art mathematical model for human ventricular tissue with fiber rotation, transmural heterogeneity, myocytes, and fibroblasts. Our mathematical model introduces fibroblasts randomly, to mimic diffuse fibrosis, in the ten Tusscher-Noble-Noble-Panfilov (TNNP) model for human ventricular tissue; the passive fibrob-lasts in our model do not exhibit an action potential in the absence of coupling with myocytes; and we allow for a coupling between nearby myocytes and fibroblasts. Our study of a single myocyte-fibroblast (MF) composite, with a single myocyte coupled to Nf fibroblasts via a gap-junctional conductance Ggap, reveals five qualitatively different responses for this composite. Our investigations of two-dimensional domains with a ran-dom distribution of fibroblasts in a myocyte background reveal that, as the percentage Pf of fibroblasts increases, the conduction velocity of a plane wave decreases until there is conduction failure. If we consider spiral-wave dynamics in such a medium we find, in two dimensions, a variety of nonequilibrium states, temporally periodic, quasiperi-odic, chaotic, and quiescent, and an intricate sequence of transitions between them; we also study the analogous sequence of transitions for three-dimensional scroll waves in a three-dimensional version of our mathematical model that includes both fiber rotation and transmural heterogeneity. We thus elucidate random-fibrosis-induced nonequilib-rium transitions, which lead to conduction block for spiral waves in two dimensions and scroll waves in three dimensions. We explore possible experimental implications of our mathematical and numerical studies for plane-, spiral-, and scroll-wave dynamics in cardiac tissue with fibrosis.
In Chapter 5 we present a detailed numerical study of the electrophysiological in-teractions between a random Purkinje network and simulated human endocardial tissue, (a) in the presence of, and (b) in the absence of existing electrical excitation in the system. We study the dependence of the activation-onset-time (ta) on the strength of coupling (Dmp) between the myocyte layer and the Purkinje network, in the absence of any external stimulus. Since the connection between the endocardial layer and the Purkinje network occurs only at discrete points, we also study the dependence of ta on the number of Purkinje-myocyte junctions (PMJs) at fixed values of Dmp, in the ab-sence of any applied excitation. We study signal propagation in the system; our results demonstrate the situations of (a) conduction block from the Purkinje layer to the endo-cardial layer, (b) anterograde propagation of the excitation from the Purkinje layer to the endocardium, (c) retrograde propagation of the excitation from the endocardium to the Purkinje layer and (d) development of reentrant circuits in the Purkinje layer that lead to formation of ectopic foci at select PMJs. We extend our study to explore the effects of Purkinje-myocyte coupling on spiral wave dynamics, whereby, we find that such coupling can lead to the distortion and breakage of the parent rotor into multiple rotors within the system; with or without internal coherence. We note that retrograde propa-gation leads to the development of reentrant circuits in the Purkinje network that help to initiate and stabilize ectopic foci. However, in some cases, Purkinje-myocyte coupling can also lead to the suppression of spiral waves. Finally we conduct four representative simulations to study the effects of transmural heterogeneity, fiber rotation and coupling with a non-penetrating Purkinje network on a three dimensional slab of cardiac tissue.
Lastly, In Chapter 6, we study reentry associated with inexcitable obstacles in the ionically-realistic TNNP model for human ventricular tissue, under the influence of high-frequency stimulation. When a train of plane waves successively impinge upon an obstacle, the obstacle splits these waves as they tend to propagate past it; the emergent broken waves can either travel towards each other, bridging the gap introduced by the obstacle at the time of splitting, or, they can travel away from each other, resulting in the growth of the gap. The second possibility eventually results in the formation of spiral waves. This phenomenon depends on frequency of the waves. At high frequency, the excitability of the tissue decreases and the broken waves have a tendency to move apart. Hence high-frequency stimulation increases the chances of reentry in cardiac tissue. We correlate the critical period of pacing that leads to reentry in the presence of an inexcitable obstacle, with the period of spiral waves, formed in the homogeneous domain, and study how the critical period of pacing depends on the parameters of the model
Electrical waves in mathematical models of human-ventricular tissue: effects of different modeling formalisms, electrophysiological factors, and heterogeneities on ventricular arrhythmias and the termination of these arrhythmias via a deep-learning approach
No full textSudden cardiac death (SCD) remains one of the significant causes of
mortality in industrialized and developing countries. SCD is often caused by
life-threatening cardiac arrhythmias like ventricular tachycardia and ventric-
ular fibrillation, which are associated with the formation of spiral, broken-
spiral (two dimensions), and scroll waves (in three dimensions) of electrical
activation in cardiac tissue. To understand, and eventually control, such
arrhythmias, it is important to carry out in vivo, ex vivo, in vitro, and in
silico studies; the last of these has become increasingly important over the
past three decades. In this thesis, I have carried out several in silico stud-
ies of state-of-the-art mathematical models for cardiac tissue which focus
mainly on four themes: (I) The effects of subcellular ion-channel modeling
on electrical-wave dynamics in cardiac tissue; in particular, I have carried
out a detailed comparison of wave dynamics in Hodgkin-Huxley and Markov-
state formalisms for the Sodium (Na) channel in some mathematical models
for human cardiac tissue. (II) The frequency and tip-trajectory of spiral
waves and its dependence on electrophysiological parameters in a realistic
mathematical model for human-ventricular tissue with and without fibrob-
lasts. (III) The arrhythmogenicity of cardiac fibrosis and its dependence
on the lacunarity parameter and Betti numbers of patterns of fibrotic tis-
sue. (IV) The efficient elimination of pathological spiral and broken-spiral
waves via a deep-learning-assisted detection and termination of spiral- and
broken-spiral waves in mathematical models for cardiac tissue.
In Chapter 2 we investigate the effects of different subcellular model-
ing formalsims for an ion-channel and on its properties, the action-potential
of a single cell, and spiral and scroll waves in two- and three-dimensional
human ventricular tissue. In particular, we compare and contrast the ex-
citation properties of cardiac myocytes and cardiac tissue modelled by (a)
a Hodgkin-Huxley-model (HHM) and (b) Markov-chain-model (MM) for-
malisms for the sodium (Na) ion channel. Specifically, we bring out the
differences between HHM and MM formalisms, for both wild-type (WT)
and mutant (MUT ) models, for ion-channel kinetics, single-myocyte action
potentials, and the spatiotemporal evolutions of spiral and scroll waves in
different mathematical models of cardiac tissue. We show that the kinetic
properties of Na ion channels are not the same for HHM and MM models; in
particular, the range of values of the trans-membrane potential V m , in which
there is a significant window current, depends significantly on these models,
so there are marked differences in the opening times of the Na ion chan-
nels, the maximal amplitude of the Na current, and the presence or absence
of a late Na current. Furthermore, these changes lead to different excita-
tion behaviours in cardiac tissue; specifically, two of the WT models showstable spiral waves, but the other one shows meandering and transiently
breaking spiral waves. Our results are based on extensive direct numerical
simulations of waves of electrical activation in these models, in two- and
three-dimensional (2D and 3D) homogeneous simulation domains and also
in domains with localised heterogeneities, either obstacles with randomly
distributed inexcitable regions or mutant cells in a wild-type background.
Our study brings out the sensitive dependence of spiral- and scroll-wave dy-
namics on these five models and the parameters that define them. We list
desiderata for a good model for the Na wild-type ion-channel; we use these
desired properties to select one of the MM models that we study.
In Chapter 3 we study the effects of different electrophysiological pa-
rameters in determining the frequency of the spiral waves in cardiac tissue
by. Spiral waves of excitation in cardiac tissue are associated with life-
threatening cardiac arrhythmias. It is, therefore, important to study the
electrophysiological factors that affect the dynamics of these spiral waves.
By using an electrophysiologically detailed mathematical model of a myocyte
(cardiac cell), we study the effects of cellular parameters, such as membrane-
ion-channel conductances, on the properties of the action-potential (AP) of
a myocyte. We then investigate how changes in these properties, specifically
the upstroke velocity and the AP duration (APD), affect the frequency ω of
a spiral wave in the mathematical model that we use for human-ventricular
tissue. We find that an increase (decrease) in this upstroke-velocity or
a decrease (increase) in the AP duration increases (decreases) the spiral-
wave frequency. We also study how other intercellular factors, such as the
fibroblast-myocyte coupling, diffusive coupling strength, and the effective
number of neighboring myocytes, modulate the ω. Finally, we demonstrate
how a spiral wave can drift to a region with a high density of inexcitable
cells called fibroblasts. Our results provide a natural explanation for the
anchoring of spiral waves in highly fibrotic regions in fibrotic hearts.
In Chapter 4 we study diffuse fibrosis (DF), interstitial fibrosis (IF),
patchy fibrosis (PF), and compact fibrosis (CF) and their arrhythmogenicity
in cardiac tissue. We use mathematcal models for DF, IF, PF, and CF to
study patterns of fibrotic cardiac tissue that have been generated by using
Perlin noise and also by a simple model. We show that the fractal dimension
D, the lacunarity L, and the Betti numbers β0 and β1 of such patterns,
which we term as fibrotic-tissue markers, can be used to characterise the
arrhythmogenicity of different types of cardiac fibrosis. We hypothesize,
and then demonstrate by extensive in silico studies of detailed mathematical
models for cardiac tissue, that the arrhytmogenicity of fibrotic tissue is high
when β0 is large and the lacunarity parameter b is small.
In Chapter 5 we devise a new defibrillation scheme that uses a convo-
lutional neural network (CNN), which we first train by using images from
our simulations of waves of electrical activation in mathematical models
for two-dimensional cardiac tissue. Unbroken- and broken-spiral waves, inpartial-differential-equation (PDE) models for cardiac tissue, are the mathe-
matical analogs of life-threatening cardiac arrhythmias, namely, ventricular
tachycardia (VT) and ventricular-fibrillation (VF). We develop (a) a deep-
learning method for the detection of unbroken- and broken-spiral waves and
(b) the elimination of such waves, e.g., by the application of low-amplitude
control currents in the cardiac-tissue context. Our method is based on a
convolutional neural network (CNN) that we train to distinguish between
patterns with spiral-waves S and without spiral-waves N S . We obtain
these patterns by carrying out extensive direct numerical simulations (DNSs)
of PDE models for cardiac tissue in which the transmembrane potential V ,
when portrayed via pseudocolor plots, displays patterns of electrical acti-
vation of types S and N S . We then utilize our trained CNN to obtain,
for a given pseudocolor image of V , a heatmap that has high intensity in
the regions where this image shows the cores of spiral waves and the associ-
ated wave fronts. Given this heatmap, we show how to apply low-amplitude
currents of 2D-Gaussian profile to eliminate spiral-waves efficiently. Our in
silico results are of direct relevance to the detection and elimination of these
arrhythmias because our elimination of unbroken or broken-spiral waves is
the mathematical analog of low-amplitude defibrillation.Department of Science and Technology (DST), India, for financial support, and Supercomputer Education and Research Centre (SERC, IISc) for computational resource
Studies of Static and Dynamic Multiscaling in Turbulence
No full textThe physics of turbulence is the study of the chaotic and irregular behaviour in driven fluids. It is ubiquitous in cosmic, terrestrial and laboratory environments. To describe the properties of a simple incompressible fluid it is sufficient to know its velocity at all points in space and as a function of time. The equation of motion for the velocity of such a fluid is the incompressible Navier–Stokes equation. In more complicated cases, for example if the temperature of the fluid also fluctuates in space and time, the Navier–Stokes equation must be supplemented by additional equations. Incompressible fluid turbulence is the study of solutions of the Navier–Stokes equation at very high Reynolds numbers, Re, the dimensionless control parameter for this problem. The chaotic nature of these solutions leads us to characterise them by their statistical properties. For example, statistical properties of fluid turbulence are characterised often by structure functions of velocity. For intermediate range of length scales, that is the inertial range, these structure functions show multiscaling. Most studies concentrate on equal-time structure functions which describe the equal-time statistical properties of the turbulent fluid. Dynamic properties can be measured by more general time-dependent structure functions. A major challenge in the field of fluid turbulence is to understand the multiscaling properties of both the equal-time and time-dependent structure functions of velocity starting from the Navier–Stokes equation. In this thesis we use numerical and analytical techniques to study scaling and multiscaling of equal-time and time-dependent structure functions in turbulence not only in fluids but also in advection of passive-scalars and passive vectors, and in randomly forced Burgers equation.CSIR (INDIA), IFCPARTypeset in LATEX by the autho
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