Biomedical subjects

Itamar Procaccia

Publications and source records attributed to Itamar Procaccia.

At least 19 recordsLinked to original sources

PubMed2004

Roughening of fracture surfaces: the role of plastic deformation.

Post mortem analysis of fracture surfaces of ductile and brittle materials on the microm-mm and the nm scales, respectively, reveal self-affine cracks with anomalous scaling exponent zeta approximately 0.8 in three dimensions and zeta approximately 0.65 in two dimensions. Attempts to use elasticity theory to explain this result failed, yielding exponent zeta approximately 0.5 up to logarithms. We show that when the cracks propagate via plastic void formations in front of the tip, followed by void coalescence, the void positions are positively correlated to yield exponents higher than 0.5.

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PubMed2004

Drag reduction by polymers in wall bounded turbulence.

We elucidate the mechanism of drag reduction by polymers in turbulent wall-bounded flows: while momentum is produced at a fixed rate by the forcing, polymer stretching results in the suppression of momentum flux to the wall. On the basis of the equations of fluid mechanics we develop the phenomenology of the "maximum drag reduction asymptote" which is the maximum drag reduction attained by polymers. Based on Newtonian information only we demonstrate the existence of drag reduction, and with one experimental parameter we reach agreement with the experimental measurements.

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PubMed2004

Algorithm for parallel Laplacian growth by iterated conformal maps.

We report an algorithm to generate Laplacian growth patterns using iterated conformal maps. The difficulty of growing a complete layer with local width proportional to the gradient of the Laplacian field is overcome. The resulting growth patterns are compared to those obtained by the best algorithms of direct numerical solutions. The fractal dimension of the patterns is discussed.

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PubMed2004

Stress field around arbitrarily shaped cracks in two-dimensional elastic materials.

The calculation of the stress field around an arbitrarily shaped crack in an infinite two-dimensional elastic medium is a mathematically daunting problem. With the exception of few exactly soluble crack shapes the available results are based on either perturbative approaches or on combinations of analytic and numerical techniques. We present here a general solution of this problem for any arbitrary crack. Along the way we develop a method to compute the conformal map from the exterior of a circle to the exterior of a line of arbitrary shape, offering it as a superior alternative to the classical Schwartz-Cristoffel transformation. Our calculation results in an accurate estimate of the full stress field and in particular of the stress intensity factors K(I) and K(II) and the T-stress which are essential in the theory of fracture.

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PubMed2004

Theory of concentration dependence in drag reduction by polymers and of the maximum drag reduction asymptote.

A simple model of the effect of polymer concentration on the amount of drag reduction in turbulence is presented, simulated, and analyzed. The qualitative phase diagram of drag coefficient versus Reynolds number (Re) is recaptured in this model, including the theoretically elusive onset of drag reduction and the maximum drag reduction (MDR) asymptote. The Re-dependent drag and the MDR are analytically explained, and the dependence of the amount of drag on material parameters is rationalized.

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PubMed2003

Eulerian statistically preserved structures in passive scalar advection.

We analyze numerically the time-dependent linear operators that govern the dynamics of Eulerian correlation functions of a decaying passive scalar advected by a stationary, forced two-dimensional Navier-Stokes turbulence. We show how to naturally discuss the dynamics in terms of effective compact operators that display Eulerian statistically preserved structures which determine the anomalous scaling of the correlation functions. In passing we point out a bonus of the present approach, in providing analytic predictions for the time-dependent correlation functions in decaying turbulent transport.

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PubMed2003

Dynamical instabilities of quasistatic crack propagation under thermal stress.

We address the theory of quasistatic crack propagation in a strip of glass that is pulled from a hot oven towards a cold bath. This problem had been carefully studied in a number of experiments that offer a wealth of data to challenge the theory. We improve upon previous theoretical treatments in a number of ways. First, we offer a technical improvement of the discussion of the instability towards the creation of a straight crack. This improvement consists in employing Padé approximants to solve the relevant Wiener-Hopf factorization problem that is associated with this transition. Next we improve the discussion of the onset of oscillatory instability towards an undulating crack. We offer a way of considering the problem as a sum of solutions of a finite strip without a crack and an infinite medium with a crack. This allows us to present a closed form solution of the stress intensity factors in the vicinity of the oscillatory instability. Most importantly we develop a dynamical description of the actual trajectory in the regime of oscillatory crack. This theory is based on the dynamical law for crack propagation proposed by Hodgdon and Sethna. We show that this dynamical law results in a solution of the actual crack trajectory in post-critical conditions; we can compute from first principles the critical value of the control parameters, and the characteristics of the solution such as the wavelength of the oscillations. We present detailed comparison with experimental measurements without any free parameters. The comparison appears quite excellent. Finally we show that the dynamical law can be translated to an equation for the amplitude of the oscillatory crack; this equation predicts correctly the scaling exponents observed in experiments.

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PubMed2003

Simple model for drag reduction.

Direct numerical simulations established that the finite-extension nonlinear-elasticity-Peterlin (FENE-P) model of viscoelastic flows exhibits the phenomenon of turbulent drag reduction which is caused in experiments by dilute polymeric additives. To gain analytic understanding of the phenomenon, we introduce in this paper a simple one-dimensional model of the FENE-P equations. We demonstrate drag reduction in the simple model, and explain analytically the main observations which include (i) reduction of velocity gradients for fixed throughput and (ii) increase of throughput for fixed dissipation.

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PubMed2003

Shell model for drag reduction with polymer additives in homogeneous turbulence.

Recent direct numerical simulations of the finite-extensibility nonlinear elastic dumbbell model with the Peterlin approximation of non-Newtonian hydrodynamics revealed that the phenomenon of drag reduction by polymer additives exists (albeit in reduced form) also in homogeneous turbulence. We use here a simple shell model for homogeneous viscoelastic flows, which recaptures the essential observations of the full simulations. The simplicity of the shell model allows us to offer a transparent explanation of the main observations. It is shown that the mechanism for drag reduction operates mainly on large scales. Understanding the mechanism allows us to predict how the amount of drag reduction depends on the various parameters in the model. The main conclusion is that drag reduction is not a universal phenomenon; it peaks in a window of parameters such as the Reynolds number and the relaxation rate of the polymer.

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PubMed2003

Drag reduction by polymers in turbulent channel flows: Energy redistribution between invariant empirical modes.

We address the phenomenon of drag reduction by a dilute polymeric additive to turbulent flows, using direct numerical simulations (DNS) of the FENE-P model of viscoelastic flows. It had been amply demonstrated that these model equations reproduce the phenomenon, but the results of DNS were not analyzed so far with the goal of interpreting the phenomenon. In order to construct a useful framework for the understanding of drag reduction we initiate in this paper an investigation of the most important modes that are sustained in the viscoelastic and Newtonian turbulent flows, respectively. The modes are obtained empirically using the Karhunen-Loéve decomposition, allowing us to compare the most energetic modes in the viscoelastic and Newtonian flows. The main finding of the present study is that the spatial profile of the most energetic modes is hardly changed between the two flows. What changes is the energy associated with these modes, and their relative ordering in the decreasing order from the most energetic to the least. Modes that are highly excited in one flow can be strongly suppressed in the other, and vice versa. This dramatic energy redistribution is an important clue to the mechanism of drag reduction as is proposed in this paper. In particular, there is an enhancement of the energy containing modes in the viscoelastic flow compared to the Newtonian one; drag reduction is seen in the energy containing modes rather than the dissipative modes, as proposed in some previous theories.

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PubMed2003

Scaling exponent of the maximum growth probability in diffusion-limited aggregation.

An early (and influential) scaling relation in the multifractal theory of diffusion limited aggregation (DLA) is the Turkevich-Scher conjecture that relates the exponent alpha(min) that characterizes the "hottest" region of the harmonic measure and the fractal dimension D of the cluster, i.e., D=1+alpha(min). Due to lack of accurate direct measurements of both D and alpha(min), this conjecture could never be put to a serious test. Using the method of iterated conformal maps, D was recently determined as D=1.713+/-0.003. In this paper, we determine alpha(min) accurately with the result alpha(min)=0.665+/-0.004. We thus conclude that the Turkevich-Scher conjecture is incorrect for DLA.

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PubMed2003

Scaling exponents in anisotropic hydrodynamic turbulence.

In anisotropic turbulence, the correlation functions are decomposed in the irreducible representations of the SO(3) symmetry group (with different "angular momenta" l). For different values of l, the second-order correlation function is characterized by different scaling exponents zeta(2)(l). In this paper, we compute these scaling exponents in a closure approximation. By linearizing the closure equations in small anisotropy we set up a linear operator and find its zero modes in the inertial interval of scales. Thus the scaling exponents in each l sector follow from solvability condition, and are not determined by dimensional analysis. The main result of our calculation is that the scaling exponents zeta(2)(l) form a strictly increasing spectrum at least until l=6, guaranteeing that the effects of anisotropy decay as power laws when the scale of observation diminishes. The results of our calculations are compared to available experiments and simulations.

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PubMed2003

Stabilization of hydrodynamic flows by small viscosity variations.

Motivated by the large effect of turbulent drag reduction by minute concentrations of polymers, we study the effects of a weakly space-dependent viscosity on the stability of hydrodynamic flows. In a recent paper [Phys. Rev. Lett. 87, 174501, (2001)], we exposed the crucial role played by a localized region where the energy of fluctuations is produced by interactions with the mean flow (the "critical layer"). We showed that a layer of a weakly space-dependent viscosity placed near the critical layer can have a very large stabilizing effect on hydrodynamic fluctuations, retarding significantly the onset of turbulence. In this paper we extend these observations in two directions: first we show that the strong stabilization of the primary instability is also obtained when the viscosity profile is realistic (inferred from simulations of turbulent flows with a small concentration of polymers). Second, we analyze the secondary instability (around the time-dependent primary instability) and find similar strong stabilization. Since the secondary instability develops around a time-dependent solution and is three dimensional, this brings us closer to the turbulent case. We reiterate that the large effect is not due to a modified dissipation (as is assumed in some theories of drag reduction), but due to reduced energy intake from the mean flow to the fluctuations. We propose that similar physics act in turbulent drag reduction.

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PubMed2003

Active and passive fields in turbulent transport: the role of statistically preserved structures.

We have recently proposed that the statistics of active fields (which affect the velocity field itself) in well-developed turbulence are also dominated by the statistically preserved structures of auxiliary passive fields which are advected by the same velocity field. The statistically preserved structures are eigenmodes of eigenvalue 1 of an appropriate propagator of the decaying (unforced) passive field, or equivalently, the zero modes of a related operator. In this paper we investigate further this surprising finding via two examples of shell models, one akin to turbulent convection in which the temperature is the active scalar, and the other akin to magnetohydrodynamics in which the magnetic field is the active vector. In the first example, all the even correlation functions of the active and passive fields exhibit identical scaling behavior. The second example appears at first sight to be a counterexample: the statistical objects of the active and passive fields have entirely different scaling exponents. We demonstrate, nevertheless, that the statistically preserved structures of the passive vector dominate again the statistics of the active field, except that due to a dynamical conservation law the amplitude of the leading zero mode cancels exactly. The active vector is then dominated by the subleading zero mode of the passive vector. Our work thus suggests that the statistical properties of active fields in turbulence can be understood with the same generality as those of passive fields.

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PubMed2002

Quasistatic brittle fracture in inhomogeneous media and iterated conformal maps: modes I, II, and III.

The method of iterated conformal maps is developed for quasistatic fracture of brittle materials, for all modes of fracture. Previous theory, that was relevant for mode III only, is extended here to modes I and II. The latter require the solution of the bi-Laplace rather than the Laplace equation. For all cases we can consider quenched randomness in the brittle material itself, as well as randomness in the succession of fracture events. While mode III calls for the advance (in time) of one analytic function, modes I and II call for the advance of two analytic functions. This fundamental difference creates different stress distribution around the cracks. As a result the geometric characteristics of the cracks differ, putting mode III in a different class compared to modes I and II.

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PubMed2002

Bi-laplacian growth patterns in disordered media.

Experiments in quasi-two-dimensional geometry (Hele-Shaw cells) in which a fluid is injected into a viscoelastic medium (foam, clay, or associating polymers) show patterns akin to fracture in brittle materials, very different from standard Laplacian growth patterns of viscous fingering. An analytic theory is lacking since a prerequisite to describing the fracture of elastic material is the solution of the bi-Laplace rather than the Laplace equation. In this Letter we close this gap, offering a theory of bi-Laplacian growth patterns based on the method of iterated conformal maps.

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PubMed2002

Transition in the fractal properties from diffusion-limited aggregation to Laplacian growth via their generalization.

We study the fractal and multifractal properties (i.e., the generalized dimensions of the harmonic measure) of a two-parameter family of growth patterns that result from a growth model that interpolates between diffusion-limited aggregation (DLA) and Laplacian growth patterns in two dimensions. The two parameters are beta that determines the size of particles accreted to the interface, and C that measures the degree of coverage of the interface by each layer accreted to the growth pattern at every growth step. DLA and Laplacian growth are obtained at beta=0, C=0 and beta=2, C=1, respectively. The main purpose of this paper is to show that there exists a line in the beta-C phase diagram that separates fractal (D<2) from nonfractal (D=2) growth patterns. Moreover, Laplacian growth is argued to lie in the nonfractal part of the phase diagram. Some of our arguments are not rigorous, but together with the numerics they indicate this result rather strongly. We first consider the family of models obtained for beta=0, C>0, and derive for them a scaling relation D=2D(3). We then propose that this family has growth patterns for which D=2 for some C>C(cr), where C(cr) may be zero. Next we consider the whole beta-C phase diagram and define a line that separates two-dimensional growth patterns from fractal patterns with D<2. We explain that Laplacian growth lies in the region belonging to two-dimensional growth patterns, motivating the main conjecture of this paper, i.e., that Laplacian growth patterns are two dimensional. The meaning of this result is that the branches of Laplacian growth patterns have finite (and growing) area on scales much larger than any ultraviolet cutoff length.

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PubMed2002

Inverse cascade regime in shell models of two-dimensional turbulence.

We consider shell models that display an inverse energy cascade similar to two-dimensional turbulence (together with a direct cascade of an enstrophylike invariant). Previous attempts to construct such models ended negatively, stating that shell models give rise to a "quasiequilibrium" situation with equipartition of the energy among the shells. We show analytically that the quasiequilibrium state predicts its own disappearance upon changing the model parameters in favor of the establishment of an inverse cascade regime with Kolmogorov scaling. The latter regime is found where predicted, offering a useful model to study inverse cascades.

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