Physics

Hello- This is not yet a paper, it is notes on what I would like to be a paper, once I improve my understanding of early rabbinic literature and philosophy of science.  While these notes do contain concepts of parallels, I prefer to avoid "parallelomania" because I feel that there are aspects of both rabbinic and applied mathematical approaches to knowledge which I have yet to see studied in the way that might make the parallel into something helpful for both fields. On the math front, such approaches are related not to the "scientific method" formalism, but to the more "art formy" methods that scientists learn from their mentors, orally, because the knowledge is not always written in books.  One example that is written in books is the use of non-convergent (asymptotic series) but I hold that even now, the *proper* use of such methods, which is problem-dependent, cannot be found in books, but must be learned via orality.

I have posted this proto-manuscript to Academia mainly as a convenient way to share with colleagues and friends, but I welcome feedback from anyone.

In this paper I will present evidence that the Babylonian rabbis who produced the Talmud also developed a practical analytical framework for solving complex problems, akin to methods in modern applied mathematics. I will argue that they did this to better and more systematically develop new knowledge that was both innovative and intellectually sound. Sound statements on difficult topics made the rabbis more persuasive and ultimately enhanced their authority. At the same time, the rabbis made this achievement an enduring one by nurturing an academic structure through a form of mentoring that was captured in the Bavli in a way that allowed it to perpetuate its own academic culture.

  The intent of this paper is not to enter into debates of the philosophy of science and Torah knowledge which are, nevertheless, mentioned on occasion. The main focus of this paper is, rather, on the nature of Babylonian rabbis’ practical techniques for solving highly complex problems, and its effectiveness as seen through the prism of modern techniques in mathematics and science. This paper also eschews any attempt to compile extensive examples of sugyot exemplifying scientific or mathematical subjects or to trace their origins. Finally, I will avoid direct engagement with the thorny issue of the exact nature of rabbinic academies (yeshivot/metivata) even as this paper will suggest a different outlook on the nature of the academy in late Sassanian Babylonia.

Transient growth properties are computed for a two-phase temporal mixing layer of immiscible fluids with interfacial tension. Large transient growth factors are found to occur at short times in parameter regimes characteristic of the primary breakup of a liquid. Optimal growth factors scale with the square of the Reynolds number, as for single-phase flow. The flow fields of optimal disturbances show liquid upflows and high-speed streamwise gas jets occurring together near the interface, suggesting transient growth as a possible mechanism for the formation of interfacial patterns. Optimal growth occurs for streamwise uniform disturbances with a spanwise wavelength proportional to the thickness of the gas boundary layer. For a coaxial jet, the predicted number of ligaments would be inversely proportional to the gas boundary layer thickness.

Accretion disk flow in a local Cartesian (or "shearing box") approximation is examined for viscous three-dimensional linear disturbances. Eigenvalue computations predict that the flow is asymptotically stable unless the rotation number falls in the range 0 < Ro < 1/2, in agreement with predictions of inviscid theory; Keplerian flow (Ro = 1/q = 2/3) is accordingly stable. Analysis of non-modal disturbances predicts large transient amplification factors, implying that the flow, although asymptotically stable, may be transiently unstable. Strong rotation, including Keplerian, two-dimensionalizes the system: the largest growth factors are found for disturbances which are uniform along the direction of the rotation axis and amplification occurs via the Orr mechanism, as in 2D shear flow. Amplification factors scale as Re 2/3 , implying very strong growth in actual disks. The implications of transient instability of rotating shear on disk turbulence are discussed.

We perform numerical simulations of two-phase liquid-gas sheared layers, with the objective of studying atomization. The Navier-Stokes equations for two-dimensional incompressible flow are solved in a periodic domain. A Volume of Fluid (VOF) method is used to track the interface. The density ratio is kept around 10. The calculations show good agreement with a fully viscous Orr-Sommerfeld linear theory 1 over several orders of magnitude of interface growth. The nonlinear development shows the growth of finger-like structures, or ligaments, and the detachment of droplets. The effect of the Weber and Reynolds numbers, the boundary layer width and the initial perturbation amplitude are discussed through a number of typical cases. Inversion of the liquid boundary layer is shown to yield more readily ligaments bending upwards and thus more likely to produce droplets.

The three-dimensional temporal instability of rotating boundary layer flows is investigated by computing classical normal modes as well as by evaluating the transient growth of optimal disturbances. The flows examined are the rotating Blasius (RB) and the rotating asymptotic suction layers (RAS), with the rotation axis normal to the basic flow plane. In agreement with an inviscid criterion, streamwise unstable modes are found in both flow cases for anti-cyclonic rotation: at high Reynolds numbers, one obtains the Rossby number unstable range 0&lt;1/Ro&lt;0.57 for RB, or 0&lt;1/Ro&lt;1 for RAS. Critical Reynolds and Rossby numbers are also determined in both instances. Moreover the dependence of transient growth with respect to wavenumbers, Rossby and Reynolds numbers is presented for both cyclonic and anti-cyclonic régimes. In particular, the peak transient growth is computed for a wide range of parameter values within the cyclonic regime and is shown to be reduced by rotation. A sc...

Improved knowledge of the magnetic field dependent flow properties of nanoparticle-based magnetic fluids is critical to the design of biomedical applications, including drug delivery and cell sorting. To probe the rheology of ferrofluid on a sub-millimeter scale, we examine the paths of 550 µm diameter glass spheres falling due to gravity in dilute ferrofluid, imposing a uniform magnetic field at an angle with respect to the vertical. Visualization of the spheres' trajectories is achieved using high resolution X-ray phase-contrast imaging, allowing measurement of a terminal velocity while simultaneously revealing the formation of an array of long thread-like accumulations of magnetic nanoparticles. Drag on the sphere is largest when the applied field is normal to the path of the falling sphere, and smallest when the field and trajectory are aligned. A Stokes drag-based analysis is performed to extract an empirical tensorial viscosity from the data. We propose an approximate physical model for the observed anisotropic drag, based on the resistive force theory drag acting on a fixed non-interacting array of slender threads, aligned parallel to the magnetic field.

Two elementary models of ocean circulation, the well-known double-gyre stream function model and a single-layer quasi-geostrophic (QG) basin model, are used to generate flow data that sample a range of possible dynamical behavior for particular flow parameters. A reservoir computing (RC) machine learning algorithm then learns these models from the stream function time series. In the case of the QG model, a system of partial differential equations with three physically relevant dimensionless parameters is solved, including Munk- and Stommel-type solutions. The effectiveness of a RC approach to learning these ocean circulation models is evident from its ability to capture the characteristics of these ocean circulation models with limited data including predictive forecasts. Further assessment of the accuracy and usefulness of the RC approach is conducted by evaluating the role of both physical and numerical parameters and by comparison with particle trajectories and with well-establis...

Paris (PArallel, Robust, Interface Simulator) is a finite volume code for simulations of immiscible multifluid or multiphase flows. It is based on the "one-fluid" formulation of the Navier-Stokes equations where different fluids are treated as one material with variable properties, and surface tension is added as a singular interface force. The fluid equations are solved on a regular structured staggered grid using an explicit projection method with a first-order or second-order time integration scheme. The interface separating the different fluids is tracked by a Front-Tracking (FT) method, where the interface is represented by connected marker points, or by a Volume-of-Fluid (VOF) method, where the marker function is advected directly on the fixed grid. Paris is written in Fortran95/2002 and parallelized using MPI and domain decomposition. It is based on several earlier FT or VOF codes such as Ftc3D, Surfer or Gerris. These codes and similar ones, as well as Paris, have been used to simulate a wide range of multifluid and multiphase flows.

We present information theoretic search strategies for single and multirobot teams to localize the source of a chemical spill in turbulent flows. In this work, robots rely on sporadic and intermittent sensor readings to synthesize information maximizing exploration strategies. Using the spatial distribution of the sensor readings, robots construct a belief distribution for the source location. Motion strategies are designed to maximize the change in entropy of this belief distribution. In addition, we show how a geophysical description of the environmental dynamics can improve existing motion control strategies. This is especially true when process and vehicle dynamics are intricately coupled with the environmental dynamics. We conclude with a summary of current efforts in robotic tracking of coherent structures in geophysical flows. Since coherent structures enables the prediction and estimation of the environmental dynamics, we discuss how this geophysical perspective can result in improved control strategies for autonomous systems.

To probe the rheology of magnetized ferrofluid on a millimeter scale, we examine the paths of glass spheres (500 micrometer diameter) falling in a bulk ferrofluid under a uniform magnetic field, applied at a fixed angle with respect to the horizontal. Visualization of the ball trajectories is accomplished by the capture of high-resolution, X-ray phase-contrast images on high-speed, digital video. The drag on each sphere is determined by measurement of its terminal velocity. The effective viscosity of the ferrofluid is observed to be anisotropic: the terminal velocity of a sphere depends on the angle of the applied magnetic field. Specifically, the drag is greater when the applied field is normal to the vertical path of the falling sphere than when parallel. We propose that this effect is due the field-induced formation of magnetic particle "threads" having diameter of a few microns (∼1000 nanoparticles) and lengths that span the container (a few millimeters). These threads form a dense parallel array aligned with the applied magnetic field and are visible in phase-contrast images of the bulk ferrofluid. We model the drag experienced by the falling ball as originating from the falling-ball Stokes flow through a fixed array of rigid threads with an inter-penetrating fluid.

The effect of an imposed magnetic field on the linear stability of immiscible two-fluid Poiseuille flow in a channel is examined for low Reynolds numbers. Surface tension acts on the interface, the fluids have different densities and viscosities, and one fluid is magnetic ͑ferrofluid͒. A Langevin function is used to model the fluid magnetization, resulting in a nonlinear permeability; the stability properties depend on this permeability relation both directly and indirectly, through the base state solution. Uniform magnetic fields applied normal or parallel to the interface both lead to an interfacial instability. Normal fields excite longer wavelength modes, generally having higher growth rates, but parallel fields can excite faster growing modes in high permeability fluids at large applied field strength. Whether or not the field stabilizes or destabilizes the flow depends on the viscosity and layer thickness ratios in a simple way, while the placement of the magnetic fluid layer does not play a major role. Growth rates predicted for realistic microchannel conditions are shown to be large enough to make ferrofluid manipulation a practical method of control.

Klein-Gordon operators with complex potentials are formulated as model systems to examine the possible impact of non-normality on the stellar pulsation eigenvalue problem. Computations of spectra, ε-pseudospectra and resolvent norms for these operators show that extended regions of complex potential, corresponding in stars to nonadiabatic regions, leads to strong non-normality. Introduction of an advective term further increases non-normal properties. Prospective model improvements and implications for resonances and for nonlinear pulsation are considered.

Strange modes have been found in the radial spectrum of many luminous stars, such as PAGB stars. The strange modes are characterized by having small amplitudes in the interior of the envelope, and egregious periods and growth-rates. It has been common belief that the strange modes are a result of strong nonadiabaticity. Recently, and perhaps surprisingly, such modes have also been found in classical Cepheid models, even though these are weakly nonadiabatic stars. Here we show that in fact there is nothing strange about these modes and that they must exist even in the adiabatic limit. They are essentially acoustic surface modes. By means of a simple change of variables and without approximation, the adiabatic linear pulsation equation for the radial displacement is reduced to a Schrödinger like equation in which the radial coordinate is the local sound traversal time. In this formulation, the narrow hydrogen partial ionization region is seen to act as a potential barrier, separating the star into two regions. Modes can be trapped either in the inner or in the surface region. Coupling through the barrier gives rise to resonances between the inner and surface regions. The strange modes are those in which the ratio of inner to surface amplitude is at a minimum. The potential problem formulation shows that strange modes exist in the adiabatic limit. As a function of the stellar parameters the relative location of the barrier changes, and this gives rise to the phenomenon of avoided level crossings along a sequence of models. The appearance of strange modes and the associated level crossings can be exhibited with an analytically solvable toy model when the potential barrier is approximated by a delta function. In the full nonadiabatic models the same trapping mechanism remains responsible for the appearance of strange modes. The unusual growth-rates are seen to also be a consequence of the relative minimum of the inner amplitude for these modes. Again the behavior of the nonadiabatic modes can be well mimicked by a simple analytical toy model.

The Space Circle is a toy whose motion is chaotic. A modified version of the device with a tunable stress parameter exhibits a variety of periodic and chaotic states. We have derived and numerically solved the equations of motion for the system and find excellent agreement with experimental results. The problems with some widely used algorithms for diagnosing experimental chaos are discussed. For the Space Circle we find that estimates of the dominant Lyapunov exponent are more robust than estimates of the fractal dimension.

We present the first algorithms that allow the estimation of non-negative Lyapunov exponents from an experimental time series. Lyapunov exponents, which provide a qualitative and quantitative characterization of dynamical behavior, are related to the exponentially fast divergence or convergence of nearby orbits in phase space. A system with one or more positive Lyapunov exponents is defined to be chaotic. Our method is rooted conceptually in a previously developed technique that could only be applied to analytically defined model systems: we monitor the long-term growth rate of small volume elements in an attractor.

Electric fields produced by coplanar point charges have often been represented by field line diagrams that depict two-dimensional slices of the three-dimensional field. Serious problems with these ``conventional'' field line diagrams (CFLDs) have been overlooked. Two of these problems, ``equatorial clumping'' and ``false monopole moment,'' occur because a two-dimensional slice lacks information vital to the accurate representation of an inherently three-dimensional field. Equatorial clumping causes most CFLDs to exhibit unphysical behavior such as irregular spacing between field lines terminating on negative charges. CFLDs can also mistakenly indicate that a neutral charge distribution has a significant monopole moment. Such phenomena make the visual estimation of local field strengths impossible and render CFLDs of little utility for representing three-dimensional fields. While these ``projection'' problems can be avoided by using two-dimensional field line diagrams to represent two-dimensional (1/r) electric fields, or by using three-dimensional field line diagrams to represent three-dimensional fields, other forms of distortion generally remain.