Analytical shock solutions at large and small Prandtl number

Exact one-dimensional solutions to the equations of fluid dynamics are derived in the large-Pr and small-Pr limits (where Pr is the Prandtl number). The solutions are analogous to the Pr = 3/4 solution discovered by Becker and analytically capture the profile of shock fronts in ideal gases. The large-Pr solution is very similar to Becker's solution, differing only by a scale factor. The small-Pr solution is qualitatively different, with an embedded isothermal shock occurring above a critical Mach number. Solutions are derived for constant viscosity and conductivity as well as for the case in which conduction is provided by a radiation field. For a completely general density- and temperature-dependent viscosity and conductivity, the system of equations in all three limits can be reduced to quadrature. The maximum error in the analytical solutions when compared to a numerical integration of the finite-Pr equations is O(1/Pr) for large Pr and O(Pr) for small Pr.Comment: 11 pages, 6 figures. Accepted for publication in Journal of Fluid Mechanics Rapid

Buoyancy instability of homologous implosions

I consider the hydrodynamic stability of imploding gases as a model for inertial confinement fusion capsules, sonoluminescent bubbles and the gravitational collapse of astrophysical gases. For oblate modes under a homologous flow, a monatomic gas is governed by the Schwarzschild criterion for buoyant stability. Under buoyantly unstable conditions, fluctuations experience power-law growth in time, with a growth rate that depends upon mean flow gradients and is independent of mode number. If the flow accelerates throughout the implosion, oblate modes amplify by a factor (2C)^(|N0| ti)$, where C is the convergence ratio of the implosion, N0 is the initial buoyancy frequency and ti is the implosion time scale. If, instead, the implosion consists of a coasting phase followed by stagnation, oblate modes amplify by a factor exp(pi |N0| ts), where N0 is the buoyancy frequency at stagnation and ts is the stagnation time scale. Even under stable conditions, vorticity fluctuations grow due to the conservation of angular momentum as the gas is compressed. For non-monatomic gases, this results in weak oscillatory growth under conditions that would otherwise be buoyantly stable; this over-stability is consistent with the conservation of wave action in the fluid frame. By evolving the complete set of linear equations, it is demonstrated that oblate modes are the fastest-growing modes and that high mode numbers are required to reach this limit (Legendre mode l > 100 for spherical flows). Finally, comparisons are made with a Lagrangian hydrodynamics code, and it is found that a numerical resolution of ~30 zones per wavelength is required to capture these solutions accurately. This translates to an angular resolution of ~(12/l) degrees, or < 0.1 degree to resolve the fastest-growing modes.Comment: 10 pages, 3 figures, accepted for publication in the Journal of Fluid Mechanics Rapid

Closed-form shock solutions

It is shown here that a subset of the implicit analytical shock solutions discovered by Becker and by Johnson can be inverted, yielding several exact closed-form solutions of the one-dimensional compressible Navier-Stokes equations for an ideal gas. For a constant dynamic viscosity and thermal conductivity, and at particular values of the shock Mach number, the velocity can be expressed in terms of a polynomial root. For a constant kinematic viscosity, independent of Mach number, the velocity can be expressed in terms of a hyperbolic tangent function. The remaining fluid variables are related to the velocity through simple algebraic expressions. The solutions derived here make excellent verification tests for numerical algorithms, since no source terms in the evolution equations are approximated, and the closed-form expressions are straightforward to implement. The solutions are also of some academic interest as they may provide insight into the non-linear character of the Navier-Stokes equations and may stimulate further analytical developments.Comment: 11 pages, 9 figures, accepted for publication in the Journal of Fluid Mechanics Rapid

“Two Wars and the Long Twentieth Century:” A Response

Drew Gilpin Faust, president of Harvard University and renowned historian of the American Civil War, authored an article in the New Yorker recently entitled “Two Wars and the Long Twentieth Century.” Taken primarily from her remarks in the Rede Lecture delivered at the University of Cambridge earlier in 2015, Faust’s article takes advantage of the proximity of the anniversaries of the First World War and the American Civil War to advocate for a dialogue of greater continuity between the two conflicts. [excerpt

Simple Waves in Ideal Radiation Hydrodynamics

In the dynamic diffusion limit of radiation hydrodynamics, advection dominates diffusion; the latter primarily affects small scales and has negligible impact on the large scale flow. The radiation can thus be accurately regarded as an ideal fluid, i.e., radiative diffusion can be neglected along with other forms of dissipation. This viewpoint is applied here to an analysis of simple waves in an ideal radiating fluid. It is shown that much of the hydrodynamic analysis carries over by simply replacing the material sound speed, pressure and index with the values appropriate for a radiating fluid. A complete analysis is performed for a centered rarefaction wave, and expressions are provided for the Riemann invariants and characteristic curves of the one-dimensional system of equations. The analytical solution is checked for consistency against a finite difference numerical integration, and the validity of neglecting the diffusion operator is demonstrated. An interesting physical result is that for a material component with a large number of internal degrees of freedom and an internal energy greater than that of the radiation, the sound speed increases as the fluid is rarefied. These solutions are an excellent test for radiation hydrodynamic codes operating in the dynamic diffusion regime. The general approach may be useful in the development of Godunov numerical schemes for radiation hydrodynamics.Comment: 16 pages, 10 figures, accepted for publication in The Astrophysical Journa

Statistical Mechanics of 2+1 Gravity From Riemann Zeta Function and Alexander Polynomial:Exact Results

In the recent publication (Journal of Geometry and Physics,33(2000)23-102) we demonstrated that dynamics of 2+1 gravity can be described in terms of train tracks. Train tracks were introduced by Thurston in connection with description of dynamics of surface automorphisms. In this work we provide an example of utilization of general formalism developed earlier. The complete exact solution of the model problem describing equilibrium dynamics of train tracks on the punctured torus is obtained. Being guided by similarities between the dynamics of 2d liquid crystals and 2+1 gravity the partition function for gravity is mapped into that for the Farey spin chain. The Farey spin chain partition function, fortunately, is known exactly and has been thoroughly investigated recently. Accordingly, the transition between the pseudo-Anosov and the periodic dynamic regime (in Thurston's terminology) in the case of gravity is being reinterpreted in terms of phase transitions in the Farey spin chain whose partition function is just a ratio of two Riemann zeta functions. The mapping into the spin chain is facilitated by recognition of a special role of the Alexander polynomial for knots/links in study of dynamics of self homeomorphisms of surfaces. At the end of paper, using some facts from the theory of arithmetic hyperbolic 3-manifolds (initiated by Bianchi in 1892), we develop systematic extension of the obtained results to noncompact Riemannian surfaces of higher genus. Some of the obtained results are also useful for 3+1 gravity. In particular, using the theorem of Margulis, we provide new reasons for the black hole existence in the Universe: black holes make our Universe arithmetic. That is the discrete Lie groups of motion are arithmetic.Comment: 69 pages,11 figures. Journal of Geometry and Physics (in press