Classical Hydrodynamics Research Articles

A Potential Flow Analogy in Fluid Film Stretching

Thin film of isothermal Newtonian fluids lying on the (x,y) plane can be extended without generating thickness nonuniformity when the film is initially uniform in thickness and the velocity field vx(x,y,t),vy(x,y,t) of stretching obeys the equations vx(x,y,t)=a(t)x+b(t)vx*(x,y), vy(x,y,t)=a(t)y+b(t)vy*(x,y), where (vx*,vy*) is identical to the two‐dimensional potential flow that satisfies the Cauchy‐Rieman equations in (x,y) coordinates, while a(t) and b(t) are arbitrary time functions. Drawing on the knowledge of classical hydrodynamics a number of interesting and physically realizable flow fields of uniform thickness film stretching can be predicted analytically by the above solution (vy,vy), which also leads to the analytical proof that thickness uniformity is maintained in any axisymmetrical stretching as long as the film is uniform initially. These findings constitute an important correction to the authors' previous statement that thickness uniformity can be maintained only when the flow field consists of uniform extension in x and y directions expressible by the equations vx=C11(t)x+C12(t)y and vy=C21(t)x+C22(t)y.

Picosecond dynamics of barrier crossing in solution: A study of the conformational change of excited state 1,1′-binaphthyl

The rates of optically induced conformational changes of excited state 1,1′-binaphthyl in a series of alcohol solvents were measured by a picosecond laser technique. Excellent agreement with experiment is found for the Kramers model for both the intermediate and high friction regimes using a frequency independent friction coefficient evaluated from classical hydrodynamics. This is the first observation of Kramers behavior in the intermediate friction regime and indicates that deviations from Kramers behavior is not a universal phenomenon in the conformational dynamics of flexible molecules in solution. It also emphasizes the possibility that the observed deviations from the Kramers model can be due to effects other than non-Markovian friction.

Profiles of minimum stress concentration for antiplane deformation of an elastic solid

A major application of the linear theory of elasticity is the description of stress concentration. Typically, the objective is to predict the maximum stress as measured by a suitable norm and to determine ways of reducing this maximum through variations in the loading and geometry. When loading is specified, the traditional approach is a trial method in which the shape is remodelled iteratively until the stress is reduced to an acceptable level. This process suggests a variational problem in which we seek minimal stress concentration, a variational problem which is unconventional because of the form assumed by the functional whose minimum is sought. For this reason, the problem does not fall within the framework of the classical calculus of variations for functionals represented by integrals. Despite the unconventional nature of the variational problem of minimum stress concentration, a degree of success has been obtained indirectly by reducing it to a kind of inverse problem which, for instance, in the case of the antiplane theory is the mathematical analogue of the two-dimensional free streamline problem of classical hydrodynamics. The intuitive idea behind this reduction is natural for stress concentration associated with a geometrical intrusion (notch, hole, inclusion, etc.). One seeks a profile which renders the stress norm uniform over the intrusion. This idea was exploited for plane deformation problems in [1-4], which serve to describe the associated inverse problem and contain interesting solutions to particular problems. None of these references offer a rigorous justification that the inverse problem solves the optimization problem. In [5], a rigorous verification is given that the inverse problem furnishes the solution to the optimization problem in the case of a void in an infinite medium subjected to plane deformation. Essentially the same analysis is used in [6] with more general stress norms, and in [7] for more general geometries. In [8], the problem of optimization of the shape of a rigid inclusion is rigorously reduced to an inverse problem whose solution is easily inferred from results in [1] or [4] (See also [9]). The three-dimensional problem of a void of optimum shape is solved in [10].

On the Burnett and higher order equations of hydrodynamics

Several derivations of the Burnett and higher order in the gradients equation of classical hydrodynamics are critically examined. It is clearly pointed out that the two available phenomenological derivations and Grad's kinetic derivation are compatible with the local equilibrium assumption and the entropy balance equation. Furthermore, the failure to fulfill these requirements when such equations are obtained from the Chapman-Enskog method for solving Boltzmann's equation is accounted for. A comparison with other more microscopic methods to obtain these equations is also provided.

Classical non-linear waves in dispersive nonlocal media, and the theory of superfluidity

The solution to nonlocal dispersive classical hydrodynamics is investigated at the fourth order of nonlinearity. At this order an extra degree of freedom appears as a result of the additive conservation of wave number in the interaction of beams of sound waves, and represents a broken symmetry. The resulting equations of motion for the background plus a distribution of sound waves are the Landau two-fluid hydrodynamics of 4He with dispersion at higher frequencies. Reasonable choices for the energy as a functional of density lead to phonon-roton type spectra. The normal fluid flow can thus be viewed either as the Stokes drift of the acoustic field or as the fluctuations of the macroscopic equations of motion of the ground state. In the case of normal dispersion the sound wave interaction is at least sixth order in the nonlinearity, and the appropriate solutions are presented. The general conclusions are unchanged though transport phenomena will be dramatically affected.

The scattering of regular surface waves by a fixed, half-immersed, circular cylinder

A train of regular surface waves is incident upon a fixed, half-immersed, circular cylinder; the waves are partially reflected and partially transmitted, and also induce hydrodynamic forces on the cylinder. In order to give a theoretical study of this problem, we make the familiar assumptions of classical hydrodynamics and then solve the linear, two-dimensional, diffraction boundary-value problem, using Ursell's multipole method. Accurate numerical results are presented (in the form of tables) for four important (complex) quantities; these are the reflection and transmission coefficients, and two dimensionless coefficients which describe the horizontal and vertical forces on the cylinder. We have also made an experimental study, in which we measured the forces on the cylinder, and the reflection coefficient. These measurements are compared with the linear theory, and also with other experimental data; discrepancies are noted and an attempt to analyse them is made. We have also measured the mean horizontal forces on the cylinder; these results are compared with the predictions of a simple formula obtained by Longuet-Higgins.

Non-linear hydrodynamics and a one fluid theory of superfluid He 4

At the fourth order of non-linearity classical hydrodynamics possesses a broken symmetry relating to the interaction of beams of sound waves. The non-linear evolution of these high-order properties of the acoustic field are identical with the two fluid hydrodynamics used to describe superfluid He 4.

Renormalized classical non-linear hydrodynamics quantum mode coupling and quantum theory of interacting phonons

Abstract Motivated by the second law a renormalization scheme is introduced within the framework of classical, non-linear hydrodynamics that enables one to directly calculate the essential results of the quantum theory of interacting phonons. Consequences for quantum mode coupling and the quantum fluctuation dissipation theorem are discussed.

Absorption of sound in 3He4He vapor mixtures

Abstract We have studied the absorption of sound in helium vapor mixtures using a broad band heat pulse technique. The results are in good accord with the predictions of classical hydrodynamics.

Renormalized classical non-linear hydrodynamics, quantum mode coupling and quantum theory of interacting phonons

Motivated by the second law a renormalization scheme is introduced within the framework of classical, unquantized non-linear hydrodynamics that enables one to directly calculate the essential results of the quantum theory of interacting phonons. Consequences for quantum mode coupling and the quantum fluctuation-dissipation theorem are discussed.

Calculation of Cylindrical Worm Gear Drives of Different Tooth Profiles

The formulae of classical hydrodynamics are not suitable for the calculation of load capacity and power loss of worm gear drives. Thus a theoretical basis had to be developed for the comparison of different tooth profiles, materials of worm and worm wheel and lubricants. The data obtained were compared with test results. It proved that the coefficient of friction is an important influence factor.

On classical hydrodynamics and collective motion as approximation to the nuclear many-body problem

Using time-dependent unitary transformations, one can cast a one-body equation of the time-dependent Hartree-Fock type into a form which is closely related to equations for a classical irrotational fluid. The hydrodynamic equation of state finds its counterpart in a stationary constrained field equation. The hydrodynamic equations in turn can be translated into a classical Hamiltonian formalism with an infinite number of generalised coordinates, which are given as all possible spatial moments of the density. The reduction to a few ones, the “natural collective coordinates” is possible by the choice of appropriate initial conditions. The lowest of the hydrodynamical frequencies can be calculated in closed form by harmonic approximations. For the quadrupole frequency a value ofω=31 A−1/3 MeV/h is obtained. As expected, the value does not agree with the experiment, but rather is in between the characteristic frequency for theβ-vibration and the isoscalar giant quadrupole vibration.

Forced precession of neutron stars

The dynamics of a neutron star subject to an external torque is investigated. The neutron star is treated as a two-component body, one component being the solid crust together with the interior plasma coupled to it by a magnetic field, the other component being the interior neutron superfluid. The latter is treated with classical hydrodynamics.

Absorption of sound in helium vapor

We have studied the absorption of sound in helium vapor using a broad band heat pulse technique. The results are in good accord with classical hydrodynamics and with experimental values of the viscosity and thermal conductivity.

Hamiltonian formalism for perfect fluids in general relativity

Schutz's Hamiltonian theory of a relativistic perfect fluid, based on the velocity-potential version of classical perfect fluid hydrodynamics as formulated by Seliger and Whitham, is used to derive, in the framework of the Arnowitt, Deser, and Misner (ADM) method, a general partially reduced Hamiltonian for relativistic systems filled with a perfect fluid. The time coordinate is chosen, as in Lund's treatment of collapsing balls of dust, as minus the only velocity potential different from zero in the case of an irrotational and isentropic fluid. A "semi-Dirac" method can be applied to quantize astrophysical and cosmological models in the framework of this partially reduced formalism. If one chooses Taub's adapted comoving coordinate system, it is possible to derive a fully reduced ADM Hamiltonian, which is equal to minus the total baryon number of the fluid, generalizing a result previously obtained by Moncrief in the more particular framework of Taub's variational principle, valid for self-gravitating barotropic relativistic perfect fluids. An unconstrained Hamiltonian density is then explicitly derived for a fluid obeying the equation of state $p=(\ensuremath{\gamma}\ensuremath{-}1)\ensuremath{\rho}$ ($1\ensuremath{\le}\ensuremath{\gamma}\ensuremath{\le}2$), which can adequately describe the phases of very high density attained in a catastrophic collapse or during the early stages of the Universe. This Hamiltonian density, shown to be equivalent to Moncrief's in the particular case of an isentropic fluid, can be simplified for fluid-filled class-$A$ diagonal Bianchi-type cosmological models and appears as a suitable starting point for the study of the canonical quantization of these models.

An approximate theoretical calculation of the spin-lattice relaxation times in fluids near—but above—the critical point

We present a theory to calculate, first, the correlation function of the physical quantity (angular momentum for spin-rotation interaction, second order spherical harmonics for the dipolar interaction), involved in magnetic spin-lattice relaxation and then the relaxation times. The time behaviour of the physical quantity h(t) is described by a generalized Langevin equation whose memory function, K(t), is assumed to be proportional to the density fluctuation correlation function, N(t). The time-independent proportionality function, α, is obtained by assuming that the relaxation time τ h (for h(t) to reach local equilibrium) is equal to the correlation time, τ n , of the density fluctuations. The density fluctuation correlation function is calculated by using the classical hydrodynamics form of the structure factor S(q, ω) [1, 2]. The definition chosen for N(t) forces us to consider only temperatures above the critical temperature, since we do not know the particle density of the liquid below the critical t...

Dynamics of spheroid-cylindrical molecules in dilute solution

The dilute solution dynamics of spheroid–cylindrical molecules, i.e., straight cylinders with oblate, spherical, or prolate hemispheroid caps at the ends, is studied in detail. The translational and rotatory diffusion coefficients and the dynamic intrinsic viscosity are evaluated numerically for short cylinders by determining the frictional force by an orthodox method of classical hydrodynamics. For long cylinders, the Oseen–Burgers procedure is shown to be valid, and the results previously obtained by it are still useful. Thus, empirical interpolation formulas for the transport coefficients above are also constructed to be applied to spheroid–cylinders of arbitrary size. The end effects on the translational and rotatory diffusion coefficients are rather small, while the effect on the zero-frequency intrinsic viscosity is remarkable, depending appreciably on the shape of the ends, though for relatively short cylinders. In general, for a rigid body of revolution having a plane of symmetry perpendicular to its axis, it is shown that the dynamic intrinsic viscosity may be expressed in terms of the zero-frequency intrinsic viscosity, the rotatory diffusion coefficient about a principal axis in the symmetry plane, and a newly defined factor, which is also associated with the rotational motion about this axis.

Bouncing Quantum Cosmologies

Spatially homogeneous cosmological models of Bianchi types I, V and IX (as well as their isotropic counterparts : the Einstein-de Sitter, open and closed Friedmann-Robertson-Walker (FRW) models, respectively), filled with a perfect fluid with an equation of state of the type p = (γ - 1 )ρ (γ = constant comprised between 1 and 2) are quantized in the framework of an ADM-type canonical quantization scheme introduced by Demaret and Moncrief. This method which generalizes a method due to Lund and originally restricted to space-times filled with a pressureless fluid (p = 0) is based on Schutz's Hamiltonian theory of a relativistic perfect fluid, extending to general relativity Seliger and Whitham's velocity-potential version of classical hydrodynamics. It leads to a Schrödinger-like equation for the wave function of the universe, which admits a true physical interpretation in terms of probability as in orthodox quantum mechanics. Contrary to Misner's early results based on a different quantization scheme, our method leads to the conclusion that, apart from a set of measure zero of models comprising FRW models as well as Bianchi models filled with a fluid with a stiff equation of state (p = ρ), all Bianchi models are non-singular, in the sense that the quantum wave function becomes zero at the classical singularity. Such a non-singular model can be interpreted as a contracting universe bouncing into an expanding universe, due to quantum gravitational fluctuations. A similar conclusion of non-singularity of a quantum Bianchi I universe has recently been obtained by Narlikar by using a different method of quantization, i.e. the method of path integration.

Comment on "A Comparison of Different Forms of Dirigible Equations of Motion"

P and Hookway discuss a seeming paradox in the different forms for the equations of motion of a dirigible or submarine. They cite two forms of the equations of motion from two different references and state: No studies that resolve the differences...have been found. The equations for the motion of a solid through a fluid are well known in classical hydrodynamics arid give a unique and unambiguous answer to the question raised by Pretty and Hookway. For plane motion of a solid of revolution, the kinetic energy of the fluid plus solid may be represented by

A self-consistent analysis for two-dimensional hydrodynamics

We study a stochastic model for 2-d incompressible fluids. We show that the kinetic equation governing the evolution of the velocity correlation function has a remarkable property which suggests that a regime of time and wave number exists over which self-consistent transport modes are exponentially decaying. The characteristic time of this regime is shorter than the characteristic time of classical hydrodynamics: it is defined by the limit k → 0, t → ∞ and k 2√In(1/ k) t finite.