Zeroth-order Equations Research Articles

Longitudinal surface curvature effects on boundary layer of a micropolar fluid

The effect of longitudinal surface curvature on steady two-dimensional incompressible laminar boundary layer of a micropolar fluid has been considered. Van Dyke's first oder perturbation analysis is applied to the full equations of motion derived in curvilinear coordinate system which facilitates to carry out boundary layer approximation for flow past a curved surface. This results into two systems of partial differential equations which are called the zeroth order and the first order boundary layer equations. The zeroth order equations are the usual boundary layer equations for a micropolar fluid. The first order equations take into account the longitudinal surface curvature effect explicitly. Similar solution of the governing equations exists if (i) the inviscid flow velocity on the surface varies linearly along the surface and (ii) the longitudinal surface curvature is constant. Numerical results are presented illustrating the dependence of the important flow quantities of both zeroth order and first order boundary layers on the micropolar fluid parameters. The results have been compared with the corresponding results for a Newtonian fluid. It has been found that the skin friction decreases and the wall couple stress increases for convex side of the surface and vice versa for the concave side.

Lepton-width suppression of vector-meson decays

Quantum-chromodynamics radiative corrections to the vector-meson lepton width are discussed. These may modify the zeroth-order equation of Van Royen and Weisskopf, thereby leading to a significant suppression, by about a factor of 2, of the calculated value of $\ensuremath{\Gamma}$. Consequences for phenomenology are examined. In particular, a prediction is made for the width of ${\ensuremath{\Upsilon}}^{\ensuremath{'}}$, as well as for widths of heavier mesons.

Geodesics in gauge supersymmetry

The exact geodesic equations of curved superspace are derived. These are expanded in terms of the metric through second order in the Fermi coordinates ${\ensuremath{\theta}}^{i}$. The zeroth-order equations are the same as Einstein's geodesic equations. The first- and second-order equations are also found to be compatible with the field equations, separately implying the proper magnitude for the Lorentz force. This compatibility holds only if the $f$ field as well as the ${L}_{\ensuremath{\mu}\ensuremath{\nu}}$ field of Arnowitt and Nath develops a superheavy mass through spontaneous symmetry breaking. This mass growth for the $f$ field has recently been shown actually to occur.

Constitutive Relations and Subsidiary Conditions in the Eikonal Approximation

The eikonal approximation is considered for propagation in an inhomogeneous, time-dependent, anisotropic environment. The usual eikonal analysis is generalized to take into account constitutive relations which contain space and time derivatives of the field quantities. The field equations are obtained for all orders of eikonal approximation. These consist of homogeneous equations for the zeroth-order, and recursion relations which connect arbitrary consecutive higher orders of approximation. An apparent inconsistency between the zeroth-order equations and the higher order ones is avoided by the development of a set of subsidiary conditions on the field approximations. The lowest order condition in this set is considered in detail, and a general relation is derived between a certain vector quantity appearing in this condition, and the tangent vector to a ray path. For an environment which is stationary and free of spatial dispersion, this relation is shown to have, as one of its consequences, a new normalization condition for the zeroth-order fields in anisotropic propagation. The relation is applied to propagation in a cold magnetoplasma, and a number of results in ionospheric propagation are recovered and extended.

Computer-aided placement for high-density chip-interconnection systems

A placement algorithm is presented for high-density chip-interconnection systems. The algorithm is novel in that the irregular nature of the pad positions and geometries of integrated-circuit chips are accommodated and used. The placement technique is based on the solution of a set of zeroth-order simultaneous equations, and is demonstrated by a simple example.

On Phythian's perturbation theory for stationary homogeneous turbulence

It is pointed out that the perturbation expansion procedure about a certain flow state requires a priori knowledge of the statistical distribution of the flow field. In Wyld's laminar perturbation procedure it was indeed natural by invoking Kraichnan's maximal randomness principle to assume the laminar flow to have a Gaussian distribution. On the other hand, one attempts to develop perturbation about a turbulent flow as in Phythian's procedure, the difficulty is that the distribution of the velocity field is just as unknown as the turbulence problem itself. It is shown that Phythian's assumption of Gaussian random force for the zeroth-order equation has the effect of deriving the direct- interaction equations under the quasi-normal hypothesis.

Magnetooptical effects in ferromagnetic metals at low temperatures

Zeroth-order and first-order Boltzmann kinetic equations are found for the frequency-dependent transverse-conductivity tensor for inelastic scattering of electrons, with an account of thermal oscillations of the scattering centers. At low temperatures this tensor, which is linear in the spin-orbit interaction, depends exponentially on the temperature. A relation is found between the magnetooptical effects in ferromagnetic metals and the topology of the Fermi surfaces; from this relation, qualitative conclusions can be drawn regarding the effect of this topology on the magnetooptical effects and regarding the sign of the anomalous magnetooptical parameter (which depends on whether hole or electronic conductivity predominates).

On the asymptotic solution of the Orr–Sommerfeld equation by the method of multiple-scales

The method of multiple-scales is used to obtain the asymptotic solution of the Orr–Sommerfeld equation. For the special case of a linear velocity profile, the solution so obtained agrees well with an approximation of the exact solution which is known. For the general case, transformations on both the dependent and independent variables are introduced to obtain a zeroth-order equation which differs from the inner equation studied so far. On the ground of the favourable comparison for the special case, the asymptotic solution constructed is expected to be uniformly valid.

Comments on the Solution to the Boltzmann Equation for a Weakly Ionized Plasma

Hydrodynamic equations describing the motion of electrons in a weakly ionized plasma are derived formally from the Boltzmann equation by means of the Chapman-Enskog procedure. Two cases are considered, each with a stationary Maxwellian distribution ascribed to the atoms. In the first, electron-electron collisions are ignored, and the electron distribution function is determined by a balance between the electron-atom collisions and the electric field. There is only one conservation law, a hydrodynamic equation for the density. In the second case, electron-electron collisions are dominant, the distribution function is a local Maxwellian, and there are five conservation laws---equations for the density, drift velocity, and temperature. The equations used previously by the author to describe low-frequency oscillations are obtained in either case if the electron-atom collision frequency is independent of velocity. Otherwise, the zeroth-order equations are still exact, but first-order corrections are required, as illustrated by the example of the velocity-independent mean free path. Our results are somewhat different from those of Davidov, who made different assumptions about the dominant collision mechanisms.

Nonlinear Plasma Waves I. Secular Behavior of Nonlinear Equations

A perturbation scheme is devised for investigating the periodic solutions of a nonlinear, second-order, ordinary differential equation with an additional small nonlinear term. It is assumed that when the additional term vanishes the solution to the remaining zeroth-order nonlinear equation is known. When this is the case, explicit expressions for the first-order solution and the first-order correction to the dispersion relation are given in terms of the known zeroth-order solution.

A Theoretical Analysis of Laminar Forced Flow and Heat Transfer About a Rotating Cone

The present paper presents analytically a method of attack on the problem of laminar forced flow and heat transfer about a rotating cone. The nonsimilar nature of the general problem requires that separate consideration be given to a slow rotating cone and a fast rotating cone, depending on the relative magnitude of the rotating speed with respect to the free-stream velocity. The Mangler transformation first reduces the problem of a slow rotating cone to one of wedge flow with a transverse velocity component. The problem is then solved by a perturbation scheme which uses the solution of wedge flow as the zeroth-order solution. The case of a fast rotating cone is solved by a series-expansion scheme which gives successive corrections to the zeroth-order solution, i.e. the solution of a rotating disk in a quiescent fluid. The zeroth-order and first-order equations for both cases are given in the present work, together with the numerical results for the special case of a cone of about 107-deg cone angle. The first-order results in both cases are shown for the drag and torque coefficients, and the local Nusselt number. Higher-order results can be obtained according to the present analysis. The effect of cone angle on the flow and heat-transfer characteristics is indicated by the comparison between the results of the 107-deg cone and those of the disk, i.e., the 180-deg cone.