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This paper deals with the problem of a relativistic stationary, axisymmetric, rigidly rotating fluid by means ofPerjes’s three-dimensional spin coefficient formalism. The spin coefficient equivalents of the Einstein and Euler equations and those of Weyl and Ricci tensors are calculated. We show that the formalism can be applied even if the cosmological constant Λ is not equal to zero.

Inviscid, transonic flows over the NACA 64A410 airfoil were calculated using a finite-difference analog of the Euler equations. Steady flow solutions were obtained for angles of attack of 0°, 2° and 4° at Mach 0.72. Unsteady calculations were made at the same Mach number and in the same general angle-of-attack range for a step change of angle of attack, a step change in pitching angular velocity, and for sinusoidal pitching oscillations about the midchord at reduced frequencies 0.2,1.0, and 5.0. (Reduced frequency, K^uC/U^.) A summary of the resulting surface pressures, shock locations, and the forces and moments is presented. The forces and moments obtained for the cases with harmonic oscillations were compared to the results from a linearized analysis which used, to approximate the indicial functions, the responses calculated for the step changes.

A number of authors have investigated conditions under which weak solutions of the initial-boundary value problem for the nonlinear wave equation will blow up in a finite time. For certain classes of nonlinearities sharp results are derived in this paper. Extensions to parabolic and to abstract operator equations are also given.

The discrete time-variant regulator problem with an infinite sum criterion is considered as a variational problem on the normed linear apace of infinite sequences (1n1) The discrete Euler equation is used to find the necessary condition that an optimal control sequence, solving this problem, must satisfy and it is shown that the optimal control is in the form of time-variant feedback gain which satisfies a discrete Riccati difference equation. The case of finite-time, time-variant regulator is treated as a special case of the infinite-time problem.

A general system of partial differential equations describing the propagation of signals of finite concentrations in a chromatographic column is derived. These equations are related to the mass-balance equations for the solutes and the carrier gas. The model used assumes that there is no temperature or pressure variation at any point in the column when the signal is eluted, and that the equilibrium between stationary and mobile phases is instantaneous. It is shown that this model, which leads to a tractable set of equations, is generally valid. The solution of this system of equations gives new insight into the phenomena which are responsible for the peak deformations and broadening in preparative scale chromatography.

An “exact” expression for the functional derivative of the distribution function of a λ-nucleon pair in nuclear matter is derived. An approximate expression is also derived by means of the Kirkwood superposition approximation. The latter expression is subsequently used to obtain the Euler equation for the correlation functionf(r1λ) of a λ-nucleon pair in nuclear matter.

The method of multiple scales is used to obtain a second-order uniformly valid expansion for the nonlinear acoustic wave propagation in a two-dimensional duct whose walls are treated with a nonlinear acoustic material. The wave propagation in the duct is characterized by the unsteady nonlinear Euler equations. The results show that nonlinear effects tend to flatten and broaden the absorption versus frequency curve, in qualitative agreement with the experimental observations. Moreover, the effect of the gas nonlinearity increases with increasing sound frequency, whereas the effect of the material nonlinearity decreases with increasing sound frequency.

The object of this paper is to relate three equations in the Newman-Penrose system of equations to the conservation laws and, hence, to the equations of motion. To do so, the corresponding result is first obtained using the Einstein equations in a null coordinate system. The Newman-Penrose equations are then analyzed. They are separated into hypersurface, propagation, supplementary, and conservation equations. When all field equations except the three conservation equations have been appropriately satisfied, the desired result follows.

The unsteady, inviscid flowfield that results when a supersonic vehicle strikes a planar oblique shock wave, though difficult to simulate experimentally, is quite easy to model and compute numerically. The complicated flowfield, which contains multiple shock wave interactions, is determined using a second-order, shock-capturing, finite-difference approach which solves the time-dependent Euler equations under a self-similar transformation. A series of numerical results for a simple two-dimensional wedge is presented which describes the entire disturbed region, including the wave structure, and shows good agreement with the available two- and three-dimensional experimental data.

A method of obtaining a first order approximation to the equations governing the motion of flow through a porous medium under non-isothermal conditions is suggested. The method starts with the governing microscopic equations of flow in the pores and averages these equations over a small volume to get the equations in terms of macroscopic quantities.

For an elastically isotropic shell of constant thickness with an analytic midsurface of zero mean curvature, it is shown that the linear Sanders—Koiter equations can be reduced exactly to two coupled fourth-order partial differential equations, involving as unknowns a stress function and the rotation of the midsurface about the normal.

An algorithm to choose the initial multipliers optimally for quasilinearization solution of optimal control problems with bounded controls and constant multipliers is proposed. It uses diagonal matrices of weighting coefficients in the performance index of an auxiliary minimization problem. This auxiliary performance index comprises the cumulative error in the system constraints and the optimum conditions in the original extremization problem. The auxiliary performance index is quadratically dependent on the multipliers for given state and control functions. The resulting variational problem leads to linear Euler equations. The computational characteristics of the proposed method are demonstrated with two numerical examples.