Showing papers in "Journal of Engineering Mathematics in 1975"

Low-Reynolds-number translation of a slender cylinder near a plane wall

Abstract: Low-Reynolds-number results are presented for the drag and induced torque on a slender circular cylinder translating near a single plane wall. Four representative situations are investigated, the principal feature of the analysis being that it is valid for all distances from the wall which are large compared with the radius of the cylinder. In particular, the results hold for distances from the wall of the same order of magnitude as the length of the cylinder. The direction and rate of rotation are given for those cases where it occurs.

Numerical solution of some ordinary differential equations occurring in plate deflection theory

Abstract: A certain fourth-order differential equation is solved numerically by the method of finite differences. Conditions on the original differential equation are given which are sufficient to quarantee that the matrix thus produced is monotone so that a straightforward error analysis is possible. This error analysis is given in detail. Examples are given which demonstrate the validity of this error analysis.

A variational approach to an unsymmetric water-wave scattering problem

Abstract: A plane surface wave train on infinitely deep water is incident upon a pair of fixed thin vertical barriers, one of which is in the surface, the other submerged. The relation between the input and output amplitudes is obtained via a variational approximation for large barrier separations. It is shown that, within this approximation, infinite spectra of totally reflected and totally transmitted waves exist if the barriers overlap, but for non-overlapping barriers this is not the case.

Bounds for the solution of a second order differential equation with mixed boundary conditions

Abstract: A finite difference scheme is given for the numerical approximation of the real solution of the second order linear differential equation, lacking the first derivative, with mixed boundary conditions. The matrix associated with the resulting system of linear equations is tridiagonal and the overall discretization error isO (h4). The derived error bound is at most four times larger than the observed maximum error in absolute value for the numerical problem considered.

A two-dimensional model for the cochlea

Abstract: A two-dimensional model for the cochlea is developed. An integral equation is derived that describes the pressure difference between the scalae. For the main quantity, the transmembrane pressure, an ordinary differential equation is obtained, which appears to be an improvement of the well-known Peterson-Bogert equation. The results are valid for all frequencies; an assumption of long or short wavelengths is not necessary at all.

Slow motion of a bubble in a viscoelastic fluid

Abstract: The motion of a deformed spherical body in a fluid medium is significantly different from the motion of an undeformed spherical body in the same medium. It is shown in this work that a bubble moving in a viscoelastic fluid takes the shaper=a+U 0η0/σa(λ 1−λ 2)(180R 3+240R 2+816R+672)P 2(cosθ)/960(1+R)3 and so one must expect the dynamics of a bubble moving in a non-Newtonian fluid to be significantly different from that of a bubble moving in a purely viscous fluid.

A new calculation of the wake of a flat plate

Abstract: A new method is presented for the calculation of the wake of a finite flat plate. The method is based upon the recent investigations of the boundary layer near the trailing edge, which led to the triple-deck structure. This multi-layered structure has now been extended to the “classical” wake, which in fact is the continuation of the lowest two layers of the triple-deck. With this new numerical formulation an accuracy of 10−3% can easily be achieved.

Finite difference techniques for ring capacitors

Abstract: This paper describes finite difference techniques used to calculate the capacitance of a ring capacitor. The determination of capacitance involves the solution of a Dirichlet boundary value problem and the calculation of the gradient of the solution obtained. Circular cylindrical coordinates are used. Nine point difference approximations are used for the Laplacian and the first derivatives of a function. If this function satisfies Laplace's equation and is sufficiently differentiable, the discretization error of each approximation isO(h 4) whereh is the maximum mesh size.

Incompressible viscous flow near the leading edge of a flat plate admitting slip

Abstract: The shear stress at the leading edge, calculated on basis of the Navier-Stokes equations and the no-slip boundary condition, approaches infinity. However, taking into account the mean free path of the molecules, which implies admitting a certain slip, the shear stress becomes inversely proportional to the square root of the Knudsen numberk ifk→0.k is defined as the ratio between the mean free path and the viscous length. The new boundary condition modifies the shear stress only within the Knudsen region of which the size is of the order of 3 to 4 times the mean free path.

On the radiation of short surface waves by a heaving circular cylinder

Abstract: A long circular cylinder half immersed in the free surface of an ideal fluid undergoes small time periodic motions. The method of matched asymptotic expansions is used to give a solution in the high frequency limit. Of particular interest are the surface waves generated by this motion, and a three term asymptotic series for their amplitude is found. It is proved that there are no eigensolutions of the infinite vertical barrier problem containing waves which are purely outgoing, and it is shown how this can be used to predict the wave amplitude to a higher order than that of the matching solution.

An isothermal theory of anisotropic rods

Abstract: To provide an isothermal, deterministic theory of anisotropic rods is the primary objective of this paper. Our starting point is the 3-D linear theory of micropolar elastodynamics. First, the governing equations of the theory are established by the use of a suitable averaging procedure together with a separation of variables solution for kinematic variables. Next, without making the usual definiteness assumption for the strain energy density, a dynamic uniqueness theorem is constructed for the solutions of the governing equations. Logarithmic convexity arguments are then used to enumerate a set of conditions sufficient for uniqueness. The theory includes the effects of warping and shearing deformations, and in fact, it incorporates as many higher order effects as deemed necessary in any special case. Also, the application of the theory is illustrated in a sample example.

Eigenvalues of a slightly stiff pendulum with a small bob

Abstract: We consider an eigenvalue problem associated with the small vibrations of a slightly stiff pendulum. In this problem, the fourth order differential equation contains two small dimensionless parameters and takes its distinctive character from the simple turning point where the coefficient of the second derivative term vanishes. This turning point always lies outside of the interval of interest. However, a significant feature is that in the parameter range corresponding to small bob mass, one endpoint lies inside the critical layer about the turning point, and hence outer expansions alone are not adequate for formation of a characteristic equation. Approximations to solutions of the governing differential equation are obtained and related by the method of matched asymptotic expansions. All outer expansions are required to be “complete” in the sense of Olver. The ordering of approximations to the characteristic equation is found to depend critically on the relative sizes of the two parameters. We obtain consistent second approximations for the square roots of the eigenvalues.

Optimal control problems with delay, the maximum principle and necessary conditions

Abstract: In this paper we consider a rather general optimal control problem involving ordinary differential equations with delayed arguments and a set of equality and inequality restrictions on state- and control variables. For this problem a maximum principle is given in pointwise form, using variational techniques. From this maximum principle necessary conditions are derived, as well as a Lagrange-like multiplier rule. Details may be found in ref. [2], together with extensions to the Hamilton-Jacobi equation and free end point problems.

Localization of non-linearities in the cochlea

Abstract: An attempt is made to locate the non-linear sources inside the cochlea. The possible sources can be divided in several classes: of no significance are the perilymph, the endolymph and the impedances of Reissner's membrane and the basilar membrane; of little significance is the motion of the mentioned membranes in their own plane; of uncertain significance are the oval window impedance and the spiral coiling of the cochlea, while the mechanism of tectorial membrane, haircells and organ of Corti is likely to be an important source of non-linearity. Moreover the widely used membrane equation is improved in the course of the work.

Plastic-elastic torsion, optimal stopping and free boundaries

Abstract: It is shown that the mathematical formulation of the plastic-elastic torsion of a cylindrical bar with cross sectionS and that of optimal stopping of Brownian motion onS with its boundary absorbing, with cost of motion per unit of time a constant and with stopping costs described by a surface of constant slope on the boundary ofS both lead to the same partial differential equation with the same free boundary conditions. The well-known membrane-sandhill analogy of Nadai is here then also a very useful interpretation of the optimal stopping problem.

Large-time inversion of certain Laplace transforms in dissipative wave propagation

Abstract: A large-time approximation for the inversion of Laplace transforms that commonly occur in dissipative wave propagation is obtained and discussed. This asymptotic approximation properly describes the bulk wave front and satisfies appropriate boundary conditions. The generality of the method is illustrated by means of examples from gas dynamics and viscoelasticity.

On the scalar scattering by a strip in a dissipative medium

Abstract: An approximate theory of the scalar scattering by a strip in a dissipative medium is established. The theory is suitable when the relations 1/Im (k)≪h≪d are satisfied by the width of the strip (2h), the smallest distance between the plane of the strip and the exciting sources (d) and the wave number (k). The expressions valid in the far-field approximation are obtained by simple manipulations. Some illustrative examples concerning the scattering of electromagnetic waves, emitted by Hertzian dipoles, by unidirectionally conducting strips are given.

The torsion problem for a circular cylinder with radial edge cracks

Abstract: A finite Mellin transform technique reduces the torsion problem for a circular cylinder with radial edge cracks to that of solving some integral equations. Expressions are found for the stress intensity factors and crack formation energy. Three particular cases are considered in detail and numerical results given.

A statistical model of the longitudinal dispersion process in turbulent flow in a channel

Abstract: The important practical problem of the dispersion of a passive contaminant in a fluid flowing through a pipe or channel of uniform cross-section is usually analysed in terms of the distribution of concentration. In this paper however a different though approximate approach is adopted which both illustrates the essential statistical nature of the process and may be quicker to employ when approximate answers are acceptable in a practical problem. A simple statistical model is proposed for the motion of a single molecule of contaminant and leads to an expression for the covariance of the velocity of the molecule in terms of the fluid velocity, and hence to a value of Taylor's longitudinal diffusivity. The model is applied to two simple flows in a channel, one of which illustrates the effect of the viscous sub-layer. Despite the number of simplifying assumptions made in constructing the model it gives results which are close to those obtained by conventional means. Ways in which the model could be adapted to give even better results are discussed.

On non-symmetric vibration of deep spherical sandwich shells

Abstract: This paper deals with the free non-symmetric vibration of deep spherical sandwich shells. The sandwich shell considered herein consists of three layers. A variational technique is utilized to obtain the equations of motion as well as the appropriate boundary conditions. The effects of transverse shear deformation and rotary inertia have been included in this analysis.

The torsional rigidity of anisotropic prisms

Abstract: Upper and lower bounds are obtained for the torsional rigidity of a prismatic cylinder of non-homogeneous anisotropic elastic material. Improvement in the bounds is obtained by expressing each bound as the quotient of two bordered determinants. Some analytical and numerical results are also presented.

Discrete solutions to engineering design problems

Abstract: A method of obtaining discrete and/or integer valued solutions to non-linear design problems is presented. The general framework is that of geometric programming which is combined with the Branch and Bound Method. Recently developed computational procedures are described and are used to demonstrate the feasibility of the above method.

Saint-Venant's problem for inhomogeneous and anisotropic solids

Abstract: The present paper is concerned with Saint-Venant's problem for inhomogeneous and anisotropic elastic cylinders when the elastic coefficients are independent of the axial coordinate. The paper points out the importance of the generalized plane strain problem in the treatment of Saint-Venant's problem.

Analysis of parameter changes in chemical systems via geometric programming

Abstract: This paper exploits the relationship of chemical equilibrium problems to geometric programming in order to study the effect of changes in certain parameters on the equilibrium solution. As a result of analyzing these problems from the geometric programming point of view we develop efficient procedures for studying the effect of changes in pressure and/or temperature on the equilibrium solution. In particular we develop methods for determining allowable ranges of change and within those ranges we develop formulas for computing the new equilibrium solution. These developments are illustrated with an example.

One-dimensional wave propagation and Fokker-Planck's equation

Abstract: The one-dimensional random wave propagation problem is analyzed. The medium is assumed to be characterized by a stationary index of refraction of the white-Gaussian process. By considering the initial value equation for the boundary value stochastic Green's function, the Fokker-Planck equation for the density function of the amplitude and phase of the reflected wave is constructed. By employing an averaging theorem due to Khas'minskii, the mean power reflected and transmitted is obtained.

Longitudinal surface curvature effect in magnetohydrodynamics

Abstract: The two-dimensional motion of an incompressible and electrically conducting fluid past an electrically insulated body surface (having curvature) is studied for a givenO (1) basic flow and magnetic field, when (i) the applied magnetic field is aligned with the velocity in the basic flow, and (ii) the applied magnetic field is within the body surface. (O(1) andO(Re−1/2) mean the first and second order approximations respectively in an expansion scheme in powers ofRe−1/2,Re being the Reynolds number.) The technique of matched asymptotic expansions is used to solve the problem. The governing partial differential equations toO(Re−1/2) boundary layer approximation are found to give similarity solutions for a family of surface curvature and pressure gradient distributions in case (i), and for uniform basic flow with analytic surface curvature distributions in case (ii). The equations are solved numerically.

Elastic wave-motion across a vertical discontinuity

Abstract: Exact solutions are obtained for the displacement field in an elastic half-space composed of two quarter spaces welded together. The configuration is excited by a plane SH wave impinging upon the discontinuity at an arbitrary angle. The application of the Kontorovich-Lebedev transform to this boundary value problem leads to two simultaneous integral equations which are solved exactly. It is shown that the discontinuity may enhance the spectral displacements up to a factor of two. The results could be applied to propagation of seismic shear waves past fault zones in the earth's crust.

Some aspects of non-uniform convergence in an elliptic singular perturbation problem

Abstract: In elliptic singular perturbation problems several types of boundary layers occur. A model problem is investigated and it is shown that, by extension of the familiar stretching technique to parameters, uniform results can be obtained.

Solutions controllable in small finite and infinitesimal theories of elastic dielectrics

Abstract: An electric field and a deformation constitute a controllable state if they can be maintained in every homogeneous, isotropic, elastic dielectric without the body force and distributed charge The controllable states possible for small finite theories of compressible elastic dielectrics are determined Also obtained are the controllable states of the classical electrostriction theory

Potentials and flows associated with a line segment

Abstract: Integral transformations are developed to construct three and five axisymmetric potentials for a needle or straight line segment. These potentials are applied to flow past a needle and one of the main results is for streaming flow past a line segment in which the fluid velocity vanishes on the boundary. This solution may also be regarded as a Stokes flow or an inviscid potential flow.