Lighthill's Method Research Articles

Two-Phase Fluctuating Flow of Dusty Viscoelastic Fluid Between Non-Conducting Rigid Plates With Heat Transfer

The objective of this article is to investigate the combined effects of the magnetic field and heat transfer on electrically conducting viscoelastic incompressible dusty fluid, flowing between two non-conducting rigid plates. The flow generation in the aforementioned fluid is caused by the oscillating pressure gradient and heat transfer. It is also assumed that all the dust particles having spherical shapes are homogeneously distributed in the fluid. In order to investigate analytical solutions, Light Hill method has been used. Furthermore, the effects of different parameters like an elastic parameter, radiation parameter, Reynolds number and Grashof number on velocity and applied shear stress have been discussed. A noteworthy relation of applied magnetic field with velocity and applied shear stresses is noticed, from the present study it is depicted that in boundary layer flow velocity increases and shear stress decreases with increase in applied magnetic field, while in ordinary course of nature, by increasing applied magnetic field decrease occurs in fluid velocity and shear stress increases.

Asymptotic solutions for laminar flow based on blood circulation through a uniformlyporous channel with retractable walls and an applied transverse magnetic field

This paper is concerned with asymptotic solutions of a nonlinear boundary value problem (BVP), which arises in a study of laminar flow in a uniformly porous channel with retractable walls and an applied transverse magnetic field. For different ranges of the control parameters (i.e. α,Re and M) arising in the BVP, four cases are considered using different singular perturbation methods. For the first case, unlike those in the existing literature, we make use of the Lighthill method and successfully construct an asymptotic solution with high-order derivatives at the center of the channel. For the second case, under large suction we consider M2=O(1) and M2=O(Re), respectively, which will further extend the applying range of asymptotic solutions. In other cases, asymptotic solutions with a boundary layer are successfully constructed. In addition, numerical solutions presented for each case agree well with asymptotic solutions, which illustrates that the asymptotic solutions constructed in this paper are more reliable. Finally, the influences of some parameters on flow field are discussed to develop a better understanding of the flow problem.

The solutions for the flow of micropolar fluid through an expanding or contracting channel with porous walls

The unsteady, two-dimensional laminar flow of an incompressible micropolar fluid in a channel with expanding or contracting porous walls is investigated. The governing equations are transformed into a coupled nonlinear two-points boundary value problem by a suitable similarity transformation. Unlike the classic Berman problem (Berman in J. Appl. Phys. 24:1232-1235, 1953), three new solutions (totally six solutions) and no-solution interval, which is one of important characteristics for the laminar flow through porous pipe with stationary wall (Terrill and Thomas in Appl. Sci. Res. 21:37-67, 1969), are found numerically for the first time. The multiplicity of the solutions is strictly dependent on the expansion ratio. Furthermore, the asymptotic solutions are constructed by the Lighthill method, which eliminates the singularity of the similarity solution, for large injection and by the matching theorem for the suction Reynolds number, respectively. The analytical solutions also are compared with the numerical ones and the results agree well.

Perturbation solutions for a micropolar fluid flow in a semi-infinite expanding or contracting pipe with large injection or suction through porous wall

Abstract We investigate an unsteady incompressible laminar micropolar flow in a semi-infinite porous pipe with large injection or suction through a deforming pipe wall. Using suitable similarity transformations, the governing partial differential are transformed into a coupled nonlinear singular boundary value problem. For large injection, the asymptotic solutions are constructed using the Lighthill method, which eliminates singularity of solution in the high order derivative. For large suction, a series expansion matching method is used. Analytical solutions are validated against the numerical solutions obtained by Bvp4c.

Computation of Acoustic Far‐Fields with the Spectral Lighthill Method

AbstractBased on Lighthill's acoustic analogy we formulate the Spectral Lighthill Method (SLM). SLM is a method for the computation of acoustic far‐fields. It uses a spatio‐temporal Fourier transform of the Lighthill stress tensor. We show that SLM is a straightforward tool for the computation of acoustic far‐fields that enhances our physical understanding of sound generation and is useful in the numerical analysis of acoustic far‐field solvers. (© 2006 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)

On Mutual Transform Between Number-Difference State and Phase StateCorresponding to Operational PhaseOperator**The project supported by the Ph. D Tutoring Programmeof the Educational Ministry of China

In the mutual transform between the number-differencestate and the phase state corresponding to the operationalphase operator we find that there exists an end-point ambiguousness. This problem canbe avoided by Lighthill's method.

A new method for explaining the generation of aerodynamic forces in flapping flight

AbstractSir James Lighthill proposed a method for estimating the hydrodynamic forces generated by slender fish, and his analysis made clear the mechanism of fish swimming. His method is based on slender body theory, and the hydrodynamic forces produced by fish are explained by their kinematics and the added mass of the body sections. More recently, numerical techniques for estimating hydrodynamic forces are making progress, but Lighthill's method is still very important and useful because it contributes to our understanding of the physics of fish swimming.We propose a new method for estimating the aerodynamic forces generated by flapping wings: the forces are explained by the added mass of vortex wake sheets. In concept, it is similar to the momentum theory for fixed wings, which was developed by Prandtl. So, when the forward velocity is large, the method agrees with the momentum theory for a fixed wing. For hovering flight, where vortex sheets are generated continuously, this method agrees with the Rankine–Froude momentum theory of propellers. Like Lighthill's method for swimming, this newly developed method provides useful insight into the physics of flapping wings. Copyright © 2001 John Wiley & Sons, Ltd.

A new method for explaining the generation of aerodynamic forces in flapping flight

Sir James Lighthill proposed a method for estimating the hydrodynamic forces generated by slender fish, and his analysis made clear the mechanism of fish swimming. His method is based on slender body theory, and the hydrodynamic forces produced by fish are explained by their kinematics and the added mass of the body sections. More recently, numerical techniques for estimating hydrodynamic forces are making progress, but Lighthill's method is still very important and useful because it contributes to our understanding of the physics of fish swimming. We propose a new method for estimating the aerodynamic forces generated by flapping wings: the forces are explained by the added mass of vortex wake sheets. In concept, it is similar to the momentum theory for fixed wings, which was developed by Prandtl. So, when the forward velocity is large, the method agrees with the momentum theory for a fixed wing. For hovering flight, where vortex sheets are generated continuously, this method agrees with the Rankine–Froude momentum theory of propellers. Like Lighthill's method for swimming, this newly developed method provides useful insight into the physics of flapping wings. Copyright © 2001 John Wiley & Sons, Ltd.

Energy and pseudomomentum of propagating disturbances on the beta-plane

The far-field amplitude of the waves generated by a steadily propagating radially symmetric disturbance on the beta-plane is calculated using Lighthill's method. From this can be obtained the fluxes of quantities such as wave energy which are radiated away from the disturbance. The radiated wave power is computed for a variety of forms of the disturbance. The rate of change of pseudomomentum in the system is also calculated: the component parallel to the motion of the disturbance is the radiated wave power divided by velocity. Results are compared to previous work and some physical issues are discussed.

A new velocity decomposition for viscous flows: lighthill's equivalent-source method revisited

A convenient decomposition for vector fields is presented that falls within the vast class of potential-vorticity decompositions of the type υ = V ¯ ϕ + w , which includes the classical Helmholtz and Clebsch decompositions. The distinguishing feature of the present decomposition is that the rotational velocity contribution w (obtained by a line-integration along a path normal to the boundary) vanishes in much of the irrotational region. As a consequence, in the irrotational region one obtains υ = V ¯ ϕ . The mathematical formulation of the problem and the application to the analysis of incompressible viscous flows around solid bodies are discussed. For flows in which the vortical region is concentrated in a thin layer surrounding the surface of the body and the wake mid-surface, the relationship between the present formulation and the classical Lighthill's equivalent-source approach is addressed. In addition, an exact extension of the Lighthill formulation is presented. The equivalence of the two approaches is established. Numerical results for the reconstruction of the two-dimensional velocity field (from a prescribed vorticity field) are presented to illustrate the differences between various approaches.

The propagation of oscillations from a point source in an anisotropic plane and half-plane with a thin coating

Two dynamical problems of the propagation of oscillations in an anisotropic medium generated by a point harmonic force are considered. In the first problem steady harmonic oscillations in an anisotropic plane are investigated. The solution can be reduced to solving a system of second-order elliptic equations for the steady part of the displacements. As the second problem, within the framework of the basic physical model [1], the dynamical contact problem for an anisotropic half-plane reinforced along its boundary by an infinite ellastic coating in the form of a thin cover is considered. The generating point force is harmonic. The solution of each of the two problems is constructed by the method of Fourier transforms. Using Lighthill's method [2] and the method of stationary phase [3], asymptotic formulae are obtained for the strains and stresses, in which surface, quasilongitudinal and quasitransverse wave!; can be distinguished explicitly.

Pseudopotential Theory for Interionic Interaction Potentials in Metals at Non-Zero Temperature. II. Temperature Dependence of Higher Order Terms in Asymptotic Expressions

Asymptotic expression for Fourier transform of a regular function f ( x ) is estimated by transforming f ( x ) into \(\int F(x-z)g(z)dz\), where F ( x ) has a singularity at x =0 and the asymptotic expression for Fourier transform of F ( x ) is estimated by the usual Lighthill's method. As in I, the transforms are applied to derive the interaction potentials in metals at non-zero temperatures. Now F ( x )= x log | x |, x 2 log | x |, and x 2 [log | x |] 2 are discussed, since they play important roles in deriving asymptotic expressions of interaction potentials V 3 [cos (2 k F r )/(2 k F r ) 3 ]+ V 4 [sin (2 k F r )/(2 k F r ) 4 at zero temperature. In a previous paper, by using F ( x )=1/ x , the temperature dependence of the first term was considered, but now that of both terms also are done. Fourier transforms of generalized function (1/ x )log | x + a | and some of related functions, for example ( x + a )( x - a )log | x + a |log | x - a | , are given.

Pseudopotential Theory for Interionic Interaction Potentials in Metals at Non-Zero Temperature. I. Elementary Derivation of Asymptotic Expressions on Basis of Hilbert Transforms

An asymptotic expression V pair.as ( r ; T ) of a pair potential between ions in metals at non-zero temperature T ≠0 is given approximately by a form V pair.as ( r ; 0) D ( r ; T ), where V pair.as ( r ; 0)= V 3 cos (2 k F r )/(2 k F r ) 3 is the Friedel oscillatory potential at T =0, D ( r ; T )= x · cosech ( x ) with x =π( k B T / E F ) k F r is a damping factor, k B the Boltzmann constant, k F and E F the Fermi wave number and the Fermi level. Lighthill's method for estimating asymptotic form of Fourier transforms of f ( x ) that includes a singular point is proved to be applicable also to transformed functions \(F(x)= \int f(x-z)g(z; T) {\rm d}z\), where g ( z ; T )→δ( z ) as T →0. The proof is done by the use of Hilbert transforms, in which f ( x )=1/ x . This modified Lighthill's method derives V pair.as ( r ; T ) in the usual way that is used in pseudopotential treatment of V pair.as ( r ; 0). Earlier works by Kohn and Vosko, March and Murray, and Flynn and Odle are examined.

Acoustic characteristics of confined jets. Part 1: Theory and numerical calculations

A theoretical model was developed to study the wall pressure and vibratory motion of an elastic pipe which is excited by a confined turbulent jet flow resulting from fluid flow through orifices in the pipe. Based on flow field measurements, the blocked surface pressure was calculated using Lighthill's method, and then used to drive the fluid-filled shell. The wall pressure and pipe wall acceleration were determined by solving the coupled fluid-solid interaction problem. The wall pressure was obained by summing the blocked surface pressure and the pressure due to the wall vibration. An amplitude-modulated convecting wave field was used to simulate the moving acoustic sources of the jet. The random nature of the turbulent jet was incorporated into the analytical model. Specifically, the acoustic pressure was assumed to result from hydrodynamic pressure fluctuations which are uncorrelated in the radial direction, but correlated in the axial direction near the jet exit. The uncorrelated pressure fluctuations in the radial direction reflect the random motion of the turbulent jet, whereas the correlated pressure fluctuations in the axial direction reflect the motion of the large-scale coherent structures near the jet exit. It has been demonstrated that the noise model is capable of relating the flow field and the acoustic field of confined jet flows. [Work supported by ONT and ONR.]

The theory of magnetohydrodynamic wave generation by localized sources. I - General asymptotic theory

The magnetohydrodynamic wave emission from several localized, periodic, kinematically specified fluid velocity fields are calculated using Lighthill's method for finding the far-field wave forms. The waves propagate through an isothermal and uniform plasma with a constant B field. General properties of the energy flux are illustrated with models of pulsating flux tubes and convective rolls. Interference theory from geometrical optics is used to find the direction of minimum fast-wave emission from multipole sources and slow-wave emission from discontinuous sources. The distribution of total flux in fast and slow waves varies with the ratios of the source dimensions l to the acoustic and Alfven wavelengths.

A perturbation method for mixed three-dimensional problems of the theory of elasticity with a complex line of boundary-condition separation

A modification of the perturbation method is proposed, based on the utilization of variational formulas and enabling asymptotic expansions (AE) to be obtained for mixed three-dimensional problems of the theory of elasticity with a complex line of boundary-condition separation. Application of Lighthill's method enables these expansions to be transformed into uniformly suitable ones. The problem for an elastic body with a slit (crack) and the contact problem of the theory of elasticity are considered separately. For the body with a slit the variational formula determines the variation of the displacement of the slit surface caused by variation in the shape of the slit contour. The effectiveness of this formula for constructing AE in problems associated with a perturbation of the shape of the slit contour is shown. Cases of slits of complex shape in an infinite body that differ slightly from a circular slit are examined in detail. A scheme for constructing similar AE is mentioned for spatial contact problems of the theory of elasticity with a complex shape of the contact area. A review of the application of perturbation methods to mixed problems in the theory of elasticity is contained in /1, 2/. The solutions of mixed spatial problems in the theory of elasticity with a complex line of boundary condition separation, obtained by using other methods, are discussed in /3–8/. The behaviour of the solution of the boundary value problem for a pseudodifferential equation (in particular, crack theory) for variation of the domain was investigated in /9/.

Wave diffraction by large offshore structures: an exact second-order theory

An exact second order theory has been formulated in this paper to calculate the wave forces on offshore structures. Lighthill's method for deep water waves has been extended to shallow water waves. Exact expression for linear velocity potential applicable to the circular cylindrical strcutures for shallow water waves have been used in these calculations. These results have been verified with those obtained by direct perturbation technique reported recently by Rahman and Heaps. It is interesting to note that both the methods yield identical results.

Uniformly valid amplitude expansions for non-Abelian gauge theories

The imposition of a particular gauge condition in a non-Abelian gauge theory may lead to large-r nonuniformities in an amplitude expansion when used in the construction of an approximate solution of the field equations of the theory. We show that for both the Yang-Mills theory and general relativity it is always possible to find a class of gauge conditions that do not suffer from this defect and at the same time lead to solvable equations in each order of the approximation employed. We further show how one can construct such gauge conditions by a method similar to Lighthill's method of strained coordinates. An example of such a construction is given for the Yang-Mills theory.

On the Damping of Second-Class Trapped Waves on a Sloping Beach

Abstract The effect of linear bottom friction on free topographic (second class) waves on a sloping beach is investigated. To handle the coastal singularity in the friction terms, Lighthill's method of strained coordinates is used to find a perturbation solution to the governing equations. Simple expressions for the decay rate are worked out for two classical beach profiles: 1) the uniformly sloping beach, and 2) the exponential depth profile due to Ball (1967). For parameter values characteristic of the north shore of Lake Ontario, which can be modelled by the Ball profile, only the gravest mode will not experience rapid attenuation due to bottom friction.

Arbitrary body, close to a wedge, in a supersonic flow

The problem of supersonic flow around bodies close to a wedge was first discussed in the two-dimensional case in [1]. The shock wave was assumed to be attached, and the flow behind it to be supersonic; taking this into account, the angle of the wedge was assumed to be arbitrary. The surface of the body was also arbitrary, provided that it was close to the surface of the wedge. In solution of the three-dimensional problem, there was first considered flow around two supporting surfaces with only slightly different angles of attack [2], and then around a delta wing [3, 4]. In all these articles, the Lighthill method was used to solve the Hilbert boundary-value problem [5, 6]. A whole class of surfaces of bodies with arbitrary edges, under the assumption that the surface of the body was cylindrical, with generatrices directed along the flow lines of the unperturbed flow behind an oblique shock wave, was discussed in [7]. In the present work, the problem is regarded for a broad class of surfaces of bodies, using a new method which generalizes the results of [8].