Muhammad Ayub

... Scr. 82 (2010) 045402 (10pp) doi:10.1088/0031-8949/82/04/045402 Diffraction of an impulsive line source with wake M Ayub, A Naeem and Rab Nawaz Department of Mathematics, Quaid-i-Azam University, 45320, Islamabad 44000, Pakistan E-mail: mayub59@yahoo.com ...

ABSTRACT The present analysis deals with diffraction of acoustic waves by an oscillating strip focusing on the exact and concise formulation of a series solution in the complex domain. The complete solution is represented by a series, the eigenfunctions of which are generalized gamma functions. An exact expression of this special function, with argument being ‘integer +1/2’, is derived. The convergence analysis of the series solution in transformed domain is discussed graphically. Finally, the scattered and total acoustic fields are obtained by exact and asymptotic evaluations of inverse Fourier transforms. The significance of the present investigation is the derivation of a high order accurate solution in a convenient form.

ABSTRACT The present analysis deals with diffraction of acoustic waves by an oscillating strip focusing on the exact and concise formulation of a series solution in the complex domain. The complete solution is represented by a series, the eigenfunctions of which are generalized gamma functions. An exact expression of this special function, with argument being ‘integer +1/2’, is derived. The convergence analysis of the series solution in transformed domain is discussed graphically. Finally, the scattered and total acoustic fields are obtained by exact and asymptotic evaluations of inverse Fourier transforms. The significance of the present investigation is the derivation of a high order accurate solution in a convenient form.

The diffraction of a cylindrical acoustic wave from a slit in a moving fluid using Myers condition (J. Sound Vib. 71, 429 (1980)) is investigated, and an improved form of the analytical solution for the diffracted field is presented. The problem is solved analytically using an integral transform, Wiener–Hopf technique, and the modified method of stationary phase. The mathematical results are well supported by graphical discussion showing how the absorbing parameter and Mach number affect the amplitude of the velocity potential.

This article presents a new stabilized finite-element formulation for convective-diffusive heat transfer. A mixed temperature and temperature-flux form is proposed that possesses better stability properties as compared to the classical Galerkin form. The issue of arbitrary combinations of temperature and temperature-flux interpolation functions is addressed. Specifically, the combinations of C ˚ interpolations that are unstable according to the Babuska–Brezzi inf-sup condition are shown to be stable and convergent within the present framework. Based on the proposed formulation, a family of 2-D elements comprising 3-and 6-node triangles and 4-and 9-node quadrilaterals has been developed. Numerical results show the good performance of the method and confirm convergence at optimal rates.

The boundary layer flow of nanofluid that is electrically conducting over a Riga plate is considered. The Riga plate is an electromagnetic actuator which comprises a spanwise adjusted cluster of substituting terminal and lasting magnets mounted on a plane surface. The numerical model fuses the Brownian motion and the thermophoresis impacts because of the nanofluid and the Grinberg term for the wall parallel Lorentz force due to the Riga plate in the presence of slip effects. The numerical solution of the problem is presented using the shooting method. The novelties of all the physical parameters such as modified Hartmann number, Richardson number, nanoparticle concentration flux parameter, Prandtl number, Lewis number, thermophoresis parameter, Brownian motion parameter and slip parameter are demonstrated graphically. Numerical values of reduced Nusselt number, Sherwood number are discussed in detail.

This study addresses the impact of convective heat and mass conditions in the peristaltic transport of fluid in a complaint wall curved channel. Formulation for flow of third grade fluid is made. Soret and Dufour effects are considered. Fluid is conducting through applied magnetic field in radial direction. Lubrication approach is employed. Solutions for stream function, temperature and concentration fields are derived. The effects of pertinent parameters in the solutions are analyzed graphically. It is found that the velocity profile is not symmetric about the central line in curved channel. The velocity and temperature are reduced by increasing magnetic field strength. The number and size of streamlines are decreased in the presence of magnetic field effect.

The effects of wall impedances on the radiation of the dominant transverse electromagnetic wave by an impedance loaded parallel-plate waveguide radiator immersed in a cold plasma have been analyzed. The solution to the governing mathematical model in cold plasma is determined while using the Wiener–Hopf technique. It is observed that the amplitude of the radiated field increases with increasing permittivity of the plasma. The work presented may be of great interest to quantify the effects of ionosphere plasma on the communicating signals between Earth station and an artificial satellite in the Earth's atmosphere.

In this study, the problem of wave scattering of an electromagnetic field in a homogeneous bi-isotropic medium by a perfectly conducting strip is theoretically analyzed. The crux of the study is a rigorous construction of a closed form solution in the complex domain. A series solution of electromagnetic plane wave diffraction problem in terms of the eigenfunctions that happen to be the generalized Gamma functions is found. In the transformed domain, the scattered field is physically interpreted by computing the convergence history, and thereby, higher order accurate solution has been obtained in complex domain in closed form.

This work aims to investigate the mode-matching (MM) and low frequency approximation (LFA) solutions of a two dimensional waveguide problem with flanged junction. The relative merits of each approach are compared for the scattering of fluid-coupled wave. The boundary value problem involving higher order derivatives at boundaries becomes a non-Sturm–Liouville problem where the use of standard orthogonality relation (OR) enables the MM solution. The derivation of LFA is made which proves to be surprisingly accurate for structure-borne mode incident. In order to validate the truncated model expansion the distribution of power in duct regions is discussed and Gibbs oscillations are incorporated by reconstruction of the normal velocity field using Lanczos filter. The structural acoustic has become a stimulating and attention-grabbing issue of this era. The interest to minimize the ducted fan noise aero engines, power station and heating, ventilation, and air conditioning (HVAC) system offers a great inspiration to engineers and scientists. Duct work is common feature in engineering structures, air craft and buildings etc. which clearly beneficial but also a channel for unwanted noise. It propagates sound at significant distances by the mechanism of reflection through the internal walls of duct and duct vibration. In order to reduce such noises some dissipative devices likewise expansion chamber, acoustic lining or silencer and absorbent material are useful. Numerous investigations [1–3] have been made to suggest some analysis for the reduction of unwanted noise. Ayub et al. [2] and Huang [3–5] considered the reactive silencer used in HVAC system for reducing ducted tonal fan noise. The duct parallel to x-axis of the inlet/outlet of the expansion chamber was taken to be bounded by membrane with varying height. Because of this variation in height of the membrane the device was tuned which gave stopband for specified frequency. The flexible channels have been discussed by Dowell and Voss [6]. They have analyzed the cavity-backed panel at the low frequency range in the presence of fluid flow. Afterwards Kang and Fuchs [7] has extensively examined their proficiency in the case of cavity-backed micro perforated membrane. Recently Lawrie and Kirby [8,9] investigated the performance of two dimensional modified reactive silencers due to their potential use of hybrid silencers devices in HAVC ducting system. Their investigation proposed that the stopband by the silencer can be broadened and/or shifted with height of the membrane.

The diffraction of a spherical acoustic wave generated by a point source from impedance slit in a moving fluid is investigated. The diffracted wave is calculated in the far field regime as a sum of fields produced by the edges of the slit and an incident field. The Myers' impedance conditions are assumed along the edges of the slit. Such conditions are well adopted for the boundaries of the impedance barriers and yield reliable predictions of the diffraction patterns. A Wiener–Hopf technique is invoked to resolve the problem in combination with Fourier transform techniques and asymptotic analysis. The appositeness of the results and the effect of pertinent physical parameters on the separated field are presented and analyzed graphically.

In this paper, the propagation and scattering of acoustic waves in a flexible wave-guide involving step discontinuity at an interface is considered. The emerging boundary value problem is non-Sturm-Liouville and is solved by employing a hybrid mode-matching technique. The physical scattering process and attenuation of duct modes versus frequency regime and change of height is studied. Moreover, the mode-matching solution is validated through a series of numerical experiments by testifying the power conservation identity and matching interface conditions.

In this study we have examined a plane wave diffraction problem by a finite plate having different impedance boundaries. The Fourier transforms were used to reduce the governing problem into simultaneous Wiener-Hopf equations which are then solved using the standard Wiener-Hopf procedure. Afterwards the separated and interacted fields were developed asymptotically by using inverse Fourier transform and the modified stationary phase method. Detailed graphical analysis was also made for various physical parameters we were interested in.

The present analysis deals with diffraction of acoustic waves by an oscillating strip focusing on the exact and concise formulation of a series solution in the complex domain. The complete solution is represented by a series, the eigenfunctions of which are generalized gamma functions. An exact expression of this special function, with argument being 'in-teger +1/2', is derived. The convergence analysis of the series solution in transformed domain is discussed graphically. Finally, the scattered and total acoustic fields are obtained by exact and asymptotic evaluations of inverse Fourier transforms. The significance of the present investigation is the derivation of a high order accurate solution in a convenient form.

The problem of sound radiation from a soft semi-infinite duct is analyzed. This duct is symmetrically placed inside another soft but infinite duct. The problem is solved when the whole fluid region inside these ducts is in motion with a constant fluid velocity. Mathematical formulation is made. The resulting problem is solved by employing integral transform and the Jones' method based on the Wiener-Hopf technique. The kernel functions for the problem have been factorized. Graphs are sketched and analyzed. It is noticed that through specific values of the involved parameters in the solution, the unwanted noise can be abated.

This work is concerned with the influence of uniform suction or injection on flow and heat transfer analysis of unsteady incompressible magnetohydrodynamic (MHD) fluid with slip conditions. The resulting unsteady problem for velocity and heat transfer is solved by means of Laplace transform. The characteristics of the transient velocity, overall transient velocity, steady state velocity and heat transfer at the walls are analyzed and discussed. Graphical results reveal that the magnetic field, slip parameter, and suction (injection) have significant influences on the velocity, and temperature distributions, which also changes the heat transfer behaviors at the two plates. The results of Fang (2004) are also recovered by keeping magnetic field and slip parameter absent.

The present analysis deals with diffraction of acoustic waves by an oscillating strip focusing on the exact and concise formulation of a series solution in the complex domain. The complete solution is represented by a series, the eigenfunctions of which are generalized gamma functions. An exact expression of this special function, with argument being 'in-teger +1/2', is derived. The convergence analysis of the series solution in transformed domain is discussed graphically. Finally, the scattered and total acoustic fields are obtained by exact and asymptotic evaluations of inverse Fourier transforms. The significance of the present investigation is the derivation of a high order accurate solution in a convenient form.

The problem of sound radiation from a soft semi-infinite duct is analyzed. This duct is symmetrically placed inside another soft but infinite duct. The problem is solved when the whole fluid region inside these ducts is in motion with a constant fluid velocity. Mathematical formulation is made. The resulting problem is solved by employing integral transform and the Jones' method based on the Wiener-Hopf technique. The kernel functions for the problem have been factorized. Graphs are sketched and analyzed. It is noticed that through specific values of the involved parameters in the solution, the unwanted noise can be abated.

The nonlinear magnetohydrodynamic (MHD) flow problem with Hall current caused by stretching surface having power law velocity distribution is solved by employing homotopy analysis method (HAM). Perturbation solution of stream function, the expression of skin friction coefficient and graphical results in absence of Hall current (Chiam, Int J Eng Sci 33 (1995), 429) are recovered as the limiting cases. It is found that unlike the solution obtained by Chiam (1995), the present results are valid for weak and large magnetic parameters.

This work is concerned with the influence of uniform suction or injection on flow and heat transfer analysis of a second order fluid. The resulting nonlinear problem for velocity is solved by means of homotopy analysis method (HAM). The comparison between the numerical solution of Hady and Gorla (Acta Mec 128 (1998), 201–208 and HAM solution is discussed with the help of numerical tables and graphs. Nonsimilar solutions to the stream function and temperature are developed. The influence of important parameters is seen on the velocity, temperature, skin friction coefficient, and temperature gradient.

The diffraction of sound from a semi-infinite soft duct is investigated. The soft duct is symmetrically located inside an acoustically lined but infinite duct. A closed-form solution is obtained using integral transform and Jones' method based on Wiener-Hopf technique. The graphical results are presented, which show how effectively the unwanted noise can be reduced by proper selection of different parameters. The kernel functions are factorized with different approaches. The results may be used to design acoustic barriers and noise reduction devices.

In this paper, we have made Wiener-Hopf analysis of an acoustic plane wave by a semi-infinite hard duct that is placed symmetrically inside an infinite soft/hard duct. The method of solution is integral transform and Wiener-Hopf technique. The imposition of boundary conditions result in a 2 × 2 matrix Wiener-Hopf equation associated with a new canonical scattering problem which is solved by using the pole removal technique. In the solution, two infinite sets of unknown coefficients are involved that satisfy two infinite systems of linear algebraic equations. These systems of linear algebraic equations are solved numerically. The graphs are plotted for sundry parameters of interest. Kernel functions are also factorized.

A new set of nonlinear equations for toroidal ion-temperature-gradient-driven (ITGD) drift-dissipative waves is derived by using Braginskii's transport model of the ion dynamics and the Boltzmann distribution of electrons in the presence of negatively charged dust grains. The temporal behaviour of the nonlinear ITGD mode is found to be governed by three nonlinear equations for the amplitudes, which is a generalization of Lorenz-and Stenflo-type equations admitting chaotic trajectories. The linear stability analysis has been presented and stationary points for our generalized mode coupling equations are also derived.

The paper presents the analytical description of diffraction phenomena of sound at the opening of a two dimensional semi-infinite acoustically soft duct. This soft duct is symmetrically located inside an infinite duct with normal impedance boundary conditions in the case where the surface acoustic impedances of the upper and lower infinite plates are different from each other. A matrix Wiener-Hopf equation associated with a new canonical scattering problem is solved explicitly. A new kernel function arose for the problem and has been factorized. The graphical results are also presented which show how effectively the unwanted noise can be reduced by proper selection of different parameters.

An analysis is performed for the slip effects on the exact solutions of flows in a generalized Burgers fluid. The flow modelling is based upon the magnetohydrodynamic (MHD) nature of the fluid and modified Darcy law in a porous space. Two illustrative examples of oscillatory flows are considered. The results obtained are compared with several limiting cases. It has been shown here that the derived results hold for all values of frequencies including the resonant frequency.

In this study, we analyzed the diffraction of the acoustic dominant mode in a parallel-plate trifurcated waveguide with normal impedance boundary conditions in the case where surface impedances of the upper and lower infinite plates are different from each other. The acoustic dominant mode is incident in a soft/hard semi-infinite duct located symmetrically in the infinite lined duct. The solution of the boundary value problem using Fourier transform leads to two simultaneous modified Wiener–Hopf equations that are uncoupled using the pole removal technique. Two infinite sets of unknown coefficients are involved in the solution, which satisfy two infinite systems of linear algebraic equations. These systems are solved numerically. The new kernel functions are factorized. Some graphical results showing the influence of sundry parameters of interest on the reflection coefficient are presented.

The problem of diffraction due to an impulse line source by an absorbing half-plane with wake using Myres' impedance condition (Myers 1980 J. Sound Vib. 71 429–34) in the presence of a subsonic fluid flow is studied. The time dependence of the field requires a temporal Fourier transform in addition to the spatial Fourier transform. The solution of the problem in the presence of wake is obtained by using Greens' function method, Fourier transform, the Wiener–Hopf technique and the modified stationary phase method. Expressions for the total far field for the trailing edge (wake present) situation are given. It is observed that the field produced by the Kutta–Joukowski condition will be substantially in excess of the field when this condition is ignored. Finally, a simple procedure is devised to calculate the inverse temporal Fourier transform. The solution for the leading edge situation can be obtained if the wake, and consequently a Kutta–Joukowski edge condition, is ignored. This can also be seen from the numerical results.

An analysis is carried out to study the magnetohydrodynamic (MHD) flow of an incompressible micropolar fluid. The flow is induced by the noncoaxial rotations of constantly accelerated porous disk and a micropolar fluid. The influence of partial slip on the flow has been taken into consideration. Numerical solution of the governing flow problem is given by means of Newton's method. The important finding in this communication is the effects of partial slip on the velocity and microrotation vector.

This article appeared in a journal published by Elsevier. The attached copy is furnished to the author for internal non-commercial research and education use, including for instruction at the authors institution and sharing with colleagues. Other uses, including reproduction and distribution, or selling or licensing copies, or posting to personal, institutional or third party websites are prohibited. In most cases authors are permitted to post their version of the article (e.g. in Word or Tex form) to their personal website or institutional repository. Authors requiring further information regarding Elsevier's archiving and manuscript policies are encouraged to visit: http://www.elsevier.com/copyright a b s t r a c t Diffraction of acoustic plane wave through a semi-infinite hard duct which is placed symmetrically inside an infinite soft/hard duct has been analyzed rigorously. Convective flow has been taken into consideration for the analysis. In this paper the applied method of solution is integral transform and Wiener-Hopf technique. The imposition of boundary conditions result in a 2 Â 2 matrix Wiener-Hopf equation associated with a new canonical scattering problem which has been solved explicitly by expansion coefficient method. The graphs are plotted for sundry parameters of interest. Kernel functions are factorized. The results have applications to duct acoustics.

The problem of unsteady free convection flow is considered for the series solution (analytic solution). The flow is induced by an infinite vertical porous plate which is accelerated in its own plane. The series solution expressions for velocity field, temperature field and concentration distribution are presented. The influence of important parameters is seen on the velocity, temperature, concentration, skin friction coefficient and temperature gradient with the help of graphs and tables. Convergence is also properly checked for different values of the important parametes for velocity field, temperature and concentration with the help of-curves.

We examine the problem of flow and heat transfer in a second grade fluid over a stretching sheet [K. Va-jravelu, T. Roper, Int. J. Nonlinear Mech. 34, 1031 (1999)]. The equations considered by Vajravelu and Roper [K. Vajravelu, T. Roper, Int. J. Nonlinear Mech. 34, 1031 (1999)], are found to be incorrect in the literature. In this paper, we not only corrected the equation but found a useful analytic solution to this important problem. We also extended the problem for hydromagnetic flow and heat transfer with Hall effect. The explicit analytic homotopy solution for the velocity field and heat transfer are presented. Graphs for the velocity field, skin friction coefficient, and rate of heat transfer are presented. Tables for the skin friction coefficient and rate of heat transfer are also presented. The convergence of the solution is also properly checked and discussed.

This paper deals with the problem of diffraction due to an impulse line source by an absorbing half plane, satisfying Myers' impedance condition (Myers, 1980 [13]) in the presence of a subsonic flow. The time dependence of the field requires a temporal Fourier transform in addition to the spatial Fourier transform. The spatial integral appearing in the solution for the diffracted field is solved asymptotically (Copson, 1967 [15]) in the far field approximation. Finally, a simple procedure is devised to calculate the inverse temporal Fourier transform.

In this paper, firstly, the far field due to a line source scattering of acoustic waves by a soft/hard half-plane is investigated. It is observed that if the line source is shifted to a large distance, the results differ from those of [16] by a multiplicative factor. Subsequently, the scattering due to a point source is also examined using the results of line source excitation. Both the problems are solved using the Wiener– Hopf technique and the steepest descent method. Some graphs showing the effects of various parameters on the diffracted field produced by the line source incidence are also plotted.

This study mainly focuses on the solution of the vertical gravity anomaly gradient and determination of its effect on the total geoid-quasigeoid separation. Due to its small effect on geoid-quasigeoid separation, the planar approximation of vertical gravity anomaly in the innermost zone has been implemented. The computation of a strongly singular integral expressing the vertical gravity anomaly gradient was used for this purpose for even order (up to n=6) of the Simpson and Newton–Cotes integration technique. The derivation of the relationships for different integration radii has been made to obtain these solutions using gridded data of free air anomaly. The comparison of relationships for the different integration radii was made in order to select an optimum radius of the integration in planar approximation for the vertical gravity anomaly gradient dependent geoid-quasigeoid separation term. The integration radii of 3.1, 6.2 and 9.3 km show an increasing behaviour towards saturation. The effect of vertical gravity anomaly gradient has similar pattern on the geoid-quasigeoid separation term towards saturation. The saturation trend for vertical gravity anomaly gradient is comparatively faster than its corresponding geoid-quasigeoid separation dependent term. The results also show that 2nd order Newton–Cotes integration is found to be comparable with the approximate linear solution for the vertical gravity anomaly gradient given by Heiskanen and Moritz. The vertical gravity anomaly dependant term has a rather small effect on geoid-quasigeoid separation in the mid elevation range and ranges from −6.18 to 2.7 mm for n=6. The findings of the study leads to the inferences that the order of integration should be selected either n=4 or n=6 for better estimates of vertical gravity anomaly gradient solution. This criterion is also valid for their effect on geoid-quasigeoid separation with planar approximation in the innermost zone for the low-to mid-range elevation areas.

In this paper we have studied the problem of diffraction of a plane wave by a finite soft–hard strip. By using the Fourier transform the boundary value problem is reduced to a matrix Wiener–Hopf equation. Using the matrix factorization of the kernel matrix, the problem is solved for two coupled equations using the Wiener–Hopf technique and the method of steepest descent. It is observed that the diffracted field is the sum of the fields produced by the two edges of the strip and an interaction field. Some graphs showing the effects of various parameters on the field produced by two edges of the strip are also plotted.