Hybrid journal (It can contain Open Access articles) ISSN (Print) 0272-4960 - ISSN (Online) 1464-3634 Published by Oxford University Press[372 journals]

Authors:Lyalinov M. Pages: 53 - 91 Abstract: In this work we study the problem of diffraction of an acoustic plane wave by a semi-infinite angular sector with impedance boundary conditions on its surface. It is studied by means of incomplete separation of variables. With the aid of Watson–Bessel integral representation the problem is reduced to a boundary value problem on the unit sphere with an operator-impedance boundary condition on a cut of the sphere. The latter problem is further studied by means of the traditional methods of extensions of sectorial sesquilinear forms. The Sommerfeld integral representation is obtained from that of Watson–Bessel with the aim to develop the far-field asymptotics. Analytic properties of the corresponding Sommerfeld transformant are also discussed. For a narrow impedance sector, an asymptotic formula for the diffraction coefficient of the spherical wave propagating from the vertex is derived. PubDate: Wed, 10 Jan 2018 00:00:00 GMT DOI: 10.1093/imamat/hxx044 Issue No:Vol. 83, No. 1 (2018)

Authors:Madhi Alsharif A. Pages: 174 - 187 Abstract: A curved liquid jet is widely used in various practical applications, such as the prilling process for generating small spherical pellets (fertilizer) and inkjet printing. A deep understanding of the mechanisms of the break-up of liquid jets and the associated flow dynamics are heavily dependent upon the nature of the fluid. In this paper, we model the viscoelastic liquid jet by using the Giesekus model. In addition, the governing equations have been reduced to 1-D by using an asymptotic approach. Then, we determine the trajectory of viscoelastic liquid curved jets. Furthermore, the nonlinear evolution equations for the jet radius and the axial velocity are solved numerically using finite differences scheme based on the Lax–Wendroff method. PubDate: Thu, 11 Jan 2018 00:00:00 GMT DOI: 10.1093/imamat/hxx040 Issue No:Vol. 83, No. 1 (2018)

Authors:Fernandez A; Spence E, Fokas A. Pages: 204 - 242 Abstract: We obtain the rigorous uniform asymptotics of a particular integral where a stationary point is close to an endpoint. There exists a general method introduced by Bleistein for obtaining uniform asymptotics in this situation. However, this method does not provide rigorous estimates for the error. Indeed, the method of Bleistein starts with a change of variables, which implies that the parameter governing how close the stationary point is to the endpoint appears in several parts of the integrand, and this means that one cannot obtain general error bounds. By adapting the above method to our particular integral, we obtain rigorous uniform leading-order asymptotics. We also give a rigorous derivation of the asymptotics to all orders of the same integral; the novelty of this second approach is that it does not involve a global change of variables. PubDate: Thu, 04 Jan 2018 00:00:00 GMT DOI: 10.1093/imamat/hxx042 Issue No:Vol. 83, No. 1 (2018)

Authors:Bastos M; Lopes P, dos Santos A. Pages: 92 - 105 Abstract: Using the reduction of a vector Riemann–Hilbert problem on the unit circle to a scalar problem on a contour in a Riemann surface, a factorization method for a class of symbols is described. The class of symbols involves outer functions and rational functions of the square root of a quotient of first degree polynomials. An application to a problem in the field of integrable systems of infinite dimension is presented. PubDate: Fri, 22 Dec 2017 00:00:00 GMT DOI: 10.1093/imamat/hxx036 Issue No:Vol. 83, No. 1 (2017)

Authors:Brosa Planella F; Please C, Van Gorder R. Pages: 106 - 130 Abstract: Understanding the solidification process of a binary alloy is important if one is to control the microstructure obtained during the casting of metals. Whilst much work has been done on the steady-state solidification problem, despite their relevance to metallurgical applications, there is less known about non-steady solidification problems and their stability. In the paper we shall consider the non-steady solidification problem in which the planar solidification front moves in a self-similar manner, in both infinite and semi-infinite planar 1D geometries. For each geometry exact solutions are known for the resulting Stefan problem. We direct our attention to the stability of each solution, demonstrating that whilst the concentration and thermal solutions remain stable, the interface corresponding to the solidification front can develop instabilities. For each geometry, we find that there are always unstable perturbations, although we observe qualitative differences in the form of the unstable perturbations for each case. These results generalize and extend several existing studies in the literature, and throw light on the instability inherent in the non-steady solidification process. PubDate: Sat, 23 Dec 2017 00:00:00 GMT DOI: 10.1093/imamat/hxx037 Issue No:Vol. 83, No. 1 (2017)

Authors:Rabieifar A; Pourseifi M, Derili H. Pages: 131 - 147 Abstract: The problem of determining the dynamic stress intensity factors (DSIFs) in a medium made by functionally graded orthotropic materials weakened by multiple axisymmetric cracks under torsional impact loading is investigated. It is assumed that the mass density and the shear modulus in the two principal directions of the functionally graded material (FGM) medium vary exponentially along the z-axis. The solution of a dynamic rotational Somigliana-type ring dislocation in an FGM orthotropic medium is obtained by using the Laplace and Hankel transforms. This solution is used to construct integral equations for a system of coaxial axisymmetric cracks, including annular and penny-shaped cracks. The integral equations are of Cauchy singular type, which are solved numerically to obtain the dislocation density on the faces of the cracks and the results are used to determine DSIFs for cracks. Numerical examples are provided to show the influences of material non-homogeneity and orthotropy as well as crack type on the DSIFs. PubDate: Wed, 27 Dec 2017 00:00:00 GMT DOI: 10.1093/imamat/hxx038 Issue No:Vol. 83, No. 1 (2017)

Authors:Liu H; Geng X. Pages: 148 - 173 Abstract: In this paper, we analyse initial-boundary problems for the vector derivative nonlinear Schrödinger equation on the semi-infinite strip $(x, t) \in (0,\infty )\times (0, T)$ via the unified transform method of Fokas. Even though additional technical complications arise in the vector case compared with scalar ones, we show that it also can be expressed in terms of the solution of a matrix Riemann–Hilbert problem. The Riemann–Hilbert problem involves a jump matrix, uniquely defined in terms of four matrix functions called spectral functions and denoted by {a(λ), b(λ), A(λ), B(λ)} that depend on the initial data and all boundary values, respectively. A key role is played by the so-called global relation which involves the known and unknown boundary values. By analysing the global relation, we present an effective characterization of the latter two spectral functions in terms of the given initial and boundary data. PubDate: Sat, 23 Dec 2017 00:00:00 GMT DOI: 10.1093/imamat/hxx039 Issue No:Vol. 83, No. 1 (2017)

Authors:Page M; Cowley S. Pages: 188 - 203 Abstract: The Kelvin–Helmholtz model for the evolution of an infinitesimally thin vortex sheet in an inviscid fluid is mathematically ill-posed for general classes of initial conditions. However, if the initial data, say imposed at t = 0, are in a certain class of analytic functions then the problem is well-posed for a finite time until a singularity forms, say at t = ts, on the vortex-sheet interface, e.g. as illustrated by Moore (1979, The spontaneous appearance of a singularity in the shape of an evolving vortex sheet. Proc. Roy. Soc. Lond. A, 365, 105–119). However, if the problem is analytically continued into the complex plane, then the singularity, or singularities, exist for t < ts away from the physical real axis. More specifically, Cowley et al. (1999, On the formation of Moore curvature singularities in vortex sheets. J. Fluid Mech., 378, 233–267) found that for a class of analytic initial conditions, singularities can form in the complex plane at t = 0+. They posed asymptotic expansions in the neighbourhood of these singularities for 0 < t ≪ 1 and found numerical solutions to the governing similarity differential equations. In this paper we obtain new exact solutions to these equations, show that the singularities always correspond to local ${\textstyle \frac {3}{2}}$-power singularities and determine both the number and precise locations of all branch points. Further, our analytical approach can be extended to a more general class of initial conditions. These new exact solutions can assist in resolving the small-time behaviour for the numerical solution of the Birkhoff–Rott equations. PubDate: Fri, 22 Dec 2017 00:00:00 GMT DOI: 10.1093/imamat/hxx041 Issue No:Vol. 83, No. 1 (2017)

Authors:Khapalov A; Lapin S. Pages: 1 - 23 Abstract: In this paper, we introduce a new modelling approach for the dynamic granular matter formation process in the form of a system of difference equations, directly tailored to the physical nature of the process at hand. Respectively, the dynamic 1D and 2D discrete models, proposed in this paper, are not constructed as numerical schemes approximating some partial differential equations (PDEs). We propose here to look for the functions describing the standing and the rolling layers of the granular matter as the limits of discrete solutions to the aforementioned model equations as the size of the mesh tends to zero. In particular, this approach allows us to differentiate between the influx of the rolling layer coming down from different directions to the corner points of the standing layer. Such points are difficult to adequately describe by means of PDEs and their straightforward numerical approximations, typically ‘ignoring’ the system's behaviour on the sets of zero measure. However, these points are critical for understanding the dynamics of formation process when the standing layer is created by the moving front of the rolling matter or when the latter is filling a cavity and/or stops rolling. The existence of distributed (infinite-dimensional) limit solutions to our discrete models as the size of the mesh tends to zero is also discussed. We illustrate our findings by numerical examples which use our models as the direct algorithm. PubDate: Sun, 04 Dec 2016 00:00:00 GMT DOI: 10.1093/imamat/hxw016 Issue No:Vol. 83, No. 1 (2016)

Authors:Wang Q; Papageorgiou D. Pages: 24 - 52 Abstract: The stability and axisymmetric deformation of two immiscible, viscous, perfect or leaky dielectric fluids confined in the annulus between two concentric cylinders are studied in the presence of radial electric fields. The fields are set up by imposing a constant voltage potential difference between the inner and outer cylinders. We derive a set of equations for the interface in the long-wavelength approximation which retains the essential physics of the system and allows for interfacial deformations to be as large as the annular gap hence accounting for possible touchdown at the inner or outer electrode. The effects of the electric parameters are evaluated initially by performing a linear stability analysis which shows excellent agreement with the linear theory of the full axisymmetric problem in the appropriate long-wavelength regime. The non-linear interfacial dynamics are investigated by carrying out direct numerical simulations of the derived long wave models, both in the absence and presence of electric fields. For non-electrified thin layer flows (i.e. one of the layers thin relative to the other) the long-time dynamics agree with the lubrication approximation results found in literature. When the liquid layers have comparable thickness our results demonstrate the existence of both finite-time and infinite-time singularities (asymptotic touching solutions) in the system. It is shown that a two-side touching solution is possible for both the non-electrified and perfect dielectric cases, while only one-side touching is found in the case of leaky dielectric liquids, where the flattened interface shape resembles the pattern solutions found in literature. Meanwhile the finite-time singular solution agrees qualitatively with the experiments of Reynolds (1965, Stability of an electrostatically supported fluid column. Phys. Fluids, 8, 161–170). PubDate: Sun, 04 Dec 2016 00:00:00 GMT DOI: 10.1093/imamat/hxw017 Issue No:Vol. 83, No. 1 (2016)