Showing papers in "Ima Journal of Applied Mathematics in 2004"

Two-dimensional computational simulation of eccentric annular cementing displacements

Abstract: We consider a two-dimensional Hele-Shaw type model for displacement flows occurring in the primary cementing of an oil well. The fluids are visco-plastic and may get stuck in the annulus if a critical pressure gradient is not exceeded. The model consists of solving a nonlinear elliptic variational inequality equation for the stream function, coupled to an equation for interface advection, or alternatively a concentration equation for the mass fraction of each fluid. The key difficulty is to accurately compute yielded and unyielded zones of the wellbore fluids, which we accomplish by use of an augmented Lagrangian method to solve the stream function equation. We validate the accuracy of our method against analytical solutions for stable steady-state displacements. We study the convergence of the interface to the steady state, showing that the apparent meta-stability is illusory. We then explore the effects of increasing eccentricity, showing that although the interface may remain stable it becomes unsteady. Initially fully mobile flows are found, but as the eccentricity increases further the narrow side fluids fail to move in the far field. The narrow side interface can progress slowly through the static fluids by a burrowing motion, but for still larger eccentricities even the interface becomes static and a narrow-side mud channel forms.

On common quadratic Lyapunov functions for stable discrete‐time LTI systems

Abstract: This paper deals with the question of the existence of weak and strong common quadratic Lyapunov functions (CQLFs) for stable discrete-time linear timeinvariant (LTI) systems. The main result of the paper provides a simple characterisation of pairs of such systems for which a weak CQLF of a given form exists but for which no strong CQLF exists. An application of this result to second order discrete-time LTI systems is presented.

Swelling-induced microchannel formation in nonlinear elasticity

Abstract: We study the effect of swelling on the axisymmetric deformation of a circular cylinder. A continuum description of swelling is established on the basis of a specific generalization of nonlinear elasticity for incompressible materials. This theory is used to analyse microchannel formation and channel growth. We show that certain kinds of material constitutive laws in conjunction with certain types of variable swelling fields are capable of inducing this formation and growth process without the aid of an externally applied traction. The solutions are presented as a bifurcation from the standard solution that involves no channel formation. The channel formation solution is found to be energetically preferable to the standard solution.

Comparative analysis of two asymptotic approaches based on integral manifolds

Abstract: A comparative analysis of the two powerful asymptotic methods, ILDM and MIM (intrinsic low-dimensional manifolds; method of invariant manifold), is presented in the paper. The two methods are based on the general theory of integral manifolds. The ILDM method is able to handle large systems of ODEs, whereas the MIM method treats systems with a limited number of unknown variables. The MIM method allows one to conduct analytical exploration of the original system and to obtain final expressions in compact form, whereas the ILDM method is a numerical approach that yields the numerical form of the desired surface. The ILDM method works well in a region where a rough splitting of the initial system exists. Regions of the phase space where splitting does not exist are problematic for the ILDM method. In these regions the MIM method provides additional information regarding the dynamical behaviour of the system. A number of simple examples are considered and analysed. It is shown that for the Semenov model (singularly perturbed system of ODEs) the ILDM method gives a surface which appears close to the first order (with respect to the corresponding small parameter) approximation of the stable (attracting) invariant manifolds. The complementary properties of the two asymptotic approaches suggests a feasible combination of the two methods, which is the subject of a future work.

A quasistatic viscoplastic contact problem with normal compliance and friction

Abstract: We consider a quasistatic contact problem between a viscoplastic body and an obstacle, the so-called foundation. The contact is modelled with normal compliance and the associated version of Coulomb's law of dry friction. We derive a variational formulation of the problem and, under a smallness assumption on the normal compliance functions, we establish the existence of a weak solution to the model. The proof is carried out in several steps. It is based on a time-discretization method, arguments of monotonicity and compactness, Banach fixed point theorem and Schauder fixed point theorem.

Electromagnetic scattering by a smooth convex impedance cone

Abstract: The problem of the diffraction of an electromagnetic plane wave by a convex cone of arbitrary smooth cross-section with impedance (Leontovich) boundary conditions is studied. The vector problem is reduced to that for the Debye potentials. By means of Kontorovich-Lebedev integrals, two spectral functions are introduced and the corresponding boundary value problem is formulated. The spectral functions for the potentials are found to satisfy the Helmholtz equations on the unit sphere and to be coupled through non-traditional boundary conditions of the impedance type with shifts on the spectral variable. The use of the Green theorem permits us to establish an integral formulation of the boundary value problem for the spectral functions. The formal asymptotic solution of the problem is then given for the case of a narrow cone. For this, two different methods are given: a method of perturbation applied to the spectral integral equations and an adaptation of the method of matching the asymptotic series in spectral domain. Both methods lead to the same closed-form result for the leading term of the scattering diagram asymptotics.

General solution in terms of complex potentials for incremental antiplane states in prestressed and prepolarized piezoelectric crystals: application to Mode III fracture propagation

Abstract: The problem considered is the antiplane deformation in prestressed and prepolarized piezoelectric crystals in equilibrium. The representation of the general solution is derived in terms of complex potentials for all piezoelectric crystal classes in which an antiplane state is possible. This generalizes earlier results obtained in respect of a specific crystal class. The general formulae are specialized to find the antiplane state generated by a Mode-III crack.

Boundary element-Landweber method for the Cauchy problem in linear elasticity

Abstract: In this paper, an iterative algorithm based on the Landweber method in combination with the boundary element method is developed for solving the Cauchy problem in isotropic linear elasticity. An efficient regularizing stopping criterion is also employed. The numerical results obtained confirm that the iterative method produces a convergent and stable numerical solution with respect to increasing the number of boundary elements and decreasing the amount of noise added into the input data, respectively.

The primal–dual active set method for a crack problem with non‐penetration

Abstract: The Lame problem in a 2D domain with a crack under a non-penetration condition is considered as a variational inequality. A primal-dual active set method is proposed as an efficient numerical solution technique and compared to a previously employed iterative method for a penalized formulation. Sufficient conditions for monotonic convergence of a discretized version of the proposed algorithm are given and numerical experiments are presented.

A rivulet of perfectly wetting fluid draining steadily down a slowly varying substrate

Abstract: We use the lubrication approximation to investigate the steady locally unidirectional gravity-driven draining of a thin rivulet of a perfectly wetting Newtonian fluid with prescribed volume flux down both a locally planar and a locally non-planar slowly varying substrate inclined at an angle to the horizontal. We interpret our results as describing a slowly varying rivulet draining in the azimuthal direction some or all of the way from the top ( = 0) to the bottom ( = ) of a large horizontal circular cylinder with a non-uniform transverse profile. In particular, we show that the behaviour of a rivulet of perfectly wetting fluid is qualitatively different from that of a rivulet of a non-perfectly wetting fluid. In the case of a locally planar substrate we find that there are no rivulets possible in 0 /2 (i.e. there are no sessile rivulets or rivulets on a vertical substrate), but that there are infinitely many pendent rivulets running continuously from = /2 (where they become infinitely wide and vanishingly thin) to = (where they become infinitely deep with finite semi-width). In the case of a locally non-planar substrate with a power-law transverse profile with exponent p > 0 we find, rather unexpectedly, that the behaviour of the possible rivulets is qualitatively different in the cases p 2 as well as in the cases of locally concave and locally convex substrates. In the case of a locally concave substrate there is always a solution near the top of the cylinder representing a rivulet that becomes infinitely wide and deep, whereas in the case of a locally convex substrate there is always a solution near the bottom of the cylinder representing a rivulet that becomes infinitely deep with finite semi-width. In both cases the extent of the rivulet around the cylinder and its qualitative behaviour depend on the value of p. In the special case p = 2 the solution represents a rivulet on a locally parabolic substrate that becomes infinitely wide and vanishingly thin in the limit /2. We also determine the behaviour of the solutions in the physically important limits of a weakly non-planar substrate, a strongly concave substrate, a strongly convex substrate, a small volume flux, and a large volume flux.

Vector functional-difference equation in electromagnetic scattering

Abstract: A vector functional-difference equation of the first order with a special matrix coefficient is analysed. It is shown how it can be converted into a Riemann-Hilbert boundary-value problem on a union of two segments on a hyper-elliptic surface. The genus of the surface is defined by the number of zeros and poles of odd order of a characteristic function in a strip. An even solution of a symmetric Riemann-Hilbert problem is also constructed. This is a key step in the procedure for diffraction problems. The proposed technique is applied for solving in closed form a new model problem of electromagnetic scattering of a plane wave obliquely incident on an anisotropic impedance half-plane (all the four impedances are assumed to be arbitrary).

The thermostat problem with a nonlocal nonlinear boundary condition

Abstract: The thermostat controller for an air-conditioning system is usually placed in a position at some distance from the unit and this can lead to large swings in temperature. This paper addresses this question by studying a paradigm - a one-dimensional heat conduction equation with and without heat loss, and where the flux of heat extracted or input by the unit is considered to be a function of the temperature at the other end. The essential results are that the system can be unstable and that this is exacerbated both by a more powerful air-conditioning unit and by more efficient insulation.

Stability of flames in an exothermic–endothermic system

Abstract: The propagation of a premixed laminar flame supported by an exothermic chemical reaction under adiabatic conditions but subject to inhibition through a parallel endothermic chemical process is considered. The temporal stability to longitudinal perturbations of any resulting flames is investigated. The heat loss through the endothermic reaction, represented by the dimensionless parameter a, has a strong quenching effect on wave propagation. The wave speed-cooling parameter (α, c) curves are determined for a range of values of the other parameters. These curves can be monotone decreasing or S-shaped, depending on the values of the parameters β, representing the rate at which inhibitor is consumed relative to the consumption of fuel, μ, the ratio of the activation energies of the reactants and the Lewis numbers. This gives the possibility of having either one, two or three different flame velocities for the same value of the cooling parameter α. For Lewis numbers close to unity, when there are three solutions, two of them are stable and one is unstable, with two saddle-node bifurcation points on the (a, c) curve. For larger values of the Lewis numbers there is a Hopf bifurcation point on the curve, dividing it into a stable and an unstable branch. The saddle-node and Hopf bifurcation curves are also determined. The two curves have a common, Takens-Bogdanov bifurcation point.

Invariant manifolds of periodic orbits for piecewise linear three‐dimensional systems

Abstract: This paper analyses the existence of invariant manifolds of periodic orbits for a specific piecewise linear three-dimensional system with two zones, whose linear parts share a pair of imaginary eigenvalues. This degenerate situation is obtained from the lack of controllability. The analysis proceeds by its reduction to a periodic one-dimensional equation for which some results of the Ambrosetti-Prodi type are given.

On interface waves in misoriented pre-stressed incompressible elastic solids

Abstract: Some relationships, fundamental to the resolution of interface wave problems, are presented. These equations allow for the derivation of explicit secular equations for problems involving waves localized near the plane boundary of anisotropic elastic half-spaces, such as Rayleigh, Scholte, or Stoneley waves. They are obtained rapidly, without recourse to the Stroh formalism. As an application, the problems of Stoneley wave propagation and of interface stability for misaligned predeformed incompressible half-spaces are treated. The upper and lower half-spaces are made of the same material, subject to the same prestress, and are rigidly bonded along a common principal plane. The principal axes in this plane do not, however, coincide, and the wave propagation is studied in the direction of the bisectrix of the angle between a principal axis of the upper half-space and a principal axis of the lower half-space.

Riemann problem for kinematical conservation laws and geometrical features of nonlinear wavefronts

Abstract: A pair of kinematical conservation laws (KCL) in a ray coordinate system (ξ, t) are the basic equations governing the evolution of a moving curve in two space dimensions. We first study elementary wave solutions and then the Riemann problem for KCL when the metric g, associated with the coordinate ξ designating different rays, is an arbitrary function of the velocity of propagation m of the moving curve. We assume that m > 1 (m is appropriately normalized), for which the system of KCL becomes hyperbolic. We interpret the images of the elementary wave solutions in the (ξ, t)-plane to the (x, y)-plane as elementary shapes of the moving curve (or a nonlinear wavefront when interpreted in a physical system) and then describe their geometrical properties. Solutions of the Riemann problem with different initial data give the shapes of the nonlinear wavefront with different combinations of elementary shapes. Finally, we study all possible interactions of elementary shapes.

A pointwise estimate on the 1-order approximation of Gεx0

Abstract: On the basis of Avellaneda & Hua-Lin (1991, Commun. Pure Appl. Math., 44 897-910), a pointwise error estimate on the 1-order approximation of Green function G e x 0 defined in R 2 is shown at first. Then based on this estimate and using asymptotic expansion method, an improved approximation of G e x 0 and its pointwise error estimate are obtained.

Shock-induced ignition of thermally sensitive explosives

Abstract: The process of planar detonation ignition, induced by a constant-velocity piston or equivalently by a shock reflected from a stationary wall, is investigated using high-resolution one-dimensional numerical simulations. The standard one-step model with Arrhenius kinetics, which models thermally sensitive explosives, is employed. Emphasis is on comparing and contrasting the results of the finite activation temperature simulations with high activation temperature asymptotic predictions and previous simulations. During the induction phase, it is shown that the asymptotic results give qualitatively good predictions. However, for parameters representative of gaseous explosives, subsequent to thermal runaway at the piston and the formation of a reaction wave, the high activation temperature asymptotic theory is qualitatively incorrect for moderately high activation temperatures. It is shown that the results are very sensitive to the value of the activation temperature, especially the distance from the piston at which a secondary shock forms and the degree of unsteadiness in the reaction wave which moves away from the piston. The dependence of the ignition evolution on the other parameters (initial shock Mach number, heat of reaction and polytropic index) is also investigated. It is shown that qualitative predictions regarding the dependence of the ignition evolution on each of the parameters can be elucidated from finite activation temperature homogeneous explosion calculations together with the high activation temperature asymptotic shock ignition results. It is found that for sufficiently strong initiating shocks the ignition evolution is qualitatively different from cases studied previously in that no secondary shock forms. For a high polytropic index, corresponding to a simple equation of state model for condensed phase explosives, the results are in much better qualitative agreement with the asymptotic theory.

Global existence and blow-up for the solutions to nonlinear parabolic–elliptic system modelling chemotaxis

Abstract: In this paper we study the global in-time and blow-up solutions for the simplified Keller-Segel system modelling chemotaxis. We prove that there is a critical number which determines the occurrence of blow-up in the two-dimensional case for 1 < p < 2. In three- or higher-dimensional cases, we show that the radial symmetrical solution will blow up if 1 < p < N N-2 (N ≥ 3) for non-negative initial value.

Blowup of solutions to generalized Riemann problem for quasilinear hyperbolic systems of conservation laws

Abstract: In this paper we study the global structure instability of Riemann solution u = U(x/t), containing at least one centred rarefaction wave, for the general n x n quasilinear hyperbolic system of conservation laws. We prove the non-existence of the global piecewise C 1 solution to a class of generalized Riemann problems, which can be regarded as a perturbation of the corresponding Riemann problem. This result shows that this kind of Riemann solution mentioned above is globally structurally unstable. Applications to quasilinear hyperbolic systems arising from physics and mechanics, particularly to surface waves in hyperelastic materials, are also given.

Lie group analysis and plane strain bending of cylindrical sectors for compressible nonlinearly elastic materials

Abstract: The plane strain bending of cylindrical sectors for two important classes of homogeneous, isotropic, compressible, nonlinearly elastic materials is considered. In contrast to the traditional approach to this problem, a double semi-inverse approach is adopted here. The strain-energy densities and the deformation fields considered each contain an arbitrary function. The deformation field is completely determined from the equations of equilibrium (as is usual) but additionally here the strain-energy function is completely determined on using Lie's linearization theorem. This theorem gives necessary and sufficient conditions for a second-order ODE to be linearizable, i.e. for a second-order equation to have a solution of the form y = ax + b, where a, b are constants and x, y are transformed variables. For harmonic materials, it has been shown in the literature that a quadrature solution can be developed. Due to the restrictive nature of an invertibility assumption inherent in this approach, however, it is shown here that very few closed-form solutions can be found in this way. The use of Lie's linearization theorem does lead to some new solutions for particular harmonic materials. This method does not, however, provide any new solutions for Varga materials. When the linearization theorem fails to determine the arbitrary function in the strain-energy density, invariance techniques are used. A new parametric solution for a specific Varga material is determined in this way.

Boundary value problems for systems of linear evolution equations

Abstract: It is shown that the new method for solving initial-boundary value problems for scalar evolution equations recently introduced by one of the authors can also be applied to systems of evolution equations. The novel step needed in this case is the construction of a scalar Lax pair by using a suitable parametrization of the dispersion relation as well as certain linear transformations. The simultaneous spectral analysis of the Lax pair yields the solution of a given initial-boundary value problem in terms of an integral in the complex spectral plane which involves an appropriate x-transform of the initial conditions and an appropriate t-transforrn of the boundary conditions. These transforms are neither the x-Fourier transform nor the t-Laplace transform, rather they are new transforms custom made for the given system of partial differential equations (PDEs) and the given domain. This method is illustrated by solving on the half-line the linearized equations governing infinitesimal deformations in a heat conducting bar.

New stress and velocity fields for highly frictional granular materials

Abstract: The idealised theory for the quasi-static flow of granular materials which satisfy the Coulomb-Mohr hypothesis is considered. This theory arises in the limit that the angle of internal friction approaches $\pi/2$, and accordingly these materials may be referred to as being `highly frictional'. In this limit, the stress field for both two-dimensional and axially symmetric flows may be formulated in terms of a single nonlinear second order partial differential equation for the stress angle. To obtain an accompanying velocity field, a flow rule must be employed. Assuming the non-dilatant double-shearing flow rule, a further partial differential equation may be derived in each case, this time for the streamfunction. Using Lie symmetry methods, a complete set of group-invariant solutions is derived for both systems, and through this process new exact solutions are constructed. Only a limited number of exact solutions for gravity driven granular flows are known, so these results are potentially important in many practical applications. The problem of mass flow through a two-dimensional wedge hopper is examined as an illustration.

Spatial decay of transient end effects for nonstandard linear diffusion problems

Abstract: In this paper we investigate the spatial decay of transient end effects for some nonstandard linear diffusion problems. We consider classes of linear parabolic equations on three-dimensional semi-infinite cylinders with non-zero boundary conditions only on the near end. If zero initial conditions were imposed, it is well-known that solutions decay exponentially with distance from the end and many results on the rate of spatial decay have been established. Here we consider a different type of initial condition that arises in regularization of ill-posed problems: namely, that the field quantity at time t = T is assumed to be proportional to that at t = 0. A spatial decay estimate for solutions of such problems is obtained with optimal decay rate explicit in the proportionality parameter. Continuous dependence on this parameter is established. The results are extended to a pseudo-parabolic equation. Binary mixtures of rigid solids are then considered. The governing PDEs are a coupled system of two parabolic equations for the temperature in each phase. Differential inequality arguments are used to obtain decay estimates for cross-sectional norms of the solution pair.

Combined axial shearing, extension, and straightening of elastic annular cylindrical sectors

Abstract: The combined axial shearing, extension, and straightening of an annular cylindrical sector is a deformation that, following Truesdell & Noll (1965, The Non-Linear Field Theories of Mechanics, Encyclopedia of Physics III/3) and Hill (1973, Z. Angew. Math. Phys., 24, 609-618), we describe in terms of two prescribed constants and two unknown functions that depend only on the radial material coordinate. Under the assumption that the material is elastic, compressible, and isotropic we show that for equilibrium in the absence of body forces the unknown functions must satisfy a system of first-order nonlinear ordinary differential equations. The system of differential equations can be decoupled for certain material classes, one of which is the class of Hadamard-Green materials. Thus, several new exact solutions are obtained and, under the assumption that the annular cylindrical sector is composed of a Hadamard-Green material that is strongly elliptic, the existence and uniqueness of solutions for two types of boundary conditions is established.

Mathematical modelling of heat transfer in a catalytic reformer

Abstract: In this work we analyse heat transfer in a system of channels connected by thin conducting walls. The channels are packed with catalytic pellets that promote exothermic catalytic combustion reactions and endothermic reforming reactions in adjacent channels. A model is developed in which the thermal conductivity and the thickness of the interconnecting wall can be used as control parameters characterizing the heat exchange between the neighbouring channels. The model is to be used as a mathematical tool to analyse design alternatives and develop accurate numerical techniques. Our objective is to study how the heat is transferred across the conducting walls and how this influences the temperature distribution in the channels. We use an asymptotic technique to do this. The structure of the walls is then examined in detail, focusing on the case when we have layered walls.

The Fokker-Planck equation and the master equation in the theory of migration

Abstract: In the theory of migration two mathematical models are regarded as very important. One is described by a quasilinear partial differential equation of parabolic type called the Fokker-Planck equation. The other is described by a nonlinear integro-partial differential equation called the master equation. Both the models are employed frequently at the same time in the theory of migration. Hence we need to investigate whether the descriptions given by the models are close to each other or not. The purpose of the present paper is to mathematically prove that if the effort required in moving is large, then the models are close to each other in the following sense: the mixed problem with the periodic boundary condition for the master equation has a unique solution that is very close to a solution of that for the Fokker-Planck equation, where the effort is a sociodynamic quantity that represents a cost incurred in moving. By making use of the result of the paper, we can apply both the models to movement of human population at the same time.

A WKB analysis of the buckling condition for a cylindrical shell of arbitrary thickness subjected to an external pressure

Abstract: We consider the plane-strain buckling of a cylindrical shell of arbitrary thickness which is made of a Varga material and is subjected to an external hydrostatic pressure on its outer surface. The WKB method is used to solve the eigenvalue problem that results from the linear bifurcation analysis. We show that the circular cross-section buckles into a non-circular shape at a value of μ 1 which depends on A 1 /A 2 and a mode number, where At and A 2 are the undeformed inner and outer radii, and μ 1 is the ratio of the deformed inner radius to A 1 . In the large mode number limit, we find that the dependence of μ 1 on A 1 /A 2 has a boundary layer structure: it is constant over almost the entire region of 0 < A 1 /A 2 < 1 and decreases sharply from this constant value to unity as A 1 /A 2 tends to unity. Our asymptotic results for A 1 - 1 = O(1) and A 1 - 1 = O(1/n) are shown to agree with the numerical results obtained by using the compound matrix method.

On the integral manifold approach to a flame propagation problem: pressure‐driven flames in porous media

Abstract: The problem of a pressure-driven flame in an inert porous medium filled with a flammable gaseous mixture is considered. In the frame of reference attached to an advancing combustion wave and after a suitable non-dimensionalization the corresponding mathematical description of the problem includes three highly nonlinear ordinary differential equations. The system is rewritten in the form of a singularly perturbed system of ordinary differential equations and is analysed analytically by the geometrical version of the asymptotic method of integral manifolds (MIM). The paper focuses on an analysis of the fine structure of the flame and its velocity on the basis of an asymptotical consideration of an arbitrary trajectory of the considered system in the phase space. It is shown that two different stages of the trajectory correspond to the two various sub-zones of the flame: the first stage (fast motion from the initial point to the slow integral) is interpreted as a preheat sub-zone and the second stage of the path corresponds to a reaction sub-zone. It is shown that an inter-zone boundary plays an important role in a determination of the flame properties: characteristics of the gaseous mixture at that point determine the flame velocity. The accepted approach of the investigation allows us to gain an analytical expression for the flame velocity. It appears that the velocity formula represents a cubic-root dependence on the Arrhenius exponent, which in turn contains the parameters of the boundary point. The theoretical predictions are found to coincide rather well with the data of direct numerical simulations.

The Reissner–Sagoci type problem for a non‐homogeneous elastic cylinder embedded in an elastic non‐homogeneous half‐space

Abstract: The problem considered is that of the torsion of a non-homogeneous elastic cylinder, which is embedded in a non-homogeneous elastic half-space (matrix) of different rigidity modulus. A rigid disc is bonded to the flat surface of the cylinder and torque is applied to the cylinder through a rigid disc. It is assumed that there is perfect bonding at the common cylindrical surface. Using integral transformation techniques the solution of the problem is reduced to dual integral equations. Later on the solution of the dual integral equations is transformed into the solution of a Fredholm integral equation of the second kind. Solving the Fredholm integral equation numerically the numerical results for torque and shear stress inside the cylinder are obtained and displayed graphically to demonstrate the effect of non-homogeneity of the elastic material on the torque and shear stress.