Contracting bubbles in Hele-Shaw cells with a power-law fluid

The problem of bubble contraction in a Hele-Shaw cell is studied for the case in which the surrounding fluid is of power-law type. A small perturbation of the radially symmetric problem is first considered, focussing on the behaviour just before the bubble vanishes, it being found that for shear-thinning fluids the radially symmetric solution is stable, while for shear-thickening fluids the aspect ratio of the bubble boundary increases. The borderline (Newtonian) case considered previously is neutrally stable, the bubble boundary becoming elliptic in shape with the eccentricity of the ellipse depending on the initial data. Further light is shed on the bubble contraction problem by considering a long thin Hele-Shaw cell: for early times the leading-order behaviour is one-dimensional in this limit; however, as the bubble contracts its evolution is ultimately determined by the solution of a Wiener-Hopf problem, the transition between the long-thin limit and the extinction limit in which the bubble vanishes being described by what is in effect a similarity solution of the second kind. This same solution describes the generic (slit-like) extinction behaviour for shear-thickening fluids, the interface profiles that generalise the ellipses that characterise the Newtonian case being constructed by the Wiener-Hopf calculation.

Linear stern waves in finite depth channels

This paper formulates an analytically tractable problem for the wake generated by a long flat bottom ship by considering the steady free surface flow of an inviscid, incompressible fluid emerging from beneath a semi-infinite rigid plate. The flow is considered to be irrotational and two-dimensional so that classical potential flow methods can be exploited. In addition, it is assumed that the draft of the plate is small compared to the depth of the channel. The linearised problem is solved exactly using a Fourier transform and the Wiener-Hopf technique, and it is shown that there is a family of subcritical solutions characterised by a train of sinusoidal waves on the downstream free surface. The amplitude of these waves decreases as the Froude number increases. Supercritical solutions are also obtained, but, in general, these have infinite vertical velocities at the trailing edge of the plate. Consideration of further terms in the expansions suggests a way of canceling the singularity for certain values of the Froude number.

Perturbation solutions for flow through symmetrical hoppers with inserts and asymmetrical wedge hoppers

Under certain circumstances, an industrial hopper which operates under the "funnel-flow" regime can be converted to the "mass-flow" regime with the addition of a flow-corrective insert. This paper is concerned with calculating granular flow patterns near the outlet of hoppers that incorporate a particular type of insert, the cone-in-cone insert. The flow is considered to be quasi-static, and governed by the Coulomb-Mohr yield condition together with the non-dilatant double-shearing theory. In two dimensions, the hoppers are wedge-shaped, and as such the formulation for the wedge-in-wedge hopper also includes the case of asymmetrical hoppers. A perturbation approach, valid for high angles of internal friction, is used for both two-dimensional and axially symmetric flows, with analytic results possible for both leading order and correction terms. This perturbation scheme is compared with numerical solutions to the governing equations, and is shown to work very well for angles of internal friction in excess of 45 degree.

Robust tracking control of autonomous underwater vehicles in the presence of disturbance inputs

An autonomous underwater vehicle (AUV) is expected to operate in an ocean in the presence of poorly known disturbance forces and moments. The uncertainties of the environment makes it difficult to apply open-loop control scheme for the motion planning of the vehicle. The objective of this paper is to develop a robust feedback trajectory tracking control scheme for an AUV that can track a prescribed trajectory amidst such disturbances. We solve a general problem of feedback trajectory tracking of an AUV in SE(3). The feedback control scheme is derived using Lyapunov-type analysis. The results obtained from numerical simulations confirm the asymptotic tracking properties of the feedback control law. We apply the feedback control scheme to different mission scenarios, with the disturbances being initial errors in the state of the AUV.

Optimal fluid injection strategies for in situ mineral leaching in two-dimensions

This paper examines the ground-water flow problem associated with the injection and recovery of certain corrosive fluids into mineral bearing rock. The aim is to dissolve the minerals in situ, and then recover them in solution. In general, it is not possible to recover all the injected fluid, which is of concern economically and environmentally. However, a new strategy is proposed here, that allows all the leaching fluid to be recovered. A mathematical model of the situation is solved approximately using an asymptotic solution, and exactly using a boundary integral approach. Solutions are shown for two-dimensional flow, which is of some practical interest as it is achievable in old mine tunnels, for example.

Numerical solution to the Saffman-Taylor finger problem with kinetic undercooling regularisation

The Saffman-Taylor finger problem is to predict the shape and,in particular, width of a finger of fluid travelling in a Hele-Shaw cell filled with a different, more viscous fluid. In experiments the width is dependent on the speed of propagation of the finger, tending to half the total cell width as the speed increases. To predict this result mathematically, nonlinear effects on the fluid interface must be considered; usually surface tension is included for this purpose. This makes the mathematical problem suffciently diffcult that asymptotic or numerical methods must be used. In this paper we adapt numerical methods used to solve the Saffman-Taylor finger problem with surface tension to instead include the effect of kinetic undercooling, a regularisation effect important in Stefan melting-freezing problems, for which Hele-Shaw flow serves as a leading order approximation when the specific heat of a substance is much smaller than its latent heat. We find the existence of a solution branch where the finger width tends to zero as the propagation speed increases, disagreeing with some aspects of the asymptotic analysis of the same problem. We also find a second solution branch, supporting the idea of a countably infinite number of branches as with the surface tension problem.

Minimising wave drag for free surface flow past a two-dimensional stern

Free surface flow past a two-dimensional semi-infinite curved plate is considered, with emphasis given to solving for the shape of the resulting wave train that appears downstream on the surface of the fluid. This flow configuration can be interpreted as applying near the stern of a wide blunt ship. For steady flow in a fluid of finite depth, we apply the Wiener-Hopf technique to solve a linearised problem, valid for small perturbations of the uniform stream. Weakly nonlinear results found using a forced KdV equation are also presented, as are numerical solutions to the fully nonlinear problem, computed using a conformal mapping and a boundary integral technique. By considering different families of shapes for the semi-infinite plate, it is shown how the amplitude of the waves can be minimised. For plates that increase in height as a function of the direction of flow, reach a local maximum, and then point slightly downwards at the point at which the free surface detaches, it appears the downstream wavetrain can be eliminated entirely.

On the empirical eigenvalue distribution of slotted amplify-and-forward relaying protocol model

In this paper we consider the case of large cooperative communication systems where terminals use the protocol known as slotted amplify-and-forward protocol to aid the source in its transmission. Using the perturbation expansion methods of resolvents and large deviation techniques we obtain an expression for the Stieltjes transform of the asymptotic eigenvalue distribution of a sample covariance random matrix of the type HH† where H is the channel matrix of the transmission model for the transmission protocol we consider. We prove that the resulting expression is similar to the Stieltjes transform in its quadratic equation form for the Marcenko-Pastur distribution.

Drug diffusion from polymeric delivery devices : a problem with two moving boundaries

An existing model for solvent penetration and drug release from a spherically-shaped polymeric drug delivery device is revisited. The model has two moving boundaries, one that describes the interface between the glassy and rubbery states of polymer, and another that defines the interface between the polymer ball and the pool of solvent. The model is extended so that the nonlinear diffusion coefficient of drug explicitly depends on the concentration of solvent, and the resulting equations are solved numerically using a front-fixing transformation together with a finite difference spatial discretisation and the method of lines. We present evidence that our scheme is much more accurate than a previous scheme. Asymptotic results in the small-time limit are presented, which show how the use of a kinetic law as a boundary condition on the innermost moving boundary dictates qualitative behaviour, the scalings being very different to the similar moving boundary problem that arises from modelling the melting of an ice ball. The implication is that the model considered here exhibits what is referred to as ``non-Fickian'' or Case II diffusion which, together with the initially constant rate of drug release, has certain appeal from a pharmaceutical perspective.

Practical stability of approximating discrete-time filters with respect to model mismatch using relative entropy concepts

This paper establishes practical stability results for an important range of approximate discrete-time filtering problems involving mismatch between the true system and the approximating filter model. Using local consistency assumption, the practical stability established is in the sense of an asymptotic bound on the amount of bias introduced by the model approximation. Significantly, these practical stability results do not require the approximating model to be of the same model type as the true system. Our analysis applies to a wide range of estimation problems and justifies the common practice of approximating intractable infinite dimensional nonlinear filters by simpler computationally tractable filters.

Practical stability with respect to model mismatch of approximate discrete-time output feedback control

This paper establishes a practical stability result for discrete-time output feedback control involving mismatch between the exact system to be stabilised and the approximating system used to design the controller. The practical stability is in the sense of an asymptotic bound on the amount of error bias introduced by the model approximation, and is established using local consistency properties of the systems. Importantly, the practical stability established here does not require the approximating system to be of the same model type as the exact system. Examples are presented to illustrate the nature of our practical stability result.

Natural convection from a plane vertical porous surface in non-isothermal surroundings

In this paper, laminar natural convection flow from a permeable and isothermal vertical surface placed in non-isothermal surroundings is considered. Introducing appropriate transformations into the boundary layer equations governing the flow derives non-similar boundary layer equations. Results of both the analytical and numerical solutions are then presented in the form of skin-friction and Nusselt number. Numerical solutions of the transformed non-similar boundary layer equations are obtained by three distinct solution methods, (i) the perturbation solutions for small � (ii) the asymptotic solution for large � (iii) the implicit finite difference method for all � where � is the transpiration parameter. Perturbation solutions for small and large values of � are compared with the finite difference solutions for different values of pertinent parameters, namely, the Prandtl number Pr, and the ambient temperature gradient n.

An effective approach to verbose queries using a limited dependencies language model

Intuitively, any ‘bag of words’ approach in IR should benefit from taking term dependencies into account. Unfortunately, for years the results of exploiting such dependencies have been mixed or inconclusive. To improve the situation, this paper shows how the natural language properties of the target documents can be used to transform and enrich the term dependencies to more useful statistics. This is done in three steps. The term co-occurrence statistics of queries and documents are each represented by a Markov chain. The paper proves that such a chain is ergodic, and therefore its asymptotic behavior is unique, stationary, and independent of the initial state. Next, the stationary distribution is taken to model queries and documents, rather than their initial distributions. Finally, ranking is achieved following the customary language modeling paradigm. The main contribution of this paper is to argue why the asymptotic behavior of the document model is a better representation then just the document’s initial distribution. A secondary contribution is to investigate the practical application of this representation in case the queries become increasingly verbose. In the experiments (based on Lemur’s search engine substrate) the default query model was replaced by the stable distribution of the query. Just modeling the query this way already resulted in significant improvements over a standard language model baseline. The results were on a par or better than more sophisticated algorithms that use fine-tuned parameters or extensive training. Moreover, the more verbose the query, the more effective the approach seems to become.