Jonathan D. Evans

Source-type solutions of the fourth-order unstable thin film equation

We consider the fourth-order thin film equation (TFE) with the unstable second-order diffusion term. We show that, for the first critical exponent where N ≥ 1 is the space dimension, the free-boundary problem the with zero contact angle and zero-flux conditions admits continuous sets (branches) of self-similar similarity solutions of the form For the Cauchy problem, we describe families of self-similar patterns, which admit a regular limit as n → 0 + and converge to the similarity solutions of the semilinear unstable limit Cahn-Hilliard equation studied earlier in [12]. Using both analytic and numerical evidence, we show that such solutions of the TFE are oscillatory and of changing sign near interfaces for all n ∈ (0, n h ), where the value characterizes a heteroclinic bifurcation of periodic solutions in a certain rescaled ODE. We also discuss the cases p ⧧ = p 0 , the interface equation, and regular analytic approximations for such TFEs as an approach to the Cauchy problem.

Unstable sixth-order thin film equation: I. Blow-up similarity solutions

We study blow-up behaviour of solutions of the sixth-order thin film equation 0,\quad p>1,\end{eqnarray*} \] SRC=http://ej.iop.org/images/0951-7715/20/8/002/non236915ude001.gif/> containing an unstable (backward parabolic) second-order term. By a formal matched expansion technique, we show that, for the first critical exponent where N is the space dimension, the free-boundary problem (FBP) with zero-height, zero-contact-angle, zero-moment, and zero-flux conditions at the interface admits a countable set of continuous branches of radially symmetric self-similar blow-up solutions where T > 0 is the blow-up time. We also study the Cauchy problem (CP) in RN × R+ and show that the corresponding self-similar family {uk(x, t)} is countable and consists of solutions of maximal regularity, which are oscillatory near the interfaces. Actually, we show that compactly supported oscillatory blow-up profiles for the CP exist for all n (0, nh), where is a heteroclinic bifurcation point for the ordinary differential equation involved. The FBP ceases to exist before, at .

Blow-up similarity solutions of the fourth-order unstable thin film equation

We study blow-up behaviour of solutions of the fourth-order thin film equation which contains a backward (unstable) diffusion term. Our main goal is a detailed study of the case of the first critical exponent where N ≥ 1 is the space dimension. We show that the free-boundary problem with zero contact angle and zero-flux conditions admits continuous sets (branches) of blow-up self-similar solutions. For the Cauchy problem in R N × R + , we detect compactly supported blow-up patterns, which have infinitely many oscillations near interfaces and exhibit a “maximal” regularity there. As a key principle, we use the fact that, for small positive n , such solutions are close to the similarity solutions of the semilinear unstable limit Cahn-Hilliard equation which are better understood and have been studied earlier [19]. We also discuss some general aspects of formation of self-similar blow-up singularities for other values of p .

Optimal timing for an indivisible asset sale

In this paper, we investigate the pricing via utility indifference of the right to sell a non-traded asset. Consider an agent with power utility who owns a single unit of an indivisible, non-traded asset, and who wishes to choose the optimum time to sell this asset. Suppose that this right to sell forms just part of the wealth of the agent, and that other wealth may be invested in a complete frictionless market. We formulate the problem as a mixed stochastic control/optimal stopping problem, which we then solve. We determine the optimal behavior of the agent, including the optimal criteria for the timing of the sale. It turns out that the optimal strategy is to sell the non-traded asset the first time that its value exceeds a certain proportion of the agents trading wealth. Further, it is possible to characterize this proportion as the solution to a transcendental equation.

Standards for renal biopsies: comparison of inpatient and day care procedures.

Abstract. There are no national standards for the adequacy and complications of percutaneous renal biopsies. We developed local standards that have been used in a prospective audit of biopsies undertaken in a tertiary pediatric nephrology unit between January 1997 and December 2000. We compared outcomes of biopsies performed on inpatients with day care procedures. A total of 251 biopsies (113 transplant) were undertaken, 114 (46%) as day care procedures. Adequate tissue for diagnosis was obtained in 245 (97.6%), with a standard set at >95%. This was also achieved for a mean number of passes in native (<3 in 80%) and transplanted (<2 in 80%) kidneys. Eleven patients (4%) developed macroscopic hematuria (standard <5%) and none required transfusion. Delay in discharge occurred in 4 patients, and a further 4 returned to the ward post discharge. There was no significant difference in complication rates between inpatient and day care patients. Our local biopsy standards were met in this audit. Such standards could provide useful comparisons between units in national audit programs, as well as permitting the monitoring of individual performance as part of clinical governance. Day care procedures benefit the patient and family, as well as significantly reducing costs.

Re-entrant corner flows of the upper convected Maxwell fluid

Steady planar flow of the upper convected Maxwell (UCM) fluid is described for re–entrant corners with angles 180°/α, where ½ ⩽ α < 1. Local to the corner we consider a class of similarity solutions associated with the inviscid flow equations which arise from the dominance of the upper convective stress derivative in the constitutive equations. These solutions, first noted by Hinch, hold in an outer (core–flow) region and give stress singularities of O(r−2(1−α)) (with r the radial distance from the corner) and a stream function behaviour of O(rnα). Here n is a parameter defining distinct solutions within this similarity class. We match such solutions to wall boundary layers, in which viscometric behaviour is retrieved. We discuss two types of boundary–layer structure. The first is a single–layer structure, previously noted by Renardy. This single layer occurs for n = 3 − α and has the viscoelastic balance of the constitutive equations holding uniformly within it. Here we complete previous analysis by considering the downstream case. The second type of boundary layer considered is a double–layer structure, which we discuss for the range 1 < n < 3 − α. Now the elastic balance of the constitutive equations holds within its main region, with a thinner region closer to the wall in which the relaxation terms are recovered. This structure extends the range of the core similarity solutions and has not been previously noted.

Re-entrant corner flows of Oldroyd-B fluids

The method of matched asymptotic expansions is used to construct solutions for the planar steady flow of Oldroyd-B fluids around re-entrant corners of angles π/α (1/2≤α<1). Two types of similarity solutions are described for the core flow away from the walls. These correspond to the two main dominant balances of the constitutive equation, where the upper convected derivative of stress either dominates or is balanced by the upper convected derivative of the rate of strain. The former balance gives the incompressible Euler or inviscid flow equations and the latter balance the incompressible Navier–Stokes equations. The inviscid flow similarity solution for the core is that first derived by Hinch (Hinch 1993 J. Non-Newtonian Fluid Mech. 50, 161–171) with a core stress singularity that depends upon the corner angle and radial distance as O(r−2(1−α)) and a velocity behaviour that vanishes as O(rα(3−α)−1). Extending the analysis of Renardy (Renardy 1995 J. Non-Newtonian Fluid Mech. 58, 83–39), this outer solution is matched to viscometric wall behaviour for both upstream and downstream boundary layers. This structure is shown to hold for the majority of the retardation parameter range. In contrast, the similarity solution associated with the Navier–Stokes equations has a velocity behaviour O(rλ) where λ∈(0,1) satisfies a nonlinear eigenvalue problem, dependent upon the corner angle and an associated Reynolds number defined in terms of the ratio of the retardation and relaxation times. This similarity solution is shown to hold as an outer solution and is matched into stress boundary layers at the walls which recover viscometric behaviour. However, the matching is restricted to values of the retardation parameter close to the relaxation parameter. In this case the leading order core stress is Newtonian with behaviour O(r−(1−λ)).

On the derivation of heterogeneous reaction kinetics from a homogeneous reaction model

A simple homogeneous reaction model in one dimension is presented in the context of silicon oxidation. We investigate two different (canonical) regimes for the oxidant diffusivity and show how these lead in the limit of fast bulk reactions to distinct sharp interface models for oxidation. The resulting heterogeneous models are moving boundary problems which correspond to the classical Stefan problem or to the Stefan problem with kinetic undercooling. The results are relevant for more general reactions but illustrate some of the peculiarities associated with silicon oxidation.

A cable model for coupled neurons with somatic gap junctions

Abstract.A cable model is presented for a pair of electrotonically coupled neurons to investigate the spatial effects of soma-somatic gap junctions. The model extends that of Poznanski et al.(1995) in which each neuron is represented by a tapered equivalent cable attached to an isopotential soma with the two somas being electrically coupled. The model is posed generally, so that both active and passive properties can be considered. In the active case a system of nonlinear integral equations is derived for the voltage, whilst in the passive case these have an exact solution that also holds for inputs modelled as synaptic reversal potentials. Analytical and numerical methods are used to examine the sensitivity of the soma potentials (in particular) to the coupling resistance.

Re-entrant corner flows of UCM fluids : the initial formation of lip vortices

We discuss here a third type of boundary-layer structure that arises in steady planar flows at re-entrant corners for the upper convected Maxwell fluid. This structure extends the class of similarity solutions that are associated with the inviscid flow equations and hold in an outer core region local to the corner. In previous work (Renardy 1995 J. Non-Newtonian Fluid Mech. 58, 83–39; Evans 2005 Proc. R. Soc. A 461, 117–142), single and double layer structures were used to the match outer core similarity solutions to wall boundary layers in which viscometric behaviour is obtained. Here a second double layer structure is discussed, which completes the range of validity for the core similarity solutions. This structure is fundamentally different to the single layer structure in that it only admits reverse flow solutions at the upstream wall, a situation of practical relevance to describe the initial formation of lip vortices in contraction flows.