Mark Bowen

The self-similar solution for draining in the thin film equation

We investigate self-similar solutions of the thin film equation in the case of zero contact angle boundary conditions on a finite domain. We prove existence and uniqueness of such a solution and determine the asymptotic behaviour as the exponent in the equation approaches the critical value at which zero contact angle boundary conditions become untenable. Numerical and power-series solutions are also presented.

VALIDATION OF A NOVEL TRANSLUMBAR ULTRASOUND TECHNIQUE FOR MEASURING RENAL DIMENSIONS IN HORSES

A reliable method for obtaining renal ultrasonographic measurements in the horse is important for diagnosis and monitoring of clinical renal disease. The aims of this prospective study were to develop and validate a novel translumbar ultrasound technique for measuring renal dimensions in horses. Six Thoroughbred or Thoroughbred part bred horses were recruited. All horses were scheduled for euthanasia due to reasons unrelated to the kidneys. Two observers recorded renal length, width, and depth; and dimensions of the cortex, medulla, pyramids, and pelvis for both kidneys in each horse using novel translumbar and conventional transabdominal ultrasound methods. The same measurements were recorded from post-mortem renal specimens. Both kidneys were consistently identified by both methods in the 15-17th intercostal spaces and paralumbar fossa. Using the translumbar technique, maximal dimensions were obtained for the left kidney in the 16th intercostal space (length 16.2 ± 2.0 cm, width 11.8 ± 0.5 cm, depth 6.4 ± 0.9 cm) and for the right kidney in the 15th intercostal space (length 16.1 ± 1.2 cm, width 13.4 ± 1.2 cm, depth 6.7 ± 0.7 cm). Renal dimensions obtained by transabdominal and translumbar projections did not differ (P > 0.05). Good correlations were found between overall renal dimensions and post-mortem measurements for both ultrasound techniques (r(2) > 0.8), but were better for the translumbar method (mean r(2) = 0.92 cf. 0.88). Good-to-excellent reliability was found for all translumbar ultrasound measurements except for the renal cortex. Reproducibility was better for the larger (overall length, width, and depth) than the smaller (cortex, medulla, and pyramids) structures. Findings indicated that translumbar ultrasonography is a valid method for measuring renal dimensions in horses.

Thin film dynamics: theory and applications

This paper is based on a series of four lectures, by the first author, on thin films and moving contact lines. Section 1 presents an overview of the moving contact line problem and introduces the lubrication approximation. Section 2 summarizes results for positivity preserving schemes. Section 3 discusses the problem of films driven by thermal gradients with an opposing gravitational force. Such systems yield complex dynamics featuring undercompressive shocks. We conclude in Section 4 with a discussion of dewetting films.

Reliability and Identification of Aortic Valve Prolapse in the Horse

BackgroundThe objectives were to determine and assess the reliability of criteria for identification of aortic valve prolapse (AVP) using echocardiography in the horse.ResultsOpinion of equine cardiologists indicated that a long-axis view of the aortic valve (AoV) was most commonly used for identification of AVP (46%; n=13). There was consensus that AVP could be mimicked by ultrasound probe malignment. This was confirmed in 7 healthy horses, where the appearance of AVP could be induced by malalignment. In a study of a further 8 healthy horses (5 with AVP) examined daily for 5 days, by two echocardiographers standardized imaging guidelines gave good to excellent agreement for the assessment of AVP (kappa>0.80) and good agreement between days and observers (kappa >0.6). The technique allowed for assessment of the degree of prolapse and measurement of the prolapse distance that provided excellent agreement between echocardiographers, days and observers (kappa/ICC>0.8). Assessments made using real-time zoomed images provided similar measurements to the standard views (ICC=0.9), with agreement for the identification of AVP (kappa>0.8).Short axis views of the AoV were used for identification of AVP by fewer respondents (23%), however provided less agreement for the identification of AVP (kappa>0.6) and only adequate agreement with observations made in long axis (kappa>0.5), with AVP being identified more often in short axis (92%) compared to long axis (76%).Orthogonal views were used by 31% of respondents to identify the presence of AVP, and 85% to identify cusp. Its identification on both views on 4 days was used to categorise horses as having AVP, providing a positive predictive value of 79% and negative predictive value of 18%. Only the non-coronary cusp (NCC) of the AoV was observed to prolapse in these studies. Prolapse of the NCC was confirmed during the optimisation study using four-dimensional echocardiography, which concurred with the findings of two-dimensional echocardiography.ConclusionsThis study has demonstrated reliable diagnostic criteria for the identification and assessment of AVP that can be used for longitudinal research studies to better define the prevalence and natural history of this condition.

Self-Similar Scaling Solutions of Differential Equations

We describe how the construction of similarity solutions of partial differential equations extends naturally from concepts in dimensional analysis. In particular, we show how to obtain self-similar solutions through scaling invariances of linear and nonlinear PDE. We give examples illustrating how similarity solutions of PDEs can be obtained from solutions of ODE problems.

Modelling in Applied Fluid Dynamics

This chapter covers the modelling of two applied problems in fluid dynamics in brief case studies. The problems, on air bearing sliders and rivulet flows in wedges, build on a common core model called lubrication theory, which is derived in the beginning of the chapter. The case studies differ in many physical aspects (compressible versus incompressible flows, and free-surface versus squeeze film flows) and mathematical techniques employed (similarity solutions versus boundary layers and linear stability analysis). They serve to illustrate the power of the modelling methods to handle a diverse range of types of problems.

Boundary Layer Theory

Solutions of singularly perturbed ordinary differential equations exhibit non-uniform convergence as \(\varepsilon \rightarrow 0\). Regular and singular solutions (also called outer and inner solutions respectively) may be needed to satisfy all of the conditions imposed in a boundary value problem. The construction of matched asymptotic expansions gives a process for obtaining the solution of the overall problem from inner and outer solutions that hold on different parts of the domain. Determination of the scaling and placement of boundary layers will be illustrated in several examples.

Fast/slow Dynamical Systems

Problems that can be written as singularly perturbed systems of first order differential equations can be solved using approaches that combine matched asymptotic expansions with phase plane analysis. The \(\varepsilon \rightarrow 0\) limit yields a separation of time-scales that reduces the overall system to different forms for the fast and slow dynamics over different intervals of time. Asymptotic matching is used to connect the fast and slow solutions and can be interpreted in terms of the geometric structure of nullclines and the structure of the phase plane.

Long-Wave Asymptotics for PDE Problems

Multi-dimensional partial differential equation problems on slender domains, having a small aspect ratio, can be analysed primarily in terms of solutions of simpler ODE problems for the variation of the solution in the slender direction. Using the aspect ratio as an asymptotic parameter, solutions having slow-variation (or “long-wave” dependence) in the wide direction can be constructed using matched asymptotics.

Reduced Models for PDE Problems

In many problems, not all details of the structure of the solution of a partial differential equation are of interest. In some cases, it is possible to obtain an essential understanding of the behaviour of the system without directly solving the full equation. This chapter outlines the method of moments which can provide information about the evolution of integrals of the solution of a PDE by only solving ODEs. A related approach is used to reduce a two-dimensional problem to a modified PDE in one-dimension exhibiting Taylor-enhanced diffusion. Examination of the Turing instability for pattern formation in reaction-diffusion systems provides a cautionary note on limitations of such reduced models.