Solutions of the Navier–Stokes Equation at Large Reynolds Number

The problem of two-dimensional incompressible laminar flow past a bluff body at large Reynolds number (R) is discussed. The governing equations are the Navier-Stokes equations. For R = ∞, the Euler equations are obtained. A solution for R large should be obtained by a perturbation of an Euler solution. However, for given boundary conditions, the Euler solution is not unique. The solution to be perturbed is the relevant Euler solution, namely the one which is the Euler limit of the Navier-Stokes solution with the same boundary conditions. For certain semi-infinite or streamlined bodies, the relevant Euler solution represents potential flow. For flow inside a closed domain a theorem of Prandtl states the relevant Euler solution has constant vorticity in each vortex. In many cases it can be determined by simultaneously considering the boundary layer equations. For flow past a bluff body, the relevant Euler solution is not known, although the free streamline flow for which the free streamline detaches smoothly from the body is a likely candidate. Even if this is correct, many unsolved problems remain. Various scalings have to be used for various regions of the flow. Possibilities of scaling for the various regions are discussed here. Special attention is paid to the region near the point of separation. A famous paper by Goldstein asserts that for an adverse smooth pressure gradient, the solution of the boundary layer equations can, in general, not be continued beyond the point of separation. Subsequent attempts by many authors to overcome the difficulty of continuation have failed. A very promising theory, going beyond conventional boundary layer theory, has recently been put forward independently by Sychev and Messiter. They assume that separation takes place in a sublayer whose thickness and length tend to zero as R tends to infinity. The pressure gradient in the sublayer is self-induced and is positive upstream of the point of separation and zero downstream. Their theory does not contradict experiments and numerical calculations, which may be reliable up to, say, R = 100, but it also shows that in this context, 100 may not be regarded as a large Reynolds number. The sublayer has the same scaling in orders of R as the sublayer at the trailing edge of a plate, found earlier by Stewartson and Messiter in studying the matching of the boundary layer solution on the plate with the Goldstein wake solution downstream of the trailing edge

Basic Concepts Underlying Singular Perturbation Techniques

In many singular perturbation problems multiple scales are used. For instance, one may use both the coordinate x and the coordinate x^* = ε^(-1)x. In a secular-type problem x and x^* are used simultaneously. This paper discusses layer-type problems in which x^* is used in a thin layer and x outside this layer. Assume one seeks approximations to a function f(x,ε), uniformly valid to some order in ε for x in a closed interval D. In layer-type problems one uses (at least) two expansions (called inner and outer) neither of which is uniformly valid but whose domains of validity together cover the interval D. To define "domain of validity" one needs to consider intervals whose endpoints depend on epsilon. In the construction of the inner and outer expansions, constants and functions of e occur which are determined by comparison of the two expansions "matching." The comparison is possible only in the domain of overlap of their regions of validity. Once overlap is established, matching is easily carried out. Heuristic ideas for determining domains of validity of approximations by a study of the corresponding equations are illustrated with the aid of model equations. It is shown that formally small terms in an equation may have large integrated effects. The study of this is of central importance for understanding layer-type problems. It is emphasized that considering the expansions as the result of applying limit processes can lead to serious errors and, in any case, hides the nature of the expansions

Note on Logarithmic Switchback Terms in Regular and Singular Perturbation Expansions

The occurrence of logarithmic switchback is studied for ordinary differential equations containing a parameter k which is allowed to take any value in a continuum of real numbers and with boundary conditions imposed at x = ε and x = ∞. Classical theory tells us that if the equation has a regular singular point at the origin there is a family of solutions which varies continuously with k, and the expansion around the origin has log x terms for a discrete set of values of k. It is shown here how nonlinearity enlarges this set so that it may even be dense in some interval of the real numbers. A log x term in the expansion in x leads to expansion coefficients containing log ε (switchback) in the perturbation expansion. If for a given value of k logarithmic terms in x and ε occur they may be obtained by continuity from neighboring values of k. Switchback terms occurred conspicuously in singular-perturbation solutions of problems posed for semi-infinite domain x ≥ ε. This connection is historical rather than logical. In particular we study here switchback terms for a specific example using methods of both singular and regular perturbations

Proof of some asymptotic results for a model equation for low Reynolds number flow

A two-point boundary value problem in the interval [ε, ∞], ε > 0 is studied. The problem contains additional parameters α ≥ 0, β ≥ 0, 0 ≤ U 0; for α = 0 an explicit construction shows that no solution exists unless k > 1. A special method is used to show uniqueness. For ε ↓ 0, k ≥ 1, various results had previously been obtained by the method of matched asymptotic expansions. Examples of these results are verified rigorously using the integral representation. For k < 1, the problem is shown not to be a layer-type problem, a fact previously demonstrated explicitly for k = 0. If k is an integer ≥ 0 the intuitive understanding of the problem is aided by regarding it as spherically symmetric in k + 1 dimensions. In the present study, however, k may be any real number, even negative

Retrospective cross sectional analysis of an acupuncture intervention for chronic pain management at Groote Schuur Hospital Pain Clinic Cape Town, South Africa

In 2015 acupuncture was introduced as an alternative intervention in the management of chronic pain, at the Chronic Pain Management Clinic of Groote Schuur Hospital, a tertiary academic hospital in Cape Town, South Africa. This study is a retrospective, cross-sectional analysis that aimed to investigate several aspects of the acupuncture intervention over a 12 month period. The main outcome measure, the Brief Pain Inventory (BPI), is a widely used, internationally validated questionnaire, containing pain intensity, pain interference, and total score. The main objective of this study was to determine if the acupuncture treatment lowered BPI scores after 6 to 9 intervention sessions. Additional objectives were to determine if there are any correlations between demographic and clinical factors and changes in BPI scores, and to describe the demographic and clinical characteristics of the study population. The data was obtained by folder reviews of 66 patients with chronic pain who were referred for acupuncture treatment between January 1, 2015 and December 31, 2015, and attended at least one treatment session. The full treatment course (6-9 sessions) was completed by 24 patients (36,3%), with an average post treatment decrease in BPI of 3,7 points. Responders (patients who obtained 2 and more point BPI decrease) comprised 70,6% of the patients who completed treatment. Decrease in BPI scores after completion of full acupuncture treatment proved to be statistically significant (p=0.002). Factors showing strongest correlation with BPI decrease were female gender and absence of medical and psychiatric co-morbidities

All in the Family: The Role of Sibling Relationships as Surrogate Attachment Figures

While several studies have analyzed the impact of mother-child attachment security on the child’s emotion regulation abilities, few studies have proposed interventions to help children improve emotion regulation abilities in the presence of an insecure mother-child attachment. This current study extends previous findings about the influence of mother-child attachment on the child’s emotion regulation abilities and contributes new research in determining whether an older sibling can moderate this effect. This study predicts that across points of assessments: 18 months, 5 years, 10 years, and 15 years, the quality of mother-child attachment security will influence the child’s performance on an emotion regulation task, such that securely attached children will demonstrate the most persistence and least distress, children with Anxious-Avoidant attachment will demonstrate the least persistence, and children with Anxious-Ambivalent will demonstrate the most distress. If, at any point, the child develops an insecure relationship with the mother and a secure relationship with the older sibling, the child’s persistence is expected to increase and the child’s distress is expected to decrease. In this way, the older sibling will serve as a surrogate attachment figure. These research findings have important implications for parenting behaviors as well as clinical practices

Notes on Stochastic Processes

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A high order compact scheme for hypersonic aerothermodynamics

A novel high order compact scheme for solving the compressible Navier-Stokes equations has been developed. The scheme is an extension of a method originally proposed for solving the Euler equations, and combines several techniques for the solution of compressible flowfields, such as upwinding, limiting and flux vector splitting, with the excellent properties of high order compact schemes. Extending the method to the Navier-Stokes equations is achieved via a Kinetic Flux Vector Splitting technique, which represents an unusual and attractive way to include viscous effects. This approach offers a more accurate and less computationally expensive technique than discretizations based on more conventional operator splitting. The Euler solver has been validated against several inviscid test cases, and results for several viscous test cases are also presented. The results confirm that the method is stable, accurate and has excellent shock-capturing capabilities for both viscous and inviscid flows

AIDS and dementia

The origin of AIDS (acquired immune deficiency syndrome) is unknown. AIDS was first recognized in the United States in 1981. Since 1981, AIDS has spread rapidly throughout the country. Initially, cases of AIDS were confined to New York, San Francisco and Los Angeles (Harris County Medical Society, 1987). AIDS is a disease caused by a virus that destroys an individual\u27s defenses against infections (United States Department of Education, 1987). It is essentially a disease of the immune system (Check, 1988). The AIDS virus, known as Human Immunodeficiency Virus, or HIV can so weaken an individual\u27s immune system that he or she cannot fight off even mild infections and eventually becomes vulnerable to life-threatening infections and cancers (United States Department of Education, 1987)