NURBS curves in direct definition of the shapes of the boundary for 2D Stokes flow problems in modified classical BIE

2018 ◽  
Vol 132 ◽  
pp. 111-126
Author(s):  
Eugeniusz Zieniuk ◽  
Krzysztof Szerszeń
Keyword(s):  
Stokes Flow ◽  
Flow Problems ◽  
Nurbs Curves ◽  
Direct Definition ◽  
Definition Of

scholarly journals Hydromechanics of low-Reynolds-number flow. Part 2. Singularity method for Stokes flows

1975 ◽  
Vol 67 (4) ◽  
pp. 787-815 ◽  
Author(s):  
Allen T. Chwang ◽  
T. Yao-Tsu Wu
Keyword(s):  
Exact Solutions ◽  
Stokes Flow ◽  
Fundamental Solutions ◽  
The Body ◽  
Circular Cylinders ◽  
Geometrical Parameters ◽  
Flow Problems ◽  
Singularity Method ◽  
Wide Range ◽  
Body Shapes

The present study further explores the fundamental singular solutions for Stokes flow that can be useful for constructing solutions over a wide range of free-stream profiles and body shapes. The primary singularity is the Stokeslet, which is associated with a singular point force embedded in a Stokes flow. From its derivatives other fundamental singularities can be obtained, including rotlets, stresslets, potential doublets and higher-order poles derived from them. For treating interior Stokes-flow problems new fundamental solutions are introduced; they include the Stokeson and its derivatives, called the roton and stresson.These fundamental singularities are employed here to construct exact solutions to a number of exterior and interior Stokes-flow problems for several specific body shapes translating and rotating in a viscous fluid which may itself be providing a primary flow. The different primary flows considered here include the uniform stream, shear flows, parabolic profiles and extensional flows (hyper-bolic profiles), while the body shapes cover prolate spheroids, spheres and circular cylinders. The salient features of these exact solutions (all obtained in closed form) regarding the types of singularities required for the construction of a solution in each specific case, their distribution densities and the range of validity of the solution, which may depend on the characteristic Reynolds numbers and governing geometrical parameters, are discussed.

Cubical Agda: A dependently typed programming language with univalence and higher inductive types

2021 ◽  
Vol 31 ◽  
Author(s):  
ANDREA VEZZOSI ◽  
ANDERS MÖRTBERG ◽  
ANDREAS ABEL
Keyword(s):  
Programming Language ◽  
Type Theory ◽  
Type Checking ◽  
Dependent Type Theory ◽  
Wide Range ◽  
General Schema ◽  
Direct Definition ◽  
Inductive Types ◽  
Definition Of ◽  
Cubical Type Theory

Abstract Proof assistants based on dependent type theory provide expressive languages for both programming and proving within the same system. However, all of the major implementations lack powerful extensionality principles for reasoning about equality, such as function and propositional extensionality. These principles are typically added axiomatically which disrupts the constructive properties of these systems. Cubical type theory provides a solution by giving computational meaning to Homotopy Type Theory and Univalent Foundations, in particular to the univalence axiom and higher inductive types (HITs). This paper describes an extension of the dependently typed functional programming language Agda with cubical primitives, making it into a full-blown proof assistant with native support for univalence and a general schema of HITs. These new primitives allow the direct definition of function and propositional extensionality as well as quotient types, all with computational content. Additionally, thanks also to copatterns, bisimilarity is equivalent to equality for coinductive types. The adoption of cubical type theory extends Agda with support for a wide range of extensionality principles, without sacrificing type checking and constructivity.

Flow Velocity Coefficient Research on the Coexistence of Pipe Flow and Flow around Pipe

2011 ◽  
Vol 204-210 ◽  
pp. 746-749
Author(s):  
Hong Bing Gao ◽  
Nan Sun ◽  
Liao Yang ◽  
Yu Ma
Keyword(s):  
Flow Field ◽  
Theoretical Basis ◽  
Flow Simulation ◽  
Flow Speed ◽  
Test Rig ◽  
Flow Problems ◽  
Relative Roughness ◽  
Velocity Coefficient ◽  
Definition Of ◽  
Test Result

Based on Darcy - Weisbach route losses equation, this test is focused on the mixed-flow problems at low speed in big flow field, designing the experiment scheme, introducing its principle, and setting up the test rig. According to the test result data, we put forward the definition of flow speed coefficient and draw its curves, and described some related issues, such as reverse pressure phenomenon，reliability of flow simulation, roughness, and so on. The results showed that the flow speed in the pipe will get higher with the flow field increasing, and smaller relative roughness pipes. The flow speed coefficient x≤1 and its curves is monotone decreasing function and will be closer to a fixed value at last with V0’s increasing. This test will provide theoretical basis for increasing fluid energy’s utilization.

Tarski

Truth ◽  
2011 ◽  
Author(s):  
Alexis G. Burgess ◽  
John P. Burgess
Keyword(s):  
Mathematical Logic ◽  
Model Theory ◽  
The State ◽  
Truth Predicate ◽  
Object Language ◽  
French And English ◽  
Direct Definition ◽  
Definition Of ◽  
Basic Features ◽  
Self Reference

This chapter offers a simplified account of the most basic features of Alfred Tarski's model theory. Tarski foresaw important applications for a notion of truth in mathematics, but also saw that mathematicians were suspicious of that notion, and rightly so given the state of understanding of it circa 1930. In a series of papers in Polish, German, French, and English from the 1930s onward, Tarski attempted to rehabilitate the notion for use in mathematics, and his efforts had by the 1950s resulted in the creation of a branch of mathematical logic known as model theory. The chapter first considers Tarski's notion of truth, which he calls “semantic” truth, before discussing his views on object language and metalanguage, recursive versus direct definition of the truth predicate, and self-reference.

Realisability semantics of parametric polymorphism, general references and recursive types

2010 ◽  
Vol 20 (4) ◽  
pp. 655-703 ◽  
Author(s):  
LARS BIRKEDAL ◽  
KRISTIAN STØVRING ◽  
JACOB THAMSBORG
Keyword(s):  
Metric Spaces ◽  
Logical Relation ◽  
Data Types ◽  
General Reference ◽  
New Approach ◽  
Parametric Polymorphism ◽  
Abstract Data ◽  
Direct Definition ◽  
Semantic Types ◽  
Definition Of

We present a realisability model for a call-by-value, higher-order programming language with parametric polymorphism, general first-class references, and recursive types. The main novelty is a relational interpretation of open types that include general reference types. The interpretation uses a new approach to modelling references.The universe of semantic types consists of world-indexed families of logical relations over a universal predomain. In order to model general reference types, worlds are finite maps from locations to semantic types: this introduces a circularity between semantic types and worlds that precludes a direct definition of either. Our solution is to solve a recursive equation in an appropriate category of metric spaces. In effect, types are interpreted using a Kripke logical relation over a recursively defined set of worlds.We illustrate how the model can be used to prove simple equivalences between different implementations of imperative abstract data types.

Application of the Cosserat Spectrum Theory to Stokes Flow

1999 ◽  
Vol 66 (3) ◽  
pp. 811-814
Author(s):  
W. Liu ◽  
A. Plotkin
Keyword(s):  
Reynolds Number ◽  
Velocity Field ◽  
Laplace Equation ◽  
Analytical Solutions ◽  
Stokes Flow ◽  
Low Reynolds Number ◽  
Flow Problems ◽  
Cosserat Spectrum ◽  
Linear Shear ◽  
Flow Past A Sphere

This paper presents an application of the Cosserat spectrum theory in elasticity to the solution of low Reynolds number (Stokes flow) problems. The velocity field is divided into two components: a solution to the vector Laplace equation and a solution associated with the discrete Cosserat eigenvectors. Analytical solutions are presented for the Stokes flow past a sphere with uniform, extensional, and linear shear freestream profiles.

scholarly journals ESSENCE OF THE SERVICE CATEGORY IN FORENSIC MERCHANDISING

2020 ◽  
Vol 2 ◽  
pp. 69-76
Author(s):  
I. Petrova ◽  
O. Dontsova
Keyword(s):  
Economic Value ◽  
Service Work ◽  
Structural Elements ◽  
Commercial Activity ◽  
Service Category ◽  
Direct Definition ◽  
Definition Of ◽  
Different Sources ◽  
Essential Service

The article states that market of services, as a field of economic activity, plays the role of a leading sector of the national economy is not without certain problems, among which was the possibility of evaluating the service within the forensic merchandising. It is noted that researchers of the concept of service paid attention to its direct definition, description of the characteristics inherent in services and identify differences between the concepts: Goods, service, work. It is established the essence of the concept of service remains debatable and insufficiently researched regarding its application in forensic merchandising. In view of this, various structural elements of the concept of service were systematized into a single holistic definition by studying scientific approaches, terminological analysis and refining the definitions of this concept from different sources. It is established that uncertainty of services is that they are processes, not material objects and do not fall under the traditional definition of goods; identified a number of properties that distinguish services from tangible goods, and identified specific features of intangible services. It is emphasized that researching on category essence of service in forensic merchandising, one cannot ignore the interpretation of this term by legislators and scholars. Analysis of the service definition in regulations has established that the generalizing concept of service for these acts is a service as an action that has a certain beneficial effect, i.e. has benefits for consumer. Scientists argue that the service usually is an intangible product, process, and ownership of it for its sale to the consumer does not arise; ownership may arise as a result of the provision of the service. The essential service characteristic is defined by scientists as economic value, which makes it an object of commercial activity, a subject of trade and an object of purchase and sale; it turns it into an economic good. It is established that service concept is most often associated with the action or activity of the service provider aimed at meeting the needs of the consumer and is realized at the time of its provision.

Dimensional Analysis and Similitude

1997 ◽  
Author(s):  
David Jon Furbish
Keyword(s):  
Dimensional Analysis ◽  
Mathematical Analysis ◽  
Average Velocity ◽  
Mathematical Formulation ◽  
Vantage Point ◽  
Slip Condition ◽  
Flow Problems ◽  
Definition Of ◽  
Clear Idea ◽  
Physical Quantities

Some fluid flow problems are sufficiently simple that they can be treated mathematically in a straightforward way, making use of definitions of physical quantities, and taking into account initial and boundary conditions. For example, our derivation of the average velocity in a conduit with parallel walls (Example Problem 3.7.1) was obtained in a straightforward way once we specified the geometry of the problem, then made use of the definition of a Newtonian fluid and the no-slip condition. Whereas this type of analysis may work for some problems, it would be misleading to think that such direct approaches to solving problems are, in principle, always possible, hinging only on one’s mathematical skills and adeptness in specifying the geometry of a problem. Herein arise two noteworthy points. First, when initially examining a problem, one can sometimes obtain a clear idea of the desired solution before attempting a formal mathematical analysis. The means to do this, as we shall see below, is supplied by dimensional analysis, and it is a strategy that ought to be adopted in many circumstances. In fact, it is worth noting that dimensional analysis underlies many of the problems presented in this text. The advantage of knowing the form of a desired solution, of course, is that one has a clear target to guide the subsequent mathematical analysis. Indeed, this is the vantage point from which many classic problems, for example Stokes’s law for settling spheres, were initially examined. Second, a complete mathematical formulation of a problem may not be possible, due to the complexity of the problem, or due to absence of information required to constrain the mathematics of the problem. As a simple example, suppose that we were unaware of the no-slip condition in our analysis of the conduit-flow problem (Example Problem 3.7.1). Our analysis in this case would have essentially ended with (3.70), with the constant of integration C undetermined. Nevertheless, we could get close to our result (3.75) for the average velocity by another way.

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