The free surface on a liquid between cylinders rotating at different speeds part II

Joseph, Daniel D.; Beavers, G. S.; Fosdick, Roger

doi:10.1007/bf00253045

Cited by 79 publications

(45 citation statements)

References 15 publications

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“…Although the second-grade model is found to predict the normal stress differences, it does not properly respond to shear thinning or thickening due to its constant apparent shear viscosity. For this reason, some experiments may be well described by the fluids of grade three or four [2,4,11]. Keeping this fact in view, the aim of the present analysis is to venture further into the regime of fourth-grade fluids.…”

Section: Introductionmentioning

confidence: 91%

Unsteady flow of a fourth-grade fluid due to an oscillating plate

Wang

2007

International Journal of Non-Linear Mechanics

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The governing non-linear high-order, sixth-order in space and third-order in time, differential equation is constructed for the unsteady flow of an incompressible conducting fourth-grade fluid in a semi-infinite domain. The unsteady flow is induced by a periodically oscillating two-dimensional infinite porous plate with suction/blowing, located in a uniform magnetic field. It is shown that by augmenting additional boundary conditions at infinity based on asymptotic structures and transforming the semi-infinite physical space to a bounded computational domain by means of a coordinate transformation, it is possible to obtain numerical solutions of the nonlinear magnetohydrodynamic equation. In particular, due to the unsymmetry of the boundary conditions, in numerical simulations non-central difference schemes are constructed and employed to approximate the emerging higher-order spatial derivatives. Effects of material parameters, uniform suction or blowing past the porous plate, exerted magnetic field and oscillation frequency of the plate on the time-dependent flow, especially on the boundary layer structure near the plate, are numerically analysed and discussed. The flow behaviour of the fourth-grade non-Newtonian fluid is also compared with those of the Newtonian fluid.

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Section: Introductionmentioning

confidence: 91%

Unsteady flow of a fourth-grade fluid due to an oscillating plate

Wang

2007

International Journal of Non-Linear Mechanics

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“…The theory of rod climbing with surface tension neglected was done by Kaye [3), Joseph and Fosdick [4], Hoffman and Gottenberg [5], and B6hme [6]. Joseph et al [7] were the first to use theory in experiments designed to measure f3. They showed that, to get the predicted shapes of the free surface to agree with measured ones, it is necessary to retain the effects of surface tension, as in (2.2).…”

Section: H(r A)= Ho(r) + H 2 (R)a 2 mentioning

confidence: 99%

“…They showed that, to get the predicted shapes of the free surface to agree with measured ones, it is necessary to retain the effects of surface tension, as in (2.2). A very accurate approximate solution of (2.2) was derived by Joseph et al [7]. When evaluated at the rod this solution gives a)4+X 2+) (2.9)…”

Section: H(r A)= Ho(r) + H 2 (R)a 2 mentioning

confidence: 99%

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Climbing constant, second-order correction of Trouton's viscosity, wave speed and delayed die swell for M1

Riccius

Chen

et al. 1990

Journal of Non-Newtonian Fluid Mechanics

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“…Joseph and R.L. Fosdick [64,65], by means of a formal series expansion in the angular velocity of the rod. However, we wish to emphasize that a rigorous proof of rod-climbing is an outstanding open question.…”

Section: Remark 211mentioning

confidence: 99%

Mathematical Problems in Classical and Non-Newtonian Fluid Mechanics

Galdi

Oberwolfach Seminars

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IntroductionBlood flow per se is a very complicated subject. Thus, it is not surprising that the mathematics involved in the study of its properties can be, often, extremely complex and challenging.The role of mathematics in the investigation of blood flow properties -as in the most part of applied sciences -is twofold and is directed toward the accomplishment of the following objectives. The first one, of a more theoretical nature, is the validation of the models proposed by engineers, and consists in securing conditions under which the governing equations possess the fundamental requirements of well-posedness, such as existence and uniqueness of corresponding solutions and their continuous dependence upon the data. The second one, of a more applied character, is to prove that these models give a satisfactory interpretation of the observed phenomena. In general, both tasks present serious difficulties in that they require the study of several different, and frequently combined, topics that include, among others, Navier-Stokes equations, non-Newtonian fluid models, nonlinear elasticity, fluid-structure interaction and multi-phase flow. It must be added that some of these topics are still at the beginning of a systematic mathematical research, whereas some others are in a continuous growth.As a matter of fact, the initiation or the methodical investigation of several of the above research areas was just motivated by problems arising in blood flow. Moreover, blood flow can also pose challenging questions in "classical" topics, questions that, in the past, happened to receive little or no attention at all. A typical example is provided by the problem of the flow of a Navier-Stokes liquid in an unbounded piping system, under a given time-periodic flow-rate, that has been "discovered" only in 2005, thanks to the work of H. Beirão da Veiga; see [7]. G.P. GaldiIt seemed to me hopeless to present and to describe in this article all relevant aspects of the mathematical analysis related to blood flow, even at an introductory level. Therefore, I preferred to concentrate on some, of the many, topics which are at the foundation of this analysis, and to point out directions for future research. More specifically, I focused on three different subjects which are the content of as many separate chapters.The first chapter deals with the study of some fundamental properties of the flow of a Navier-Stokes liquid in a piping system, which can be either unbounded or bounded. Here, I have concentrated the analysis mostly on steady-state and time-periodic motions, and on their attainability. There are several reasons for this choice. On the one hand, because these types of motions are the most "elementary" to occur in the arterial and venuous system, and, on the other hand, because the initial-boundary value problem in an unbounded piping system has been investigated in full detail in the most recent article [86], to which I refer the interested reader.The second chapter is dedicated to the mathematical analysis of certain non-Newtonian...

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The free surface on a liquid between cylinders rotating at different speeds part II

Cited by 79 publications

References 15 publications

Unsteady flow of a fourth-grade fluid due to an oscillating plate

Unsteady flow of a fourth-grade fluid due to an oscillating plate

Climbing constant, second-order correction of Trouton's viscosity, wave speed and delayed die swell for M1

Mathematical Problems in Classical and Non-Newtonian Fluid Mechanics

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