Viscous and resistive eddies near a sharp corner
TL;DR: In this paper, it was shown that when either or both of the boundaries is a rigid wall and when the angle between the planes is less than a certain critical angle, any flow sufficiently near the corner must consist of a sequence of eddies of decreasing size and rapidly decreasing intensity.
Abstract: Some simple similarity solutions are presented for the flow of a viscous fluid near a sharp corner between two planes on which a variety of boundary conditions may be imposed. The general flow near a corner between plane boundaries at rest is then considered, and it is shown that when either or both of the boundaries is a rigid wall and when the angle between the planes is less than a certain critical angle, any flow sufficiently near the corner must consist of a sequence of eddies of decreasing size and rapidly decreasing intensity. The ratios of dimensions and intensities of successive eddies are determined for the full range of angles for which the eddies exist. The limiting case of zero angle corresponds to the flow at some distance from a two-dimensional disturbance in a fluid between parallel boundaries. The general flow near a corner between two plane free surfaces is also determined; eddies do not appear in this case. The asymptotic flow at a large distance from a corner due to an arbitrary disturbance near the corner is mathematically similar to the above, and has comparable properties. When the fluid is electrically conducting, similarity solutions may be obtained when the only applied magnetic field is that due to a line current along the intersection of the two planes; it is shown that the effect of such a current is to widen the range of corner angles for which eddies must appear.
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TL;DR: In this article, the authors propose a definition of vortex in an incompressible flow in terms of the eigenvalues of the symmetric tensor, which captures the pressure minimum in a plane perpendicular to the vortex axis at high Reynolds numbers, and also accurately defines vortex cores at low Reynolds numbers.
Abstract: Considerable confusion surrounds the longstanding question of what constitutes a vortex, especially in a turbulent flow. This question, frequently misunderstood as academic, has recently acquired particular significance since coherent structures (CS) in turbulent flows are now commonly regarded as vortices. An objective definition of a vortex should permit the use of vortex dynamics concepts to educe CS, to explain formation and evolutionary dynamics of CS, to explore the role of CS in turbulence phenomena, and to develop viable turbulence models and control strategies for turbulence phenomena. We propose a definition of a vortex in an incompressible flow in terms of the eigenvalues of the symmetric tensor ${\bm {\cal S}}^2 + {\bm \Omega}^2$ are respectively the symmetric and antisymmetric parts of the velocity gradient tensor ${\bm \Delta}{\bm u}$. This definition captures the pressure minimum in a plane perpendicular to the vortex axis at high Reynolds numbers, and also accurately defines vortex cores at low Reynolds numbers, unlike a pressure-minimum criterion. We compare our definition with prior schemes/definitions using exact and numerical solutions of the Euler and Navier–Stokes equations for a variety of laminar and turbulent flows. In contrast to definitions based on the positive second invariant of ${\bm \Delta}{\bm u}$ or the complex eigenvalues of ${\bm \Delta}{\bm u}$, our definition accurately identifies the vortex core in flows where the vortex geometry is intuitively clear.
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TL;DR: In this paper, a flat fluid interface moving steadily over a flat solid is modeled with the creeping flow approximation, which turns out to be self-consistent, and the role of long-range forces are explored with the aid of the lubrication flow approximation.
1,419 citations
TL;DR: In this article, the authors review the status of the understanding of fluid flow phenomena particular to microdevices and emphasize the use of MEMS as sensors and actuators for flow diagnosis and control.
Abstract: Manufacturing processes that can create extremely small machines have been developed in recent years. Microelectromechanical systems (MEMS) refer to devices that have characteristic length of less than 1 mm but more than 1 micron, that combine electrical and mechanical components and that are fabricated using integrated circuit batch-processing techniques. Electrostatic, magnetic, pneumatic and thermal actuators, motors, valves, gears, and tweezers of less than 100-μm size have been fabricated. These have been used as sensors for pressure, temperature, mass flow, velocity and sound, as actuators for linear and angular motion and as simple components for complex systems such as micro-heat-engines and micro-heat-pumps The technology is progressing at a rate that fa r exceeds that of our understanding of the unconventional physics involved in the operation as well as the manufacturing of those minute devices. The primary objective of this article is to critically review the status of our understanding of fluid flow phenomena particular to microdevices. In terms of applications, the paper emphasizes the use of MEMS as sensors and actuators for flow diagnosis and control.
1,197 citations
Cites background from "Viscous and resistive eddies near a..."
...Other examples where sUp-flow must be admitted include comer flows (Moffatt, 1964; Koplik and Banavar, 1995) and extrusion of polymer melts from capillary tubes (Pearson and Petrie, 1968; Richardson, 1973; Den, 1990)....
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...Other examples where slip-flow must be admit-ibility coefficient a and bulk expansion coefficient b are much ted include corner flows (Moffatt, 1964; Koplik and Banavar,smaller compared to those for gases....
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TL;DR: In this paper, the authors present results from molecular dynamics simulations of newtonian liquids under shear which indicate that there exists a general nonlinear relationship between the amount of slip and the local shear rate at a solid surface.
Abstract: Modelling fluid flows past a surface is a general problem in science and engineering, and requires some assumption about the nature of the fluid motion (the boundary condition) at the solid interface. One of the simplest boundary conditions is the no-slip condition1,2, which dictates that a liquid element adjacent to the surface assumes the velocity of the surface. Although this condition has been remarkably successful in reproducing the characteristics of many types of flow, there exist situations in which it leads to singular or unrealistic behaviour—for example, the spreading of a liquid on a solid substrate3,4,5,6,7,8, corner flow9,10 and the extrusion of polymer melts from a capillary tube11,12,13. Numerous boundary conditions that allow for finite slip at the solid interface have been used to rectify these difficulties4,5,11,13,14. But these phenomenological models fail to provide a universal picture of the momentum transport that occurs at liquid/solid interfaces. Here we present results from molecular dynamics simulations of newtonian liquids under shear which indicate that there exists a general nonlinear relationship between the amount of slip and the local shear rate at a solid surface. The boundary condition is controlled by the extent to which the liquid ‘feels’ corrugations in the surface energy of the solid (owing in the present case to the atomic close-packing). Our generalized boundary condition allows us to relate the degree of slip to the underlying static properties and dynamic interactions of the walls and the fluid.
1,144 citations
TL;DR: In this article, the phase interface in a capillary and the spreading of viscous fluid drops on solid surfaces are solved, and the dependence of this angle on the velocity with allowance for capillary forces is determined.
Abstract: Fluid motion along a smooth, solid surface is examined when the free surface forms a final visible angle with the solid boundary. The dependence of this angle on the velocity with allowance for capillary forces is determined. The Reynolds number is small. The problem of the motion of the phase interface in a capillary and the spreading of viscous fluid drops on solid surfaces are solved. Experimental results are explained. Up to now, not only were analytical results lacking in this field, but also there was not even a precise formulation of the problem (see the review in [1]).
1,074 citations
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TL;DR: In this paper, the steady motion of a viscous fluid was studied in terms of the two-dimensional steady motion, and the authors proposed a steady motion model for viscous fluids.
Abstract: (1915). L. The two-dimensional steady motion of a viscous fluid. The London, Edinburgh, and Dublin Philosophical Magazine and Journal of Science: Vol. 29, No. 172, pp. 455-465.
340 citations
01 Jul 1949
TL;DR: In this article, it was shown that the most important term in the stream-function is of the form rmf(θ), where r, θ are plane polar coordinates.
Abstract: 1. In a steady two-dimensional motion of viscous liquid in the sharp corner formed by the rigid straight boundaries θ = 0, α, where r, θ are plane polar coordinates, it is found that, near enough to the corner, the most important term in the stream-function is of the form rmf(θ). The index m is evaluated in §§ 2–4 for values of α between 360 and 90°, and is found to be complex if α is less than about 146°; the limiting form of the stream-function when α is small is considered in § 5.
183 citations
90 citations