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Similarity solution

About: Similarity solution is a research topic. Over the lifetime, 2074 publications have been published within this topic receiving 59790 citations.


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TL;DR: In this article, the authors investigated the potential limitations of hydrodynamic lubrication theory in micro-systems by theoretical and computational methods and found that the damping force is proportional to the inverse of the fluid thickness to the first power.
Abstract: Classical hydrodynamic lubrication theory has been one of the most successful and widely used theories in all of engineering and applied science. This theory predicts that the force resisting the squeezing of a fluid between two parallel plates is inversely proportional to the cube of the fluid thickness. However, recent reports on liquid squeeze film damping in microsystems appear to indicate that experimentally measured damping force is proportional to the inverse of the fluid thickness to the first power—a large fundamental discrepancy from classical theory. This paper investigates potential limitations of lubrication theory in microsystems by theoretical and computational methods. The governing equations for a Newtonian incompressible fluid are solved subject to two-dimensional, parallel surface squeezing by an open-source computational fluid dynamics program called parallel hierarchic adaptive stabilized transient analysis (PHASTA), and by a classical similarity solution technique. At low convective Reynolds numbers, the damping force is determined as a function of the ratio of a reference film thickness H to a reference direction B along the film. Good agreement with classical lubrication theory is found for aspect ratios H/B as high as 1 despite the fact that lubrication theory requires that this ratio be "small. " A similarity analysis shows that when instantaneous convective Reynolds number is of order 10―100 (a range present in experiment), calculated damping deviates significantly from lubrication theory. This suggests that nonlinearity associated with high Reynolds numbers could explain the experimentally observed discrepancy in damping force. Dynamic analysis of beams undergoing small vibrations in the presence of a liquid medium further supports this finding.

9 citations

Journal ArticleDOI
TL;DR: In this article, the boundary layer flow and heat transfer of an electrically conducting viscous fluid over a stretching sheet is dealt with, where Lie-group method is applied for determining the symmetry reductions for the governing equations by reducing the number of independent variables in the given system of partial differential equations by one, leading to a system of non-linear ordinary differential equation.
Abstract: This work deals with the boundary layer flow and heat transfer of an electrically conducting viscous fluid over a stretching sheet. Lie-group method is applied for determining the symmetry reductions for the governing equations by reducing the number of independent variables in the given system of partial differential equations by one, leading to a system of non-linear ordinary differential equation. The resulting system is then solved numerically using shooting method coupled with Runge-Kutta scheme. Effects of various values of physical parameters on the horizontal and vertical velocities, temperature profiles, wall heat transfer and the wall shear stress (skin friction), have been studied and the results are plotted. Furthermore, a comparison between the present results with existing numerical and homotopy methods has been reported and we found that they are in a good agreement.

9 citations

Journal ArticleDOI
TL;DR: In this article, it was shown that the Oldroyd-B model has a limiting Weissenberg number for the continuous squeezing flow of oldroyd type fluids in a two-dimensional wedge.
Abstract: The paper is concerned with the continuous squeezing flow of Oldroyd-type fluids in a two-dimensional wedge. The flow mimics the lubrication action in a squeezing flow and is important in that there exists a similarity solution for any simple fluid. We are only concerned with Oldroyd-type fluids, however. It is shown by using a parameter continuation method that the Oldroyd-B model has a limiting Weissenberg number. The Phan Thien/Tanner model does not have this limiting Weissenberg number.

9 citations

Journal ArticleDOI
TL;DR: In this paper, the Navier-Stokes equations were solved using matched asymptotic expansions and a variable-spaci ng, finite difference method with dynamic mesh modification in a corner delimited by a vertical heated semi-infinite plate and a second room temperature plate forming an arbitrary angle.
Abstract: Laminar natural convection flow in a corner delimited by a vertical heated semi-infinite plate and a second room-temperature plate forming an arbitrary angle is studied both analytically and numerically. Analytically, repeated use of the matched asymptotic expansions technique shows that the boundary layer along the vertical plate is described, to first order, by Polhausen's classical similarity solution; this buoyancy-induced flow drives an outer irrotational flow which, in turn, provides the motive force for a self-similar viscous boundary layer along the second plate. In the numerical approach, a variable-spaci ng, finite difference method with dynamic mesh modification permits a solution of the Navier-Stokes equations to be obtained in the whole field, confirming the validity of the approximate analytical model.

9 citations

Journal ArticleDOI
TL;DR: In this paper, the problem of the small-scale yielding (SSY) plane-strain asymptotic fields for the interfacial free-edge joint singularity is examined in detail, and comparisons are made with the inter-dimensional crack tip.
Abstract: The problem of the small-scale yielding (SSY) plane-strain asymptotic fields for the interfacial free-edge joint singularity is examined in detail, and comparisons are made with the interfacial crack tip. The geometries are idealized as isotropic elasto-plastic materials with Ramberg-Osgood power-law hardening properties bonded to a rigid elastic substrate. The resulting fields are shown to be singular and are presented in terms of radial and angular distributions of stress and displacement, and as idealized plastic slip-line sectors. A fourth-order Runge-Kutta numerical method provides solutions to fundamental equations of equilibrium and compatibility that are verified with those of a highly focused finite element (FE) analysis. It is shown that, as in the case of the crack, the asymptotic singular fields are only dependent on the hardening parameter and only a small range of interfacial mode-mix ratios are permitted. The order for the stress singularity may be formulated in terms of the hardening parameter and the elastic solution for incompressible material. The rigid-slip-line field for the interfacial free-edge joint is presented, and it is shown that there is some significant similarity between the asymptotic fields of the deviatoric polar stresses for the joint and the crack-tip having an elastic wedge sector.

9 citations


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Performance
Metrics
No. of papers in the topic in previous years
YearPapers
202313
202238
202141
202045
201947
201850