The Recirculatory Flow Induced by a Laminar Axisymmetric Jet Issuing From a Wall

1987 ◽  
Vol 109 (3) ◽  
pp. 237-241 ◽  
Author(s):  
W. Schneider ◽  
E. Zauner ◽  
H. Bo¨hm
Keyword(s):  
Point Sources ◽  
Creeping Flow ◽  
Outer Flow ◽  
Axisymmetric Jet ◽  
Submerged Jet ◽  
Viscous Forces ◽  
Recirculatory Flow ◽  
Self Similar ◽  
Infinite Wall ◽  
Main Region

The laminar, axisymmetric, submerged jet issuing from a plane, infinite wall perpendicular to the jet axis is considered at very large distance from the nozzle. Based on previous results of an asymptotic analysis, an approximate analytical solution for the complete flow field is obtained. The structure of the far field is discussed by considering various regions the size of which depends strongly on the Reynolds number. The main region is a toroidal eddy in which both inertial and viscous forces are of importance. Closer to the nozzle there is a slender jet flow with slowly varying momentum flux together with a self-similar viscous outer flow. At larger distances, the flow resembles the creeping flow due to point sources of momentum and mass, with the former decaying more rapidly than the latter as infinity is approached. Analytical predictions of the location of the eddy center compare favorably with experimental and numerical results.

Transition in the axisymmetric jet

1981 ◽  
Vol 104 ◽  
pp. 369-386 ◽  
Author(s):  
Brian J. Cantwell
Keyword(s):  
Reynolds Number ◽  
Autonomous System ◽  
Analytic Solution ◽  
Creeping Flow ◽  
Stable Node ◽  
Particle Path ◽  
Axisymmetric Jet ◽  
Flow Limit ◽  
Self Similar ◽  
Particle Paths

The unsteady laminar flow from a point source of momentum is considered. Dimensional considerations lead to a formulation of the problem which is self-similar in time. Three limiting cases are examined. In the limit t → ∞ the solution corresponds to the classic steady solution first discovered by Landau (1944). The limit t → 0 was examined recently by Sozou & Pickering (1977) and was shown to correspond to the flow from an unsteady dipole of linearly increasing strength. More recently Sozou (1979) determined an analytic solution for the creeping flow limit Re → 0. In the present work, unsteady particle trajectories for each of these cases are examined by reducing the particle path equations to an autonomous system with the Reynolds number as a parameter. Transition of the jet is examined as a bifurcation of this system. In the case of the creeping-flow solution, the particle-path pattern exhibits a structure which is not easily discerned in any of the other variables which govern the flow. For sufficiently small Reynolds number the particle paths converge to a single stable node which lies on the axis of the jet. At a Reynolds number of 6·7806 the pattern bifurcates to a saddle lying on the axis of the jet plus two stable nodes lying symmetrically to either side of the axis. At a Reynolds number of 10·09089 the pattern bifurcates a second time to form a saddle and two stable foci.

scholarly journals Axisymmetric spreading of surfactant from a point source

2017 ◽  
Vol 832 ◽  
pp. 777-792 ◽  
Author(s):  
Shreyas Mandre
Keyword(s):  
Fluid Velocity ◽  
Radial Distance ◽  
Domain Size ◽  
Fluid Interface ◽  
Viscous Forces ◽  
Constant Rate ◽  
Adsorbed Phase ◽  
Self Similar ◽  
Marangoni Stress ◽  
Dissolved Phase

Guided by computation, we theoretically calculate the steady flow driven by the Marangoni stress due to a surfactant introduced on a fluid interface at a constant rate. Two separate extreme cases, where the surfactant dynamics is dominated by the adsorbed phase or the dissolved phase, are considered. We focus on the case where the size of the surfactant source is much smaller than the size of the fluid domain, and the resulting Marangoni stress overwhelms the viscous forces so that the flow is strongest in a boundary layer close to the interface. We derive the resulting flow in a region much larger than the surfactant source but smaller than the domain size by approximating it with a self-similar profile. The radially outward component of fluid velocity decays with the radial distance $r$ as $r^{-3/5}$ when the surfactant spreads in an adsorbed phase, and as $r^{-1}$ when it spreads in a dissolved phase. Universal flow profiles that are independent of the system parameters emerge in both the cases. Three hydrodynamic signatures are identified to distinguish between the two cases and verify the applicability of our analysis with successive stringent tests.

Two-dimensional sink flow of a stratified fluid contained in a duct

1972 ◽  
Vol 53 (2) ◽  
pp. 329-349 ◽  
Author(s):  
Jorg Imberger
Keyword(s):  
Layer Thickness ◽  
Schmidt Number ◽  
Critical Distance ◽  
Force Balance ◽  
Viscous Force ◽  
Order Unity ◽  
Outer Flow ◽  
Viscous Forces ◽  
Primary Focus ◽  
Inertia Forces

A reservoir is assumed to be filled with water which has a linear variation of density with depth. The geometry of the boundaries is simplified to a parallel walled duct with the line sink at the centre of the fluid. The primary focus is on partitioning the flow into distinct flow regimes and predicting the withdrawal-layer thickness as a function of the distance from the sink; the predictions are verified experimentally.For fluids with a Schmidt number of order unity, the withdrawal layer is shown to be composed of distinct regions in each of which a definite force balance prevails. The outer flow, where inertia forces are neglected, changes from a parallel uniform flow upstream to a symmetric self-similar withdrawal layer near the sink. For distances from the sink smaller than a critical distance, dependent on the flow parameters, inertia forces become of equal importance to buoyancy and viscous forces. The equations valid in this inner region are derived. Using the inner limit of the outer flow as the upstream boundary condition, these inner equations are solved approximately for the withdrawal-layer thickness by an integral method. The inner and outer variations of δ, the withdrawal-layer thickness, are combined to yield a composite solution and it is seen that the inclusion of inertia forces yields layers thicker than those obtained from a strict buoyancy-viscous force balance. In terms of the inner variables the only parameter remaining is the Schmidt number.Laboratory experiments were carried out to verify the theoretical conclusions. The observed withdrawal-layer thicknesses were shown to be closely predicted by the integral solution. Furthermore, the data could be represented in terms of the inner variables by a single curve dependent only on the Schmidt number.

Creeping Flow Through an Axisymmetric Sudden Contraction or Expansion

2001 ◽  
Vol 124 (1) ◽  
pp. 273-278 ◽  
Author(s):  
Sourith Sisavath ◽  
Xudong Jing ◽  
Chris C. Pain ◽  
Robert W. Zimmerman
Keyword(s):  
Pressure Drop ◽  
Expansion Ratio ◽  
Excess Pressure ◽  
Creeping Flow ◽  
Numerical Studies ◽  
Equivalent Length ◽  
Sudden Contraction ◽  
Circular Orifice ◽  
Flow Through ◽  
Infinite Wall

Creeping flow through a sudden contraction/expansion in an axisymmetric pipe is studied. Sampson’s solution for flow through a circular orifice in an infinite wall is used to derive an approximation for the excess pressure drop due to a sudden contraction/expansion in a pipe with a finite expansion ratio. The accuracy of this approximation obtained is verified by comparing its results to finite-element simulations and other previous numerical studies. The result can also be extended to a thin annular obstacle in a circular pipe. The “equivalent length” corresponding to the excess pressure drop is found to be barely half the radius of the smaller tube.

Stokes flow past bubbles and drops partially coated with thin films. Part 2. Thin films with internal circulation – a perturbation solution

1983 ◽  
Vol 132 ◽  
pp. 295-318 ◽  
Author(s):  
Robert E. Johnson ◽  
S. S. Sadhal
Keyword(s):  
Thin Films ◽  
Fluid Layer ◽  
Cell Structure ◽  
Creeping Flow ◽  
Immiscible Fluid ◽  
Fluid Viscosity ◽  
Perturbation Scheme ◽  
Bulk Fluid ◽  
Viscous Forces ◽  
Surface Tension Forces

In the present study we examine the steady axisymmetric creeping flow due to the motion of a liquid drop or a bubble which is partially covered by a thin immiscible fluid layer or film. The analysis is based on the assumption that surface-tension forces are large compared with viscous forces which deform the drop, and that the circulation in the film is weak. The latter assumption is satisfied provided that the film-fluid viscosity is not too small. A perturbation scheme based on the thinness of the fluid layer is used to construct the solution.One of the principal results is an expression for the drag force on the complex drop. We also find that the extent to which the drop or bubble is covered the film has a maximum value depending on the magnitude of the driving force on the film. In addition, we find the rather interesting result that when the ratio of the primary drop viscosity and bulk fluid viscosity is greater than ½, the circulation within the film may have a double-cell structure.

scholarly journals Axial creeping flow in the gap between a rigid cylinder and a concentric elastic tube

2016 ◽  
Vol 806 ◽  
pp. 580-602 ◽  
Author(s):  
S. B. Elbaz ◽  
A. D. Gat
Keyword(s):  
Fluid Layer ◽  
Elastic Shell ◽  
Gravity Current ◽  
Travelling Wave ◽  
Elastic Tube ◽  
Creeping Flow ◽  
Shear Stresses ◽  
Length Scales ◽  
Rigid Cylinder ◽  
Viscous Forces

We examine transient axial creeping flow in the annular gap between a rigid cylinder and a concentric elastic tube. The gap is initially filled with a thin fluid layer. We employ an elastic shell model and the lubrication approximation to obtain governing equations for the elastohydrodynamic interaction. At long axial length scales viscous forces are balanced by elastic tension, while at shorter length scales the viscous–elastic balance is achieved by means of an interplay between elastic bending, tension and shear stresses. Based on a viscous gravity current analogy in the tensile–viscous regime, we devise propagation laws for displacement flows which are induced by a variety of boundary conditions and examine different limits of the prewetting thickness. Next we focus on the moving elastohydrodynamic contact line at the edge of a penetrating film. A uniform matched asymptotic solution connecting the interior tension-based region with a boundary layer region near the propagation front is presented. Finally, a constructive example is shown in which isolated moving deformation patterns are created and superimposed to form a travelling wave displacement field. The presented interaction between viscosity and elasticity may be applied to fields such as soft robotics and micro-scale or larger swimmers by allowing for the time-dependent control of an axisymmetric compliant boundary.

Wirebond Deformation During Molding of IC Packages

1995 ◽  
Vol 117 (1) ◽  
pp. 14-19 ◽  
Author(s):  
A. A. O. Tay ◽  
K. S. Yeo ◽  
J. H. Wu ◽  
T. B. Lim
Keyword(s):  
Gold Wire ◽  
Creeping Flow ◽  
The Finite Element Method ◽  
Melt Flow ◽  
Hardening Material ◽  
Transfer Molding ◽  
Viscous Forces ◽  
Parametric Studies ◽  
Short Circuits ◽  
Very High

During the transfer molding of IC packages, wirebonds are deformed by the action of flow-induced viscous forces acting along them. Excessive deformation of wirebonds could give rise to short circuits and bond pull-outs. In this paper, the deformation of gold wirebonds during transfer molding of IC packages is studied using the finite element method. Hitherto, only elastic deformation of wirebonds has been considered. In this paper, a more realistic elasto-plastic large-deflection model is employed. The gold wire is assumed to be made of a bilinear strain hardening material. It is shown that plastic deformation in the wirebond can occur even if the melt flowrate is not very high. However, wirebond deflection may still be within acceptable limits even though certain portions of the wirebond have yielded plastically. The deformation of parabolic wirebonds under the action of melt flow, both normal and parallel to the plane of the wirebond, is also studied. The melt flow within the cavity is simulated assuming creeping flow. Parametric studies of the effects of wirebond dimensions, namely bond height, span and wire diameter, on wirebond deformation are also carried out.

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