Flow in a channel with pulsating walls

In this paper calculations are made of the two-dimensional flow field of an incompressible viscous fluid in a long parallel-sided channel whose walls pulsate in a prescribed way. The study covers all values of the unsteadiness parameter α and the steady-streaming Reynolds number. The wall motion is, in general, assumed to be of small amplitude and sinusoidal. Particular attention is given to the steady component of the flow at second order in the amplitude parameter ε. The results for the corresponding problem in axisymmetric geometry are given in an appendix.Next the following problem is considered: the calculation of the wall motion which will result, in response to prescribed unsteady pressures imposed at the ends of the channel and outside its walls, if the walls are assumed to respond elastically to variations in transmural pressure. It is found that the system has a natural frequency of oscillation, and that resonance will occur if this frequency is close to a multiple of the frequency of the external pressure fluctuations. Finally the preceding work is applied in a discussion of blood flow in the coronary arteries of large mammals.

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Asymmetrical Flow Driving in Two-Sided Lid-Driven Square Cavity with Antiparallel Wall Motion

MATEC Web of Conferences ◽

10.1051/matecconf/202033001009 ◽

2020 ◽

Vol 330 ◽

pp. 01009

Author(s):

El Amin Azzouz ◽

Samir Houat

Keyword(s):

Wall Motion ◽

Velocity Ratio ◽

Two Dimensional ◽

Square Cavity ◽

Driven Cavity ◽

Dimensional Flow ◽

Fluid Properties ◽

Two Dimensional Flow ◽

Vertical Walls ◽

Volume Method

The two-dimensional flow in a two-sided lid-driven cavity is often handled numerically for the same imposed wall velocities (symmetrical driving) either for parallel or antiparallel wall motion. However, in this study, we present a finite volume method (FVM) based on the second scheme of accuracy to numerically explore the steady two-dimensional flow in a two-sided lid-driven square cavity for antiparallel wall motion with different imposed wall velocities (asymmetrical driving). The top and the bottom walls of the cavity slide in opposite directions simultaneously at different velocities related to various imposed velocity ratios, λ = -2, -6, and -10, while the two remaining vertical walls are stationary. The results show that varying the velocity ratio and consequently the Reynolds ratios have a significant effect on the flow structures and fluid properties inside the cavity.

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Centrifugal instability of an oscillatory flow over periodic ripples

Journal of Fluid Mechanics ◽

10.1017/s002211209000060x ◽

1990 ◽

Vol 217 ◽

pp. 1-32 ◽

Cited By ~ 25

Author(s):

Tetsu Hara ◽

Chiang C. Mei

Keyword(s):

Oscillatory Flow ◽

Sandy Beach ◽

Three Dimensional ◽

Small Region ◽

Oscillating Flow ◽

Two Dimensional ◽

Steady Streaming ◽

Centrifugal Instability ◽

Two Dimensional Flow ◽

Finite Slope

An oscillating flow over a sandy beach can initiate and enhance the formation of bed ripples, with crests perpendicular to the direction of the ambient oscillation. Under certain circumstances, bridges may develop to span adjacent ripple crests, resulting in a brick pattern. It has been suggested that the onset of this transition is due to a three-dimensional centrifugal instability of an otherwise two-dimensional flow over periodic long-crested ripples. Here we analyse theoretically such an instability by assuming that the ripples are rigid and smooth. Two complementary cases are studied. We first consider a weak ambient oscillation over ripples of finite slope in Case (i). The three-dimensional disturbance is found to be localized in a small region either along the crests or along the troughs. In Case (ii) we analyse finite oscillations over ripples of mild slope. The region influenced by the instability is now comparable with a ripple wavelength and the unstable disturbance along adjacent ripples may interact with each other. Four types of harmonic and subharmonic instabilities are found. The associated steady streaming close to the ripple surface shows various tendencies of possible sand accumulations, some of which appear to be qualitatively relevant to the initiation of brick-patterned ripples.

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Streaming from a sphere due to a pulsating source

Journal of Fluid Mechanics ◽

10.1017/s0022112090001367 ◽

1990 ◽

Vol 210 ◽

pp. 459-473 ◽

Cited By ~ 16

Author(s):

Norsarahaida Amin ◽

N. Riley

Keyword(s):

Reynolds Number ◽

Point Source ◽

Two Dimensional ◽

Simple Experiment ◽

Steady Streaming ◽

Reynolds Number Flow ◽

Plane Jets ◽

Circular Jets ◽

Two Dimensional Flow ◽

Number Flow

The steady streaming outside the Stokes shear-wave layer, which forms on the surface of a sphere when placed close to an oscillatory point source, is considered. Particular attention is devoted to the case of high streaming Reynolds-number flow. Thin circular jets, analogous to the plane jets known to occur in two-dimensional flow, are predicted and visualized by means of a simple experiment.

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On the evolution of packets of long surface waves

Proceedings of the Royal Society of London Series A - Mathematical and Physical Sciences ◽

10.1098/rspa.1975.0110 ◽

1975 ◽

Vol 344 (1638) ◽

pp. 427-433 ◽

Cited By ~ 34

Keyword(s):

Nonlinear Waves ◽

Small Amplitude ◽

Vries Equation ◽

Three Dimensional ◽

Finite Depth ◽

Two Dimensional ◽

Direction Transverse ◽

Amplitude Parameter ◽

First Approximation ◽

Poisson Type

Three dimensional inviscid nonlinear waves on the surface of water of finite depth are examined in the limit of long waves. It is shown that small amplitude waves having a suitably slow variation in the direction transverse to that of propagation satisfy a two dimensional analogue of the well known Korteweg-de Vries equation when the parameter Δ =ε /h 2 k 2 is finite; where ε is an amplitude parameter, h is the depth and k is the wavenumber. When Δ is small this analogue is reduced, to first approximation, to a scaled form of the nonlinear Schrödinger-Poisson type equations adumbrated by Davey & Stewartson (1974).

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Three-dimensional streaming flow in confined geometries

Journal of Fluid Mechanics ◽

10.1017/jfm.2015.336 ◽

2015 ◽

Vol 777 ◽

pp. 408-429 ◽

Cited By ~ 14

Author(s):

Bhargav Rallabandi ◽

Alvaro Marin ◽

Massimiliano Rossi ◽

Christian J. Kähler ◽

Sascha Hilgenfeldt

Keyword(s):

Time Scales ◽

Vortex Motion ◽

Three Dimensional ◽

Two Dimensional ◽

Hamiltonian Description ◽

Steady Streaming ◽

Dimensional Flow ◽

Time Flow ◽

Two Dimensional Flow ◽

Streaming Flows

Steady streaming vortex flow from microbubbles has been developed into a versatile tool for microfluidic sample manipulation. For ease of manufacture and quantitative control, set-ups have focused on approximately two-dimensional flow geometries based on semi-cylindrical bubbles. The present work demonstrates how the necessary flow confinement perpendicular to the cylinder axis gives rise to non-trivial three-dimensional flow components. This is an important effect in applications such as sorting and micromixing. Using asymptotic theory and numerical integration of fluid trajectories, it is shown that the two-dimensional flow dynamics is modified in two ways: (i) the vortex motion is punctuated by bursts of strong axial displacement near the bubble, on time scales smaller than the vortex period; and (ii) the vortex trajectories drift over time scales much longer than the vortex period, forcing fluid particles onto three-dimensional paths of toroidal topology. Both effects are verified experimentally by quantitative comparison with astigmatism particle tracking velocimetry (APTV) measurements of streaming flows. It is further shown that the long-time flow patterns obey a Hamiltonian description that is applicable to general confined Stokes flows beyond microstreaming.

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Two-Dimensional Flow of Liquid During the Late Stage of Solidification of Alloys

Journal of Applied Mechanics ◽

10.1115/1.3422670 ◽

1972 ◽

Vol 39 (1) ◽

pp. 65-70 ◽

Cited By ~ 6

Author(s):

R. H. Tien

Keyword(s):

Liquid Flow ◽

Interdendritic Liquid ◽

Bubble Formation ◽

External Pressure ◽

Two Dimensional ◽

Strong Function ◽

Cu Alloy ◽

Solid Region ◽

Two Dimensional Flow ◽

Late Stages

This is a mathematical analysis of the two-dimensional interdendritic liquid flow occurring during late stages of solidification, i.e., when the liquidus front reaches the center of the slab. The liquid flow considered is that caused by density change accompanying freezing. The liquid-solid region is considered a porous medium where the liquid permeability is a function of the solid fraction. Analytical solution for the two-dimensional velocity and pressure fields in solidifying Al-4.5 percent Cu alloy is obtained from a mathematical model which includes the concept of conservation of mass and Darcy’s law to correlate the pressure and velocity. Assuming that there is no gas-bubble formation and that the solidified metal in the mushy zone remains rigid, the liquid moving with a creeping velocity in the interdendritic regions is under tension (when no external pressure is applied). The magnitude of this tension increases with increasing depth of the solidifying ingot and is a strong function of the cooling rate. For example, a negative pressure of the order of 100 atm in the interdendritic liquid of solidifying Al-4.5 percent Cu alloy is estimated at ∼90 percent solidification. On the basis of the present analysis, an estimate can be made of pressures required to suppress blow-hole formation during the later stages of freezing arising from the solidification shrinkage.

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