Far-field disturbance flow induced by a small non-neutrally buoyant sphere in a linear shear flow

2010 ◽  
Vol 643 ◽  
pp. 449-470 ◽  
Author(s):  
EVGENY S. ASMOLOV ◽  
FRANÇOIS FEUILLEBOIS
Keyword(s):  
Shear Flow ◽  
The Other ◽  
Small Reynolds Number ◽  
Small Sphere ◽  
Cylindrically Symmetric ◽  
Linear Shear ◽  
Longitudinal Coordinate ◽  
Field Disturbance ◽  
Neutrally Buoyant ◽  
Rate Gradient

The disturbance flow due to the motion of a small sphere parallel to the streamlines of an unbounded linear shear flow is evaluated at small Reynolds number using the method of matched expansions. Decaying laws are obtained for all velocity components in a far inviscid region and viscous wakes. The z component (in the direction of the shear-rate gradient) of the disturbance velocity is cylindrically symmetric in the inviscid region. It decays with the distance r from the sphere like r−5/3, while the y component (in the direction of vorticity) decays like r−4/3. The widths of two viscous wakes, located upstream and downstream of the sphere, grow with the longitudinal coordinate x as yw ~ zw ~ |x|1/3. The maximum x and z components of the velocity are located in the wake cores; they scale like |x|−2/3 and |x|−1 respectively. For two particles interacting through their outer regions, the migration velocity of each particle is the sum of the velocity of an isolated particle and of a disturbance velocity induced by the other one. Particles placed in the normal or transversal directions repel each other. When each particle is located in a wake of the other one, they may either attract or repel each other.

A note on the rate of heat or mass transfer from a small particle freely suspended in a linear shear field

1980 ◽  
Vol 98 (2) ◽  
pp. 299-304 ◽  
Author(s):  
Andreas Acrivos
Keyword(s):  
Mass Transfer ◽  
Nusselt Number ◽  
Shear Flow ◽  
Small Particle ◽  
Arbitrary Shape ◽  
Reynolds Numbers ◽  
Small Sphere ◽  
Linear Shear ◽  
Small Reynolds Numbers ◽  
Freely Suspended

It is shown that for a small sphere freely suspended in a linear shear flow at small Reynolds numbers, the Nusselt number N is given by $N = \{1 - \alpha P^{\frac{1}{2}} + o(P^{\frac{3}{2}})\}^{-1}$, where P is the Péclet number. For any given type of shear flow, the numerical value of the constant α can be obtained from a general expression derived by Batchelor (1979). The corresponding result for a particle of arbitrary shape is N/N0 = {1 − αN0P½ + O(P3/2)}−1, where N0 is the Nusselt number for pure conduction.

The lift on a small sphere in wall-bounded linear shear flows

1993 ◽  
Vol 246 ◽  
pp. 249-265 ◽  
Author(s):  
John B. McLaughlin
Keyword(s):  
Shear Flow ◽  
Closed Form ◽  
Lift Force ◽  
Closed Form Solution ◽  
Analytical Form ◽  
Form Solution ◽  
Rigid Sphere ◽  
Small Sphere ◽  
Bounded Linear ◽  
Linear Shear

This paper presents a closed-form solution for the inertial lift force acting on a small rigid sphere that translates parallel to a flat wall in a linear shear flow. The results provide connections between results derived by other workers for various limiting cases. An analytical form for the lift force is derived in the limit of large separations. Some new results are presented for the disturbance flow created by a small rigid sphere translating through an unbounded linear shear flow.

scholarly journals Time-evolving bubbles in two-dimensional Stokes flow

1995 ◽  
Vol 301 ◽  
pp. 325-344 ◽  
Author(s):  
Saleh Tanveer ◽  
Giovani L. Vasconcelos
Keyword(s):  
Surface Tension ◽  
Shear Flow ◽  
Exact Solutions ◽  
Stokes Flow ◽  
Mathematical Structure ◽  
The Other ◽  
Two Dimensional ◽  
Slow Viscous Flow ◽  
Expanding Circle ◽  
Linear Shear

A general class of exact solutions is presented for a time-evolving bubble in a two-dimensional slow viscous flow in the presence of surface tension. These solutions can describe a bubble in a linear shear flow as well as an expanding or contracting bubble in an otherwise quiescent flow. In the case of expanding bubbles, the solutions have a simple behaviour in the sense that for essentially arbitrary initial shapes the bubble its asymptote is expanding circle. Contracting bubbles, on the other hand, can develop narrow structures (‘near-cusps’) on the interface and may undergo ‘breakup’ before all the bubble fluid is completely removed. The mathematical structure underlying the existence of these exact solutions is also investigated.

A note on the instabilities of a horizontal shear flow with a free surface

2000 ◽  
Vol 406 ◽  
pp. 337-346 ◽  
Author(s):  
L. ENGEVIK
Keyword(s):  
Surface Tension ◽  
Free Surface ◽  
Shear Flow ◽  
Velocity Profile ◽  
Froude Number ◽  
The Other ◽  
Algebraic Equations ◽  
Horizontal Shear ◽  
Analytical Treatment ◽  
Unstable Region

The instabilities of a free surface shear flow are considered, with special emphasis on the shear flow with the velocity profile U* = U*0sech2 (by*). This velocity profile, which is found to model very well the shear flow in the wake of a hydrofoil, has been focused on in previous studies, for instance by Dimas & Triantyfallou who made a purely numerical investigation of this problem, and by Longuet-Higgins who simplified the problem by approximating the velocity profile with a piecewise-linear profile to make it amenable to an analytical treatment. However, none has so far recognized that this problem in fact has a very simple solution which can be found analytically; that is, the stability boundaries, i.e. the boundaries between the stable and the unstable regions in the wavenumber (k)–Froude number (F)-plane, are given by simple algebraic equations in k and F. This applies also when surface tension is included. With no surface tension present there exist two distinct regimes of unstable waves for all values of the Froude number F > 0. If 0 < F [Lt ] 1, then one of the regimes is given by 0 < k < (1 − F2/6), the other by F−2 < k < 9F−2, which is a very extended region on the k-axis. When F [Gt ] 1 there is one small unstable region close to k = 0, i.e. 0 < k < 9/(4F2), the other unstable region being (3/2)1/2F−1 < k < 2 + 27/(8F2). When surface tension is included there may be one, two or even three distinct regimes of unstable modes depending on the value of the Froude number. For small F there is only one instability region, for intermediate values of F there are two regimes of unstable modes, and when F is large enough there are three distinct instability regions.

scholarly journals Fluid-Structure Energy Transfer of a Tensioned Beam Subject to Vortex-Induced Vibrations in Shear Flow

2011 ◽  
Author(s):  
Remi Bourguet ◽  
Michael S. Triantafyllou ◽  
Michael Tognarelli ◽  
Pierre Beynet
Keyword(s):  
Energy Transfer ◽  
Shear Flow ◽  
Cross Flow ◽  
Reynolds Numbers ◽  
Structural Vibrations ◽  
Fluid Structure ◽  
Vortex Induced Vibrations ◽  
Linear Shear ◽  
Structural Responses ◽  
Lock In

The fluid-structure energy transfer of a tensioned beam of length to diameter ratio 200, subject to vortex-induced vibrations in linear shear flow, is investigated by means of direct numerical simulation at three Reynolds numbers, from 110 to 1,100. In both the in-line and cross-flow directions, the high-wavenumber structural responses are characterized by mixed standing-traveling wave patterns. The spanwise zones where the flow provides energy to excite the structural vibrations are located mainly within the region of high current where the lock-in condition is established, i.e. where vortex shedding and cross-flow vibration frequencies coincide. However, the energy input is not uniform across the entire lock-in region. This can be related to observed changes from counterclockwise to clockwise structural orbits. The energy transfer is also impacted by the possible occurrence of multi-frequency vibrations.

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