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.

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.

Some problems concerning the production of a linear shear flow using curved wire-gauze screens

1976 ◽  
Vol 76 (4) ◽  
pp. 689-709 ◽  
Author(s):  
I. P. Castro
Keyword(s):  
Shear Flow ◽  
Incompressible Fluid ◽  
Experimental Work ◽  
Arbitrary Shape ◽  
Practical Problem ◽  
Analytic Solution ◽  
Linear Shear ◽  
Wire Gauze ◽  
The Common ◽  
Small Shear

The flow of an incompressible fluid through a curved wire-gauze screen of arbitrary shape is reconsidered. Some inconsistencies in previously published papers are indicated and the various approximations and linearizations (some of which are necessary for a complete analytic solution) are discussed and their inadequacies demonstrated. Attention is concentrated on the common practical problem of calculating the screen shape required to produce a linear shear flow and experimental work is presented which supports the contention that the theoretical solutions proposed by Elder (1959)–subsequently discussed by Turner (1969) and Livesey & Laws (1973)-and Lau & Baines (1968) are inadequate, although, for the case of small shear, Elder's theory appears to be satisfactory. Since, in addition, there are inevitable difficulties concerning both the value of the deflexion coefficient appropriate to any particular screen and inhomogeneities in the screen itself, it is concluded that the preparation of a curved screen to produce the commonly required moderate to large linear shear flow is bound to be somewhat empirical and should be attempted with caution.

Motion of a spherical microcapsule freely suspended in a linear shear flow

1980 ◽  
Vol 100 (4) ◽  
pp. 831-853 ◽  
Author(s):  
D. Barthès-Biesel
Keyword(s):  
Shear Flow ◽  
Stokes Equations ◽  
Dynamic Equilibrium ◽  
Simple Shear Flow ◽  
Regular Perturbation ◽  
Highly Nonlinear ◽  
Linear Shear ◽  
Freely Suspended ◽  
Newtonian Viscous Fluid ◽  
Limiting Case

The motion of a spherical microcapsule freely suspended in a simple shear flow is studied. The particle consists of a thin elastic spherical membrane enclosing an incompressible Newtonian viscous fluid. The motions of the internal liquid and of the suspending fluid are both described by Stokes equations. On the deformed surface of the membrane, continuity of velocities is imposed together with dynamic equilibrium of viscous and elastic forces. Since this problem is highly nonlinear, a regular perturbation solution is sought in the limiting case where the deviation from sphericity is small. In particular, the nonlinear theory of large deformation of membrane shells is expanded up to second-order terms. The deformation and orientation of the microcapsule are obtained explicitly in terms of the magnitude of the shear rate, the elastic coefficients of the membrane, the ratio of internal to external viscosities. It appears that the very viscous capsules are tilted towards the streamlines, whereas the less viscous particles are oriented at nearly 45° to the streamlines. The tank-treading motion of the membrane around the liquid contents is predicted by the model and appears as the consequence of a solid-body rotation superimposed upon a constant elastic deformation.

scholarly journals Numerical analysis of the angular motion of a neutrally buoyant spheroid in shear flow at small Reynolds numbers

2015 ◽  
Vol 92 (6) ◽  
Author(s):  
T. Rosén ◽  
J. Einarsson ◽  
A. Nordmark ◽  
C. K. Aidun ◽  
F. Lundell ◽  
...  
Keyword(s):  
Numerical Analysis ◽  
Shear Flow ◽  
Reynolds Numbers ◽  
Angular Motion ◽  
Small Reynolds Numbers ◽  
Neutrally Buoyant

Shear-induced incipient motion of a single sphere on uniform substrates at low particle Reynolds numbers

2017 ◽  
Vol 825 ◽  
pp. 284-314 ◽  
Author(s):  
J. R. Agudo ◽  
C. Illigmann ◽  
G. Luzi ◽  
A. Laukart ◽  
A. Delgado ◽  
...  
Keyword(s):  
Shear Flow ◽  
Numerical Study ◽  
Reynolds Numbers ◽  
Angle Of Repose ◽  
External Parameter ◽  
Incipient Motion ◽  
Substrate Topography ◽  
Linear Shear ◽  
Effective Level ◽  
Single Sphere

We study the incipient motion of single spheres in steady shear flow on regular substrates at low particle Reynolds numbers. The substrate consists of a monolayer of regularly arranged fixed beads, in which the spacing between beads is varied resulting in different angles of repose and exposures of the particle to the main flow. The flow-induced forces and the level of flow penetration into the substrate are determined numerically. Since experiments in this range had shown that the critical Shields number is independent of inertia but strongly dependent on the substrate geometry, the particle Reynolds number was fixed to 0.01 in the numerical study. Numerics indicates that rolling motion is always preferred to sliding and that the flow penetration is linearly dependent on the spacing between the substrate particles. Besides, we propose an analytical model for the incipient motion. The model is an extension of Goldman’s classical result for a single sphere near a plain surface taking into account the angle of repose, flow orientation with respect to substrate topography and shielding of the sphere to the linear shear flow. The effective level of flow penetration is the only external parameter. The model, applied to triangular and quadratic arrangements with different spacings, is able to predict the dependence of the critical Shields number on the geometry and on the orientation of the substrate. The model is in very good agreement with numerical results. For well-exposed particles, we observed that the minimum critical Shields number for a certain angle of repose does not depend sensitively on the considered arrangement. At large angles of repose, as expected in fully armoured beds, the model is consistent with experimental results for erodible beds at saturated conditions.

The resistance to a particle of arbitrary shape in translational motion at small Reynolds numbers

1963 ◽  
Vol 17 (4) ◽  
pp. 561-595 ◽  
Author(s):  
H. Brenner ◽  
R. G. Cox
Keyword(s):  
Arbitrary Shape ◽  
Translational Motion ◽  
Reynolds Numbers ◽  
The Body ◽  
Additional Force ◽  
Particle Dimension ◽  
Small Reynolds Numbers ◽  
Maximum Particle ◽  
Arbitrary Body ◽  
Uniform Stream

Assuming that the Stokes flow past an arbitrary particle in a uniform stream is known for any three non-coplanar directions of flow, then the force on the body to O(R), for any direction of flow, is given explicitly in terms of these Stokes velocity fields. The Reynolds number (R) based on the maximum particle dimension is assumed small. For bodies with certain types of symmetry it suffices merely to know the Stokes resistance tensor for the body in order to calculate this force. In this case the resulting formula is identical to that of Brenner (1961) and Chester (1962). However, for bodies devoid of such symmetry, their formula is incomplete—there being an additional force at right angles to the uniform stream which remains invariant under a reversal of the flow at infinity. As this additional force is a lift force, it follows that the Brenner-Chester formula furnishes the correct drag on a body of arbitrary shape; moreover, this drag is always reversed to at least O(R) by a reversal of the uniform flow at infinity.Exactly analogous formulae are derived using the classical Oseen equations, and it is shown that although this gives both the correct vector force on bodies with the above types of symmetry and the correct drag on bodies of arbitrary shape, it gives in general an incorrect lift component for completely arbitrary particles.Finally, the singular perturbation result for the force on an arbitrary body is extended to terms of O(R2log R). This higher-order contribution to the force is given explicitly in terms of the Stokes resistance tensor, and has the property of being reversed by a reversal of the flow at infinity, regardless of the geometry of the body.These results are collected in the Summary at the end of the paper.

Drag and lift forces on a rotating sphere in a linear shear flow

1999 ◽  
Vol 384 ◽  
pp. 183-206 ◽  
Author(s):  
RYOICHI KUROSE ◽  
SATORU KOMORI
Keyword(s):  
Shear Flow ◽  
Lift Force ◽  
Fluid Velocity ◽  
Reynolds Numbers ◽  
Fluid Shear ◽  
Rotating Sphere ◽  
High Particle ◽  
Linear Shear ◽  
Lift Forces ◽  
Drag And Lift

The drag and lift forces acting on a rotating rigid sphere in a homogeneous linear shear flow are numerically studied by means of a three-dimensional numerical simulation. The effects of both the fluid shear and rotational speed of the sphere on the drag and lift forces are estimated for particle Reynolds numbers of 1[les ]Rep[les ]500.The results show that the drag forces both on a stationary sphere in a linear shear flow and on a rotating sphere in a uniform unsheared flow increase with increasing the fluid shear and rotational speed. The lift force on a stationary sphere in a linear shear flow acts from the low-fluid-velocity side to the high-fluid-velocity side for low particle Reynolds numbers of Rep<60, whereas it acts from the high-velocity side to the low-velocity side for high particle Reynolds numbers of Rep>60. The change of the direction of the lift force can be explained well by considering the contributions of pressure and viscous forces to the total lift in terms of flow separation. The predicted direction of the lift force for high particle Reynolds numbers is also examined through a visualization experiment of an iron particle falling in a linear shear flow of a glycerin solution. On the other hand, the lift force on a rotating sphere in a uniform unsheared flow acts in the same direction independent of particle Reynolds numbers. Approximate expressions for the drag and lift coefficients for a rotating sphere in a linear shear flow are proposed over the wide range of 1[les ]Rep[les ]500.

The lift force on a spherical bubble in a viscous linear shear flow

1998 ◽  
Vol 368 ◽  
pp. 81-126 ◽  
Author(s):  
DOMINIQUE LEGENDRE ◽  
JACQUES MAGNAUDET
Keyword(s):  
Reynolds Number ◽  
Shear Rate ◽  
Shear Flow ◽  
Lift Force ◽  
Lift Coefficient ◽  
Reynolds Numbers ◽  
Bubble Surface ◽  
Vorticity Field ◽  
Linear Shear ◽  
High Reynolds

The three-dimensional flow around a spherical bubble moving steadily in a viscous linear shear flow is studied numerically by solving the full Navier–Stokes equations. The bubble surface is assumed to be clean so that the outer flow obeys a zero-shear-stress condition and does not induce any rotation of the bubble. The main goal of the present study is to provide a complete description of the lift force experienced by the bubble and of the mechanisms responsible for this force over a wide range of Reynolds number (0.1[les ]Re[les ]500, Re being based on the bubble diameter) and shear rate (0[les ]Sr[les ]1, Sr being the ratio between the velocity difference across the bubble and the relative velocity). For that purpose the structure of the flow field, the influence of the Reynolds number on the streamwise vorticity field and the distribution of the tangential velocities at the surface of the bubble are first studied in detail. It is shown that the latter distribution which plays a central role in the production of the lift force is dramatically dependent on viscous effects. The numerical results concerning the lift coefficient reveal very different behaviours at low and high Reynolds numbers. These two asymptotic regimes shed light on the respective roles played by the vorticity produced at the bubble surface and by that contained in the undisturbed flow. At low Reynolds number it is found that the lift coefficient depends strongly on both the Reynolds number and the shear rate. In contrast, for moderate to high Reynolds numbers these dependences are found to be very weak. The numerical values obtained for the lift coefficient agree very well with available asymptotic results in the low- and high-Reynolds-number limits. The range of validity of these asymptotic solutions is specified by varying the characteristic parameters of the problem and examining the corresponding evolution of the lift coefficient. The numerical results are also used for obtaining empirical correlations useful for practical calculations at finite Reynolds number. The transient behaviour of the lift force is then examined. It is found that, starting from the undisturbed flow, the value of the lift force at short time differs from its steady value, even when the Reynolds number is high, because the vorticity field needs a finite time to reach its steady distribution. This finding is confirmed by an analytical derivation of the initial value of the lift coefficient in an inviscid shear flow. Finally, a specific investigation of the evolution of the lift and drag coefficients with the shear rate at high Reynolds number is carried out. It is found that when the shear rate becomes large, i.e. Sr=O(1), a small but consistent decrease of the lift coefficient occurs while a very significant increase of the drag coefficient, essentially produced by the modifications of the pressure distribution, is observed. Some of the foregoing results are used to show that the well-known equality between the added mass coefficient and the lift coefficient holds only in the limit of weak shears and nearly steady flows.

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