General stability of memory-type thermoelastic Timoshenko beam acting on shear force

In this paper, we consider a linear thermoelastic Timoshenko system with memory effects where the thermoelastic coupling is acting on shear force under Neumann–Dirichlet–Dirichlet boundary conditions. The same system with fully Dirichlet boundary conditions was considered by Messaoudi and Fareh (Nonlinear Anal TMA 74(18):6895–6906, 2011, Acta Math Sci 33(1):23–40, 2013), but they obtained a general stability result which depends on the speeds of wave propagation. In our case, we obtained a general stability result irrespective of the wave speeds of the system.     

History force on a sphere in a weak linear shear flow

A numerical study of history forces acting on a spherical particle in a linear shear flow, over a range of finite Re, is presented. In each of the cases considered, the particle undergoes rapid acceleration from Re₁ to Re₂ over a short-time period. After acceleration, the particle is maintained at Re₂ in order to allow for clean extraction of drag and lift kernels. Good agreement is observed between current drag kernel results and previous investigations. Furthermore, ambient shear is found to have little influence on the drag kernel. The lift kernel is observed to be oscillatory, which translates to a non-monotonic change in lift force to the final steady state. In addition, strong dependence on the start and end conditions of acceleration is observed. Unlike drag, the lift history kernel scales linearly with Reynolds number and shear rate. This behavior is consistent with a short-time inviscid evolution. A simple expression for the lift history kernel is presented.     

The orientation dynamics of small prolate and oblate spheroids in linear shear flows

The present work deals with the stable orientation of oblate and prolate spheroids in general steady linear flows and with the mode of convergence to these stable orientations. The orientation dynamics is governed by the Jeffery equation. The stable orientations are either fixed points or limit cycles in the orientation space. The type of stable orientation depends on whether the eigenvalues of the linear part of Jeffery equation are real or complex. We define prolate and oblate spheroids to be equivalent if the aspect ratio of one is the reciprocal of the other. We show that, in a given flow, equivalent oblate and prolate spheroids possess the same number of fixed points and limit cycles of which only one is stable. If they possess only fixed points, then their corresponding stable fixed points are orthogonal. If they possess one fixed point and one limit cycle each, then the stable fixed point of one is orthogonal to the plane of the limit cycle of the other. The rate of convergence to these attractors is important to consideration of the orientations in time-space varying flow fields. We show that non-normal growth (NNG) of the distance to these attractors may delay the convergence by several characteristic shear time scales. We derive conditions for occurrence of NNG and explicit expressions for the maximal duration of the growth. We consider a specific case of which the vorticity is a stable orientation of prolate spheroids. We analyze the conditions that imply monotonic or, conversely, non-monotonic convergence to this orientation due to NNG. We thereby find the corresponding conditions for convergence of the equivalent oblate spheroids to their attractors, normal to the vorticity. We show that the convergence is monotonic if the vorticity is parallel to the strain tensor’s largest eigenvector, but that NNG occurs if the vorticity is parallel to the strain tensor’s intermediate eigenvector. The NNG duration decreases with increasing vorticity-strain ratio and with the strain intermediate eigenvalue approaching the largest eigenvalue.     

Sliding of fine particles on the slip surface of rising gas bubbles: Resistance of liquid shear flows

In this paper a model was developed to describe the shear flow resistance force and torque acting on a fine particle as it slides on the slip surface of a rising gas bubble. The shear flow close to the bubble surface was predicted using a Taylor series and the numerical data obtained from the Navier–Stokes equations as a function of the polar coordinates at the bubble surface, the bubble Reynolds number, and the gas hold-up. The particle size was considered to be sufficiently small relative to the bubble size that the bubble surface could be locally approximated to a planar interface. The Stokes equation for the disturbance shear flows was solved for the velocity components and pressure using series of bispherical coordinates and the boundary conditions at the no-slip particle surface and the slip bubble surface. The solutions for the disturbance flows were then used to calculate the flow resistance force and torque on the particle as a function of the separation distance between the bubble and particle surfaces. The resistance functions were determined by dividing the actual force and torque by the corresponding (Stokes) force and torque in the bulk phase. Finally, numerical and simplified analytical rational approximate solutions for force correction factors for sliding particles as a function of the (whole range of the) separation distance are presented, which are in good agreement with the exact numerical result and can be readily applied to more general modelling of the bubble–particle interactions.     

Multi-frequency vortex-induced vibrations of a long tensioned beam in linear and exponential shear flows

The multi-frequency vortex-induced vibrations of a cylindrical tensioned beam of aspect ratio 200, free to move in the in-line and cross-flow directions within first a linearly and then an exponentially sheared current are investigated by means of direct numerical simulation, at a Reynolds number equal to 330. The shape of the inflow profile impacts the spectral content of the mixed standing-traveling wave structural responses: narrowband vibrations are excited within the lock-in area, which is limited to a single region lying in the high flow velocity zone, for the linear shear case; in contrast, the lock-in condition occurs at several spanwise locations in the exponential shear case, resulting in broadband responses, containing a wide range of excited frequencies and spatial wavenumbers. The broadband in-line and cross-flow vibrations occurring for the exponential shear current have a phase difference that lies within a specific range along the entire span; this differs from the phase drift noted for narrowband responses in linear shear flow. Lower vibration amplitudes, time-averaged and fluctuating in-line force coefficients are observed for the exponential shear current. The cross-flow force coefficient has comparable magnitude for both inflow profiles along the span, except in zones where the broadband vibrations are under the lock-in condition but not the narrowband ones. As in the narrowband case, the fluid forces associated with the broadband responses are dominated by high frequencies related to high-wavenumber vibration components. Considerable variability of the effective added mass coefficients along the span is noted in both cases.     

Spatially periodic suspensions of convex particles in linear shear flows. II. Rheology

Following the purely kinematical developments of Part 1, a rigorous analysis is presented of the “almost” time-periodic low Reynolds number hydrodynamics of a spatially periodic suspension of identical convex particles in a Newtonian liquid undergoing a macroscopically homogeneous linear shear flow. By considering the case of a single particle within a unit cell of the instantaneous spatially periodic configuration, the quasistatic dynamical analysis of this infinite-particle system is effected in much the same way as for a single particle suspended in an unbounded fluid. This is accomplished via the introduction of a partitioned hydrodynamic Stokes resistance matrix, linearly relating the force, couple and stresslet on the particle in the unit cell to the translational and rotational particle-(mean) suspension slip velocities and the mean rate-of-strain of the suspension. In contrast with the unbounded fluid case for a given geometry of the individual particles, the (purely geometric) elements of the resistance matrix depend upon the instantaneous lattice configuration.These dynamic quasistatic calculations for a given instantaneous lattice conformation, in particular that for the stresslet, are then appropriately averaged over both space and time for the class of almost time-periodic, lattice-reproducing, flows discussed in Part I. (In actually performing the time average, an important distinction is drawn between the ergodic and deterministic shear processes whose kinematical basis was laid in Part I.) In turn, this averaged dynamical information is translated into knowledge of the rheological properties of the macroscopically homogeneous suspension.A rigorous asymptotic, lubrication-theory analysis is performed during the course of an illustrative calculation of the rheological properties of a concentrated suspension of almost-touching spheres in a simple shear flow. Contrary to the findings of a previous heuristic treatment of this same lubrication-theory problem—one that ignores evolutionary variations in the instantaneous geometrical configuration of the spatially periodic suspension as the shear proceeds—the time-average properties of the suspension are found to be nonsingular in the limit.Finally, brief comments are offered on potential extensions of the scheme to include nonlinear phenomena, such as nonNewtonian fluids and inertial effects.     

Vortex structures in shear flows

Large-scale vortex structures in shear flows are investigated. An effective method of describing these objects, which makes it possible to go beyond the framework of weak nonlinearity, is proposed. This is especially important in investigating spatially isolated structures. Evolution-type equations describing the shape of the vortex structures are obtained and their steady-state solutions are examined. A detailed classification of the structures in two-dimensional and cylindrical channels is given. Attention is drawn to the qualitative similarity of some of these structures to the well-known structures in real turbulent flows (wall eddies, turbulent slugs). It is established that isolated vortex structures in a pipe whose radius is fairly large as compared with their transverse dimension have kinematic characteristics similar to those of Hill vortices. The prospects of the method are discussed.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 2, pp. 65–75, March–April, 1989.The author wishes to thank G. S. Golitsyn for his interest and G. I. Barenblatt and V. M. Gryanik for discussing the results and offering useful advice.     

Localized disturbances in parallel shear flows

The development of localized disturbances in parallel shear flows is reviewed. The inviscid case is considered, first for a general velocity profile and then in the special case of plane Couette flow so as to bring out the key asymptotic results in an explicit form. In this context, the distinctive differences between the wave-packet associated with the asymptotic behavior of eigenmodes and the non-dispersive (inviscid) continuous spectrum is highlighted. The largest growth is found for three-dimensional disturbances and occurs in the normal vorticity component. It is due to an algebraic instability associated with the lift-up effect. Comparison is also made between the analytical results and some numerical calculations.Next the viscous case is treated, where the complete solution to the initial value problem is presented for bounded flows using eigenfunction expansions. The asymptotic, wave-packet type behaviour is analyzed using the method of steepest descent and kinematic wave theory. For short times, on the other hand, transient growth can be large, particularly for three-dimensional disturbances. This growth is associated with cancelation of non-orthogonal modes and is the viscous equivalent of the algebraic instability. The maximum transient growth possible to obtain from this mechanism is also presented, the so called optimal growth.Lastly the application of the dynamics of three dimensional disturbances in modeling of coherent structures in turbulent flows is discussed.     

Aerofoil section characteristics in shear flows

Summary The technique of Glauert's image method for a single source is applied to determine the image system of a single vortex in shear flows of arbitrary velocity profile. The aerofoil section characteristics are obtained analytically by the extension of the image system for a single vortex and for a single source to those for vortex and source distributions. Numerical calculations are made and the results show the effect on the aerofoil section characteristics of vorticity in flow fields which have been obtained by combining linear shear flows, by comparison with those obtained in uniform flow.

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Übersicht Das Glauertsche Bildverfahren für eine Einzelquelle wird verwendet, um das Bildsystem eines Einzelwirbels in einer Scherströmung von beliebigem Gesclwindigkeitsprofil zu ermitteln. Die Kennwerte für einen Tragflügelquerschnitt werden analytisch dadurch erhalten, daß das Bildsystem für einen Einzelwirbel und eine Einzelquelle zu einer Verteilung von Wirbeln und Quellen erweitert wird. Die Ergebnisse numerischer Lösungen zeigen die Abhängigkeit der Tragflügelkennwerte von der Verwirbelung in Strömungsfeldern mit linearen Scherströmungen und den Vergleich mit den entsprechenden, in gleichförmigen Strömungen erhaltenen Werten.

     

Spatially periodic suspensions of convex particles in linear shear flows. I. Description and kinematics

Preparatory to a subsequent dynamical study in Part II, aimed at calculating the rheological properties of geometrically-ordered models of concentrated suspensions, a purely kinematical study is here presented of the motion of a mobile spatially periodic array of identical convex particles, typically spheres, participating in a macroscopically homogeneous linear shear flow to which the suspension as a whole is subjected. The geometrical configuration of such particle-lattice suspensions is shown to evolve temporally in a manner dependent upon the initial lattice configuration and the specific bulk shearing motion to which the suspension is subjected. Under certain circumstances the particle-lattice system is found to reproduce itself periodically in time—or, less stringently—“almost” periodically. Precise circumstances under which this occurs are exhaustively delineated for the entire class of two-dimensional isochoric spatially homogeneous shearing motions, parametrized by a scalar λ expressing the relative amounts of shear and vorticity present in the flow. This investigation is performed for both two- and three-dimensional lattices. (Eventual time averaging of the local, instantaneous, dynamical, interstitial fluid properties of these almost self-reproducing systems in Part II furnishes the rheological properties of the suspension.) Using concepts borrowed from Minkowski's geometry of numbers, calculations are outlined for establishing the maximum volume fraction of suspended particles that is kinematically possible for each shearing motion. This is observed to be always less than would obtain in a comparable static system.     

Distinguishing shear banding from shear thinning in flows with a shear stress gradient

Shear banding occurs in complex fluids that exhibit a non-monotonic constitutive instability, such as wormlike micelles, and potentially also in polymeric fluids with presumably monotonic constitutive behavior. However, velocity profiles for shear thinning fluids in geometries possessing a stress gradient, such as Taylor-Couette flow, could be misidentified as shear banding. To address this, we present a model-free experimental procedure to distinguish shear banding from strong shear thinning using high-resolution velocimetry. The approach is developed and validated using simulations using the d-Giesekus model and is based upon the behavior of the width of the apparent interface between the high and low shear rate regions. It is then tested using experimental data for model wormlike micellar solutions. The method allows shear banding to be distinguished from shear thinning in cases where this difference is otherwise indistinguishable. As a by-product, it also provides an estimate of the stress diffusivities for shear banding fluids.     

Lift force exerted on a spherical particle in a laminar boundary layer

A spherical particle moving in an unbounded viscous shear flow is acted upon by a lift force [1, 2] which results from taking the inertial terms into account in the equations of motion. When the particle moves at the bottom of a laminar boundary layer the magnitude of the force differs from that obtained in [1, 2], The problem of determining the lift force exerted on the particle as a function of its distance from the wall has been solved by matched asymptotic expansions. The magnitude of the force is expressed in terms of a multiple integral which can be evaluated numerically.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 5, pp. 66–71, September–October, 1989.In conclusion, the author wishes to thank M. N. Kogan, N. K. Makashev, and A. Yu. Boris for useful discussions.     

Rarefied gas effect in hypersonic shear flows

Recently,as aerodynamics was applied to flying vehicles with very high speed and flying at high altitude,the numerical simulation based on the Navier-Stokes(NS)...     

Coherent structures in countercurrent axisymmetric shear flows

The dynamical behaviors of coherent structures in countercurrent axisymmetric shear flows are experimentally studied.The forward velocity U1 and the velocity ratio R=(U1-U2)/(U1+U2),where U2 denotes the suction velocity,are consldered as the control parameters.Two kinds of vortex structures,i.e.,axisymmetric and helical structures,were discovered with respect to different reginmes in the R versus U1 diagram .In the case of U1 rangjing from 3 to 20m/s and R from 1 to 3,the axisymmetric structures plan an important role.Based on the dynamical behaviors of axisymmetric structures,a critical forward velocity U1cr=6.8m/s was defined,subsequently,the subcritical velocity regime:U1>U1cr and the supercritical velocity regime:U1<U1er,In the subcritical velocity regine,the flow system contains shear layer self-excited oscillations in a certain range of the velocity ratio with respect to any forward velocity.In the supercritical velocity regime,the effect of the velocity ratio could be explained by the relative movement and the spatial evolution of the axisymmetric structure undergoes the following stages:(1) Kelvin-Helmholtz instability leading to vortex rolling up,(2) first time vortex agglomeration.(3) jet colunn self-excited oscillation,(4) shear layer self-excited oscillation,(5)“ordered tearing“,(6) turbulence in the case of U1<4m/s (the “ordered tearing“ does not exist when U1>4m/s),correspondingly,the spatial evolution of the temporal asymptotic behavior of a dynamical system can be described as follows:(1) Hopt bifurcation,(5) chaos(“weak turbulence“)in the case of U1<4m/s(superharmonic bifurcation does not exist when U1>4m/s).The proposed new terms,superharmonic and reversed superbarmonic bifurcations,are characterized of the frequency doubling rather than the period doubling.A kind of unfamiliar vortices referred to as the helical structure was discovered experimentally when the forward velocity around 2m/s and the velocity range from 1.1 to 2.3,There are two base frequencies contained in the flow system and they could coexist as indicated by the Wigner-Ville-Distribution and the temporal asymptotic behavior of the dynamical system corresponding to the helical vortex could be described as 2-torus as indicted by the 3D reconstructed phase trajectory and correlation dimension.The scenario of the spatial evolution of helical structures could be described as follows:the jet column is separated into two parts at a certain spatial location and they entangle each other to form the helical vortex until the occurrence of those separated jet columns to reconnect further downstream with the result that the flow system evolves into turbulence in a catastrophic form.Correspondingly,the dynamical system evolves directly into 2-tiorus through the supercritical Hopf bifurcation followed by a transition from a quasi-periodic attractor to a strange attractor.In the case of U1=2m/s,the parametric evolution of the temporal asymptotic behavior of the dynamical system as the velocity ratio increases from 1 to 3 could be described as follows:(1)2-torus(Hopf bifurcation),(2) limit cycle(reversed Hopf bifurcation),(3) strange attractor (subbarmonic bifurcation).