Influence of rotatory inertia and transverse shear on stochastic instability of the cross-ply laminated beam

The stochastic instability problem associated with an axially loaded cross-ply laminated beam is formulated. The effects of shear deformation and rotatory inertia are included in the present formulations. The beam is subjected to time-dependent deterministic and stochastic forces. By using the direct Liapunov method, bounds for the almost sure instability of beams as a function of viscous damping coefficient, variance of the stochastic force, ratio of principal lamina stiffnesses, shear correction factor, number of layers, mode numbers and geometrical ratio, are obtained. Numerical calculations are performed for the Gaussian process with a zero mean and variance σ² as well as for harmonic process with an amplitude A.     

A problem of unsymmetrical bending of shear-deformable circular ring plates

Summary The classical solution of the problem of a circular ring plate which is clamped along its outer edge and loaded along its inner edge by the rotation about a diameter of a rigid circular inclusion is generalized so as to account for the effect of transverse shear deformability. Numerical results for overall stiffness, for bending and twisting moments and for transverse shear stress resultants are obtained for several values of the ratio of outer radius r ₀to inner radius r _ias a function of a transverse shear deformability parameter which for a transversely homogenous plate of thickness h is of the form

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Zur unsymmetrischen Biegung schubverformbarer Kreisringplatten

übersicht Die klassische Lösung des Problems der Kreisringplatte, die an ihrem Äu\eren Rande eingespannt ist und deren innerer Rand belastet ist durch die Rotation um einen Durchmesser von einer starren kreisförmigen Einschlie\ung, wird verallgemeinert durch die Beachtung des Einflusses der transversalen Schubverformbarkeit. Numerische Resultate werden erhalten für die Steifigkeit des Systems, für Biege- und Torsionsmomente, und für QuerkrÄfte, für verschiedene Werte des VerhÄltnisses von Au\enradius r ₀zu Innenradius r _i,als Funktion eines Schubverformbarkeitsparameters . Dieser Parameter hat für den Fall einer transversal homogenen Platte von der Dicke h die Form .

     

A solution of the E-ε model for nearly homogeneous turbulence with a mean shear

An analytical solution of the E- model for the downstream evolution of a stationary and nearly homogeneous turbulent shear flow is presented. In case that the turbulent time scale has adjusted itself to the time scale imposed by the shear, an asymptotic solution can be derived from the full solution, which shows that both E and increase downstream exponentially. By comparing this asymptotic solution with experimental data a value for the unknown constant c _l , in the -equation, is derived. Moreover, we find an expression for the downstream development of the variance of a scalar, which is also compared with experimental data. The analytical solution shows that a homogeneous shear flow with a uniform velocity gradient can only be obtained if the shear is sufficiently small. In the experiments this condition is not always satisfied. A discussion is given on how a nearly homogeneous shear flow can be obtained over a limited downstream interval by changing the initial conditions in E and , and a comparison is made with experimental data. Finally it is shown that better transverse homogeneity can be obtained by taking an exponential velocity profile instead of a linear profile.     

Hall effects on the magnetohydrodynamic shear flow past an infinite porous flat plate subjected to uniform suction or blowing

An analysis is made of Hall effects on the steady shear flow of a viscous incompressible electrically conducting fluid past an infinite porous plate in the presence of a uniform transverse magnetic field. It is shown that for suction at the plate, steady shear flow solution exists only when S²<Q, where S and Q are the suction and magnetic parameters, respectively. The primary flow velocity decreases with increase in Hall parameter m. But the cross-flow velocity first increases and then decreases with increase in m. Similar results are obtained for variation of the induced magnetic field with m. It is further found that for blowing at the plate, steady shear flow solution exists only when , where S₁ is the blowing parameter.     

Transport of a Passive Scalar in a Stratified Porous Medium

A uniform and horizontal head gradient J is applied to a stratified formation whose given random conductivity K is function of the vertical coordinate x ₃ only. K is assumed to be stationary and of finite integral scale I _v. By Darcy's law, the velocity field V ₁(x ₃)=JK depicts a fluctuating shear flow. A solute body is injected instantaneously in the formation. In a Lagrangean framework, the second spatial moment of the mean concentration C(x,t) can be related to the one-particle trajectories variance X ₁₁(t,Pe) where Pe = V₁I_v/D _dT and _dT is the transverse pore-scale dispersion coefficient. X ₁₁ was determined in the past by Matheron and de Marsily (1980). The present study is concerned with determining the local concentration variance _C ² , that depends on the two-particles trajectories covariance Z ₁₁(t). The latter is derived exactly and langle Crangle and _C ² are determined by assuming normal or lognormal probability distribution of trajectories. The results are illustrated for small and very large (ergodic) solute plumes. For large travel time the concentration coefficient of variation at the center of the plume tends asymptotically to a constant value, unlike formations with finite horizontal correlation length of the hydraulic conductivity. The results may serve for benchmarking of numerical codes and in applications for short travel distances in highly anisotropic formations.     

Generation and manipulation of uniform shear with the use of screens

A procedure for the generation of uniformly sheared turbulent flow in the wind-tunnel using separate channels with adjustable blockage is described in detail. Measurements of the reduction of mean shear in a flow crossing transverse grids and gauzes with uniform solidities are also reported: for small flow disturbances, i.e. small relative screen element thickness, d/h, the shear reduction factor, A, was found comparable to predictions, A p; however, for d/h > 0.01, the ratio A/A _p decreased rapidly with increasing d/h. Results on the effect of screen insertion on the upstream shear and the combined effects of multiple screens are briefly discussed.List of symbols A measured shear reduction factor - A _P predicted shear reduction factor - A _p predicted shear reduction factor at large Reynolds numbers - A _prod product of individual grid reduction factors - d rod diameter; screen element thickness - h height of test section - K pressure loss coefficient - K pressure loss coefficient at large Reynolds numbers - M screen spacing; mesh size - P pressure drop across screen - R Reynolds number based on screen element thickness - U ₁ far upstream mean velocity - U ₁ ^* far downstream mean velocity - U _c centreline mean velocity - U _ch mean velocity in central channel - U _ref reference upstream velocity - x ₁ co-ordinate axes, i = 1, 2, 3 - deflection coefficient - shear parameter - fluid density - screen solidity W. G. Rose, In Memoriam.     

The effect of leaching on mixing in porous media with a single heterogeneity

This paper presents an exploratory study of the effect of leaching on mixing in a porous medium containing a single heterogeneity to investigate the effect of the heterogeneity and time-dependent pore structure on dispersion. A percolation-convection simulation (PCS) model is used along with laboratory model experiments to study the mixing. The results show that mixing changes when the pores of the models are leached and that there is a change in regime influence during leaching. The simulation represents the mixing through a first leach for homogeneous media and for heterogeneous media with significant changes in permeability. If the pore structure is changing with time, prediction of mixing must include effects of heterogeneity and regime influence. Although the experimental results are representative of idealized laboratory sized systems they provide insight into the effects of leaching in heterogeneous media. Further the simulation may be useful on a field scale.Nomenclature b molecular weight, gm/mol - C concentration, mol/cm³ - C ₀ initial concentration, mol/cm³ - d _rms root-mean-squared distance, cm - d ₅₀ 50% grain size, cm - D channel depth, cm - f _n fraction of input tracer in effluent at time t _n - K ₁ permeability of flow field outside of heterogeneity, cm² - k ₂ permeability of heterogeneity, cm² - k _S reaction rate constant, cm/min - K _L microscopic dispersion coefficient in the longitudinal direction, cm²/sec - K _O overall dispersion coefficient in the longitudinal direction, cm²/sec - K _T microscopic dispersion coefficient in the transverse direction, cm²/sec - L length of channel, cm - n exponent for velocity - P pressure, N/M² - Pe Peclet number, Lv/K _O - P _ext local pressure outside heterogeneity, N/M² - P _int local pressure inside heterogeneity, N/M² - Q volumetric flow rate, cm³/sec - R channel half width, cm - t time, sec - W _c channel width, cm - W _c0 initial channel width, cm - v interstitial fluid velocity, cm/sec - v _k macroscopic velocity in transverse direction, cm/sec - v _y macroscopic velocity in longitudinal direction, cm/sec - v fluid velocity entering the medium, cm/sec - x _i transverse location of parcel at time t _i, cm - y _i longitudinal location of parcel at time t _i, cm - x microscopic movement in transverse direction, cm - y microscopic movement in longitudinal direction, cm Greek Letters t time increment, sec - ₀ overall dispersivity, cm - ² longitudinal variance of the distribution, cm² - porosity - _B bulk density, gm/cm³ - fractional grade of leachable material Currently with Center for Naval Analysis.     

Influence of superimposed steady shear flow on the dynamic properties of non-Newtonian fluids

Summary Experiments in which an oscillatory shear flow is superimposed on a steady-state circular shear flow between a cone and a plate were performed on non-Newtonian solutions by means of aWeissenberg Rheogoniometer. The steady-state shear stress and in a first approximation also the normal stress difference arising from the steady shear flow appear not to be influenced by the superimposed oscillatory flow. On the other hand, the dynamic moduli as obtained from the oscillatory parts of shear stress and shear flow are highly dependent on the superimposed steady rate of shear. The absolute value of the complex shear modulus decreases and the phase difference between oscillatory shear stress and shear flow increases in all cases and for all frequencies if the superimposed shear rate is increased. Consequently, this phase difference can become equal to and even larger than /2. Between the angular frequency ₀ at which the phase difference is /2 and the steady shear rateq the relation ₀= 1/2,q was experimentally found to exist in most cases. These dynamic results cannot be described by the current theories of viscoelasticity. The large and fast deformations imposed on the material should explicitly be taken into account.     

The effect of different X-wire calibration schemes on some turbulence statistics

The effective-angle and look-up-table calibrations are used to reduce X-wire datasets in two flows: decaying grid turbulence at R50 and the far field of a circular jet at R400 with a mean streamwise velocity of nearly 6 ms^–1 in each case. A static check, which involves mean values of the velocity components, indicates that the effective-angle calibration incorrectly reduces the range of the estimated velocity components and is therefore likely to be less reliable than the look-up-table approach. The two methods produce significantly different results for both small- and large-scale statistics of the streamwise and, more especially, transverse velocity fluctuations. For grid turbulence, the mean-squared temporal derivatives of the streamwise and transverse velocities are larger by 3% and 19%, respectively, when estimated with the look-up-table method compared to the effective-angle calibration. The increase in the mean-squared transverse velocity derivative yields results in closer agreement with isotropy. At velocities below 6 ms^–1, the effective-angle is a quantity that strongly depends on the velocity magnitude and direction.     

The stokes problems for a conducting fluid with a suspension of particles

The Stokes problems of an incompressible, viscous, conducting fluid with embedded small spherical particles over an infinite plate, set into motion in its plane by impulse and by oscillation, in the presence of a transverse magnetic field, are studied. The velocities of the fluid and of the particles and the wall shear stress are obtained. The stress is found to increase due to the particles and the magnetic field, with the effect of the particles diminishing as the field strength is increased.Nomenclature H ₀ strength of the imposed magnetic field - k density ratio of particles to fluid (per unit volume of flow field) - m _e ² H ₀ ² / - t time - y co-ordinate normal to the plate - u fluid velocity - v particle velocity - _e magnetic permeability of the fluid - kinematic viscosity of the fluid - electric conductivity of the fluid - fluid density - particle relaxation time - frequency of oscillation of the plate     

Rheological properties of liver actin solutions

G-actin prepared from pork liver and purified to give a product of 95–98% purity was polymerized to F-actin in a rotational dynamic viscometer. The rheological properties were investigated during the course of polymerisation and after polymerisation. G andG measured at frequencies around 0.1 Hz increased strongly during the first 3 hours of the reaction and then slowly approached a constant value in the range of 10–50 mPa. The rise during polymerisation was more than two orders of magnitude. When a solution of F-actin was subjected to steady shear at a high shear rate for a short period of time and subsequently dynamic measurements were performed, then a considerably smallerG (about 70% smaller) was observed immediately after cessation of the steady shear.G then increased with time and approached its initial value.When measuring the viscosity of F-actin solutions as a function of shear rate, a strong shear thinning effect was seen which did not vanish even at shear rates as low as 0.001 s^–1. Even at this low shear rates, a pronounced yield maximum was observed before reaching the steady state. Oscillatory experiments showed a remarkably weak frequency dependence ofG. The results imply that F-actin solutions are largely structured forming a weak temporary network which can be easily destroyed by application of high shear rates. It seems most likely that the destruction occurs by a reversible, shear induced depolymerisation process.     

Three views of viscoelasticity for Cox–Merz materials

A slight rearrangement of the classical Cox and Merz rule suggests that the shear stress value of steady shear flow, , and complex modulus value of small amplitude oscillatory shear, G^∗(ω) = (G^′2 + G^″2)^1/2, are equivalent in many respects. Small changes of material structure, which express themselves most sensitively in the steady shear stress, τ, show equally pronounced in linear viscoelastic data when plotting these with G^∗ as one of the variables. An example is given to demonstrate this phenomenon: viscosity data that cover about three decades in frequency get stretched out over about nine decades in G^∗ while maintaining steep gradients in a transition region. This suggests a more effective way of exploiting the Cox–Merz rule when it is valid and exploring reasons for lack of validity when it is not. The τ −G^∗ equivalence could also further the understanding of the steady shear normal stress function as proposed by Laun.     

On-line measurement of elasticity and viscosity in flowing polymeric liquids

A new slit-die rheometer (the Stressmeter) for on-line and sample measurement of the viscosity, , and the first normal stress difference, N ₁, in steady shear flow for molten polymers and other high-viscosity liquids is described. Two liquid-filled transverse slots, located in one die wall near the center station, give pressures P ₂ and P ₃ from whose difference the wall shear stress is calculated. In the other die wall at a location opposite the center of the P ₂ slot is a flush-mounted transducer, giving a pressure P ₁. N ₁ is calculated from the hole pressure P ^* = P ₁–P ₂. A metering pump, used to measure the flow rate Q, is supplied with melt either from an extruder (online mode) or from a pressurized sample cylinder (sample mode). The wall shear rate is calculated from Q and ; the Weissenberg-Rabinowitsch correction and a new small-viscous-heating-correction algorithm (affecting ) are used. Viscous heating corrections are small; entrance and exit errors are negligible. The instrument is tested by comparing its results with those obtained from cone-plate and capillary rheometers. Measurement ranges extend to = 200 kPa, = 3000 s^–1, and temperature = 250°C.Dedicated to Prof. Dr. J. Meissner on the occasion of his retirement from the chair of Polymer Physics at the Eidgenössische Technische Hochschule (ETH) Zürich, Switzerland     

Displacement effect of square-ended pitot tubes in shear flows

Measurements in uniformly sheared flows indicate that the displacement of total-pressure tube readings due to shear is roughly constant, even for values of the shear parameter smaller than previously believed.List of symbols D tube outer diameter - d tube inner diameter - h wind-tunnel height - K tube shear parameter - K _r reference tube shear parameter - L characteristic scale of turbulence - P pressure - P _D total pressure indicated by tube of diameter D - P ₀ free stream total pressure - P _r total pressure indicated by reference tube - P _s pressure indicated by static pressure tube - U mean velocity - U _c centerline mean velocity - U _D mean velocity indicated by tube of diameter D - U _r mean velocity indicated by reference tube - u r.m.s. velocity - y transverse coordinate - empirical coefficient - wind-tunnel shear parameter - displacement - fluid density     

Numerical analysis of the transient conjugated heat transfer in a circular duct with a power-law fluid

We treat numerically in this paper, the transient analysis of a conjugated heat transfer process in the thermal entrance region of a circular tube with a fully developed laminar power-law fluid flow. We apply the quasi-steady approximation for the power-law fluid, identifying the suitable time scales of the process. Thus, the energy equation in the fluids is solved analytically using the well-known integral boundary layer technique. This solution is coupled to the transient energy equation for the solid where the transverse and longitudinal heat conduction effects are taken into account. The numerical results for the temporal evolution of the average temperature of the tube wall, _av, is plotted for different nondimensional parameters such as conduction parameter, , the aspect ratios of the tube, and ₀ and the index of power-law fluid, n.     

Rupture of pastes under shear

Summary The flow behaviour of pastes with high concentration of solids (80% by weight and more) was examined in the range of shear rates between 1 and 600 sec^–1. Apart from showing a thixotropic behaviour, some of the pastes responded by an irreversible rupture when a critical shear rate was exceeded.

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Zusammenfassung Das Fließverhalten von Pasten mit einer hohen Konzentration an fester Phase (80% und mehr) wurde in einem Schergeschwindigkeitsbereich zwischen 1 und 600 sec^–1 untersucht. Es wurde gefunden, daß manche Pasten außer Thixotropie auch ein irreversibles Bruchverhalten zeigen, wenn eine kritische Schergeschwindigkeit überschritten wird.

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M torque - M _i initial torque - M _e equilibrium or final torque - M _m maximum recoverable torque - H gap between the plates - R radius of parallel plate - t time - t _e time to reach the equilibrium torque - shear rate (s^–1) - shear rate at the radiusR - shear stress - _R shear stress at the radiusR - angular velocity On leave from the University of Naples, Istituto di Principi di Ingegneria Chimica, Naples, Italy.With 7 figures and 1 table     

An alternative approach to the linear theory of viscoelasticity and some characteristic effects being distinctive of the type of material

In the introduction some postulates on which the linear theory of viscoelasticity is based are recalled, and the postulate of passivity is substituted by a stronger postulate called detailed passivity.Next, a symmetric formulation of this theory is presented which is founded in a well-balanced way on the limiting properties of elasticity and viscosity. This leads to the introduction of the basic functions of creep compliance J ⁺(t) and stressing viscosity ⁺(t) associated to one another, whereas the basic functions retardation fluidity ⁺(t) and relaxation modulus G ⁺(t) emerge as their time derivatives. Correspondingly, four complex basic functions are defined as their Carson transforms.In addition to the proper retardation and relaxation terms, these basic functions contain the non-disappearing constants of either instantaneous compliance J ₀ or instantaneous viscosity ₀ and also of either ultimate fluidity or ultimate modulus G . Therefrom ensues a classification of linear viscoelastic materials into four types: instantaneous elasticity or viscosity is allowed to combine with ultimate viscosity or elasticity. The latter alternative, signifying fluidlike or solidlike materials, leads, of course, to a quite different behavior in many situations; however, remarkable distinctive features are associated to the first one as well.A few respective examples are outlined: 1) propagation of shear waves in a half-space with periodic and step-shaped excitation, 2) dissipation of work in a torsional vibration damper, and 3) shear flow between two parallel porous plates with injection and suction.Finally, materials with viscous initial behavior are defended against the notion that they be of no or almost no real significance.Delivered as a Plenary Lecture at the Fourth European Rheology Conference, Seville (Spain), 4–9 September 1994. The herein only outlined topics are taken from a recently pulished monograph (Geisekus, 1994) in which complete derivations of the results and more detailed discussions are given.Dedicated to Professor K. Walters on the occasion of his 60th birthday.     

Flow-induced oscillation of two circular cylinders in tandem arrangement at low Re

Results are presented for flow-induced vibrations of a pair of equal-sized circular cylinders of low nondimensional mass (m^*=10) in a tandem arrangement. The cylinders are free to oscillate both in streamwise and transverse directions. The Reynolds number, based on the free-stream speed and the diameter of the cylinders, D is 100 and the centre-to-centre distance between the cylinders is 5.5D. The computations are carried out for reduced velocities in the range 2≤U^*≤15. The structural damping is set to zero for enabling maximum amplitudes of oscillation. A stabilized finite element method is utilized to carry out the computations in two dimensions. Even though the response of the upstream cylinder is found to be qualitatively similar to that of an isolated cylinder, the presence of a downstream cylinder is found to have significant effect on the behaviour of the upstream cylinder. The downstream cylinder undergoes very large amplitude of oscillations in both transverse and streamwise directions. The maximum amplitude of transverse response of the downstream cylinder is quite similar to that of a single cylinder at higher Re beyond the laminar regime. Lock-in and hysteresis are observed for both upstream and downstream cylinders. The downstream cylinder undergoes large amplitude oscillations even beyond the lock-in state. The phase between transverse oscillations and lift force suffers a 180 jump for both the cylinders almost in the middle of the synchronization regime. The phase between the transverse response of the two cylinders is also studied. Complex flow patterns are observed in the wake of the freely vibrating cylinders. Based on the phase difference and the flow patterns, the entire flow range is divided into five sub-regions.     

Measurement of the Coefficient of Transverse Dispersion in Flow Through Packed Beds for a Wide Range of Values of the Schmidt Number

Experimental values of the coefficient of transverse dispersion (D _T) were measured with the system 2-naphthol/water, over a range of temperatures between 293K and 373K, which corresponds to a range of values of viscosity () between 2.83×10^–4 Ns/m² and 1.01×10^–3 Ns/m² and of molecular diffusion coefficient (D _m) between 1.03×10^–9 m²/s and 5.49×10^–9 m²/s. Since the density () of water is close to 10³ kg/m³, the corresponding variation of the Schmidt number (Sc=/D _m) was in the range 1000 – 50. More than 200 experimental values of the transverse dispersion coefficient were obtained using beds of silica sand with average particle sizes (d) of 0.297 and 0.496mm, operated over a range of interstitial liquid velocities (u) between 0.1mm/s and 14mm/s and this gave a variation of the Reynolds number (Re=du/) between 0.01 and 3.5.Plots of the dimensionless coefficient of transverse dispersion (D _T/D _m) vs. the Peclet number (Pe_m=ud/D _m) based on molecular diffusion bring into evidence the influence of Sc on transverse dispersion. As the temperature is increased, the value of Sc decreases and the values of D _T/D _m gradually approach the line corresponding to gas behaviour (i.e. Sc 1), which is known to be well approximated by the equation D _T/D _m=1/+ud/12D _m, where is the tortuosity with regard to diffusion.     

An empirical equation of the relative viscosity of polymer melts filled with various inorganic fillers

Summary Based on Maron-Pierce's equation, an empirical equation was suggested, which relates the relative viscosity ( _r ) of the polymer melt filled with various inorganic filler, such as glass fiber, carbon fiber, talc, precipitated- and natural-calcium carbonate powder, and glassy small sphere, to the volume fraction () of the filler. The equation is _r = (1 –/A)^–2, whereA is a parameter relating to the packing geometry of the filler, which is similar to the parameter ₀ in Maron-Pierce's equation. In the equation _r is defined as the ratio of the viscosity of the filledsystem to that of the medium at the same shear stress not the shear rate. The applicability of the equation is above the shear stress about 10⁴ dyne/cm². The equation has a simple form and is considered to have a practical utility for filled-polymer melt systems.With 2 figures and 1 table