Oscillating radial flow between parallel plates

An analysis is presented for laminar radial flow due to an oscillating source between parallel plates. The source strength varies according to Q=Q ₀ cos ωt, and the solution is in the form of an infinite series in terms of a reduced Reynolds number, R _a ^* =Q ₀/4πνa/(r/a)². (Q ₀ = amplitude of source strength, ω = frequency, a = half distance between plates, r = radial coordinate, t = time, and ν = kinematic viscosity.) The results are valid for small values of R _a ^* and all values of the frequency Reynolds number, α=ωa ²/ν. The effects of the parameters R _a ^* and α are discussed.     

Terminal velocity of a Taylor drop in a vertical pipe

A scaling analysis based on the field equations for two phases and the jump conditions at the interface is carried out to deduce a balance of forces acting on a Taylor drop rising through stagnant liquid in a vertical pipe. The force balance is utilized to deduce a functional form of an empirical correlation of terminal velocity of the Taylor drop. Undetermined coefficients in the correlation are evaluated by making use of available correlations for two limiting cases, i.e. extremely high and low Reynolds number Taylor bubbles in large pipes. Terminal velocity data obtained by interface tracking simulations are also used to determine the coefficients. The proposed correlation expresses the Froude number Fr as a function of the drop Reynolds number Re_D, the Eötvös number Eo_D and the viscosity ratio μ^*. Comparisons between the correlation, simulations and experimental data confirm that the proposed correlation is applicable to Taylor drops under various conditions, i.e., 0.002 < Re_D < 4960, 4.8 < Eo_D < 228, 0 ? μ^* ? 70, 1 < N < 14700, −12 < log M < 4, and d/D < 1.6, where N is the inverse viscosity number, M the Morton number, d the sphere-volume equivalent drop diameter and D the pipe diameter.     

Self-similar collapse of 2D and 3D vortex filament models

In this article, a very simple toy model for a candidate blow-up solution of the Euler equation by Boratav and Pelz (vortex dodecapole) is investigated. In this model, vortex tubes are replaced with straight vortex filaments of infinitesimal thickness, and the entire motion is monitored by tracing the motion of a representative point on one vortex filament. It is demonstrated that this model permits a self-similar collapse solution which provides the time dependence of the length scale as (t _c ? t)^1/2, (t < t _c), where the collapse time t _c depends on the initial configuration. From the conservation of circulation, this time dependence implies that vorticity ω scales as (t _c ? t) ^?1, which agrees with the one observed in the direct numerical (pseudo spectral) simulations of the vortex dodecapole. Finally, possible modification of the model is considered.     

Effect of the turbulent viscosity relaxation time on the modeling of nozzle and jet flows

Certain modifications of three-equation turbulence models are proposed. They are intended for increasing the accuracy of the calculations of turbulent flows in nozzles with boundary layer separation and in supersonic jets with complicated shock wave structures. Basing on the idea of the inclusion of flow prehistory in terms of an additional relaxation equation for nonequilibrium turbulent viscosity we propose three modifications of the k-ω-µ _t model based on the k-ω model and a version of the k- ?-µ _t turbulence model. In these modifications we introduce an additional dependence of the nonequilibrium turbulent viscosity relaxation time on different physical parameters which can be important near the point of boundary layer separation from the nozzle wall, such as viscous effects and effects of large gradients of the mean velocity and the kinetic energy of turbulence (turbulent pressure). The comparison of the results of the calculations with the experimental data shows that all the proposed versions of the three-equation models make it possible to improve the accuracy of the calculations of turbulent flows in nozzles and jets.     

Static wall layers in plane channel displacement flows

We study steady and pulsating displacement flows of a Bingham fluid by a Newtonian fluid, along a plane channel. For sufficiently large yield stress a static residual wall layer can result during the displacement. The flow is parameterised by the Reynolds number (Re), the Bingham number (B) and the viscosity ratio (M). Perhaps intuitively, thicker layers are found with larger M and at lower Re. The residual layer is formed on the advective timescale of the displacement but drains on a slower timescale governed by M. For larger M truly stationary layers are only found for large t when the layer has thinned sufficiently to become static. Increased Re results in increased energy production locally around the finger. For large enough Re the energy production can play a significant role in yielding the fluid. As the energy production rate increases it also becomes focused around the corner or shoulder region of the front, and spreads axially along the initial part of the residual layer. This causes fluid to yield increasingly far behind the front and allows for the layer to thin. As B increases the static layer tends to decrease (see also [1], [2]). At small Re the static layer thickness appears to be independent of M. At large Re the layer thickness is dependent on M and decreases asymptotically to a constant value as B → ∞.For pulsating displacement flow rates, Q(t) = 2(1 + Asin  ωt) : A ∈ [0, 1) we study two ranges: ωRe ? 2π and ωRe ? 2π. For the viscous regime (ωRe ? 2π) a pseudo-steady 1D model predicts that the residual layer should remain static for 3(1 + Asin  ωt) < MB. In practice we find that partial mobilisation of the residual layer occurs even when this inequality is satisfied, but not if MB becomes significantly larger than 3(1 + A). For ωRe ? 2π we mobilise the layer for significantly larger values of MB and at smaller A, than in the viscous regime. This effect is traced to the occurrence of out-of-phase velocity fluctuations in the displacing fluid within a wall layer close to the interface.     

Impact energy absorption by a rate-dependent locking target

The impact by an elastic cylindrical piston on a thin plate-like target resting on a rigid foundation is considered. The relationship between force F acting on the target and displacement x is given by F=kx+q dx/dt provided dx/dt≥0 and 0≤x<d (k, q and d≥0). When x=d locking occurs, and F can assume any value ≥kd without increase in x. The displacement is assumed to be completely irreversible. The motion of the impactor is assumed to be governed by the elementary wave equation and, since the target is thin, wave motion in the target is neglected. The energy W=εFdx and its components W _k=kεx dx (the energy absorbed in a corresponding quasistatic process) and W _q=qε(dx/dt)² dt (the excessive energy because of the rate-dependence) are determined in terms of the impact energy as functions of non-dimensional parameters representing k, q and d. With the aid of diagrams, it is shown under what circumstances locking occurs, and under what circumstances W _k or W _q, or both, are large.     

Steady and transient shear flows of polystyrene solutions I: Concentration and molecular weight dependence of non-dimensional viscometric functions

Start up from rest and relaxation from steady shear flow experiments have been performed on monodisperse polystyrene solutions with molecular weight ranging from 1.3 × 10⁵ to 1.6 × 10⁶ and concentration c ranging from 5% to 40%. A method of reduced variables based on the use of a characteristic time τ_w is proposed. τ_w is defined as the product of zero shear viscosity with the steady state elastic compliance.Reduced steady and transient viscometric functions so obtained depend on the ratio M/M_e (where M_e is the entanglement molecular weight). Limiting forms are obtained when M/M_e ? 18. In steady flow, a simple correlation is found between shear and normal stresses.In stress relaxation experiments, independent of shear rate, the long-time behaviour can be characterised by a single relaxation time τ₁, which is identical for shear and normal stresses. τ₁ can be simply related to the zero shear rate viscosity and the limiting elastic compliance.     

Dissipative interface waves and the transient response of a three-dimensional sliding interface with Coulomb friction

We investigate the linearized response of two elastic half-spaces sliding past one another with constant Coulomb friction to small three-dimensional perturbations. Starting with the assumption that friction always opposes slip velocity, we derive a set of linearized boundary conditions relating perturbations of shear traction to slip velocity. Friction introduces an effective viscosity transverse to the direction of the original sliding, but offers no additional resistance to slip aligned with the original sliding direction. The amplitude of transverse slip depends on a nondimensional parameter η=c_sτ₀/μv₀, where τ₀ is the initial shear stress, 2v₀ is the initial slip velocity, μ is the shear modulus, and c_s is the shear wave speed. As η→0, the transverse shear traction becomes negligible, and we find an azimuthally symmetric Rayleigh wave trapped along the interface. As η→∞, the inplane and antiplane wavesystems frictionally couple into an interface wave with a velocity that is directionally dependent, increasing from the Rayleigh speed in the direction of initial sliding up to the shear wave speed in the transverse direction. Except in these frictional limits and the specialization to two-dimensional inplane geometry, the interface waves are dissipative. In addition to forward and backward propagating interface waves, we find that for η>1, a third solution to the dispersion relation appears, corresponding to a damped standing wave mode. For large-amplitude perturbations, the interface becomes isotropically dissipative. The behavior resembles the frictionless response in the extremely strong perturbation limit, except that the waves are damped. We extend the linearized analysis by presenting analytical solutions for the transient response of the medium to both line and point sources on the interface. The resulting self-similar slip pulses consist of the interface waves and head waves, and help explain the transmission of forces across fracture surfaces. Furthermore, we suggest that the η→∞ limit describes the sliding interface behind the crack edge for shear fracture problems in which the absolute level of sliding friction is much larger than any interfacial stress changes.     

Decay of Almost Periodic Solutions¶of Conservation Laws

We consider the asymptotic behavior of solutions of systems of inviscid or viscous conservation laws in one or several space variables, which are almost periodic in the space variables in a generalized sense introduced by Stepanoff and Wiener, which extends the original one of H. Bohr. We prove that if u(x,t) is such a solution whose inclusion intervals at time t, with respect to ?>0, satisfy l _epsiv;(t)/t→0 as t→∞, and such that the scaling sequence u ^T (x,t)=u(T x,T t) is pre-compact as t→∞ in L _loc ¹(? ^{d +1} ₊, then u(x,t) decays to its mean value \(\), which is independent of t, as t→∞. The decay considered here is in L ¹ _loc of the variable ξ≡x/t, which implies, as we show, that \(\) as t→∞, where M _x denotes taking the mean value with respect to x. In many cases we show that, if the initial data are almost periodic in the generalized sense, then so also are the solutions. We also show, in these cases, how to reduce the condition on the growth of the inclusion intervals l _?(t) with t, as t→∞, for fixed ? > 0, to a condition on the growth of l _?(0) with ?, as ?→ 0, which amounts to imposing restrictions only on the initial data. We show with a simple example the existence of almost periodic (non-periodic) functions whose inclusion intervals satisfy any prescribed growth condition as ?→ 0. The applications given here include inviscid and viscous scalar conservation laws in several space variables, some inviscid systems in chromatography and isentropic gas dynamics, as well as many viscous 2 × 2 systems such as those of nonlinear elasticity and Eulerian isentropic gas dynamics, with artificial viscosity, among others. In the case of the inviscid scalar equations and chromatography systems, the class of initial data for which decay results are proved includes, in particular, the L ^∞ generalized limit periodic functions. Our procedures can be easily adapted to provide similar results for semilinear and kinetic relaxations of systems of conservation laws.     

Rheology of suspensions of spherical particles in a newtonian and a non-newtonian fluid

Properties of suspensions of spherical glass beads (25–38 μm dia.) in a Newtonian fluid and a non-Newtonian (NBS Fluid 40) fluid were measured at volume fractions, φ, of 0%, 10%, 20% and 30%. Measurements were made using a modified and computerized Weissenberg Rheogoniometer. Properties measured included steady shear viscosity, η($\text{γ}\text{.}$), first normal stress difference, N₁($\text{γ}\text{.}$), linear viscoelastic properties, η′(ω) and G′(ω), shear stress relaxation, σ^? ($\text{γ}\text{.}$, t), and growth, σ⁺($\text{γ}\text{.}$, t) and normal stress relaxation, N₁^?($\text{γ}\text{.}$, t).For a the Newtonian fluid, increasing φ causes both η and η′ to increase, with η′ showing a slight frequency dependence. Both N₁ and G′ are zero and stress relaxation and growth occur essentially instantaneously. For the NBS fluid, both η and η′ increse with φ at all $\text{γ}\text{.}$ and ω, respectively, the increase being greater as $\text{γ}\text{.}$ and ω approach zero. N₁ and G′ are less affected by the presence of the particles than η and η′ with the effect on G′ being more pronounced than on N₁. For fixed $\text{γ}\text{.}$, stress relaxation and growth exhibit greater non-linear effects as φ is increased. A model for predicting a priori the linear viscoelastic properties for suspensions was found to yeild reasonable estimates up to φ = 20%.     

Bounds on the distribution of extreme values for the stress in composite materials

Suitable macroscopic quantities beyond effective elastic properties are used to assess the distribution of stress within a composite. The composite is composed of N anisotropic linearly elastic materials and the length scale of the microstructure relative to the loading is denoted by ε. The stress distribution function inside the composite λ^ε(t) gives the volume of the set where the norm of the stress exceeds the value t. The analysis focuses on the case when 0<ε?1. A rigorous upper bound on lim_ε→0λ^ε(t) is found. The bound is given in terms of a macroscopic quantity called the macro stress modulation function. It is used to provide a rigorous assessment of the volume of over stressed regions near stress concentrators generated by reentrant corners or by an abrupt change of boundary loading.     

Instabilities in Dynamic Anti-plane Sliding of an Elastic Layer on a Dissimilar Elastic Half-Space

The stability of dynamic anti-plane sliding at an interface between an elastic layer and an elastic half-space with dissimilar elastic properties is studied. Friction at the interface is assumed to follow a rate- and state-dependent law, with a positive instantaneous dependence on slip velocity and a rate weakening behavior in the steady state. The perturbations at the interface are of the form exp(ikx ₁+pt), where k is the wavenumber, x ₁ is the coordinate along the interface, p is the time response to the perturbation and t is time. A key feature of the problem is that interfacial waves both in freely slipping contact as well as in bonded contact exist for the problem. Attention is focused on the role of the interfacial waves on slip stability. Instabilities are plotted in the $\operatorname{Re} (pL/V_{o})$ versus $\operatorname{Im} (p/|k|c_{s})$ plane, where L is a length scale in the friction law, V _o is the unperturbed slip velocity and c _s is the shear wave speed of the layer. Stability of both rapid and slow slip is studied. The results show one mechanism by which instabilities occur is the destabilization by friction of the interfacial waves in freely slipping contact/bonded contact. This occurs even in slow sliding, thus confirming that the quasi-static approximation is not valid for slow sliding. The effect of material properties and layer thickness on the stability results is studied.     

Temperature correlations with vorticity and velocity in a turbulent cylinder wake

This work aims to understand the difference in the correlations between the fluctuating temperature and the vorticity from that between the fluctuating temperature and the velocity in a turbulent cylinder near wake. Measurements are made at x/d = 10, 20 and 40, where x is the streamwise distance from the cylinder axis and d is the cylinder diameter, with a Reynolds number of 2.5×10³ based on d and the free-stream velocity. The three components of the fluctuating velocity vector u_i(i = 1, 2 and 3), vorticity vector ω_i (i = 1, 2 and 3), and temperature θ in the plane of the mean shear are measured simultaneously with a multi-wire probe consisting of four X-hotwires and four cold wires. It is found that at x/d = 10, both correlations between u_iand θ and between ω_i and θ predominantly take place at St = 0.21, due to the concentric distribution of the Kármán vortices and the heat. With increasing x/d, the correlation between ω_i (i = 1, 2 and 3) and θ drops rapidly, as a result of the weakened Kármán vortices; in contrast, the correlation between u₁ and θ increases appreciably, largely due to an enhanced correlation between u₁ and θ at low frequencies or scales of motions larger than the Kármán vortex. The slowly decreasing (along x) two-point autocorrelations of u₁ and θ suggest that the very-large-scale motions (VLSMs) found in wall flows occur also in the turbulent wake and are responsible for the high correlation between u₁ and θ at low frequencies.     

On the use of stretched-exponential functions for both linear viscoelastic creep and stress relaxation

The use of the stretched-exponential function to represent both the relaxation function g(t)=(G(t)-G _∞)/(G ₀-G _∞) and the retardation function r(t) = (J _∞+t/η-J(t))/(J _∞-J ₀) of linear viscoelasticity for a given material is investigated. That is, if g(t) is given by exp (?(t/τ)^β), can r(t) be represented as exp (?(t/λ)^µ) for a linear viscoelastic fluid or solid? Here J(t) is the creep compliance, G(t) is the shear modulus, η is the viscosity (η^?1 is finite for a fluid and zero for a solid), G _∞ is the equilibrium modulus G _e for a solid or zero for a fluid, J _∞ is 1/G _e for a solid or the steady-state recoverable compliance for a fluid, G ₀= 1/J ₀ is the instantaneous modulus, and t is the time. It is concluded that g(t) and r(t) cannot both exactly by stretched-exponential functions for a given material. Nevertheless, it is found that both g(t) and r(t) can be approximately represented by stretched-exponential functions for the special case of a fluid with exponents β=µ in the range 0.5 to 0.6, with the correspondence being very close with β=µ=0.5 and λ=2τ. Otherwise, the functions g(t) and r(t) differ, with the deviation being marked for solids. The possible application of a stretched-exponential to represent r(t) for a critical gel is discussed.     

The effect of 1,3:2,4-<Emphasis Type="Italic">bis</Emphasis>-<Emphasis Type="Italic">O</Emphasis>-(<Emphasis Type="Italic">p</Emphasis>-methylbenzylidene)-<Emphasis Type="SmallCaps">d</Emphasis>-sorbitol (PDTS) on uniaxial elongational viscosity of polypropylene

We investigated the dynamic viscoelasticity and elongational viscosity of polypropylene (PP) containing 0.5 wt% of 1,3:2,4-bis-O-(p-methylbenzylidene)-d-sorbitol (PDTS). The PP/PDTS system exhibited a sol–gel transition (T _gel) at 193 °C. The critical exponent n was nearly equal to 2/3, in agreement with the value predicted by a percolation theory. This critical gel is due to a three-dimensional network structure of PDTS crystals. The elongational viscosity behavior of neat PP followed the linear viscosity growth function 3η ⁺ (t), where η ⁺ (t) is the shear stress growth function in the linear viscoelastic region. The elongational viscosity of the PP/PDTS system also followed the 3η ⁺ (t) above T _gel but did not follow the 3η ⁺ (t) and exhibited strong strain-softening behavior below T _gel. This strain softening can be attributed to breakage of the network structure of PDTS with a critical stress (σ _c) of about 10⁴ Pa.     

Effects of aspect ratio on the drag of a wall-mounted finite-length cylinder in subcritical and critical regimes

The effect of cylinder aspect ratio (??H/d, where H is the cylinder height or length, and d is the cylinder diameter) on the drag of a wall-mounted finite-length circular cylinder in both subcritical and critical regimes is experimentally investigated. Two cases are considered: a smooth cylinder submerged in a turbulent boundary layer and a roughened cylinder immersed in a laminar uniform flow. In the former case, the Reynolds number Re _d (??U _?? d/??, with U _?? being the free-stream velocity and ?? the fluid viscosity) was varied from 2.61?×?10⁴ to 2.87?×?10⁵, and two values of H/d (2.65 and 5) were examined; in the latter case, Re _d ?=?1.24?×?10⁴?C1.73?×?10⁵ and H/d?=?3, 5 and 7. In the subcritical regime, both the drag coefficient C _D and the Strouhal number St are smaller than their counterparts for a two-dimensional cylinder and reduce monotonously with decreasing H/d. The presence of a turbulent boundary layer causes an early transition from the subcritical to critical regime and considerably enlarges the Re _d range of the critical regime. No laminar separation bubble occurs on the finite-length cylinder immersed in the turbulent boundary layer, and consequently, the discontinuity is not observed in the C _D?CRe _d and St?CRe _d curves. In the roughened cylinder case, the Re _d range of the critical regime grows gradually with decreasing H/d, while the C _D crisis becomes less obvious. In both cases, H/d has a negligible effect on the critical value of Re _d at which transition occurs from the subcritical to critical regime.     

Phase behavior and effects of microstructure on viscoelastic properties of a series of polylactides and polylactide/poly(ε-caprolactone) copolymers

Dynamic viscoelastic measurements were combined with differential scanning calorimetry (DSC) and atomic force microscopy (AFM) analysis to investigate the rheology, phase structure, and morphology of poly(l-lactide) (PLLA), poly(ε-caprolactone) (PCL), poly(d,l-lactide) (PDLLA) with molar composition l-LA/d-LA?=?53:47, and poly(l-lactide-co-ε-caprolactone) (PLAcoCL) with molar composition l-LA/CL?=?67:33. After melt conformation, both copolymers PDLLA and PLAcoCL were found to be amorphous whereas PLLA and PCL presented partial crystallinity. The copolymers and PCL were considered as thermorheologically simple according to the rheological methods employed. Therefore, data from different temperatures could be overlapped by a simple horizontal shift (a _T) on elastic modulus, G′, and loss modulus, G′, versus frequency graph, generating the corresponding master curves. Moreover, these master curves showed a dependency of G″≈ω and G′≈ω ² at low frequencies, which is a characteristic of homogeneous melts. For the first time, fundamental viscoelastic parameters, such as entanglement modulus G _N ⁰ and reptation time τ _d, of a PLAcoCL copolymer were obtained and correlated to chain microstructure. PLLA, by contrast, was unexpectedly revealed as a thermorheologically complex liquid according to the failure observed in the superposition of the phase angle (δ) versus the complex modulus (G*); this result suggests that the narrow window for rheological measurements, chosen to be close to the melting point centered at 180 °C thus avoiding thermal degradation, was not sufficient to assure an homogeneous behavior of PLLA melts. The understanding of the melt rheology related to the chain microstructural aspects will help in the understanding of the complex phase structures present in medical devices.     

Two new methods for the approximate determination of the initial coefficient of the first normal stress difference

Two methods for determining the initial coefficient of the first normal stress difference are presented. They are based on the evaluation of the steady viscosity function η($\text{γ}\text{.}$) and the viscosity function η⁺($\text{γ}\text{.}$, t) at the start-up of a flow with a very small rate of deformation $\text{γ}\text{.}$ < $\text{γ}\text{.}$₀. For the functions η($\text{γ}\text{.}$) and η⁺($\text{γ}\text{.}$), equations are given which can be used for a simple evaluation of the integral relationships obtaiend for ψ₁₀. The values for ψ₁₀ calculated by the two methods are compared with values obtained by the well-known methods via measurement of the ψ₁($\text{γ}\text{.}$) or η″(ω)/ω functions and extrapolation to zero). Both methods give values which are in satisfactory agreement with the experimental values.     

PIV measurement of separated flow over a blunt plate with different chord-to-thickness ratios

The influence of the chord-to-thickness ratio (c/t) on the spatial characteristics of the separated shear layer over a blunt plate and the leading-edge vortices embedded in the separated shear layer was studied extensively using planar particle image velocimetry (PIV). Three systems corresponding to different shedding modes were chosen for the comparative study: c/t=3, 6 and 9. The Reynolds number based on the plate's thickness (t) was Re_t=1×10³. A gigapixel CCD camera was used to acquire images with a spatial resolution of 0.06t×0.06t in the measurement range of 9.5t×4.5t. Distributions of statistical quantities, such as the streamline pattern, streamwise velocity fluctuation intensity, shear stress and reverse flow intermittency, showed that the separated shear layer in the system with c/t=3 did not reattach to the plate's surface, while the near‐wake behind the trailing edge was highly unstable because the energetic leading-edge vortices were shed into the wake. The separated shear layer of the system with c/t=6 periodically reattached to the plate's surface, which resulted in intensified fluctuations of the near wake behind the trailing edge. In the longest system (c/t=9), the separated shear layer always reattached to the plate's surface far upstream from the trailing edge, which did not induce large fluctuations of the near wake. Furthermore, the proper orthogonal decomposition (POD) was extensively employed to filter the original velocity fields spatially to identify the large-scale vortices immersed in the separated shear layer easily. The distribution of the v-v correlation coefficients of the spatially filtered flow fields reflected the organized large-scale vortices in the three systems. The number of alternations of the positive and negative correlation coefficients across the flow field were determined to be 1, 2 and 3 for the systems with c/t=3, 6 and 9, respectively; this is in agreement with the shedding mode of each system. The distribution of the swirling strength of the separated shear layer accurately determined the positions and structures of the large-scale vortices formed above the plate surface.