Low Wavenumber Limit Research Articles

Gyro-averaging operators with magnetic field inhomogeneity

We derive expressions for the gyro-averaging operator that is applicable to electrostatic fluctuations in a spatially inhomogeneous magnetic field. Both low and high wavenumber limits are considered. The gyro-averaging operator for the former case is represented by sums of Bessel functions with different orders. A simplified expression is provided as a Padé approximant in the low wavenumber limit. This form could be used in practical computations based on the gyrofluid formulation. In the high wavenumber limit, we find that the operator naturally involves fractional derivatives whose physical interpretations are yet to be explored. Discussions are made of a potential impact of this asymptotic expression in the high wavenumber limit.

Dynamical chemistry: non-equilibrium effective actions for reactive fluids

We present two approaches for describing chemical reactions taking place in fluid phase. The first method mirrors the usual derivation of the hydrodynamic equations of motion by relating conserved—or to account for chemical reactions, non-conserved—currents to local-equilibrium parameters. The second method involves a higher-brow approach in which we attack the same problem from the perspective of non-equilibrium effective field theory (EFT). Non-equilibrium effective actions are defined using the in–in formalism on the Schwinger–Keldysh contour and are therefore capable of describing thermal fluctuations and dissipation as well as quantum effects. The non-equilibrium EFT approach is especially powerful as all terms in the action are fully specified by the symmetries of the system; in particular the second law of thermodynamics does not need to be included by hand, but is instead derived from the action itself. We find that the equations of motion generated by both methods agree, but the EFT approach yields certain advantages. To demonstrate some of these advantages we construct a quadratic action that is valid to very small distance scales—much smaller than the scales at which ordinary hydrodynamic theories break down. Such an action captures the full thermodynamic and quantum behavior of reactions and diffusion at quadratic order. Finally, taking the low-frequency and low-wavenumber limit, we reproduce the linearized version of the well-known reaction–diffusion equations as a final coherence check.

Stability of surfactant-laden liquid film flow over a cylindrical rod

The stability of surfactant-laden liquid film flow over a cylindrical rod is examined in creeping flow limit using standard temporal linear stability analysis. The clean film flow configuration (i.e., in absence of surfactant) is well-known to become unstable owing to Rayleigh-Plateau instability of cylindrical liquid interfaces. Previous studies demonstrated that for a static liquid film (i.e., zero basic flow) coating a rod, the presence of interfacial surfactant decrease the growth of Rayleigh-Plateau instability, but is unable to suppress it completely. Further, the presence of interfacial surfactant is known to introduce an additional mode, referred to as surfactant mode in the present work. To the best of our knowledge, the stability of surfactant mode has not been analyzed in the context of cylindrical film flows. Thus, we reexamined the stability of surfactant-laden cylindrical liquid film flow to analyze the stability behavior of the above said two modes when the basic flow is turned on. The present study reveals that the incorporation of basic flow in stability analysis leads to the complete suppression of Rayleigh-Plateau instability due to the presence of interfacial surfactants as compared to the partial suppression obtained for a stationary liquid film. Three nondimensional parameters appear for this problem: Bond number (denoted as Bo) which characterizes the strength of basic flow, Marangoni number (denoted as Ma) which signifies the presence of surfactant, and ratio of rod radius to film thickness denoted as S. In creeping flow limit, the characteristic equation is quadratic with one root belonging to Rayleigh-Plateau mode and the other to surfactant mode. We first carried out an asymptotic analysis to independently capture the eigenvalues corresponding to both the modes in limit of long-wave disturbances. The long-wave results show that the Rayleigh-Plateau instability is completely suppressed on increasing the Marangoni number above a critical value while the surfactant mode always remains stable in low wave-number limit. The continuation of long-wave results to arbitrary wavelength disturbances show that the suppression of Rayleigh-Plateau instability mode still holds, however, the surfactant mode becomes unstable at sufficiently high values of Marangoni number. Further, this surfactant mode instability shifts toward low wave numbers with critical Marangoni number for instability scaling with wave number in a particular fashion. We used this scaling and carried out an asymptotic analysis to capture this instability in low wave-number limit. Depending on S and Bo, we observed the existence of a stable gap in terms of Ma where both the eigen-modes remain stable. Our results indicate that for a given Bond number, the width of stable gap in terms of Ma decreases with decrease in S and the stable gap vanishes when S is sufficiently small. The effect of increasing Bond number (or equivalently, the strength of basic flow) is found to be stabilizing for the film flow configuration.

Modelling wall pressure fluctuations under a turbulent boundary layer

The derivation of the wave vector-frequency (w-f) spectrum of wall pressure fluctuations below a turbulent boundary layer developed over a rigid flat plate is re-considered. The Lighthill's equation for pressure fluctuations is derived in a frame of reference fix with respect to the plate, at low Mach numbers, and transformed into the convected frame moving with the flow. To model the source terms of the Lighthill equation, it is assumed that in the inertial range, the turbulence is locally isotropic in the convected frame. The w-f spectrum of isotropic turbulence is obtained from symmetry considerations by extending the isotropy to space time, based on the concept of sweeping velocity. The resulting solution for the pressure w-f spectrum contains a term (the mean shear-turbulence term) which does not fulfill the Kraichnan Philipps theorem, due to the form of the selected turbulent velocity spectrum. The viscous effects are accounted for by a cut-off depending on wall distance; this procedure allows extending the model beyond the inertial range contribution. The w-f pressure spectrum is derived and compared to the experimental low wavenumber data of Farabee and Geib (1991) [8] and Bonness et al. (2010) [5], for which a good agreement is obtained. The derived expression is also compared to Chase theoretical model Chase (1987) [6] and found to agree well in the vicinity of the convective ridge of the subsonic domain and to differ significantly both in supersonic and subsonic low wavenumber limits. The pressure spectrum derived from the model and its scaling are discussed and compared to experimental data and to the empirical model of Goody (2002) [23], which results from the compilation of a large set of experimental data. Very good agreement is obtained, except at vanishing frequencies where it is claimed that the experimental results lack of significance due to the limited size of the experimental facilities. This hypothesis supported by the results obtained from numerical simulations.

Inertio-gravity waves under the non-traditional -plane approximation: singularity in the large-scale limit

The low horizontal wavenumber limit of waves on a plane under a constant rotation rate (the so-called $f$-plane) is degenerate: all wave frequencies asymptotically approach the inertial frequency. This degeneracy has no serious consequence when the rotation axis is perpendicular to the plane (traditional $f$-plane approximation). However, when the rotation axis is tilted from the vertical direction (non-traditional $f$-plane approximation), we encounter a type of ‘singularity’ in the sense that each term of the Taylor expansion of the wave frequency in the horizontal Coriolis parameter diverges in the limit of low horizontal wavenumber. Such a drastic change of the solution behaviour by adding the horizontal Coriolis parameter in the low horizontal wavenumber limit is rather counter-intuitive, because the conventional scale analysis suggests that the horizontal Coriolis effect is negligible in this limit. However, the degeneracy of the system makes this effect critical with a need for considering higher-order terms. Two possible asymptotic limits are proposed for resolving this degeneracy. One of them, as it turns out, amounts to a representation of the wave frequency by a Taylor expansion in the horizontal wavenumber.

Static structural signatures of nearly jammed disordered and ordered hard-sphere packings: Direct correlation function.

The nonequilibrium process by which hard-particle systems may be compressed into disordered, jammed states has received much attention because of its wide utility in describing a broad class of amorphous materials. While dynamical signatures are known to precede jamming, the task of identifying static structural signatures indicating the onset of jamming have proven more elusive. The observation that compressing hard-particle packings towards jamming is accompanied by an anomalous suppression of density fluctuations (termed "hyperuniformity") has paved the way for the analysis of jamming as an "inverted critical point" in which the direct correlation function c(r), rather than the total correlation function h(r), diverges. We expand on the notion that c(r) provides both universal and protocol-specific information as packings approach jamming. By considering the degree and position of singularities (discontinuities in the nth derivative) as well as how they are changed by the convolutions found in the Ornstein-Zernike equation, we establish quantitative statements about the structure of c(r) with regards to singularities it inherits from h(r). These relations provide a concrete means of identifying features that must be expressed in c(r) if one hopes to reproduce various details in the pair correlation function accurately and provide stringent tests on the associated numerics. We also analyze the evolution of systems of three-dimensional monodisperse hard spheres of diameter D as they approach ordered and disordered jammed configurations. For the latter, we use the Lubachevsky-Stillinger (LS) molecular dynamics and Torquato-Jiao (TJ) sequential linear programming algorithms, which both generate disordered packings, but can show perceptible structural differences. We identify a short-ranged scaling c(r)∝-1/r as r→0 that accompanies the formation of the delta function at c(D) that indicates the formation of contacts in all cases, and show that this scaling behavior is, in this case, a consequence of the growing long rangedness in c(r), e.g., c∝-1/r^{2} as r→∞ for disordered packings. At densities in the vicinity of the freezing density, we find striking qualitative differences in the structure factor S(k) as well as c(r) between TJ- and LS-generated configurations, including the early formation of a delta function at c(D) in the TJ algorithm's packings, indicating the early formation of clusters of particles in near contact. Both algorithms yield structure factors that tend towards zero in the low-wave-number limit as jamming is approached. Correspondingly, we observe the expected power-law decay in c(r) for large r, in agreement with previous theoretical work. Our work advances the notion that static signatures are exhibited by hard-particle packings as they approach jamming and underscores the utility of the direct correlation function as a sensitive means of monitoring for the appearance of an incipient rigid network.

Time-Resolved Light Scattering Study on the Kinetics of the Liquid-Liquid Transition in Triphenyl Phosphite.

There is experimental evidence suggesting the existence of a liquid-liquid transition (LLT) in a single-component liquid. However, none of this evidence is free from controversy, including the case of a molecular liquid, triphenyl phosphite, which we study here. Furthermore, the kinetics of LLT has been largely unexplored. Here we study the phase-transition dynamics of triphenyl phosphite in a supercooled liquid state by means of time-resolved polarized and depolarized small-angle light scattering to clarify whether the transition is a liquid-liquid transition (LLT) or merely nanocrystal formation. A part of this study was recently reported in another of our papers [Shimizu, R.; Kobayashi, M.; Tanaka, H. Phys. Rev. Lett. 2014, 112, 125702]. A detailed analysis of our experimental results of light scattering and the comparison with heat evolution during LLT have revealed the following facts. The polarized scattering from domains has a finite (nonzero) intensity in the low-wavenumber limit, and the time evolution of its average intensity is almost proportional to the square of the heat-releasing rate. The depolarized scattering intensity monotonically increases in the process of LLT during isothermal annealing above the spinodal temperature TSD but exhibits a peak below TSD. On the basis of these results, we suggest that the primary process is LLT, whose order parameter is of a nonconserved nature, but accompanies nanocrystal formation. In the NG-type LLT, the sharp interface between liquid II droplets and the liquid I matrix promotes nanocrystal formation there, whereas much less nanocrystal formation is induced in the SD-type LLT due to the lack of such sharp interfaces.

On the torsional wave dispersion in a hollow sandwich circular cylinder made from viscoelastic materials

The dispersion of torsional wave propagation in the sandwich hollow cylinder made of linear viscoelastic materials is investigated. The investigations are carried out within the scope of the piecewise homogeneous body model by utilizing the exact field equations of viscoelasto-dynamics. The mechanical relations of the materials are described through fractional exponential operators. The analytical expression for the dispersion equation is obtained and the algorithm is developed for its numerical solution. The influence of the viscosity of the layers of the sandwich hollow cylinder is studied through the rheological parameters which characterize the characteristic creep time and long-term values of the elastic constants. It is assumed that the materials of the inner and outer cylinders are the same. Numerical results (dispersion curves) are obtained and the influence of the mentioned rheological parameters on these curves is investigated for the following two cases: attenuation of the torsional waves is dispersive (Case 1) and attenuation of the torsional waves is non dispersive (Case 2). The analytical expression is obtained for the low wavenumber limit values of the torsional wave propagation velocity which is valid in Case 1. As a result of the numerical investigations, in particular, it is established that in the case where the rheological parameters of the layers are the same, the viscosity of the layers’ materials causes the torsional wave propagation velocity to increase.

Evidence of liquid-liquid transition in triphenyl phosphite from time-resolved light scattering experiments.

Here, we study the phase transition kinetics in a supercooled liquid state of triphenyl phosphite by means of time-resolved polarized and depolarized light scattering to address a long-standing controversy on its mechanism, i.e., whether the phenomenon is primarily induced by liquid-liquid transition (LLT) or by nanocrystal formation. We find that the polarized scattering intensity exhibits a peak as a function of time, and its low wave number limit is nonzero for any annealing temperatures, both of which strongly indicate the nonconserved nature of an order parameter governing the transition. We also observe evolution of depolarized scattering. Above the spinodal temperature TSD, the depolarized scattering intensity monotonically increases with time since it is dominated by scattering from nanocrystallites, which are continuously formed during the process. Below TSD, on the other hand, it exhibits a distinct peak as a function of time as the polarized scattering intensity does. This appearance of the peak suggests that dielectric tensor fluctuations responsible for the depolarized scattering mainly come from isotropic density fluctuations and not from nanocrystallites, supporting the occurrence of LLT.

Torsional wave dispersion in a finitely pre-strained hollow sandwich circular cylinder

This paper studies torsional wave dispersion in a three-layered (sandwich) hollow cylinder with finite initial strains. The investigations are carried out within the scope of the piecewise homogeneous body model with the use of the three-dimensional linearized theory of elastic waves in initially stressed bodies. The mechanical relations of the materials of the cylinders are described through their harmonic potential. The analytical expression is obtained for the low wavenumber limit values of the torsional wave propagation velocity. The numerical results on the influence of the initial stretching or compression of the cylinders along the torsional wave propagation direction are presented and discussed.

Wave propagation along a non-principal direction in a compressible pre-stressed elastic layer

The dispersive behavior of small amplitude waves propagating along a non-principal direction in a pre-stressed, compressible elastic layer is considered. One of the principal axes of stretch is normal to the elastic layer and the direction of propagation makes an angle θ with one of the in-plane principal axes. The dispersion relations which relate wave speed and wavenumber are obtained for both symmetric and anti-symmetric motions by formulating the incremental boundary value problem for a general strain energy function. The behavior of the dispersion curves for symmetric waves is for the most part similar to that of the anti-symmetric waves at the low and high wavenumber limits. At the low wavenumber limit, depending on the pre-stress and propagation angle, it may be possible for both the fundamental mode and the next lowest mode to have finite phase speeds, while other higher modes have an infinite phase speed. At the high wavenumber limit, the phase speeds of the fundamental mode and the higher modes tend to the Rayleigh surface wave speed and the limiting wave speeds of the layer, respectively. Numerical results are presented for a Blatz–Ko material and the effect of the propagation angle is clearly illustrated.

Role of wall deformability on interfacial instabilities in gravity-driven two-layer flow with a free surface

The linear stability of gravity-driven flow of two superposed Newtonian liquid layers down a deformable, inclined, wall is analyzed in order to examine the effect of wall deformability on the interfacial instabilities in the system. There are three distinct interfacial modes in this composite system, viz., gas-liquid (GL), liquid-liquid (LL), and liquid-solid (LS) modes. For a rigid-wall, the GL interface becomes unstable above a critical Reynolds number, while the stability of the LL interface depends on the relative placement of the liquid layers. When the more viscous liquid is adjacent to rigid surface, the LL mode becomes unstable beyond a critical Reynolds number (Re), while it becomes unstable even at Re=0 when the less viscous liquid is next to rigid-wall. Our asymptotic results show that solid deformability has a stabilizing effect on both GL and LL modes in the low-wavenumber limit when the more viscous liquid layer is near the deformable wall. Numerical results reveal that both the GL and LL interfacial instabilities can be suppressed for all wavenumbers when the solid layer becomes sufficiently deformable. With further increase in solid deformability, all three interfacial modes become unstable. However, the parameters characterizing the solid (shear modulus, thickness, and solid viscosity) can be chosen such that the GL and LL interfaces remain stable (which are otherwise unstable in flow down a rigid incline) at all wavenumbers without the destabilization of LS interface. When the thickness of the top (less viscous) liquid layer is greater, it is more difficult to obtain stable flow configuration by manipulating the solid parameters. When the less viscous liquid is adjacent to the deformable surface, solid deformability always has a destabilizing effect on LL interfacial mode, and it is not possible to simultaneously stabilize both GL and LL interfaces for this configuration.

The finite-product method in the theory of waves and stability

This paper presents a method of analysing the dispersion relation and field shape of any type of wave field for which the dispersion relation is transcendental. The method involves replacing each transcendental term in the dispersion relation by a finite-product polynomial. The finite products chosen must be consistent with the low-frequency, low-wavenumber limit; but the method is nevertheless accurate up to high frequencies and high wavenumbers. Full details of the method are presented for a non-trivial example, that of anti-symmetric elastic waves in a layer; the method gives a sequence of polynomial approximations to the dispersion relation of extraordinary accuracy over an enormous range of frequencies and wavenumbers. It is proved that the method is accurate because certain gamma-function expressions, which occur as ratios of transcendental terms to finite products, largely cancel out, nullifying Runge’s phenomenon. The polynomial approximations, which are unrelated to Taylor series, introduce no spurious branches into the dispersion relation, and are ideal for numerical computation. The method is potentially useful for a very wide range of problems in wave theory and stability theory.

Time-harmonic wave propagation in a pre-stressed compressible elastic bi-material laminate

The dispersive behaviour of time-harmonic waves propagating along a principal direction in a perfectly bonded pre-stressed compressible elastic bi-material laminate is considered. The dispersion relation which relates wave speed and wavenumber is obtained by formulating the incremental boundary value problem and the use of the propagator matrix technique. At the low wavenumber limit, depending on the pre-stress, both the fundamental mode and the next lowest mode may have finite phase speeds. For the higher modes which have infinite phase speeds in the low wavenumber region, an expression to determine the cut-off frequencies is obtained. At the high wavenumber limit, the phase speeds of the fundamental mode and higher modes tend to phase speeds of the surface wave, the interfacial wave or the limiting phase speed of the composite. For numerical examples, either a two-parameter compressible neo-Hookean material or a two-parameter compressible Varga material is assumed.

Wave propagation in pre-stressed imperfectly bonded compressible elastic layered composites

The dispersive behavior of in-plane time-harmonic waves propagating in a pre-stressed imperfectly bonded incompressible symmetric layered composite, has been analyzed recently, symmetric waves by Leungvichcharoen and Wijeyewickrema [Leungvichcharoen, S., Wijeyewickrema, A.C., 2003. Dispersion effects of extensional waves in pre-stressed imperfectly bonded incompressible elastic layered composites. Wave Motion 38, 311–325] and anti-symmetric waves by Leungvichcharoen et al. [Leungvichcharoen, S., Wijeyewickrema, A.C., Yamamoto, T., 2004. Anti-symmetric waves in pre-stressed imperfectly bonded incompressible elastic layered composites. International Journal of Solids and Structures 41, 6873–6894]. In the present paper, the corresponding problems for a symmetric layered composite consisting of compressible isotropic elastic materials are considered. The imperfect interface is simulated by a shear-spring type resistance model, which can also accommodate the extreme cases of perfectly bonded and fully slipping interfaces. The dispersion relation is obtained by formulating the incremental boundary-value problem and using the propagator matrix technique. Similar to the case of the incompressible layered composite, the dispersion relations for symmetric and anti-symmetric waves in the compressible layered composite differ from each other only through the elements of the propagator matrix associated with the inner layer. The behavior of the dispersion curves for symmetric waves is for the most part similar to that of anti-symmetric waves at the low and high wavenumber limits. At the low wavenumber limit, depending on the pre-stress, for perfectly bonded and imperfect interface cases, a finite phase speed may exist only for the fundamental mode while other higher modes have an infinite phase speed. However, for a fully slipping interface, at the low wavenumber limit it may be possible for both the fundamental mode and the next lowest mode to have finite phase speeds. For the higher modes which have infinite phase speeds in the low wavenumber region, an expression to determine the cut-off frequencies is obtained. At the high wavenumber limit, the phase speeds of the fundamental mode and the higher modes tend to the phase speeds of the surface wave or the interfacial wave or the limiting phase speed of the composite. For numerical examples, either a compressible two-parameter neo-Hookean material or a compressible two-parameter Varga material is assumed, and the effect of imperfect interfaces on both kind of waves is clearly evident in the numerical results.

Elastohydrodynamic Suppression of Free-Surface Instabilities in Annular Liquid Film Flow Outside Wires and Inside Tubes

The linear stability of gravity-driven annular liquid film flow outside wires and inside tubes is analyzed when the rigid surface is replaced by a deformable (neo-Hookean) solid wall. On a rigid surface, an annular liquid thread becomes unstable due to a Rayleigh-type capillary instability even in the absence of flow. In the presence of flow, the annular liquid film becomes unstable due to a flow-driven, free-surface instability, which occurs over and above the curvature-induced capillary instability. In this paper, a low-wavenumber perturbation analysis is first used to elucidate the effect of wall deformability on the free-surface instability. It is shown that the free-surface instability is completely stabilized in the low-wavenumber limit when the wall is made sufficiently deformable. A numerical method is subsequently used to determine the stability of the system at arbitrary wavenumbers. Results from the numerical solution reveal that the prediction of instability suppression at low wavenumbers extends to finite wavenumbers as well. However, as the solid wall is made deformable even further, the free surface is destabilized at finite wavenumbers by the deformability; in addition, the liquid-solid interface could also become unstable when the solid deformability becomes high. It is demonstrated, however, that there is a sufficient range of shear modulus of the solid where the annular flow is stable at all wavenumbers. The results of this study have implications in wire-coating operations, as well as in biological settings such as closure of lung airways.

Experimental measurements of spectral subgrid-scale Prandtl number in a heated turbulent wake flow, EDQNM predictions and 4/3-law

For turbulent flows including scalar transport, the spectral diffusivity as well as the spectral Prandtl number characterizes the scalar-variance transfer rate from any wavenumber κ to all scales smaller than a specified filter scale. Classical theories predict a plateau behavior at inertial-range wavenumbers with deviations of the diffusivity at scales close to the cutoff wavenumber. However, to date these quantities have not been measured in high Reynolds number flows. To examine realistic turbulence at a moderately high Reynolds number, in this study an experimental measurement technique previously applied to obtain spectral eddy viscosities (Cerutti et al. 2000 Journal of Fluid Mechanics 421 307–338) is extended and applied to the case of scalar eddy diffusivity. We show that the subgrid-scale (SGS) dissipation spectrum of scalar variance can be expressed as the co-spectrum of negative SGS scalar flux and filtered scalar gradient. Using local isotropy, the result is expressed in terms of the streamwise components of the SGS scalar flux and filtered scalar gradient that can be measured in experiments. Using an array of four X-type hot-wire and four I-type cold-wire probes, two-dimensional box-filtered velocities and temperatures in the streamwise (invoking Taylor's hypothesis) and cross-stream directions are obtained at the centerline of a heated wake flow, at a Taylor-scale Reynolds number of 350. From the radial dissipation spectra, the spectral eddy viscosity and Prandtl number are evaluated. The measured spectral eddy viscosity, diffusivity and Prandtl number (the ratio of the spectral eddy viscosity to diffusivity) decrease near the filter scale. They also decrease with decreasing wavenumber near the integral scale of turbulence. Measurement results are compared with predictions from a classical two-point closure (eddy-damped quasi-normal Markovian, EDQNM) applied with a Gaussian filter and finite integral scale. Measurement and predictions are in qualitative agreement (decreasing trends observed near the filter wavenumber), but the details differ especially in the low wavenumber limit. This could be due to the assumption of isotropy which does not hold for the experiment at large scales. We also find significant sensitivity of the EDQNM predictions to free parameters of the eddy-damping term. Analysis in physical space shows that the two-point correlation between the SGS scalar flux and filtered scalar plays an important role to predict a mixed third-order structure function in the von Kármán–Howarth–Kolmogorov equation for the resolved scalar field (the LES equivalent of the 4/3-law).

Stability of two-layer viscoelastic plane Couette flow past a deformable solid layer: implications of fluid viscosity stratification

The linear stability of two-layer plane Couette flow of upper-convected Maxwell (UCM) fluids of thicknesses ( 1 − β ) R and β R , with viscosities μ a and μ b , and relaxation times τ a and τ b past a linear viscoelastic solid layer (of thickness H R , shear modulus G , and viscosity η w ) is determined using a combination of low wavenumber asymptotic analysis and a numerical method. The asymptotic analysis is used to determine the effect of the deformable solid layer on the two-fluid interfacial instability due to elasticity and viscosity stratification in the low wavenumber limit. The asymptotic results show that as the solid layer is made more deformable, the two-fluid interfacial instability could be completely stabilized in the long wave limit, when the non-dimensional parameter Γ = V μ b / ( G R ) ∼ O ( 1 ) exceeds a certain critical value. Here, V is the dimensional velocity of the moving top plate. In marked contrast with the previous results obtained for matched fluid viscosities [V. Shankar, J. Non-Newtonian Fluid Mech., 117 (2004) 163], the present asymptotic results show that the stabilizing or destabilizing nature of the solid layer on the two-fluid interfacial instability is controlled only by the fluid viscosity ratio and is independent of the relaxation times of the two UCM fluids. In general, it is found that the solid layer has a stabilizing effect for β < 0.5 , μ r < 1 and β > 0.5 , μ r > 1 , while it has both stabilizing and destabilizing effects (depending on its thickness H, and the fluid thickness ratio β ) for β > 0.5 , μ r < 1 and β < 0.5 , μ r > 1 . In the absence of the solid layer, the two-fluid interfacial mode is unstable or stable depending on the ratio of relaxation times between the two fluids and the thickness ratio β . It is thus possible, under appropriate combinations of relaxation times, viscosities, and thicknesses of the two fluids, to stabilize (destabilize) the two-fluid interfacial mode by the deformable solid layer while it is unstable (stable) in the absence of the solid layer. Another important result from the present low wavenumber (denoted by k ) analysis is that the non-dimensional solid elasticity parameter Γ required to stabilize or destabilize the two-fluid interfacial mode is an O ( 1 ) quantity in the low- k -limit when μ a ≠ μ b , with numerical values of Γ significantly smaller than 1 for H ∼ O ( 1 ) . Whereas when μ a = μ b our asymptotic analysis (as well as the earlier study) shows that Γ ∝ k − 1 for k ≪ 1 . The results from the low- k -asymptotic analysis are continued numerically to finite values of k and Reynolds number, and the numerical results confirm that the stabilization of the two-fluid interfacial mode by the solid layer extends to finite values of k . However, for short wavelength fluctuations with k ≫ 1 , the fluid velocity perturbations are localized near the two-fluid interface, so the solid layer has no effect on these fluctuations. These short wavelength unstable modes can be stabilized only by the presence of a sufficiently strong interfacial tension between the two fluids. Thus, the present study shows that the viscosity mismatch between the two fluids profoundly changes the stabilizing or destabilizing effect of the deformable solid layer on the two-fluid interfacial mode in viscoelastic fluids when compared to the conclusions reached using matched fluid viscosities.

Anti-symmetric waves in pre-stressed imperfectly bonded incompressible elastic layered composites

The effect of an imperfect interface on the dispersive behavior of in-plane time-harmonic symmetric waves in a pre-stressed incompressible symmetric layered composite, was analyzed recently by Leungvichcharoen and Wijeyewickrema (2003). In the present paper the corresponding case for time harmonic anti-symmetric waves is considered. The bi-material composite consists of incompressible isotropic elastic materials. The imperfect interface is simulated by a shear-spring type resistance model, which can also accommodate the extreme cases of perfectly bonded and fully slipping interfaces. The dispersion relation is obtained by formulating the incremental boundary-value problem and using the propagator matrix technique. The dispersion relations for anti-symmetric and symmetric waves differ from each other only through the elements of the propagator matrix associated with the inner layer. The behavior of the dispersion curves for anti-symmetric waves is for the most part similar to that of symmetric waves at the low and high wavenumber limits. At the low wavenumber limit, depending on the pre-stress for perfectly bonded and imperfect interface cases, a finite phase speed may exist only for the fundamental mode while other higher modes have an infinite phase speed. However, for a fully slipping interface in the low wavenumber region it may be possible for both the fundamental mode and the next lowest mode to have finite phase speeds. For the higher modes which have infinite phase speeds in the low wavenumber region an expression to determine the cut-off frequencies is obtained. At the high wavenumber limit, the phase speeds of the fundamental mode and the higher modes tend to the phase speeds of the surface wave or the interfacial wave or the limiting phase speed of the composite. The bifurcation equation obtained from the dispersion relation yields neutral curves that separate the stable and unstable regions associated with the fundamental mode or the next lowest mode. Numerical examples of dispersion curves are presented, where when the material has to be prescribed either Mooney–Rivlin material or Varga material is assumed. The effect of imperfect interfaces on anti-symmetric waves is clearly evident in the numerical results.

Dispersion effects of extensional waves in pre-stressed imperfectly bonded incompressible elastic layered composites

The effect of an imperfect interface, on time-harmonic extensional wave propagation in a pre-stressed symmetric layered composite is considered. The bimaterial composite consists of incompressible isotropic elastic materials. The shear spring type resistance model employed to simulate the imperfect interface can accommodate the extreme cases of perfect bonding and a fully slipping interface. The dispersion relation obtained by formulating the incremental boundary-value problem and the use of the propagator matrix technique, is analyzed at the low and high wavenumber limits. For the perfectly bonded and imperfect interface cases in the low wavenumber region, only the fundamental mode has a finite phase speed, while other higher modes have an infinite phase speed when the dimensionless wavenumber approaches zero. However, for the fully slipping interface in the low wavenumber region, both the fundamental mode and the next lowest mode have finite phase speeds. In the high wavenumber region, when the dimensionless wavenumber tends to infinity, the phase speeds of the fundamental mode and the higher modes depend on the phase speeds of the surface and interfacial waves and on the limiting phase speed of the composite. An expression to determine the cut-off frequencies is obtained from the dispersion relation. Numerical examples of dispersion curves are presented, where when the material has to be prescribed either Mooney–Rivlin material or Varga material is assumed. The effect of the imperfect interface is clearly evident in the numerical results.