Uni-modal destabilization of a visco-elastic floating ice layer by wind stress

A destabilization by wind stress of a homogeneous visco-elastic ice layer of infinite horizontal extent and finite thickness floating on a water layer of finite depth is studied. The water is assumed to be weakly compressible; the viscous dissipation in the water layer is shown to be negligible compared to that in the ice layer. In the model, we assume that a homogeneous wind shear stress is applied to the upper surface of the ice layer at near field and the compression within the ice layer is fixed below the maximum admissible value, i.e. below the value above which ice can no longer be treated as an elastic material. The effect of viscosity is shown to stabilize the acoustic mode and all the unstable seismic modes that in a purely elastic model, treated by Brevdo and Il'ichev [Brevdo, L., Il'ichev, A., 2001. Multi-modal destabilization of a floating ice layer by wind stress. Cold Reg. Sci. Technol. 33, 77–89], possess unbounded growth rates for a growing wave number. The buckling mode is unstable in the domain of parameters considered. In all the cases treated, the model is marginally absolutely stable. The localized unstable disturbances propagate against the wind. The spatially amplifying waves in the model amplify in the direction opposite to the wind direction. The stability results are well approximated by those in a Kirchoff–Love thin ice plate model.     

Dynamics of viscoelastic liquid filaments: Low capillary number flows

Many applications of viscoelastic free surface flows requiring formation of drops from small nozzles, e.g., ink-jet printing, micro-arraying, and atomization, involve predominantly extensional deformations of liquid filaments. The capillary number, which represents the ratio of viscous to surface tension forces, is small in such processes when drops of water-like liquids are formed. The dynamics of extensional deformations of viscoelastic liquids that are weakly strain hardening, i.e., liquids for which the growth in the extensional viscosity is small and bounded, are here modeled by the Giesekus, FENE-P, and FENE-CR constitutive relations and studied at low capillary numbers using full 2D numerical computations. A new computational algorithm using the general conformation tensor based constitutive equation [M. Pasquali, L.E. Scriven, Theoretical modeling of microstructured liquids: a simple thermodynamic approach, J. Non-Newtonian Fluid Mech. 120 (2004) 101–135] to compute the time dependent viscoelastic free surface flows is presented. DEVSS-TG/SUPG mixed finite element method [M. Pasquali, L.E. Scriven, Free surface flows of polymer solutions with models based on conformation tensor, J. Non-Newtonian Fluid Mech. 108 (2002) 363–409] is used for the spatial discretization and a fully implicit second-order predictor–corrector scheme is used for the time integration. Inertia, capillarity, and viscoelasticity are incorporated in the computations and the free surface shapes are computed along with all the other field variables in a fully coupled way. Among the three models, Giesekus filaments show the most drastic thinning in the low capillary number regime. The dependence of the transient Trouton ratio on the capillary number in the Giesekus model is demonstrated. The elastic unloading near the end plates is investigated using both kinematic [M. Yao, G.H. McKinley, B. Debbaut, Extensional deformation, stress relaxation and necking failure of viscoelastic filaments, J. Non-Newtonian Fluid Mech. 79 (1998) 469–501] and energy analyses. The magnitude of elastic unloading, which increases with growing elasticity, is shown to be the largest for Giesekus filaments, thereby suggesting that necking and elastic unloading are related.     

Bounds for the effective properties of heterogeneous plates

This paper presents new bounds for heterogeneous plates which are similar to the well-known Hashin–Shtrikman bounds, but take into account plate boundary conditions. The Hashin–Shtrikman variational principle is used with a self-adjoint Green-operator with traction-free boundary conditions proposed by the authors. This variational formulation enables to derive lower and upper bounds for the effective in-plane and out-of-plane elastic properties of the plate. Two applications of the general theory are considered: first, in-plane invariant polarization fields are used to recover the “first-order” bounds proposed by Kolpakov [Kolpakov, A.G., 1999. Variational principles for stiffnesses of a non-homogeneous plate. J. Meth. Phys. Solids 47, 2075–2092] for general heterogeneous plates; next, “second-order bounds” for n-phase plates whose constituents are statistically homogeneous in the in-plane directions are obtained. The results related to a two-phase material made of elastic isotropic materials are shown. The “second-order” bounds for the plate elastic properties are compared with the plate properties of homogeneous plates made of materials having an elasticity tensor computed from “second-order” Hashin–Shtrikman bounds in an infinite domain.     

Effect of air gap and order of plates on ballistic resistance of two layered armor

The high velocity normal impact of a three-dimensional rigid conical impactor penetrating into a two-layered ductile armor with an air gap is studied using a simplified model for an impactor–armor interaction. The goal of the study is to investigate analytically the dependence of the ballistic resistance of the armor on the order of the plates in the armor and on the width of an air gap between the plates. It is found that the ratio between the values of a single parameter depending on the material properties of the plates determines this dependence in both cases. This parameter characterizes the properties of the material of the plate; for the most widely used models of impactor–armor interaction, it is the ratio of the distortion pressure to the density of the plate.     

Boundary conditions for stokes flows near a porous membrane

A theoretical development is carried out to model the boundary conditions for Stokes flows near a porous membrane, which, in general, allows non-zero slip as well as normal flow at the surface. Two types of models are treated: an infinitesimally thin plate with a periodic array of circular apertures and a series of parallel slits. For Stokes flows, the mean normal flux and slip velocity are proportional to the pressure difference across the membrane and the average shear stress at the membrane, respectively. The appropriate proportionality constants which depend on the membrane geometry are calculated as functions of the porosity. An interesting feature of the results is that the slip at the membrane has, in general, a direction different from that of the applied shear for these models.     

Application of the Discrete Fourier Transform to Study the Dynamics of Wave Packets

A method for solving equations that describe the dynamics of wave packets of the Tollmien–Schlichting waves in the boundary layer is proposed. The method of splitting the initial problem into the linear and nonlinear parts at each time step is used. The linear part is resolved by using an equation for spectral components of the wave packet with a subsequent Fourier transform from the space of wavenumbers to the physical space. A system of ordinary differential equations is solved in the physical space. The Fourier transform is performed by means of the library procedure of the fast Fourier transform. As examples, the problems solved were the linear dynamics of the wave packet concentrated in the vicinity of the instability region (i.e., a set of wave vectors in the space of wavenumbers for which the imaginary part of the eigenfrequency of the Tollmien–Schlichting waves is positive) and the nonlinear dynamics of the wave packet overlapping the instability region.     

Conical flow breakdown in non-free shock-wave/boundary-layer interaction

The non-free interaction between a shock wave and the boundary layer on a swept plate set at incidence in the undisturbed flow is studied using different experimental methods including special laser techniques for visualizing supersonic conical gas flows. It is shown that under shock-layer conditions the non-free interaction can lead to conical flow breakdown before the incident shock reaches the leading edge of the plate.Translated from Izvestiya Rossiiskoi Academii Nauk, Mekhanika Zhidkosti i Gaza, No. 6, 2004, pp. 45–58. Original Russian Text Copyright © 2004 by Zubin and Ostapenko.     

Instability of two-layer film flow along an inclined surface

In [1–3] the method of expansion in a small wave number is used to investigate stability of two-layer flows; the results are valid for the neutral curves and in their neighborhood. Here, the eigenvalue problem is solved numerically, the wave disturbances are considered over the entire region of instability and the effect of the governing parameters on the characteristics of the most unstable disturbances is established.Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No.2, pp. 10–18, March–April, 1992.     

Theoretical study of the effect of liquid desiccant mass flow rate on the performance of a cross flow parallel-plate liquid desiccant-air dehumidifier

A computer simulation using MATLAB is investigated to predict the distribution of air stream parameters (humidity ratio and temperature) as well as desiccant parameters (temperature and concentration) inside the parallel plate absorber. The present absorber consists of fourteen parallel plates with a surface area per unit volume ratio of 80 m²/m³. Calcium chloride as a liquid desiccant flows through the top of the plates to the bottom while the air flows through the gap between the plates making it a cross flow configuration. The model results show the effect of desiccant mass flow rate on the performance of the dehumidifier (moisture removal and dehumidifier effectiveness). Performance comparisons between present cross-flow dehumidifier and another experimental cross-flow dehumidifier in the literature are carried out. The simulation is expected to help in optimizing of a cross flow dehumidifier.     

A 1D-Averaged Model for Stable and Unstable Miscible Flows in Porous Media with Varying Péclet Numbers and Aspect Ratios

This paper deals with reducing the number of spatial dimensions of the models used to solve stable and unstable miscible flows in saturated and homogeneous porous media. Unstable miscible displacements occur when a fluid displaces another fluid of higher viscosity, with which it can fully mix. Stable flows occur if the displaced fluid is less viscous than the displacing one. First, a 1D-averaged model is identified, capable of accurately describing unstable flows at high Péclet numbers. Second, another 1D-averaged model is determined, capable of accurately predicting miscible displacements at low Péclet numbers. Third, a new model is proposed, for any Péclet number and for both stable and unstable flows, as a combination of the previous two models. This combination involves three parameters whose values depend on the dimensionless numbers of the problem, namely, the viscosity ratio M, the Péclet number P_e, the aspect ratio A, and the dispersion length ratio ε. These parameters are computed for several values of M, P_e, A with ε=1 by matching results from direct 2D simulations, obtained from a numerical model previously validated against experimental data. It is found that a 1D-averaged model combining an extended version of the Todd–Longstaff model and the diffusive term of the 1D-miscible model forms an accurate general model for miscible displacements in homogeneous porous media. This paper also provides a large set of data computed from high-resolution 2D simulations of unstable miscible displacements.     

Nonlinear Oscillations of Viscoelastic Rectangular Plates

Nonlinear oscillations of viscoelastic simply supported rectangular plates are studied by assuming the Voigt–Kelvin constitutive model. Using Hamilton's principle in conjunction with the kinematics associated with Kirchhoff's plate model, the governing equations of motion including the effect of damping are represented in terms of the transversal deflection and a stress function. Utilizing the Bubnov–Galerkin method, the nonlinear partial differential equations are reduced to an ordinary differential equation which is studied geometrically by approximate construction of the Poincaré maps. Explicit expressions are given for periodic solutions.     

Dynamic stress concentrations in thick plates with two holes based on refined theory

Based on complex variables and conformal mapping, the elastic wave scat- tering and dynamic stress concentrations in the plates with two holes are studied by the refined dynamic equation of plate bending. The problem to be solved is changed to a set of infinite algebraic equations by an orthogonM function expansion method. As examples, under free boundary conditions, the numerical results of the dynamic moment concen- tration factors in the plates with two circular holes are computed. The results indicate that the parameters such as the incident wave number, the thickness of plates, and the spacing between holes have great effects on the dynamic stress distributions. The results are accurate because the refined equation is derived without any engineering hypothese.     

Effects of partial crack–face contact for the bending of thin shell structures

The physical occurrence that crack surfaces are in contact at the compressive edges when a flat or a shell is subjected to a bending load has been recognized. This article presents a theoretical analysis of crack–face contact effect on the stress intensity factor of various shell structures such as spherical shell, cylindrical shell containing an axial crack, cylindrical shell containing a circumferential crack and shell with two non-zero curvatures, under a bending load. The formulation of the problem is based on the shear deformation theory, incorporating crack–face contact by introducing distributed force at the compressive edge. Material orthotropy is concerned in this analysis. Three-dimensional finite element analysis (FEA) is conduced to compare with the theoretical solution. It is found that due to curvature effect crack–face contact behavior in shells differs from that in flat plates, in that partial contact of crack surfaces may occur in shells, depending on the shell curvature and the nature of the bending load. Crack–face contact has significant influence on the stress intensity factor and it increases the membrane component but decreases the bending component.     

Nonequilibrium stretching dynamics of dilute and entangled linear polymers in extensional flow

We propose an extension of the FENE-CR model for dilute polymer solutions [M.D. Chilcott, J.M. Rallison, Creeping flow of dilute polymer solutions past cylinders and spheres, J. Non-Newtonian Fluid Mech. 29 (1988) 382–432] and the Rouse-CCR tube model for linear entangled polymers [A.E. Likhtman, R.S. Graham, Simple constitutive equation for linear polymer melts derived from molecular theory: Rolie–Poly equation, J. Non-Newtonian Fluid Mech. 114 (2003) 1–12], to describe the nonequilibrium stretching dynamics of polymer chains in strong extensional flows. The resulting models, designed to capture the progressive changes in the average internal structure (kinked state) of the polymer chain, include an ‘effective’ maximum contour length that depends on local flow dynamics. The rheological behavior of the modified models is compared with various results already published in the literature for entangled polystyrene solutions, and for the Kramers chain model (dilute polymer solutions). It is shown that the FENE-CR model with an ‘effective’ maximum contour length is able to describe correctly the hysteretic behavior in stress versus birefringence in start-up of uniaxial extensional flow and subsequent relaxation also observed and computed by Doyle et al. [P.S. Doyle, E.S.G. Shaqfeh, G.H. McKinley, S.H. Spiegelberg, Relaxation of dilute polymer solutions following extensional flow, J. Non-Newtonian Fluid Mech. 76 (1998) 79–110] and Li and Larson [L. Li, R.G. Larson, Excluded volume effects on the birefringence and stress of dilute polymer solutions in extensional flow, Rheol. Acta 39 (2000) 419–427] using Brownian dynamics simulations of bead–spring model. The Rolie–Poly model with an ‘effective’ maximum contour length exhibits a less pronounced hysteretic behavior in stress versus birefringence in start-up of uniaxial extensional flow and subsequent relaxation.     

Forward roll coating flows of viscoelastic liquids

Roll coating is distinguished by the use of one or more gaps between rotating cylinders to meter and apply a liquid layer to a substrate. Except at low speed, the two-dimensional film splitting flow that occurs in forward roll coating is unstable; a three-dimensional steady flow sets in, resulting in more or less regular stripes in the machine direction. For Newtonian liquids, the stability of the two-dimensional flow is determined by the competition of capillary and viscous forces: the onset of meniscus nonuniformity is marked by a critical value of the capillary number. Although most of the liquids coated industrially are non-Newtonian polymeric solutions and dispersions, most of the theoretical analyses of film splitting flows relied on the Newtonian model. Non-Newtonian behavior can drastically change the nature of the flow near the free surface; when minute amounts of flexible polymer are present, the onset of the three-dimensional instability occurs at much lower speeds than in the Newtonian case.Forward roll coating flow is analyzed here with two differential constitutive models, the Oldroyd-B and the FENE-P equations. The results show that the elastic stresses change the flow near the film splitting meniscus by reducing and eventually eliminating the recirculation present at low capillary number. When the recirculation disappears, the difference of the tangential and normal stresses (i.e., the hoop stress) at the free surface becomes positive and grows dramatically with fluid elasticity, which explains how viscoelasticity destabilizes the flow in terms of the analysis of Graham [M.D. Graham, Interfacial hoop stress and instability of viscoelastic free surface flows, Phys. Fluids 15 (2003) 1702–1710].     

Modeling of failure wave propagation in impact compression of brittle materials (Glasses)

For a plane brittle failure wave in a material loaded by an elastic compression wave, an analytic solution is obtained on the basis of a model with a Drucker-Prager type condition and a given value of the failure wave propagation velocity. The wave failure model supplemented with threshold criteria is generalized to two-dimensional flows in breaking plates. Collision of glass plates with a rigid wall is modelled and three- and two-wave failure structures in the plate are obtained. Some data on the leading elastic wave decay and the arrest of the failure wave under the action of the overtaking unloading that propagates from the region of spread of the fractured material near the wall are obtained.     

On the application of en-methods to three-dimensional boundary-layer flows

Extension of the eⁿ-method from two-dimensional to three-dimensional boundary-layer flows has not been straightforward. Confusion has centred on whether to use temporal or spatial stability theories, conversion between the two approaches, and the choice of integration path. The aim of this study is to clarify the confusion about the direction and magnitude of maximum growth in convectively unstable three-dimensional non-parallel boundary layers. To this end, the time-asymptotic response of the boundary layer to an impulsive point excitation is considered. Since all frequencies and all wavenumbers are excited by an impulsive point source, the most amplified component of the response is equivalent to the result of maximizing the growth over arbitrary choices of harmonic point excitation; the standard eⁿ-approach. The impulse response is calculated using a spatial steepest-descent method, which is distinct from the earlier Cebeci–Stewartson method. It is necessary to allow both time and spanwise distance to become complex during integration, but with the constraint that both are real at the end point. This method has been applied to the two-dimensional Blasius boundary layer, for which validation of the method is more straightforward, and also to a three-dimensional Falkner–Skan–Cooke (with non-zero pressure gradient and sweep) boundary layer. Dimensional frequencies and spanwise wavenumbers of propagating components are kept constant (although not necessarily real), as is physically relevant to steady flows with spatial inhomogeneity in the chordwise direction only. With this method a spatial approach is taken without having to make a priori choices about the value of disturbance frequency or wavenumber. Further, purely by choosing a downstream observation point, it is possible to find the maximum-amplitude component directly without having to calculate the entire impulse response (or wave packet). If the flow is susceptible to more than one convective instability mode, provided the modes are separated in the frequency–wavenumber space, separate n-factors can be calculated for each mode. Wave-packet propagation in the Ekman layer (a strictly parallel three-dimensional boundary layer) is also discussed to draw comparisons between the conditions for maximum growth in parallel and non-parallel boundary layers.     

Detonation Wave Propagation in Rotational Gas Flows

This paper studies the propagation of detonation and shock waves in vortex gas flows, in which the initial pressure, density, and velocity are generally functions of the coordinate — the distance from the symmetry axis. Rotational axisymmetric flow having a transverse velocity component in addition to a nonuniform longitudinal velocity is considered. The possibility of propagation of Chapman–Jouguet detonation waves in rotating flows is analyzed. A necessary conditions for the existence of a Chapman–Jouguet wave is obtained.     

Strong recondensation in a one-component and a two-component rarefied gas for an arbitrary value of Knudsen number

As is known, surface phenomena such as evaporation, absorption, and reflection of molecules from the surface of a body depend strongly on its temperature [1–5]. This leads to the establishment of a flow of a substance between two surfaces maintained at different temperatures (recondensation). The phenomenon of recondensation was studied in kinetic theory comparatively long ago. However, up to the present, only the case of small mass flows in a onecomponent gas has been investigated completely [3,4]. Meanwhile it is clear that by the creation of appropriate conditions we can obtain considerable flows of the recondensing substance, so that the mass-transfer rate will be of the order of the molecular thermal velocity. Such a numerical solution of the problem with strong mass flows along the normal to the surface for small Knudsen numbers for a model Boltzmann kinetic equation was obtained in [7]. In this study we numerically solve the problem of strong recondensation between two infinite parallel plates over a wide range of Knudsen numbers for a one-component and a two-component gas, on the basis of the model Boltzmann kinetic equation [6] for a one-component gas and the model Boltzmann kinetic equation for a binary mixture in the form assumed by Hamel [8], for a ratio of the plate temperatures equal to ten. We also investigate the effect of the relative plate motion on the recondensation flow.Moscow. Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 5, pp. 130–138, September–October, 1972.     

A timestepper approach for the systematic bifurcation and stability analysis of polymer extrusion dynamics

We discuss how matrix-free/timestepper algorithms can efficiently be used with dynamic non-Newtonian fluid mechanics simulators in performing systematic stability/bifurcation analysis. The timestepper approach to bifurcation analysis of large-scale systems is applied to the plane Poiseuille flow of an Oldroyd-B fluid with non-monotonic slip at the wall, in order to further investigate a mechanism of extrusion instability based on the combination of viscoelasticity and non-monotonic slip. Due to the non-monotonicity of the slip equation the resulting steady-state flow curve is non-monotonic and unstable steady states appear in the negative-slope regime. It has been known that self-sustained oscillations of the pressure gradient are obtained when an unstable steady state is perturbed [M.M. Fyrillas, G.C. Georgiou, D. Vlassopoulos, S.G. Hatzikiriakos, A mechanism for extrusion instabilities in polymer melts, Polymer Eng. Sci. 39 (1999) 2498–2504].Treating the simulator of a distributed parameter model describing the dynamics of the above flow as an input–output “black-box” timestepper of the state variables, stable and unstable branches of both equilibrium and periodic oscillating solutions are computed and their stability is examined. It is shown for the first time how equilibrium solutions lose stability to oscillating ones through a subcritical Hopf bifurcation point which generates a branch of unstable limit cycles and how the stable periodic solutions lose their stability through a critical point which marks the onset of the unstable limit cycles. This implicates the coexistence of stable equilibria with stable and unstable periodic solutions in a narrow range of volumetric flow rates.