Solution existence conditions for elastoplastic constitutive models of granular materials

In this paper, the conditions of solution existence for stress rates under given strain rates are investigated. The focus of the solution existence investigation is on the non-associated flow rule and elastic stress–strain relationship. Granular materials characterized with strong non-associated plastic flows are used as a particular example for analysis. Various flow rules for granular materials are analyzed, including Rowe’s, Roscoe’s flow rules and their modified versions. In the elastic stress–strain relationships of materials, the effects of Poisson’s ratio on solution existence are investigated. Both isotropic and anisotropic elasticity are considered. Given a granular material and its states, it is found that there exists a critical Poisson’s ratio for a particular non-associated flow rule. When the Poisson’s ratio of a material is above this critical Poisson’s ratio, its constitutive model is susceptible to solution non-existence. It is suggested that special attentions should be paid to the selection of material Poisson’s ratio and non-associated flow rule to ensure the existence of elastoplastic solutions.     

Existence criteria for reflection configurations in shock–vortex interactions

The length scale criteria is widely accepted as an explanation for transition and hence existence of different shock wave reflection configurations in pseudo-steady flows. However, there has not been any attempt to validate this criteria using information obtained from a time-dependent numerical simulation. A high resolution time-dependent numerical simulation in pseudo-steady flow is carried out in the present work. Time-dependent numerical data is used to calculate flow features in a laboratory frame of reference to verify validity of the length scale criteria for existence of different shock wave reflection configurations in pseudo-steady flow. This analysis is then extended to the study of unsteady shock wave reflection configurations in shock–vortex interactions. It is shown that the existence of regular reflection (RR) and Mach reflection (MR) configurations in an unsteady flowfield resulting from shock–vortex interactions can also be explained locally based on limiting conditions similar to that prescribed by the length scale criteria for pseudo-steady flow.

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Spanwise characteristics of the separated flow in a suddenly expanding duct

Summary The secondary flow due to the growth of the streamwise vortices near the side walls serves to diminish the spanwise uniformity of the time-mean flow properties. In the region adjacent to the side walls, momentum mixing is enhanced due to the existence of the secondary flow and the separated shear layer spreads faster. There is a corresponding increase in the non-coherent turbulence in this region near the side walls. The increased spreading rates and overall turbulence in the shear layer, in turn, tend to suppress the rolling-up of the separated shear layer into organized structures. This effect is rapidly carried into the core two-dimensional flow region as the streamwise vortex grows under the influence of the adverse pressure gradient. The surface visualizations provide further evidence of the existence of secondary flows near the side walls.     

Orientation instability of shear flow of a nematic liquid crystal

The instability of shear flow of a nematic liquid crystal layer is studied. The case when the orientation vector and the flow velocity vector are parallel is considered. It is shown that the orientation instability of this flow is possible if the anchoring boundary condition is weak and if the splay-bend constants in the Frank energy are taken into account. For this type of instability, periodic structures are possible to appear. Their wave vector belongs to the plane of flow and is perpendicular to the velocity vector. The medium parameters are estimated on the basis of the existence condition for this instability. The period of the appearing periodic structures is evaluated.     

Two dimensional Stokes flows with slip-stick boundary conditions

We derive the necessary condition for the existence of a solution of the biharmonic equation for general boundary conditions. Making use of this result, we derive a necessary condition in terms of the basic flow, for the existence of a solution for Stokes flow past a circular cylinder with slip-stick boundary conditions. We state and prove a circle theorem when the basic flow satisfies the necessary condition.     

On the existence of indeterminate solutions to the equations of motion under non-associated flow

Under certain conditions, an indeterminate solution exists to the equations of motion for dynamic elastic–plastic deformation of materials using constitutive laws based on non-associated flow that suggests that an initially unbounded dynamic perturbation in the stress can develop from a quiescent state on the yield surface. The existence of this indeterminate solution has been alleged to discourage use of non-associated flow rules for both dynamic and quasi-static analysis theoretically. It is shown in this paper that the indeterminate solution that may solve the equations of motion is intrinsically dynamic, and it determinately goes to zero in the quasi-static limit regardless of other indeterminate parameters. Consequently, the existence of this unstable dynamic solution has no impact on stability and use of non-associated flow rules for analysis of the quasi-static problem. More importantly, for dynamic applications, it is also shown that the indeterminate solution solves the equations of motion only if critical restrictions are applied to the constitutive equations such that the effective modulus during loading is constant and the direction of the perturbation is unidirectional over a finite time interval. It is shown that common components of the constitutive laws used in metal forming and deformation analysis are inconsistent with these restrictions. So, these common models can be generalized to include non-associated flow for analysis of the dynamic problem without concern that the solution will become indeterminate.     

Experimental and numerical study of a spiral structure at the periphery of an aspirated rotor–stator cavity

The purpose of this paper is to study the range of existence, the process of transition and the phase velocity of the spiral structure in an aspirated rotor–stator cavity. Experience shows that for a given flow rate and rotation, a whole range of azimuthal wave numbers are possible. Some are highly stable while others on the fringes of this range are subject to multiple transitions that depend on the fluctuations of the flow. Numerical simulation offers the advantage of enabling control over the wave number and the disturbance of the flow. Both approaches enable us to better understand the dynamics of this instability.     

Motion of Elastic Thin Films by Anisotropic Surface Diffusion with Curvature Regularization

In this paper, we prove short time existence, uniqueness, and regularity for a surface diffusion evolution equation with curvature regularization in the context of epitaxially strained two-dimensional films. This is achieved by using the H ^?1-gradient flow structure of the evolution law, via De Giorgi??s minimizing movements. This seems to be the first short time existence result for a surface diffusion type geometric evolution equation in the presence of elasticity.     

Estimation of velocity eigenfunction and vorticity distributions from the timeline visualization technique

For the case of quasi-periodic flow, it is demonstrated that use of the hydrogen bubble timeline method leads to reasonable estimates of the eigenfunction of the streamwise velocity fluctuation. Both amplitude and phase distributions across an unstable wake flow are well-approximated. It is shown that the vorticity extrema, as well as the degree of concentration of vorticity, are in good agreement with those calculated from linear stability theory. A critical assessment is given of the possible uncertainties associated with this technique: the existence of a finite, but unknown cross-stream velocity component; bubble rise due to buoyancy effects; wake defect created downstream of the bubble wire; and resolution of the digitized image. Furthermore, the uncertainty in the streamwise velocity, arising from existence of a finite cross-stream velocity component, is actually less than that corresponding to a single-element hot film probe over certain regimes of operation.     

Elliptic cylinder in inviscid shear flow

We examine the flow of a plane parallel inviscid stream about an elliptic contour. There is vorticity far ahead of the body because of nonuniformity of the velocity profile. In the case of a small vorticity parameter the velocity profile will be parabolic. In contrast with [1] and [2], we assume the existence of additional circulational flow around the contour. The magnitude of this flow circulation is determined from the condition under which the flow leaves the trailing edge of the body (the analog of the Chaplygin-Zhukovskii postulate in potential flow).The results obtained in this study can be used, in particular, to evaluate the flow past a two-dimensional body in the wake behind another body.The author wishes to thank G. A. Dombrovskii for his interest in this study.     

Diffusion velocity for a three-dimensional vorticity field

A discussion is presented on the existence of a diffusion velocity for the vorticity vector that satisfies extensions of the Helmholtz vortex laws in a three-dimensional, incompressible, viscous fluid flow. A general form for the diffusion velocity is derived for a complex-lamellar vorticity field that satisfies the property that circulation is invariant about a region that is advected with the sum of the fluid velocity and the diffusion velocity. A consequence of this property is that vortex lines will be material lines with respect to this combined velocity field. The question of existence of diffusion velocity for a general three-dimensional vorticity field is shown to be equivalent to the question of existence of solutions of a certain Fredholm equation of the first kind. An example is given for which it is shown that a diffusion velocity satisfying this property does not, in general, exist. Properties of the simple expression for diffusion velocity for a complex-lamellar vorticity field are examined when applied to the more general case of an arbitrary three-dimensional flow. It is found that this form of diffusion velocity, while not satisfying the condition of circulation invariance, nevertheless has certain desirable properties for computation of viscous flows using Lagrangian vortex methods. The significance and structure of the noncomplex-lamellar part of the viscous diffusion term is examined for the special case of decaying homogeneous turbulence.     

Existence of Stationary Supersonic Flows Past a Pointed Body

In this paper we study the mathematical aspects of the stationary supersonic flow past a non-axisymmetric curved pointed body. The flow is described by a steady potential flow equation, which is a quasilinear hyperbolic equation of second order. We prove the local existence of the solution to this problem with a pointed shock attached at the tip of the pointed body, provided the pointed body is a perturbation of a circular cone, and the vertex angle of the approximate cone of the pointed body is less than a critical value. The solution is smooth in between the shock and the surface of the body. Consequently, such a structure of flow near the tip of the pointed body and its stability is verified mathematically. Accepted October 13, 2000?Published online January 22, 2001     

Low Reynolds number flow inside straight micro channels with irregular cross sections

This paper presents an analysis of the compound effect of finite temperature differences and fluid friction on the existence of an optimum laminar flow regime in singly connected micro channels with complex free flow area cross sections. A widespread conviction has been established that the two competing irreversibility sources in a channel flow with heat transfer lead to the existence of an optimum flow regime. The results presented in this paper clearly shows the opposite. When an objective function is represented by the entropy generation rate per unit heat capacity rate of the fluid stream, the thermodynamic optimum flow regime represents a rather rare occurrence in the laminar region of irregularly shaped ducts. The presence of an extremum is more probable for very small diameters, the ones of an order of magnitude of O(≤10⁻³ m). The analysis is performed for selected ranges of relevant geometric, flow, and thermal parameters of a set of straight micro channels with irregular free flow area cross-sections. The following geometries of the free flow area cross section were investigated: (i) sine duct, (ii) circular duct, (iii) elliptical duct, (iv) moon-shaped ducts, and (v) four-cuspped duct. The range of Reynolds numbers has been established between O(10²) and O(10⁴). The existence of the objective function minimum is confirmed for ducts with an irregular cross section only for very small hydraulic diameters. These minima are relatively weak, and as a general rule, the sets of optimum parameters are close to the onset of turbulence or possibly even in the transitional or turbulent regions. Received on 10 November 1998     

Existence of Pulsating Roll-Waves for the Saint Venant System

The phenomenon of roll-waves occurs when shallow water flows down open inclined channels. This flow is described by the Saint Venant’s equations with a friction term due to Chezy. In the case of a flat bottom, their existence (as entropic and periodic travelling waves) follows from a classical work due to DRESSLER [6]. The aim of this paper is to prove the existence of roll-waves when the bottom is modulated by a small periodic perturbation. Following JIN and KATSOULAKIS [15], we first compute a Burgers-type equation which possesses “pulsating” roll-waves (the wave speed oscillates around an average velocity). We prove, in a mathematically rigorous fashion, the existence of these solutions.     

Statistics of contact force network in dense granular matter

We investigate the distribution of contact forces in a static granular system and in annular shear granular flow, using the discrete element method, and considering the influences of both packing fraction and friction coefficient. We find the existence of a critical packing fraction. If the packing fraction is lower than this critical value, all contact forces in granular system vanish. For shear granular flow, the critical packing fraction is significantly smaller than that for static granular system. The ...     

Inviscid instability of two-fluid free surface flow down an incline

The inviscid temporal stability analysis of two-fluid parallel shear flow with a free surface, down an incline, is studied. The velocity profiles are chosen as piecewise-linear with two limbs. The analysis reveals the existence of unstable inviscid modes, arising due to wave interaction between the free surface and the shear-jump interface. Surface tension decreases the maximum growth rate of the dominant disturbance. Interestingly, in some limits, surface tension destabilises extremely short waves in this flow. This can happen because of the interaction with the shear-jump interface. This flow may be compared with a corresponding viscous two-fluid flow. Though viscosity modifies the stability properties of the flow system both qualitatively and quantitatively, there is qualitative agreement between the viscous and inviscid stability analysis when the less viscous fluid is closer to the free surface.     

Champ dynamique dans une cellule d'échangeur à vortex : 1. Régime d'entrée laminaire

Numerical simulation and LDV measurements are performed for laminar inlet flow condition in a vortex exchange chamber with isothermal conditions. Newtonian and non-Newtonian fluids were used. Experimental measurements are compared to results of numerical computations with good agreement. Both show the existence of a secondary flow generated by hydrodynamical instabilities due to streamline curvature. The main vortex flow appears then to be restricted by this secondary flow.     

Isentropic Gas Flow for the Compressible Euler Equation in a Nozzle

We study the motion of isentropic gas in a nozzle. Nozzles are used to increase the thrust of engines or to accelerate a flow from subsonic to supersonic. Nozzles are essential parts for jet engines, rocket engines and supersonicwind tunnels. In the present paper, we consider unsteady flow, which is governed by the compressible Euler equation, and prove the existence of global solutions for the Cauchy problem. For this problem, the existence theorem has already been obtained for initial data away from the sonic state, (Liu in Commun Math Phys 68:141–172, 1979). Here, we are interested in the transonic flow, which is essential for engineering and physics. Although the transonic flow has recently been studied (Tsuge in J Math Kyoto Univ 46:457–524, 2006; Lu in Nonlinear Anal Real World Appl 12:2802–2810, 2011), these papers assume monotonicity of the cross section area. Here, we consider the transonic flow in a nozzle with a general cross section area. When we prove global existence, the most difficult point is obtaining a bounded estimate for approximate solutions. To overcome this, we employ a new invariant region that depends on the space variable. Moreover, we introduce a modified Godunov scheme. The corresponding approximate solutions consist of piecewise steady-state solutions of an auxiliary equation, which yield a desired bounded estimate. In order to prove their convergence, we use the compensated compactness framework.     

A simplified method proposal for practical determination of aperiodic two-phase flow instability

A short outline of the instability phenomenon is given. Two methods are explained for the problem solution, i.e. the practical determination of the existence or not of an aperiodic instability and, in the case of existence, what to do to eliminate it. The first method substantially comes from that used by boiler designers. A second, new method is suggested: this is a manual, simple method. The two methods are compared: the new one is reliable, and has the great advantage of the simplicity and the possibility of specifying the diaphragm at the pipe mouth for eliminating the two-phase flow instability.     

Particle dynamics in dense shear granular flow

The particle dynamics in an annular shear granular flow is studied using the discrete element method, and the influences of packing fraction, shear rate and friction coefficient are analyzed. We demonstrate the existence of a critical packing fraction exists in the shear granular flow. When the packing fraction is lower than this critical value, the mean tangential velocity profile exhibits a rate-independent feature. However, when the packing fraction exceeds this critical value, the tangential velocity profile becomes rate-dependent and varies gradually from linear to nonlinear with increasing shear rate. Furthermore, we find a continuous transition from the unjammed state to the jammed state in a shear granular flow as the packing fraction increases. In this transforming process, the force distribution varies distinctly and the contact force network also exhibits different features.