Showing papers by "Kumbakonam R. Rajagopal published in 1999"

Intelligent cruise control systems and traffic flow stability

Abstract: There are two kinds of stability associated with traffic flow problems – string stability (or car-following stability) and traffic flow stability. We provide a clear distinction between traffic flow stability and string stability, and such a distinction has not been recognized in the literature, thus far. String stability is stability with respect to intervehicular spacing; intuitively, it ensures the knowledge of the position and velocity of every vehicle in the traffic, within reasonable bounds of error, from the knowledge of the position and velocity of a vehicle in the traffic. String stability is analyzed without adding vehicles to or removing vehicles from the traffic. On the other hand, traffic flow stability deals with the evolution of traffic velocity and density in response to the addition and/or removal of vehicles from the flow. Traffic flow stability can be guaranteed only if the velocity and density solutions of the coupled set of equations is stable, i.e., only if stability with respect to automatic vehicle following and stability with respect to density evolution is guaranteed. Therefore, the flow stability and critical capacity of any section of a highway is dependent not only on the vehicle following control laws and the information used in their synthesis, but also on the spacing policy employed by the control system. Such a dependence has practical consequences in the choice of a spacing policy for adaptive cruise control laws and on the stability of the traffic flow consisting of vehicles equipped with adaptive cruise control features on the existing and future highways. This critical dependence is the subject of investigation here.

The effect of the slip boundary condition on the flow of fluids in a channel

Abstract: The assumption that a liquid adheres to a solid boundary (“no-slip” boundary condition) is one of the central tenets of the Navier-Stokes theory. However, there are situations wherein this assumption does not hold. In this paper we investigate the consequences of slip at the wall on the flow of a linearly viscous fluid in a channel. Usually, the slip is assumed to depend on the shear stress at the wall. However, a number of experiments suggests that the slip velocity also depends on the normal stress. Thus, we investigate the flow of a linearly viscous fluid when the slip depends on both the shear stress and the normal stress. In regions where the slip velocity depends strongly on the normal stress, the flow field in a channel is not fully developed and rectilinear flow is not possible. Also, it is shown that, in general, traditional methods such as the Mooney method cannot be used for calculating the slip velocity.

On the thermomechanics of shape memory wires

Abstract: The thermomechanical behavior of a shape memory wire is modeled based on a theory that takes cognizance of the fact that the body can possess multiple natural configurations [1]. The constitutive equations are developed by first constructing the form of the Helmholtz potential (based on different modes of energy storage), and dissipation mechanisms. The internal energy includes contributions from the strain energy, the latent energy, the interfacial energy and thermal energy. The entropy of the system includes the"entropy jump" associated with the phase transition.¶The role of the rate of mechanical dissipation as a mechanism for entropy generation and its importance in describing the hysteretic behavior is brought out by considering the difference between hysteretic and non-hysteretic (dissipation-less) behavior.¶Finally, simple linear or quadratic forms are assumed for the various constitutive functions and the full shape memory response is modeled. A procedure for the determination of the constants is also indicated and the constants for two systems (CuZnAl and NiTi) are calculated from published experimental data (see [2, 3]). The predictions of the theory show remarkable agreement with the experimental data. However, some of the results predicted by the theory are different from the experimental results reported in Huo and Muller [2] We discuss some of the issues regarding this discrepancy and show that there appears to be some internal inconsistency between the experimental data reported in Figure 6 and Figure 9 of Huo and Muller [2] (provided they represent the same sample).

Some simple flows of a Johnson-Segalman fluid

Abstract: Unlike most other fluid models, the Johnson-Segalman fluid allows for a non-monotonic relationship between the shear stress and rate of shear in a simple shear flow for certain values of the material parameter. This has been used for explaining a phenomenon such as “spurt”. Here, we study three simple flows of a Johnson-Segalman fluid with a view towards understanding its response characteristics. We find that boundary conditions can have a very interesting effect on the regularity of the solution; changing them continuously leads to solutions that change their regularity. First, we consider the flow through a circular pipe and find solutions that have discontinuous velocity profiles which have been used to explain the phenomenon of “spurt” (cf. [10], [11]). Second, we consider the flow past an infinite porous plate and show that it will not admit solutions which have discontinuous velocity gradients, the solutions being necessarity smooth. Lastly, we study Poiseuille flow in a concentric annulus with porous boundaries. While “spurt” could be explained alternatively by allowing for “stick-slip” at the wall, the Johnson-Segalman model seems particularly suited in describing the appearance of “shear-layers” (cf. [13]).

A constitutive equation for non-linear electro-active solids

Abstract: Electro-active solids are solids that are either infused with electrorheological fluids or embedded with electrically conducting particles, the body as a whole however conducting negligible current. In this paper, we provide a mathematical framework, within the context of continuum mechanics, for the study of electro-active solids. The theory assumes that the body can be considered as a continuum, in the sense of homogenization, which is isotropic, incompressible, elastic and is capable of responding to an electric field. Appealing to standard techniques in continuum mechanics, we obtain a constitutive relation for the stresses in terms of the deformation and electric field. This is used in a study of triaxial extension, simple shear and anisotropy induced by the electric field.

Remarks on “On a Generalized Nonlinear K –ε Model and the Use of Extended Thermodynamics in Turbulence”

Abstract: In [1], the author, an expert on turbulence modeling, has commented, amongst other issues, on a paper by Huang and Rajagopal [2]. It appears from his comments that the author has either misread the paper or has misunderstood some statements in [2]. These comments here will, I hope, set matters straight. In [2] Huang and Rajagopal appeal to a similarity between turbulent flow of a Newtonian fluid and the laminar flow of a non-Newtonian fluid (see [3]) to develop models in turbulence. It is possible that such an analogy should not be invoked, but there seems to be no reason a priori why such an analogy should not be used. At the very least there seems to be no reason to discard such an idea without testing it carefully, in view of the similarity in the secondary flow structures in the two cases. The paper [2] was primarily such an exploration, and it turns out that many of the models that are used in turbulence can indeed be generated by such a method and a specific model generated in [2] embeds many models that have been used in turbulence. In pointing to the inadequacy of an approximation to a model used in [2] to describe relaxation effects, Speziale gives the impression that the models described in [2] are all incapable of describing relaxation effects. This is not correct. In [2] we have developed a model that can stress-relax (see (3.34) and (3.35) of [2]):

Aggregation of a class of linear, interconnected dynamical systems

Abstract: We introduce the notion of a "meaningful" average of a collection of dynamical systems as distinct from an "ensemble" average. Such a notion is useful for the study of a variety of dynamical systems such as traffic flow, power systems, and econometric systems. We also address the associated issue of the existence and computation of such an average for a class of interconnected, linear, time invariant dynamical systems. Such an "average" dynamical system is not only attractive from a computational perspective, but also represents the average behavior of the interconnected dynamical system. The problem of analysis and control of hierarchical, large scale control systems can be simplified by approximating the lower level dynamics of such systems with such an average dynamical system.

Mechanics of electrorheological materials

Abstract: Publisher Summary The modeling of the flows of electrorheological fluids is restricted to one-dimension. Experimental evidence suggests that electrorheological fluids thicken significantly on the application of the electrical field, respond in a Bingham like fashion with the yield depending on the applied field, and develop normal stress differences and stress relax. The response of electrorheological materials is also significantly affected by the thermal conditions. The fiber like structures that are formed on the application of the electrical field suggest that such fluids ought to be modeled as anisotropic fluids. This leads to an additional level of complexity that should be introduced after a better understanding of such materials is achieved. However, this aspect of the modelling is crucial and has to be reckoned with, if the behavior of electrorheological fluids is to be captured. The electrorheological response of liquid crystals has also been studied where the material is anisotropic even before the application of the electric field. The purpose of documenting the model is to merely show general, principles of continuum mechanics can be used to develop general enough models that can describe the response characteristics of electrorheological fluids. General three dimensional continnum models that can describe the response of electrorheological materials such as thickening, stress-relaxation, normal stress differences, and yield, are presented in the chapter.