Showing papers by "Kumbakonam R. Rajagopal published in 2008"

On the thermodynamics of fluids defined by implicit constitutive relations

Abstract: In this paper, we develop a thermodynamically consistent theory for describing the response of nonlinear viscous fluids whose constitutive equations are of the form f (T, D) = 0. We show that such constitutive equations which include classical constitutive equations wherein the stress is expressed explicitly in terms of the kinematical quantities, provide a rich class of physically meaningful fluid response functions which allows us to describe a wider range of material behavior, including that of a general class of incompressible fluids, incompressible fluids with pressure dependent viscosity, and Bingham (or pseudoplastic) materials.

A review of mathematical models for the flow of traffic and some recent results

Abstract: After reviewing a variety of macroscopic models based on the continuum approach, cellular automata models, models based on the kinetic theory of gases as well as microscopic models, and delineating their usefulness and deficiencies, we present a model based on a discrete system of dynamical systems. We evaluate the efficacy of the model by comparing the predictions of the model with real time data from monitoring facilities on a highway in Austin, Texas.

Flows of Incompressible Fluids subject to Navier's slip on the boundary

Abstract: We establish closed form analytical solutions for the flows of a generalized fluid of complexity two, which includes the Navier-Stokes fluid, power-law fluid and the second-grade fluid as special subclasses, in special geometries under the assumption that the flows meet Navier slip conditions at the boundary. The boundary conditions allow the extremes of no-slipping and no-sticking. We also allow for different boundary conditions at different parts of the boundary, which leads to interesting consequences with regard to the solutions. The dependence of the form of the solutions on the boundary conditions is discussed in some detail.

A thermodynamic framework for a mixture of two liquids

Abstract: In this study, we extend a thermodynamic framework that has been used with some success for describing the response of a variety of single constituent continua. Using the thermodynamic framework, we obtain a model for the mixture of two compressible fluids that has a much simpler structure than the model obtained earlier within the context of mixture theory. We also investigate the response of a mixture of two fluids that is constrained to have a constant volume, using the same thermodynamic framework.

A quasi-correspondence principle for Quasi-Linear viscoelastic solids

Abstract: In this paper we show that the correspondence principle that allows one to obtain solutions to boundary-initial value problems for Linear viscoelastic solids from solutions to that for a linearized elastic solid can be extended, in many circumstances, to the case of the Quasi-Linear viscoelastic solids introduced by Fung. We illustrate the ability to generalize the correspondence principle by considering a variety of problems including torsion, transverse loading of beams and several problems that involve a single non-zero stress component. This extension is however not possible for certain classes of problems and we present a specific example where the correspondence principle breaks down. The correspondence principle between Linear elasticity and Linear viscoelasticity also breaks down under certain conditions, however the correspondence between the solutions for Linear viscoelasticity and Quasi-Linear viscoelasticity is even more fragile in that it breaks down while the classical correspondence works, and hence we refer to the correspondence as a quasi-correspondence principle.

A semi-inverse problem of flows of fluids with pressure-dependent viscosities

Abstract: A flow is said to be controllable if such a flow can be engendered by the application of appropriate surface tractions. The question, in which fluids are such flows possible leads to the inverse problem of determining constitutive relations, as such flows in general will not be possible in all materials. Thus the determination of the appropriate constitutive class in which certain types of deformations are possible is an inverse problem. A very special subclass of controllable flows is defined through semi-inverse methods. Such semi-inverse solutions are not possible in all bodies but only in special bodies and thus yet constitute an inverse problem. Here, we use a semi-inverse method to consider the flow of fluids whose viscosity depends on the pressure, glowing down an inclined plane, and a flow between two parallel plates. We illustrate that we have an inverse problem with regard to choices of constitutive relations by showing that such flows are only possible in fluids with special viscosity–pressure ...

Discretization of an unsteady flow through a porous solid modeled by darcy's equations

Abstract: The system of unsteady Darcy's equations considered here models the time-dependent flow of an incompressible fluid such as water in a rigid porous medium. We propose a discretization of this problem that relies on a backward Euler's scheme for the time variable and finite elements for the space variables. We prove a priori error estimates that justify the optimal convergence properties of the discretization and a posteriori error estimates that allow for an efficient adaptivity strategy both for the time steps and the meshes.

On the Modeling of the Viscoelastic Response of Embryonic Tissues

Abstract: Forgacs et al. have developed a continuum model for cellular aggregates that has at its basis the work of Steinberg and co-workers concerning the coalescence of cells that takes into account surface tension effects, to describe the response of embryonic tissues. Here, we propose a solid model that reflects the fact that such embryonic tissues in the bulk (not at the cellular level) behave like solids rather than as fluids. The model stems from a robust thermodynamic framework and we show that our model fits the experimental data exceptionally well.

Thermodynamics of hypoelasticity

Abstract: A thermodynamic framework for hypoelasticity is constructed based on a modified Gibbs Function which depends on the stress deviator and specific volume. In the hypoelastic equations considered here, the stress deviator is obtained from a rate equation involving the corotational derivative and coefficients which depend on the invariants of the stress deviator as well as specific volume. Non-negative entropy production is enforced, leading to a non-dissipative condition in the sense that entropy is produced only by heat transfer. The resulting conditions lead to relations among the coefficients in the rate equation. These relations underdetermine the coefficients, so that there is leeway in specifying these coefficients. An example of a set of rate equations which satisfy the thermodynamics is presented.

On the modeling of entropy producing processes

Abstract: A general thermodynamic framework is presented for the study of the response of bodies undergoing entropy producing processes. In general, in such processes the natural configuration of a body, i.e., the configuration that the body would take on the removal of all external stimuli, changes. The fact that material symmetry of the body in these various natural configurations could be different a llows one to model the response of bodies that cannot be described by traditional models that are in place. It is assume d that the processes take place in a manner such that the rate at which entropy is produced is maximized. Knowing how the material stores energy, produces entropy, conducts heat, absorbs or emits radiation, etc., allows one t o determine the constitutive equation for the stress and other relevant quantities. The fact that the body's natural configuration changes and the form for the stress response from the natural configuration changes, leads to a lot of chal lenges with regard to the development of analytical as well as numerical methods for the study of the response of bodies.

Development of a Diagnostic System for Air Brakes in Trucks

Abstract: The purpose of this Innovations Deserving Exploratory Analysis (IDEA) project was to enhance the safety of trucks by developing diagnostic systems that assess the health of brake systems. Brake systems in trucks are crucial for ensuring the safety of vehicles and passengers on the roadways. Most trucks in the U.S. are equipped with S-cam air brake systems and they are sensitive to maintenance. Moreover, any incident involving trucks results in material damage, which can have serious environmental and economic repercussions, and sometimes, the loss of life, which cannot be measured in economic terms. This project consisted of three stages. In the first stage, the authors developed leak detection algorithms based on a mathematical model that predicts the evolution of the pressure in the brake chambers, without leak, in response to the brake pedal input. The second stage included the estimation of push rod stroke in the absence of a leak. The third stage included the development, experimental implementation, testing and evaluation of the prototype diagnostic system in the presence of leaks. Tests of the prototype diagnostic system conducted on the test bench at Texas A&M University indicated that the prototype diagnostic system accurately predicts the "severity" of the leak (characterized by the "effective area" of the cross-section of the leak) and reasonably predicts the pushrod stroke in the presence of leaks. The underlying algorithms required the development and corroboration of a mathematical model for pressure transients in the presence of a leak. The model used orifice equations for describing the mass flow rate of air leaking and incorporated the balance of mass equation. Experiments indicate that the resulting mathematical model successfully predicts brake pressure.

Modeling of Entropy Producing Processes

Abstract: In all processes concerning real materials, entropy is produced While undergoing such processes, the natural configuration of a body (the configuration that the body would take on the removal of all external stimuli) changes It is also possible that the material symmetry of the body in these various natural configurations could be different Usually, one requires that all processes meet the second law of thermodynamics namely that the rate of entropy production is non‐negative In this study we make a more stringent requirement, namely that the rate of entropy production is maximized A framework is developed wherein knowing how the material stores energy, produces entropy, conducts heat, absorbs or emits radiation, etc, allows one to determine the constitutive equation for the stress and other relevant quantities