Showing papers by "Kumbakonam R. Rajagopal published in 2005"

A Model for the Formation and Lysis of Blood Clots

Abstract: Both biochemical and mechanical factors have to be taken into account if a meaningful model for the formation, growth, and lysis of clots in flowing blood is to be developed. Most models that are curr

Mathematical Issues Concerning the Navier–Stokes Equations and Some of Its Generalizations

Abstract: This chapter primarily deals with internal, isothermal, unsteady flows of a class of incompressible fluids with both constant and shear or pressure dependent viscosity that includes the Navier–Stokes fluid as a special subclass. We begin with a description of fluids within the framework of a continuum. We then discuss various ways in which the response of a fluid can depart from that of a Navier–Stokes fluid. Next, we introduce a general thermodynamic framework that has been successful in describing the disparate response of continua that includes those of inelasticity, solid-to-solid transformation, viscoelasticity, granular materials, blood and asphalt rheology, etc. Here, it leads to a novel derivation of the constitutive equation for the Cauchy stress for fluids with constant, or shear and/or pressure, or density dependent viscosity within a full thermomechanical setting. One advantage of this approach consists in a transparent treatment of the constraint of incompressibility. We then concentrate on the mathematical analysis of three-dimensional unsteady flows of fluids with shear dependent viscosity that includes the Navier–Stokes model and Ladyzhenskaya's model as special cases. We are interested in the issues connected with mathematical self-consistency of the models, i.e., we are interested in knowing whether (1) flows exist for reasonable, but arbitrary initial data and all instants of time, (2) flows are uniquely determined, (3) the velocity is bounded and (4) the long-time behavior of all possible flows can be captured by a finite-dimensional, small (compact) set attracting all flow trajectories exponentially. For simplicity, we eliminate the choice of boundary conditions and their influence on the flows by assuming that all functions are spatially periodic with zero mean value over a periodic cell. All these results can however be extended to internal flows wherein the tangential component of the velocity satisfies Navier's slip at the boundary. Most of the results also hold for the no-slip boundary condition. While the mathematical consistency understood in the above sense for the Navier–Stokes model in three dimensions has not been established as yet, we will show that Ladyzhenskaya's model and some of its generalization enjoy all above characteristics for a certain range of parameters. We also discuss briefly further results related to generalizations of the Navier–Stokes equations.

Competition Between Radial Expansion and Thickening in the Enlargement of an Intracranial Saccular Aneurysm

Abstract: Rupture of intracranial saccular aneurysms is the leading cause of spontaneous subarachnoid hemorrhage, which results in significant morbidity and mortality. Although many have suggested that saccular aneurysms enlarge and rupture due to mechanical instabilities, our recent nonlinear analyses suggest that at least certain classes of aneurysms do not exhibit a quasi-static limit point instability or dynamic instabilities in response to periodic loading. Based on an increased understanding of the ubiquitous role of growth and remodeling within the vasculature and recent histopathological data on saccular aneurysms, it is hypothesized that a stress-mediated regulation of collagen turnover causes their enlargement. There is a need, however, for a theoretical framework to explore this and competing hypotheses. In this paper, we present a 2-D constrained mixture model for growth and remodeling of an ellipsoidally shaped saccular aneurysm and numerically simulate enlargement and changes in material symmetry in the aneurysmal wall. Results suggest that ellipsoidal aneurysms tend toward spherical shapes, and a competition between radial expansion and wall thickening plays a critical role in determining the stability of an enlarging lesion.

On steady flows of fluids with pressure– and shear–dependent viscosities

Abstract: There are many technologically important problems such as elastohydrodynamics which involve the flows of a fluid over a wide range of pressures. While the density of the fluid remains essentially constant during these flows whereby the fluid can be approximated as being incompressible, the viscosity varies significantly by several orders of magnitude. It is also possible that the viscosity of such fluids depends on the shear rate. Here we consider the flows of a class of incompressible fluids with viscosity that depends on the pressure and shear rate. We establish the existence of weak solutions for the steady flows of such fluids subjected to homogeneous Dirichlet boundary conditions and to specific body forces that are not necessarily assumed to be small. A novel aspect of the study is the manner in which we treat the pressure that allows us to establish its compactness, as well as that of the velocity gradient. The method draws upon the physics of the problem, namely that the notion of incompressibility is an idealization that is attained by letting the compressibility of the fluid to tend to zero.

On the nature of constraints for continua undergoing dissipative processes

Abstract: When dealing with mechanical constraints, it is usual in continuum mechanics to enforce ideas that stem from the seminal work of Bernoulli and D'Alembert and require that internal constraints do no...

On Fully Developed Flows of Fluids with a Pressure Dependent Viscosity in a Pipe

Abstract: Stokes recognized that the viscosity of a fluid can depend on the normal stress and that in certain flows such as flows in a pipe or in channels under normal conditions, this dependence can be neglected. However, there are many other flows, which have technological significance, where the dependence of the viscosity on the pressure cannot be neglected. Numerous experimental studies have unequivocally shown that the viscosity depends on the pressure, and that this dependence can be quite strong, depending on the flow conditions. However, there have been few analytical studies that address the flows of such fluids despite their relevance to technological applications such as elastohydrodynamics. Here, we study the flow of such fluids in a pipe under sufficiently high pressures wherein the viscosity depends on the pressure, and establish an explicit exact solution for the problem. Unlike the classical Navier-Stokes solution, we find the solutions can exhibit a structure that varies all the way from a plug-like flow to a sharp profile that is essentially two intersecting lines (like a rotated V). We also show that unlike in the case of a Navier-Stokes fluid, the pressure depends both on the radial and the axial coordinates of the pipe, logarithmically in the radial coordinate and exponentially in the axial coordinate. Exact solutions such as those established in this paper serve a dual purpose, not only do they offer solutions that are transparent and provide the solution to a specific but simple boundary value problems, but they can be used also to test complex numerical schemes used to study technologically significant problems.

On the Diffusion of Fluids Through Solids Undergoing Large Deformations

Abstract: In this paper, we investigate the problem of diffusion of a Newtonian fluid through a non-linearly elastic solid undergoing large deformations, within the framework of mixture theory. Our aim is to delineate the effect of various boundary conditions on the diffusion process. We find that the results are quite insensitive to the boundary conditions in that the predictions agree exceedingly well with the experiments (whether it is the saturation boundary condition used by Rajagopal, Wineman and Gandhi, the traction splitting boundary condition used by Rajagopal and Tao, the natural boundary condition used by Baek and Srinivasa, or the fact that the chemical potential is continuous across the boundary), provided that an appropriate choice is made for the drag.

Simulation of fiber spinning including flow-induced crystallization

Abstract: Kannan et al. [J. Rheol. 46, 979–999 (2002)] studied the problem of fiber spinning for polymers that are largely atactic in nature, within a thermodynamic framework developed by Rao and Rajagopal [Z. Angew. Math. Phys. 53, 365–406 (2002)], where significant crystallization does not take place. Here, we develop a model within the same framework that can take into account flow-induced crystallization and the anisotropy of the crystalline part of the semicrystalline polymer. We find that the predictions of the model agree very well with experimental data.

Heat-Induced Changes in the Finite Strain Viscoelastic Behavior of a Collaagenous Tissue

Abstract: Supra-physiological temperatures are increasingly being used to treat many different soft tissue diseases and injuries. To identify improved clinical treatments, however, there is a need for better information on the effect of the mechanics on the thermal damage process as well as the effect of the incurred damage on the subsequent mechanical properties. In this paper, we report the first biaxial data on the stress relaxation behavior of a collagenous tissue before and after thermal damage. Based on a two-dimensional finite strain viscoelastic model, which incorporates an exponential elastic response, it is shown that the thermal damage can significantly decrease the characteristic time for stress relaxation and the stress residual.

Inflation, Extension, Torsion and Shearing of an Inhomogeneous Compressible Elastic Right Circular Annular Cylinder:

Abstract: We study the inflation, extension, torsion and shearing of an isotropic inhomogeneous compressible annular right circular cylinder. Current approaches to homogenization that appeal to an equivalence in the stored energies could lead to serious errors in the estimate for stresses in a inhomogeneous body as stresses depend on the derivatives of the stored energy with respect to the deformation gradient. This is a serious drawback as many a time failures are determined by the stresses. The study demonstrates that, in particular, great caution should be exercised in homogenization, especially if an inhomogeneous body is to be approximated by a homogeneous body belonging to the same class. Comparison of local measures, such as stresses, reveal that their values in the case of the inhomogeneous body and its homogeneous counterpart can be both qualitatively and quantitatively far apart. Even the differences in global measures like the axial load, torque, etc., are found to be significant between the inhomogeneou...

On the propagation of waves through porous solids

Abstract: This note reexamines Biot's model for the propagation of acoustic waves in a material such as cohensionless sand, infused with a fluid, within the context of mixture theory. Instead of the standard entropy equation that is used in mixture theory, an inequality for the viscous dissipation is employed here due to a conceptual difficulty that one encounters in applying the standard equation to a mixture of sand and a fluid. The wave equations are reformulated by taking the velocity field, instead of the displacement, for the fluid as a primary quantity. By recognizing and thereby exploiting the dependence of the stored energy of the sand on the pore fluid pressure and choosing an appropriate form for the rate of dissipation, a set of governing equations are obtained which are equivalent to those derived by Biot [J. Acoust. Soc. Am. 28(1956) 168, 179; J. Appl. Phys. 33(1962) 1482]. A differential equation for the pore fluid pressure is derived and the effects of drag and virtual mass are dealt with in a unified fashion. The procedure allows us to develop generalizations to Biot's equations in a rational manner.

On the role of the eshelby energy-momentum tensor in materials with multiple natural configurations

Abstract: We discuss the connection between two important parallel developments in continuum mechanics that has gone unnoticed despite intense activity in both these seemingly disparate areas: the first stems from the pioneering work of Eckart on the role of the evolution of “natural configurations” in the inelastic response of solids; the second stems from the seminal work of Eshelby concerning the energy-momentum tensor associated with the driving forces that arise as a consequence of inhomogeneities and defects during the deformations of solids. We show that a variety of driving forces manifest themselves as a consequence of the evolution of natural configurations, depending on the particular process under consideration. Our study makes it clear that no new balance laws need be invoked in order to accommodate such driving forces, the usual balances laws of mechanics being sufficient.

A note on the flows of inhomogeneous fluids with shear-dependent viscosities

Abstract: Inhomogeneous fluids have not been studied with the intensity that they deserve. In fact, many studies that are supposedly concerned with the response of inhomogeneous fluids are not directed at inhomogeneous fluids, and this stems from not recognizing the fact that the properties of a fluid varying in its current configuration does not mean that the fluid is inhomogeneous. Here, we show that mild variations in the properties of the fluid which might warrant it being approximated as a homogeneous fluid with average properties could lead to significant errors in the computation of both global and local quantities, associated with the flow.

Information flow and its relation to the stability of the motion of vehicles in a rigid formation

Abstract: It is known in the literature on automated highway systems that information flow can significantly affect the propagation of errors in spacing in a collection of vehicles. This paper investigates this issue further for a homogeneous collection of vehicles, where in the motion of each vehicle is modeled as a point mass. The structure of the controller employed by the vehicles is as follows: U/sub i/(s)=C(s)/spl Sigma/ /sub j/spl isin/si/(X/sub i/ - X/sub j/ - L/sub ij//s) where U/sub i/(s) is the (Laplace transformation of) control action for the i/sup th/ vehicle, L/sub ij/is the position of the i/sup th/ vehicle, L/sub ij/ is the desired distance between the i/sup th/ and the j/sup th/ vehicles in the collection, C(s) is the controller transfer function and S/sub i/ is the set of vehicles that the i/sup th/ vehicle can communicate with directly. This paper further assumes that the information flow is undirected, i.e., i/spl isin/S/sub j//spl harr/j/spl isin/S/sub i/, and the information flow graph is connected. We consider information flow in the collection, where each vehicle can communicate with a maximum of q(n) vehicles, such that q(n) may vary with the size n of the collection. We first show that C(s) cannot have any zeroes at the origin to ensure that relative spacing is maintained in response to a reference vehicle making a maneuver where its velocity experiences a steady state offset. We then show that if the control transfer function C(s) has one or more poles located at the origin of the complex plane, then the motion of the collection of vehicles will become unstable if the size of the collection is sufficiently large. These two results imply that C(0)/spl ne/0 and C(0) is well defined. We further show that if q(n)/sup 3//n/sup 2//spl rarr/0 as n /spl rarr//spl infin/ then there is a low frequency sinusoidal disturbance of at most unit amplitude acting on each vehicle such that the maximum errors in spacing response increase at least as O (/spl radic/(n/sup 2/)/q(n)/sup 3/). A consequence of the results presented in this paper is that the maximum of the error in spacing and velocity of any vehicle can be made insensitive to the size of the collection only if there is at least one vehicle in the collection that communicates with at least O(n/sup 2/3/) other vehicles in the collection.

A note on a correspondence principle in nonlinear viscoelastic materials

Abstract: We show that models for the nonlinear viscoelastic response of solids generated on the basis of a correspondence principle developed by Schapery(1984) do not satisfy the balance of angular momentum for large deformations. This principle, which is valid if the displacement gradients are sufficiently small, has been used in several papers to develop models to describe the fracture of viscoelastic solids, and these studies need to be reexamined in the light of this note.

Modelling constant displacement rate experiments of asphalt concrete using a thermodynamic framework

Abstract: This study is concerned with the constitutive modeling of asphalt concrete mixtures. The response of the asphalt concrete pavement depends on its internal structure. The internal structure of the asphalt concrete mixture evolves during the loading process. Here, we develop a single constituent model for asphalt concrete mixture by associating different natural configurations (stress-free configurations) with distinct internal structures of the body. The evolution of the natural configurations is determined using a thermodynamic criterion, namely the maximization of the rate of dissipation. Making appropriate assumptions concerning the manner in which the body stores and dissipates energy, the constitutive relations for the stress is deduced. Constant displacement rate experiments are carried out at different confinement pressures on asphalt concrete specimens made of two different aggregates—granite and limestone. The efficacy of the model in predicting the mechanical response of asphalt concrete mixtures...

On the Bending of Shape Memory Wires

Abstract: In recent years there has been considerable interest in describing the response of shape-memory wires, thin films, beams, and other related structures in view of their technological relevance as parts of a plethora of devices such as actuators, stents, dental implants, and so on Several analyses of the response of such materials have been carried out, but many of them have severe shortcomings because they neglect to pay adequate attention to the dissipative nature of such bodies Other studies, in addition to the aforementioned deficiency, make kinematic assumptions that are flawed We present a theoretical framework for the study of the finite deformation bending of shape-memory wires that takes cognizance of the dissipative nature of the body under consideration and also utilizes physically reasonable kinematic assumptions in the development of a finite deformation counterpart of the Bernoulli–Euler beam theory The predictions of the theory are in keeping with the experimental results that are available

Simulation of the film blowing process for semicrystalline polymers

Abstract: We simulate the film blowing process using a fully thermodynamic model developed to study crystallization in polymers. The model stems from a general thermodynamic framework that is particularly well suited to describing dissipative processes during which the symmetry of the material can change; thus, it provides a good basis for studying the crystallization process in polymers. The polymer melt is modeled as a rate-type viscoelastic fluid and the crystalline solid polymer is modeled as an anisotropic elastic solid. The initiation criterian, marking the onset of crystallization and equations governing the crystallization kinetics, arise naturally in this setting in terms of the appropriate thermodynamic functions. The mixture region, wherein the material transitions from a melt to a semicrystalline solid, is modeled as a mixture of a viscoelastic fluid and an elastic solid. The anisotropy of the crystalline phase and consequently that of the final solid depends on the deformation in the melt duri...

Simulation of Flow Induced Crystallization in Fiber Spinning Using the Radial Resolution Approximation

Abstract: In this article we develop a more accurate model for describing flow induced crystallization within the thermomechanical framework created by Rao and Rajagopal [1, 2] to describe the problem of polymer crystallization in general and the problem of fiber spinning in particular. We incorporate the changes in the material symmetry during crystallization to study the problem of fiber spinning. This study differs from the recent studies by Kannan and Rajagopal [3, 4] in that a more appropriate two dimensional form for the balance of energy is used here, while in the previous studies by the same authors a one dimensional approximation is employed. The model incorporates the effects due to melt viscoelasticity, drag on the fiber, gravity, inertia effects, the cooling of the fiber, the initiation of crystallization (that depends on both the temperature and deformation), flow induced crystallization and the anisotropy of the crystalline phase of the semicrystalline solid. The predictions of the model are compared ...