Kumbakonam R. Rajagopal

Bio: Kumbakonam R. Rajagopal is an academic researcher from Texas A&M University. The author has contributed to research in topics: Constitutive equation & Viscoelasticity. The author has an hindex of 77, co-authored 659 publications receiving 23443 citations. Previous affiliations of Kumbakonam R. Rajagopal include Kent State University & University of Wisconsin-Madison.

Uniqueness and drag for fluids of second grade in steady motion

Abstract: For incompressible fluids of second grade that are compatible with the Clausius-Duhem inequality, non-uniqueness of steady flows with small Reynolds number (i.e. creeping flows) is possible provided the ‘absorption number’ is also small. We discuss this uniqueness question, generalize a well-known theorem of Tanner concerning how solutions of the Stokes equations may be used to generate solutions of the creeping flow equations for fluids of second grade, and give a new uniqueness theorem appropriate to a class of problems for the steady creeping flow of fluids of second grade. Under the conditions for uniqueness, we obtain a simple formula for the drag force on a fixed body which is immersed in a fluid of second grade which is undergoing uniform creeping flow. For bodies with certain geometric symmetries, the non-Newtonian nature of the fluid has no effect upon the drag.

Modeling anisotropic fluids within the framework of bodies with multiple natural configurations

Abstract: This is a follow up of a paper on a thermodynamic framework for rate type models [J. Non-Newtonian Fluid Mech. 88 (2000) 207] published in this journal. The previous paper used the notion that certain materials have multiple natural configurations and that their response can be characterized as a class of elastic responses from an evolving set of natural configurations, and used this framework to model the behavior of a class of visoelastic fluids that are isotropic with regard to their viscous as well as their elastic response. Here, we extend the framework to the modeling of anisotropic fluids . Anisotropic fluids are invariably modeled within the framework of director theories, and such theories require boundary conditions for the directors for the resolution of boundary value problems. Here, we present an approach to the modeling of anisotropic fluids, which is not a director theory; no balance laws for directors are posited nor is there a notion of a director body force, director (or cosserat) stress or director kinetic energy. Thus, the present approach does not require specifying any additional boundary conditions other than that usually specified for viscous fluids, even for flows that involve spatially inhomogeneous fields. Moreover, the framework is based on sound thermodynamical footing, the evolution of the natural configurations being determined by the rate of dissipation of the material. To delineate the efficacy of the theory, we solve a problem associated with a shearing flow of the fluid in which we discuss the tumbling and alignment of certain vectors that represent the axes of anisotropy of the fluid and which may be associated with rod-like structures in the fluid.

Flow of an Oldroyd-B fluid due to a stretching sheet in the presence of a free stream velocity

Abstract: We consider the flow of an Oldroyd-B fluid occupying the space over an elastic sheet, due to the stretching of the sheet, in the presence of a constant free-stream velocity. By introducing a similarity transformation for the velocity field, as well as the three components of the stress tensor, the governing equations are reduced to a system of coupled non-linear ordinary differential equations. The resulting equations are solved numerically and by means of a perturbation in the Weissenberg number. A comparison of the two solutions shows exceedingly good agreement.

On the development and generalizations of Cahn–Hilliard equations within a thermodynamic framework

Abstract: We provide a thermodynamic basis for the development of models that are usually referred to as “phase-field models” for compressible, incompressible, and quasi-incompressible fluids. Using the theory of mixtures as a starting point, we develop a framework within which we can derive “phase-field models” both for mixtures of two constituents and for mixtures of arbitrarily many fluids. In order to obtain the constitutive equations, we appeal to the requirement that among all admissible constitutive relations that which is appropriate maximizes the rate of entropy production (see Rajagopal and Srinivasa in Proc R Soc Lond A 460:631–651, 2004). The procedure has the advantage that the theory is based on prescribing the constitutive equations for only two scalars: the entropy and the entropy production. Unlike the assumption made in the case of the Navier–Stokes–Fourier fluids, we suppose that the entropy is not only a function of the internal energy and the density but also of gradients of the partial densities or the concentration gradients. The form for the rate of entropy production is the same as that for the Navier–Stokes–Fourier fluid. As observed earlier in Heida and Malek (Int J Eng Sci 48(11):1313–1324, 2010), it turns out that the dependence of the rate of entropy production on the thermodynamical fluxes is crucial. The resulting equations are of the Cahn–Hilliard–Navier–Stokes type and can be expressed both in terms of density gradients or concentration gradients. As particular cases, we will obtain the Cahn–Hilliard–Navier–Stokes system as well as the Korteweg equation. Compared to earlier approaches, our methodology has the advantage that it directly takes into account the rate of entropy production and can take into consideration any constitutive assumption for the internal energy (or entropy).

On the thermomechanics of materials that have multiple natural configurations

Abstract: Many bodies, both solid and fluid, are capable of being stress-free in numerous configurations that are not related to each other through a rigid body motion. Moreover, it is possible that these bodies could have different material symmetries in these different stress-free “natural” configurations. In order to describe the response of such bodies, it is necessary to know the manner in which these “natural” configurations evolve as well as a class of response functions for the stress that are determined by kinematical quantities that are measured from these evolving natural configurations. In this review article, we provide a framework to describe the mechanics of such bodies whose “natural configurations” evolve during a thermodynamic process. The framework is capable of describing a variety of responses and has been used to describe traditional metal plasticity, twinning, traditional viscoelasticity of both solids and fluids, solid-to-solid phase transitions, polymer crystallization, response of multi-network polymers, and anisotropic liquids. The classical theories of elastic solids and viscous fluids are included as special cases of the framework. After a review of the salient features of the framework, we briefly discuss the status of viscoelasticity, traditional plasticity, twinning and solid to solid phase transitions within the context of the framework.

A new constitutive framework for arterial wall mechanics and a comparative study of material models

Abstract: In this paper we develop a new constitutive law for the description of the (passive) mechanical response of arterial tissue. The artery is modeled as a thick-walled nonlinearly elastic circular cylindrical tube consisting of two layers corresponding to the media and adventitia (the solid mechanically relevant layers in healthy tissue). Each layer is treated as a fiber-reinforced material with the fibers corresponding to the collagenous component of the material and symmetrically disposed with respect to the cylinder axis. The resulting constitutive law is orthotropic in each layer. Fiber orientations obtained from a statistical analysis of histological sections from each arterial layer are used. A specific form of the law, which requires only three material parameters for each layer, is used to study the response of an artery under combined axial extension, inflation and torsion. The characteristic and very important residual stress in an artery in vitro is accounted for by assuming that the natural (unstressed and unstrained) configuration of the material corresponds to an open sector of a tube, which is then closed by an initial bending to form a load-free, but stressed, circular cylindrical configuration prior to application of the extension, inflation and torsion. The effect of residual stress on the stress distribution through the deformed arterial wall in the physiological state is examined. The model is fitted to available data on arteries and its predictions are assessed for the considered combined loadings. It is explained how the new model is designed to avoid certain mechanical, mathematical and computational deficiencies evident in currently available phenomenological models. A critical review of these models is provided by way of background to the development of the new model.

The physics of debris flows

Abstract: Recent advances in theory and experimen- tation motivate a thorough reassessment of the physics of debris flows. Analyses of flows of dry, granular solids and solid-fluid mixtures provide a foundation for a com- prehensive debris flow theory, and experiments provide data that reveal the strengths and limitations of theoret- ical models. Both debris flow materials and dry granular materials can sustain shear stresses while remaining stat- ic; both can deform in a slow, tranquil mode character- ized by enduring, frictional grain contacts; and both can flow in a more rapid, agitated mode characterized by brief, inelastic grain collisions. In debris flows, however, pore fluid that is highly viscous and nearly incompress- ible, composed of water with suspended silt and clay, can strongly mediate intergranular friction and collisions. Grain friction, grain collisions, and viscous fluid flow may transfer significant momentum simultaneously. Both the vibrational kinetic energy of solid grains (mea- sured by a quantity termed the granular temperature) and the pressure of the intervening pore fluid facilitate motion of grains past one another, thereby enhancing debris flow mobility. Granular temperature arises from conversion of flow translational energy to grain vibra- tional energy, a process that depends on shear rates, grain properties, boundary conditions, and the ambient fluid viscosity and pressure. Pore fluid pressures that exceed static equilibrium pressures result from local or global debris contraction. Like larger, natural debris flows, experimental debris flows of ;10 m 3 of poorly sorted, water-saturated sediment invariably move as an unsteady surge or series of surges. Measurements at the base of experimental flows show that coarse-grained surge fronts have little or no pore fluid pressure. In contrast, finer-grained, thoroughly saturated debris be- hind surge fronts is nearly liquefied by high pore pres- sure, which persists owing to the great compressibility and moderate permeability of the debris. Realistic mod- els of debris flows therefore require equations that sim- ulate inertial motion of surges in which high-resistance fronts dominated by solid forces impede the motion of low-resistance tails more strongly influenced by fluid forces. Furthermore, because debris flows characteristi- cally originate as nearly rigid sediment masses, trans- form at least partly to liquefied flows, and then trans- form again to nearly rigid deposits, acceptable models must simulate an evolution of material behavior without invoking preternatural changes in material properties. A simple model that satisfies most of these criteria uses depth-averaged equations of motion patterned after those of the Savage-Hutter theory for gravity-driven flow of dry granular masses but generalized to include the effects of viscous pore fluid with varying pressure. These equations can describe a spectrum of debris flow behav- iors intermediate between those of wet rock avalanches and sediment-laden water floods. With appropriate pore pressure distributions the equations yield numerical so- lutions that successfully predict unsteady, nonuniform motion of experimental debris flows.

Solving Ordinary Differential Equations

Abstract: Initial-value ordinary differential equation solution via variable order Adams method (SIVA/DIVA) package is collection of subroutines for solution of nonstiff ordinary differential equations. There are versions for single-precision and double-precision arithmetic. Requires fewer evaluations of derivatives than other variable-order Adams predictor/ corrector methods. Option for direct integration of second-order equations makes integration of trajectory problems significantly more efficient. Written in FORTRAN 77.

Hyperelastic modelling of arterial layers with distributed collagen fibre orientations

Abstract: Constitutive relations are fundamental to the solution of problems in continuum mechanics, and are required in the study of, for example, mechanically dominated clinical interventions involving soft biological tissues. Structural continuum constitutive models of arterial layers integrate information about the tissue morphology and therefore allow investigation of the interrelation between structure and function in response to mechanical loading. Collagen fibres are key ingredients in the structure of arteries. In the media (the middle layer of the artery wall) they are arranged in two helically distributed families with a small pitch and very little dispersion in their orientation (i.e. they are aligned quite close to the circumferential direction). By contrast, in the adventitial and intimal layers, the orientation of the collagen fibres is dispersed, as shown by polarized light microscopy of stained arterial tissue. As a result, continuum models that do not account for the dispersion are not able to capture accurately the stress–strain response of these layers. The purpose of this paper, therefore, is to develop a structural continuum framework that is able to represent the dispersion of the collagen fibre orientation. This then allows the development of a new hyperelastic free-energy function that is particularly suited for representing the anisotropic elastic properties of adventitial and intimal layers of arterial walls, and is a generalization of the fibre-reinforced structural model introduced by Holzapfel & Gasser (Holzapfel & Gasser 2001 Comput. Meth. Appl. Mech. Eng. 190, 4379–4403) and Holzapfel et al. (Holzapfel et al. 2000 J. Elast. 61, 1–48). The model incorporates an additional scalar structure parameter that characterizes the dispersed collagen orientation. An efficient finite element implementation of the model is then presented and numerical examples show that the dispersion of the orientation of collagen fibres in the adventitia of human iliac arteries has a significant effect on their mechanical response.